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¸1sChapter 1 Section 1 1. Find the sum of all positive integers between 84 and 719 which are exactly divisible by 5. Answer: 50,800 2. What is the sum of the geometric progression if there are 4 geometric means between 3 and 729? Answer: 1092 3. Find the 50th term of a geometric progression if the 20th term is 1200 and the 30th term is also 1200. Answer: 1200 4. The arithmetic mean of two numbers is 4, and their harmonic mean is 15/4. Find the numbers. Answer: 3 & 5 5. Find the 30th term of the A.P 4, 7, 10,… Answer: 91 6. Find the sum of the infinite geometric progression 6, -2, 2/3. Answer: 9/2 7. Find the sum of the first 100 positive integers exactly divisible by 7. Answer: 35,350 8. If the 5th term in A.P. is 17 and the 3rd term is 10, what is the 8th term? Answer: 27.5 9. There are 4 geometric mean between 3 and 729, Find the sum of the G.P. Answer: 1092 10. The seventh term of an arithmetic sequence is 5 and the twelfth term is -15. What is the first term of this sequence? Answer: 29

1. The positive values of x so that 4x, 5x + 4, 3x2 – 1will be in arithmetic progression is: Answer: 3 2. There are line (9) arithmetic means between 11 and 51. The sum of the progression is: Answer: 341 3. Find the quadratic mean of the numbers 8,9,9,13. Answer: 9.94 4 .Find the 100th term of the sequence 1.01, 1.00, 0.99… Answer: 0.02 5. The geometric mean of a and b is: Answer: √ab 6. The fourth term of a geometric progression is 189 and the sixth term is 1701, the 8th term is: Answer: 15309 7. The geometric mean and the harmonic mean of two numbers are 12 and 36/5 respectively. What are the numbers? Answer: 36 & 4 8. If x, 4x+8, 30x +24 are in geometrical progression, find the common ratio. Answer: 6 9. The 1st, 4th, 8th terms of an A.P. are themselves geometric progression (G.P.). What is the common ratio of the G.P.? Answer: 4/3 10. What value of x makes the three terms x, x/(x + 1) and 3x/[(x + 1)(x + 2)] those of a geometric sequence? Answer: -1/2

Section 2 1. Determine the sum of the first 4 terms of the sequence whose general term is given by 3n -2. Answer: 112 2. Solve for x in the following equation: x + 3x + 5x + … + 49x = 625. Answer: 1 3. Determine the 5th term of the sequence whose sum of n terms is given by 2n+3 – 5. Answer: 128 4. Determine x so that 2x + 1, x2 + x + 1, 3x2 – 3x + 3 are consecutive terms of an arithmetic progression. Answer: 2 5. Find the ratio of an infinite geometric series of the sum is 2 and the first term is ½. Answer: ¾ 6. The sum of a geometric series is follows: S = 1.0 + 1.1 + 1.21 + 1.1331 + … up to 50 th term, Answer: 1163.91 7. The sum of the three numbers in AP is 33, if the sum of their squares is 461, find the numbers. Answer: 4, 11, 18 8. Three numbers are in AP such that the sum of the first and third is 12 and the product of the first and second is 24. Find the largest of three numbers. Answers: 8 9. The arithmetic mean of 6 numbers is 17. If two numbers are needed to the progression, the new set of number will have an arithmetic mean of 19. What are two numbers if their difference is 4? Answer: 23, 27 10. The first term of the arithmetic progression (AP) is 6 and the 10th term is 3 times the second number. What is the common difference? Answer: 2

1. The motion of a particle through a certain medium is such that it moves two thirds as far each second as in the preceding second. If it moves 6m of the first second, how far will it move before coming to rest? Answer: 18 2. A man who is on diet losses 24 lb. in 3 months, 16 lb. is the next 3 months and so on for a long time. What is the maximum total weight loss? Answer: 72 3. The first term of an arithmetic progression (AP) is 6 and the 10th term is 3 times the second number. What is the common difference. Answer: 2 4. The 10th term of the series a, a-b, a-2b, … is: Answer: 1-9b 5. The arithmetic mean of 80 numbers is 55. If two numbers namely 250 and 850 are removed what is the arithmetic mean of the remaining numbers? Answer: 42.31 6. The sum of five arithmetic means between 34 and 42 is: Answer: 190 7. There are 6 geometric means between 4 and 8748. Find the sum if all terms. Answer: 13120 8. Determine x so that x, 2x+7, 10x-7 will form a geometric progression. Answer: 7 9. The sum of all numbers between 0 and 10000 which is exactly divisible by 77 is: answer: 645645 10. There are seven arithmetic means between 3 and 35. Find the sum of all terms. Answer: 171

1. The sum of the progression 5, 8, 11, 14 … is 1025. How many terms are there? Answer: 25 2. The arithmetic mean of two numbers is 5, and their harmonic mean is 24/5. Find the numbers. Answer: 4 & 6 3. Find the sum of the numbers divisible by 6 which lie between 75 and 190 Answer: 2508 4. How many term of the progression 4, 7, 10, 13, … must be taken so that the sum will be 69 Answer: 6 5. If the third of a GP is 20 and the 6th term is 160, what is the first term? Answer: 5 6. Find the 15th term of the progression 1/4, 1/7, 1/10, … Answer: 1/46 7. The fourth term of the geometric progression is 189 and the sixth term is 1701, the 8th term is: Answer: 15309 8. The sum of the three in AP is 33, if the sum of their squares is 461, find the numbers. Answer: 4, 11, 18 9. The motion of a particle through a certain medium is such that it moves two thirds as far each second as in the preceding second. If it moves 6m of the first second, how far will it move before coming to rest. 10. There are line (9) arithmetic means between 11 and 51. The sum of progression is: Answer: 341

Section 3 1. If n is a positive integer such that n!/(n – 2)! = 342, find n. Answer: 19 2. A critical number c of a function is a number in the domain of f such that: Answer: f’(c) = 0 or f’ (c) is undefined 3. What is the range of the function y=x=2/3? Answer: y>0 4. How many integers are there in the solution set of |x – 2| ≤ 5? Answer: 11 5. How many positive integers are there in the solution set of x/x – 2 > 5? Answer: 0 6. What is the domain of the function f(x) = 3x3-7? Answer: X ≥ 1.33 7. The number of integers that satisfy the inequality x2 + 48 < 16x is: Answer: 7 8. Given: A = {2,6,10}. Which of the following is equal to the given set A? Answer: All are correct 9. Let V be the set of letters in the English alphabet. The |V| is ______. Answer: 26 10. Given A = {11,12,13} and B = {11,13,15}. Find A-B. Answer: {12}

1. If set Y is the set of all positive integers less than 10 and set B is a subset of set Y, set B may be ______. Answer: {6,7,8} 2. Given A = {1,2} and B = {a, b}. Find AxB. Answer: {(1,a), (1,b), (2,a), (2,b)} 3. Given A = {1,2,3} and B = {a, b, c. Find the cardinality of AxB. Answer: 9 4. Given A = {1,2,3} and B = {1,3,5}. Find AB. Answer: {1,2,3,5} 5. Given A = {1,2,3} and B = {1,3,5}. Find AB. Answer: {1,3} 6. Given A = {a,b,c}, B = {a,c,e}, and C = {e,f,g}. Which of the given sets are disjoint? Answer: A & C 7. Given A = {11,12,13} and B = {11,13,15}. Find A –B. Answer: {12} 8. Given A = {31,32,33} and B = {31,33,35}. Find B\As Answer: {35} 9. Let A be the set of positive integers than 10 with a universal set of all positive integers. Find A. Answer: {1,2,3,4,5,6,7,8,9,10} 10. Let A = {0,2,4,6,8,10}, B = {0,1,2,3,4,5,6} and C = {4,5,6,7,8,9,10}. What is (A B) C? Answer: {0,2,4,5,6,7,8,9,10}

Section 4 1. Let A={4, 6, 7, 8, 9}, B={6, 7, 8, 25}. Determine A XOR B. Answer: {4, 9, 25} 2. Let A = {a, b, c}, B = {b, c, d} and C = {b, c, e}. Find (A ∪ B) ∩ (A ∪ C). Answer: {a, b, c} 3. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find A – B. Answer: {1, 2, 4, 5} 4. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find B – A. Answer: {0, 6} 5. Find the power set of A = {a, b}, where a and b are distinct elements. Answer: {Ø, {a}, {b}, {a, b}} 6. Let A = {41, 42, 43, 44} and B = {43, 44, 45, 46}. Find the symmetric difference of A and B. Answer: {41, 42, 45, 46} 7. Let A={0,2,4,6,8,10}, B={0,1,2,3,4,5,6} and C={4,5,6,7,8,9,10}. What is A ∩ B∩ C? Answer: {4,6} 8. Find the floor and ceiling functions, respectively, of 7/8. Answer: 0, 1 9.If Z is the set of odd positive integers less than 10, then, the cardinality of Z is: Answer: 5 10.If a and b are both even numbers, which of the following could be and odd integer? Answer: (a + 1)/(b + 1)

1.Given the sets: A={a,c,e}; B={c,e,a} and C={a,c,c,c,e}. Which of the following statements is false? Answer: A≠C 2. Given the sets: A={3,4,5,6,7} and B={4,5,6}. Which of the following statements is false? Answer: A⊆ B 3. Given Set A={-4,-5,1,3,13}, if we have the relationship C ⊂ A. What should set C be? Answer: {-4, 1, 13} 4. Find the cardinality of the power set of {8, 13, 21, 57}. Answer: 16 5. Which of the following is a null set? Answer: {x|x is an integer such that x^2 = 2} 6.How many elements does the set Ƿ(Ƿ(Ø)) has/have? Answer: 2 7. Given |G| is 9 and |H| is 3. All of the given are possible values for the cardinality of the union of set G and set H, except __. Answer: 7 8. Given |P| is 12 and |Q| is 7. All of the given are possible values for the cardinality of the intersection. Answer: 9 9. Let f1 and f2 be functions from R to R such that f1(x) = x2 and f2(x) = x – x2. What is the function f1f2. Answer: x3- x4 10. Let {an} be a sequence that satisfies the recurrence relation an = an-1 – an-2 for n=2,3,4… and suppose that a0 = 3 and a1 = 5. What is a2? Answer: 2

SECTION 5 1

List the member of Set W, where W = {x | x is the square of an integer and x <30}.

Answer: {0, 1, 4, 9, 16, 25} 2

Find the cardinality of A^2 if A = {0, 1, 3}.

Answer: 9 3

Find the truth set of the given predicate: P(x) = x^2 < 3. The domain is the set of integers.

Answer: {-1, 0, 1} 4

Let f = 0.5 + g, where g is the ceiling function of 3/2. Determine the floor function of f.

Answer: 2 5. Let f = 0.5 x g, where g is the floor function of 5/2. Find the value of the floor function of f. Answer: 1 6. Let A = {g, h, i, j, k} and B = {11, 12, 13, 14} with f(g) = 12, f(h) = 11, f(i) = 14, f(j) = 11 and f(k) = 14. The image of the subset S = {h, i, j} is ________. Answer: {11, 14} 7. Let X = {1, 2, 3} and Y = {a, b, c, d}. Given a function H from X to Y. Which of the following shows a one-to-one function? Answer: H(1) = c, H(2) = a, H(3) = d 8. Let X = {1, 2, 3, 4} and Y = {a, b, c}. Given a function H from X to Y. Which of the following shows an onto function? Answer: H(1) = c, H(2) = b, H(3) = a, H(4) = a 9. Let f be the function from {a, b, c, d} to {1, 2, 3, 4}. Which of the following definitions will make f bijective? Answer: f(a) = 4, f(b) = 1, f(c) =3, f(d) = 2 10. A function is bijective if it is both one-to-one and onto. Answer: One-to-one and onto

1. Let f be the function from {p, q, r} to {11, 12, 13} such that f(p) = 12, f(q) = 13 and f(r) = 11. Find the inverse of f. Answer: f-1(11) = r, f-1(12) = p, f-1(13) = q 2. Find the inverse of the given function described as follows: Let f: Z → Z such that f(x) = x + 1. Answer: f-1(y) = y – 1 3. Let Set A = (-1, 0] and Set B = [0 ,1). Find the complement of Set A if the universal set is the set R of all real numbers. Answer: ( -∞ , -1 ] ∪ ( 0, ∞ ) 4. An engineering system has two components. Let us define the following events: A = first component is good; A’ = first component is defective; B = second component is good; B’ = second component is defective Describe the following events in terms of A, A’, B, and B’: (a) at least one component is good (b) one is good and one is defective. Answer: A∪B, AB’∪A’B 5. Under which conditions is xy/x-y negative? Answer: x < y < 0 6. The set O of odd positive integers less than 10 can be expressed by _____________. Answer: {1, 3, 5, 7, 9} 7. What is the Cartesian product of A = {1, 2} and B = {a, b}? Answer: {(1, a), (2, a), (1, b), (2, b)} 8. What is the cardinality of the set of odd positive integers less than 10? Answer: 5 9. Which of the following two sets are equal? Answer: A = {1, 2, 3} and B = {2, 1, 3} 10. What is the Cardinality of the Power set of the set {0, 1, 2}. Answer: 8

SECTION 6 1. The complement of the set A is _____________. Answer: U – A 2. The bit string for the set {2, 4, 6, 8, 10} (with universal set of natural numbers less than or equal to 10) is ____________________. Answer: 0101010101 3. Let Ai = {i, i+1, i+2,…}. Then set {n, n+1, n+2, n+3,…} is the _________ of the set Ai. Answer: Intersection 4. The bit strings for the sets are 1111100000 and 1010101010. The union of these sets is ___________. Answer: 1111101010 5. The set difference of the set A with null set is __________. Answer: A 6. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then number of elements in A U B is: Answer: 4 7. Let the set A is {1, 2, 3} and B is { 2, 3, 4}. Then number of elements in A ∩ B is: Answer: 2 8. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then the set A – B is: Answer: {1} 9. In which of the following sets A- B is equal to B – A. Answer: A={1, 2, 3}, B ={2, 3, 1} 10. Let A be set of all prime numbers, B be the set of all even prime numbers, C be the set of all odd prime numbers, then which of the following is true? Answer: All of the mentioned

1 If A has 4 elements B has 8 elements then the minimum and maximum number of elements in A U B are respectively. Answer: 8, 12 2

If A is {{Φ}, {Φ, {Φ}}, then the power set of A has how many element?

Answer: 4 3

Which sets are not empty?

Answer: { x: x is a prime number less than 5 and is odd} 4

The shaded area of figure is best described by,

Answer: A ∩ B 5

The shaded area of figure is best described by,

Answer: A U B – B 6

If n(A)=20 and n(B)=30 and n(A U B) = 40 then n(A ∩ B) is:

Answer: 10 7

The shaded area of figure is best described by,

Answer: Venn Diagram

Answer: B – (A ∩ B) – (C ∩ B) 8. Let A : All badminton player are good sportsperson. B: All person who plays cricket are good sportsperson. Let X denotes set of all badminton players, Y of all cricket players, Z of all good sportsperson. Then which of the following statements is correct? Answer: Z contains both X and Y 9 If n(A)=10 , n(B)=30,n(C)=50 and if set A,B,C are pairwise disjoint then which of the following is correct? Answer: All of the mentioned 10

In the given figure the if n(A)=20,n(U)=50,n(C)=10 and n(A∩B)=5 then n(B)=?

Answer: 35

SECTION 7 1

The shaded area of figure is best described by,

Answer: A U B – (A ∩ B) 2

Let C and D be two sets then which of the following statements are true?

1) C U D = D U C

2) C ∩ D = D ∩ C Answer: Both of these statements 3

If set C is {1, 2, 3, 4} and C – D = Φ then set D can be:

Answer: {1, 2, 3, 4} 4

Let C and D be two sets then C – D is equivalent to:

Answer: C ∩ D’ 5

For two sets C and D the set (C – D) ∩ D will be:

Answer: Φ 6

Which of the following statement regarding sets is false.

Answer: (A U B)’ =A’ U B’ 7

Let C = {1,2,3,4} and D = {1, 2, 3, 4} then which of the following hold not true in this case.

Answer: C ∩ D = C – D 8

If C’ U (D ∩ E’) is equivalent to:

Answer: (C ∩ ( D’ U E))’

9 Let Universal set U is {1, 2, 3, 4, 5, 6, 7, 8} ,(Complement of A) A’ is {2, 5, 6, 7}, A ∩ B is {1, 3, 4} then the set B’ will surely have of which of the element. Answer: 8 10

Let a set be A then A ∩ φ and A U φ are respectively.

Answer: φ, A

1 If in sets A, B, C, the set B ∩ C consists of 8 elements, set A ∩ B consists of 7 elements and set C ∩ A consists of 7 elements then the minimum element in set A U B U C will be: Answer: 4,845 2

Let set A ={1, 2} and C be {3, 4} then A X B (Cartesian product of set A and B) is:

Answer: {(1, 3) , (2, 4), (1, 4) , (2, 3) } 3

If set A has 4 elements and B has 3 elements then set n(A X B) is:

Answer: 12 4

If set A has 3 elements then number of elements in A X A X A are:

Answer: 27 5

Which of the following statements regarding sets is false?

Answer: A X B = B X A 6

If n(A X B) = n(B X A) = 36 then which of the following may hold true?

Answer: n(A)=6, n(b)=6 7

Let the sets be A, B, C, D then (A ∩ B) X (C ∩ D) is equivalent to:

Answer: (A X C) ∩ (B X D) 8 B) is:

If set A and B have 3 and 4 elements respectively then the number of subsets of set (A X

Answer: 4096 9

If set A X B=B X A then which of the following sets may satisfy.

Answer: A={1, 2} , B={2, 1} 10

If a set contains 3 elements then the number of subsets is:

Answer: 8

Section 8 1 At a meeting, after everyone had shaken hands with everyone else, it was found that 66 handshakes were exchanged. How many were at the meeting? Answer: Union set 2

If a set is empty then number of subsets will be:

Answer: 1 3

If the number of subsets of a set are 4 then the number of elements in that sets are:

Answer: 2 4

Let a set be A={1, 2, 3} then the number of subsets containing two elements will be:

Answer: 3 5

Let the set be A= {a , b, c, {a,b}} then which of the following is false.

Answer: {a} Є A 6 3, is:

If A={1, 2, 3, 4} ,then the number of the subsets of A that contain the element 2 but not

Answer: 4 7 Let A(1), A(2), A(3), A(100) be 100 sets such that number of elements in A(i)=i+1 and A(1) is subset of A(2), A(2)is subset of A(3), A(99) is subset of A(100). The the number of elements in union of the all the sets are: n(A(1) U A(2) U A(3) U A(100)): Answer: 101 8 A function is said to be ______________ if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. Answer: One-to-one 9

The value of ⌊1/2. ⌊5/2⌋ ⌋ is ______________.

Answer: 1 10

Which of the following function f: Z X Z → Z is not onto?

Answer: f(a, b) = |b|

1 The domain of the function that assign to each pair of integers the maximum of these two integers is ___________. Answer: Z+ X Z 2 Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = 3x + 4. Then the composition of f and g is ____________. Answer: 6x + 9 3

The inverse of function f(x) = x3 + 2 is ____________.

Answer: f -1 (y) = (y – 2) 1/3 4

The g -1({0}) for the function g(x)= ⌊x⌋ is ___________.

Answer: {x | 0 ≤ x ≤ 1} 5

What is domain of function f(x)= x1/2 ?

Answer: [0, ∞) 6

What is range of function f(x) = x-1 which is defined everywhere on its domain?

Answer: (-∞, ∞) 7

If f(x) = 2x then range of the function is:

Answer: (0, ∞) 8

If f(x) = x2 + 4 then range of f(x) is given by:

Answer: [4, ∞) 9

An injection is a function which is:

Answer: one-one 10. what is domain of function f(x)=x^-1 for it to be defined everywhere on domain? Answer: (-∞, ∞)-{0}

Section 9 1

A mapping f : X -> Y is one one if:

Answer: If f(x1) = f(x2) then x1 = x2 for all x1, x2 in X. 2 A function is defined by mapping f : A -> B such that A contains m elements and B contains n elements and m ≤ n then number of one one functions are: Answer: nCm x m! 3 A function is defined by mapping f: A -> B such that A contains m elements and B contains n elements and m>n then number of one one functions are: Answer: 0 4 Set A has 3 elements and set B has 4 elements then number of injections defined from A to B are? Answer: 24 5 A function is defined by mapping f : A -> B such that A contains m elements and B contains n elements and 1≤n≤m then number of onto functions are: Answer: r=1∑r=nnCr (-1)n-r rm 6 A function is defined by mapping f: A->B such that A contains m elements and B contains n elements and m > n then number of bijections are: Answer: 0 7

A floor function map a real number to:

Answer: greatest previous integer 8

A ceil function map a real number to:

Answer: smallest following integer 9

A function f(x) is defined as f(x) = x – [x], where [.] represents GIF then:

Answer: f(x) will be fractional part of x 10

Floor(2.4) + Ceil(2.9) is equal to:

Answer: 5

1 If x, and y are positive numbers both are less than one, then maximum value of floor(x + y) is? Answer: 1 2 is?

If x, and y are positive numbers both are less than one, then maximum value of ceil(x + y)

Answer: 2 3

If X = Floor(X) = Ceil(X) then:

Answer: X is a Integer 4

A function f(x) is defined from A to B then f -1 is defined:

Answer: from B to A 5

If f is a function defined from R to R , is given by f(x) = 3x – 5 then f –1(x) is given by:

Answer: (x+5)/3 6

If f is a function defined from R to R , is given by f(x) = x2 then f –1(x) is given by:

Answer: does not exist since it is not a bijection 7

The cardinality of the set A = {1, 2, 3, 4, 6} is:

Answer: 5 8

If A is a subset of B:

Answer: Cardinality of B is greater than A 9

If A is a subset of B and B is a subset of C, then cardinaity of A U B U C is equal to:

Answer: Cardinality of C 10

If A is any statement, then which of the following is a tautology?

Answer: A ∨ ¬A

Section 10 1

If A is any statement, then which of the following is not a contradiction?

Answer: A ∨ F 2 A compound proposition that is neither a tautology nor a contradiction is called a ___________. Answer: Contingency 3

¬ (A ∨ q) ∧ (A ∧ q) is a ___________.

Answer: Contradiction 4

(A ∨ ¬A) ∨ (q ∨ T) is a __________.

Answer: Tautology 5

A → (A ∨ q) is a __________.

Answer: Tautology 6

The contrapositive of p → q is the proposition:

Answer: ¬q → ¬p 7

The inverse of p → q is the proposition:

Answer: ¬p → ¬q 8

The converse of p → q is the proposition:

Answer: q → p 9 What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?” Answer: If the home team wins, then it is not drizzling 10 What is the converse of the conditional statement “If it ices today, I will play ice hockey tomorrow. Answer: “I will play ice hockey tomorrow only if it ices today.”

1 What are the contrapositive of the conditional statement “I come to class whenever there is going to be a test. Answer: “If I do not come to class, then there will not be a test.” 2 What are the inverse of the conditional statement “ A positive integer is a composite only if it has divisors other than 1 and itself.” Answer: “If a positive integer is not composite, then it has no divisors other than 1 and itself.” 3 What are the converse of the conditional statement “When Raj stay up late, it is necessary that Raj sleep until noon.” Answer: “If Raj sleep until noon, then Raj stay up late.” 4 What are the contrapositive of the conditional statement “Medha will find a decent job when she labour hard.”? Answer: “If Medha will not find a decent job, then she not labour hard.” 5 What are the inverse of the conditional statement “If you make your notes, it will be a convenient in exams.” Answer: “If you do not make notes, then it will not be a convenient in exams.” 6 The compound propositions p and q are called logically equivalent if ________ is a tautology. Answer: p ↔ q 7

p → q is logically equivalent to:

Answer: ¬p ∨ q 8

p ∨ q is logically equivalent to:

Answer: ¬p → q 9

¬ (p ↔ q) is logically equivalent to:

Answer: p↔¬q 10 What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?” Answer: If the home team wins, then it is not drizzling

1

p ∧ q is logically equivalent to:

Answer: ¬ (p → ¬q) 2

Which of the following statement is correct?

Answer: All of mentioned 3

p ↔ q is logically equivalent to:

Answer: (p → q) ∧ (q → p) 4

(p → q) ∧ (p → r) is logically equivalent to:

Answer: p → (q ∧ r) 5

(p → r) ∨ (q → r) is logically equivalent to:

Answer: (p ∧ q) → r 6

¬ (p ↔ q) is logically equivalent to:

Answer: p ↔ ¬q 7

Let P (x) denote the statement “x >7.” Which of these have truth value true?

Answer: P (9) 8 The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people. Answer: ∀x(C(x) → F (x)) 9 The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people. Answer: ∃x (F (x) ∧ P (x)) 10 Let domain of m includes all students , P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English. Answer: No student spends more than 2 hours in playing polo

Section 11 1

Factor the expression x2 + 6x + 8 as completely as possible.

Answer: (x+4)(x+2) 2

For all real numbers x, the minimum value of 1 + 2cos(4x) is:

Answer: -1 3

Find the value of x which will satisfy the equation √(x-2) / √x = 1.

Answer: 4 4

If x = 2.0001, which of the following expressions has the largest value?

Answer: 2/(x - 2) 5

Solve: √(2x-5)-√(x-2)=2.

Answer: 27 6

If 16 is 4 more than 4x, find x-1.

Answer: 2 7

Give the factors of: a2-x2

Answer: (a+x)(a-x) 8

(a-b)3 is equivalent to which of the following?

Answer: a3-3a2b+3ab2-b3 9

Reduce the following complex fraction into simple functions.

Answer: (a-2)/(a-4) 10

Reduce the following complex fraction into simple fractions.

Answer: -x+y/x+y

1

Find the value of x which will satisfy the equation √x - 2/ √x =1.

Answer: 4 2

Solve for a in the equation: a = 64x4y.

Answer: 43x+y 3

Simplify: 3x - 3x-1 - 3x-2.

Answer: 5×3x-2 4

Which of the following is true?

Answer: 55+55+55+55+55=56 5

Simplify:∛2x4 -∛16x4 +2∛54x4.

Answer: 5∛2x4 6

Solve for x: 3x5x+1 = 6x+2.

Answer: 2.1544 7

Factor the expression x3 + 8.

Answer: (x+2)(x2-2x+4) 8

Factor the expression (x4 – y4) as completely as possible.

Answer: (x2+y2)(x+y)(x-y) 9

Simplify:

Answer: a-6b7 10

Solve for x: 37x+1 = 6561.

Answer: 1

Section 12 1

If 3a = 7b, then 3a2/7b2 =?

Answer: 7/3 2

If x to the ¾ power equals 8, then x equals:

Answer: 16 3

If 33y = 1, what is the value of y/33?

Answer: 0 4

Find the value of x that will satisfy the following expression: √(x-2)=-√x+2.

Answer: x = 9/4 5

Which of the following is equivalent to:

Answer: Figure 4

6

Simplify the following:

7a+2 – 8(7a+1) + 5(7a) + 49(7a-2). Answer: -7a 7

Solve for x:

Answer: 16/25 & 0

8

Factor the expression: 16 – 10x + x2.

Answer: (x-8)(x-2) 9

Which of the following is not an identity?

Answer: 2(x-1)+3(x+1) = 5x+4 10

Solve for x:

Answer: Any value

1

Resolve (x+2)/(x2-7x+12) into partial fraction.

Answer: 6/(x-4)-5/(x-3) 2 If the angles of the triangle are 2x, x + 15, and 2x + 15, find the smallest of the angle in mills. Answer: 800 mils 3

If 1/x + 1/y = 3 and 2/x – 1/y = 1. Then x is equal to:

Answer: 3/4 4

If 3x = 4y then (3x2)/(4y2 ) is equal to:

Answer: 4/3 5

What is the smallest value of x that satisfies the equation: x(x + 4) = -3.

Answer: -3 6

If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y = ?

Answer: 6 7

If a = 3, then 2 / (1/7 + 1/a) = ?

Answer: 21 / 5 8

If -3/(a - 3) = 3/(a + 2), then a = ?

Answer: ½ 9

Solve the z if the equation is 4 x 10-5 = z.

Answer: 0.00004 10

If 3a = 7b, then 3a2/7b2 =?

Answer: 7/3

Section 13 1

If (x+3) : 10=(3x-2): 8, find (2x-1).

Answer: 3 2

The linear distance between -4 and 17 on the number line is:

Answer: 20 3

For what value of k the equation below has no value of x: 2x + 3 = x - 2kx – 5.

Answer: -0.5 4

Find the value of A for which the equation A(2x+3)-(x-4) = 3x+10 is an identity.

Answer: 2 5

Solve for x if 8y = 3x - 11.

Answer: (8y + 11)/3 6

If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y = ?

Answer: 6 7

If f(x) = x2 + x + 1, then f(x) – f(x-1)=.

Answer: 2x 8

Solve the simultaneous equations: 3x – y = 6; 9x – y = 12.

Answer: x = 1; y = -3 9

Solve algebraically: 4x2 + 7y2 = 32 11y2 – 3x2 = 41.

Answer: y = 4, x = ±1 and y = -4, x = ±1 10

Find k so that the expression kx2 – 3kx + 9 is a perfect square.

Answer: 4

1

Which of the following quadratic equations will have two real and distinct roots?

Answer: 6x2-61x +143 = 0 2

The roots of the equation 2x2 – 3x + 20 = 0 are.

Answer: complex and unequal 3

Which of the following quadratic trinomial is NOT factorable?

Answer: 6x2- 52x – 60 4

The value of k for which the roots of 8x2+8kx+3k+2 = 0 are real and equal is:

Answer: 2 5

The degree of the polynomial f(x, y, z) = 7x3y2-4xz5+2x2y is:

Answer: 3 6

The least common multiple of (x-2)2 and x2+x-6 is:

Answer: (x-2)2(x+3) 7

Given f(x) = (x – 4) (x + 3) + 4, when f(x) is divided by x – k, the remainder is k. Find k.

Answer: 4 8

The equation y = (x-1)/ (x+2) is not defined at x =?

Answer: -2 9

If the expression x3 + 2hx - 2 is equal to 6 when x = -2, what is the value of h?

Answer: -4 10 From the equation 12x3 – 8x2 + kx + 18 = 0, find the value of k if one root is the negative of the other. Answer: -27

Section 14 1

Simplify:

Answer: y5/2 /x 2

Simplify:

Answer: 1/x2y7z3 3

Simplify:

Answer: α-6b7

4

Simplify:

Answer: 1/x3y 5

4xy – 4x2 –y2 is equal to:

Answer: –(2x-y)2 6

Find the quotient of 3x5 – 4x3 + 2x2 + 36x + 48 divided by x3 – 2x2 + 6.

Answer: 3x2+ 6x + 8 7

The quotient of (x5 +32) by (x+2) is:

Answer: x4– 2x3+ 4x2– 8x + 16 8

Find the sum and product of roots of the equation x3 + 2x2 – 23x – 60 = 0.

Answer: -2, 60 9 = 0.

Find the equation whose roots are two times the roots of the equation x3 – 6x2 + 11x – 6

Answer: x3– 12x2+ 44x – 48 = 0 10

If one root of 9x^2 – 6x + k = 0 exceed the other by 2, find the value of k.

Answer: -8

1

If ax3 + bx2 + cx + d is divided by x - 2, then the reminder is equal to.

Answer: 8a + 4b + 2c + d 2 For the remainder of the division of x3 - 2x2 + 3kx + 18 by x - 6 to be equal to zero, k must be equal to: Answer: -9 3

If one root of 9x^2-6x+k=0 exceeds the other by 2, find the value of k.

Answer: -8 4

Find k in the equation 4x2 + kx + 1 = 0 so that it will only have one real root.

Answer: 4 5

Given f(x) = (x+3)(x-4) + 4 when divided by (x-k), the remainder is k. Find k.

Answer: 4 6

The polynomial x3 + 4x2 -3x + 8 is divided by x-5. What is the remainder?

Answer: 218 7

In the quadratic equation Ax2 + Bx + C = 0, the product of the roots is:

Answer: C/A 8

In the equation 3x2 + 4x + (2h – 5) = 0, find h if the product of the roots is 4.

Answer: 17/2 9 Find the value of k in the quadratic equation (2k + 2) x2 + (4 – 4k) x + k – 2 = 0 so that the roots are reciprocal of each other. Answer: -4 10

If one root of 9x^2 – 6x + k = 0 exceed the other by 2, find the value of k.

Answer: -8

Section 15 1

If (x -3) is a factor of the polynomial x4 – 4x3 – 7x2 + kx + 24, what is the value of k?

Answer: 22 2

Find the sum of the roots 5x2 -10x + 2 = 0.

Answer: 2 3

Which of the following quadratic equations will have two real and distinct roots?

Answer: 6x2-61x +143 = 0 4

The roots of the equation 6x2 – 61x + 143 = 0 are:

Answer: real and distinct 5

Find the value of k that will make x2 – 28x +k have equal roots.

Answer: 196 6

If x:y:z =4: - 3:2 and 2x+4y-3z = 20, find the value of x.

Answer: -8 7

The roots of the equation 2x2 – 3x + 20 = 0 are:

Answer: complex and unequal 8

If x3+3x2+(5+k)x+2-k is divided by x+1 and the remainder is 3, then the value of k is:

Answer: -2 9

Which of the following quadratic trinomial is NOT factorable?

Answer: 6x2- 52x – 60 10

The value of k for which the roots of 8x2+8kx+3k+2 = 0 are real and equal is:

Answer: 2 1

The degree of the polynomial f(x, y, z) = 7x3y2-4xz5+2x2y is:

Answer: 3 2

The least common multiple of (x-2)2 and x2+x-6 is:

Answer: (x-2)2(x+3) 3

In the equation x^2 + x = 0, one root is equal to:

Answer: none of these choices

4

The roots of x^2 + x + 1 = 0 are:

Answer: none of these choices 5

(a - b)^3 = ?

Answer: a^3 - 3a^2b + 3ab^2 - b^3 6

The equation whose roots are the reciprocals of the roots of 2x2 - 3x - 5 = 0 is:

Answer: 5x2+ 3x - 2 = 0 7 The maximum possible number of positive roots that the equation x4 + 2x3 – 3x2 +bx –5 = 0 can have if b represent a positive real number is: Answer: 3 8

The roots of a quadratic equation are 1/3 and ¼. What is the equation?

Answer: 12x2– 7x + 1 = 0 9

The equation whose roots are the reciprocals of the roots of the equation 2x2 -3x - 5 = 0.

Answer: 5x2+3x–2=0 10

The remainder when 2x4-kx-15x2-3x-2 is divided by (x-3) is 4. What is the value of k?

Answer: 4

1

If x3+3x2+(5+k)x+2-k is divided by x+1 and the remainder is 3, then the value of k is____.

Answer: -2 2

Solve: 5x3 – 5x2 – 10x = 0.

Answer: 0,-1,2 3

Solve: 25y2 – 3 = 0.

Answer: +/- √3/5 4

Solve for the quadratic equation below:x2– 6x +1=0.

Answer: 3 ± √8 5

Solve for the equation below: t5-9t3= 0.

Answer: 0, 3, -3 6

Solve for the quadratic equation below:

(z-2)2 -36 =0; Answer: -4, 8 7

Solve for x in: y= 10/(3-7x).

Answer: 3y-10/7y 8

Find the sum and product of roots of the equation x3 + 2x2 – 23x – 60 = 0.

Answer: -2, 60 9

In the equation 3x2 + 4x + (2h – 5) = 0, find h if the product of the roots is 4.

Answer: 17/2 10

If (x -3) is a factor of the polynomial x4 – 4x3 – 7x2 + kx + 24, what is the value of k?

Answer: 22

Section 1.16 1. In how many ways can a party of 6 people be seated on a row of 6 seats if a certain 2 refuse to sit next to each other? Answer: 480 ways 2. A number between 1 and 10000 is randomly selected. What is the probability that it will be divisible by 4 and 5? Answer: 0.05 3. In how many ways can 9 different books be arranged on a shelf so that 3 of the books are never all together? Answer: 332,640 4. How many 4-digits even numbers can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if each digit is to be used only once in each number? Answer: 2,520 5. A guy has 8 flowers of different variety. In how many ways can he select 2 or more flowers to form a bouquet? Answer: 247 6. At a conference, after everyone had shaken hands with everyone else, it was found that 45 handshakes were exchanged. How many were at the conference? Answer: 10 7. When two dice are thrown, what is the probability that the sum of the two faces shown is 6? Answer: 5/36 8. Two dice are tossed. What is the probability that the sum of the two dice is greater than 3? Answer: 11/12 9. How many ways can 3 men and 4 women be seated on a bench if the women to be together? Answer: 576 10. In how many ways can a caravan of 9 covered wagons from India be arranged in a circle? Answer: 40320

11. Two balls are drawn one at a time from a basket containing 4 black balls and 5 white balls. If the first ball is returned before the second ball is drawn, find the probability that both balls are black. Answer: 0.198

12.In how many ways can 3 acacias, 4 narras, and 2 mahoganies be arranged along a property line if one does not distinguish among trees of the same kind? Answer: 1260 13. Allen picked 10 differently colored beads. How many different bracelets can he make if he is going to use only 6 of the 10 beads per bracelet? Answer: 12600 14. Suppose a die is rolled three times. What is the probability that all three rolls are the same? Answer: 1/36 15. Ricky and George each throw dice. If Ricky gets a sum of four what is the probability that George will get less than of four? Answer: 1/12 16. One box contains four cards numbered 1, 3,5,and 6. Another box contains three cards numbered 2, 4, and 7. One card is drawn from each bag. Find the probability that the sum is even. Answer: 5/12 17. The probability of getting credit in an examination is 1/3. If three students are selected at random, what is the probability that at least one of them got a credit? Answer: 19/27 18. In how many ways can a hostess select six luncheon guests from 10 women if she is to avoid having particular two of them together at the luncheon? Answer: 140 19. One letter is taken from each of the words PARALLEL and LEVEL at random. What is the probability of getting the same letter? Answer: 1/5 20. A card is chosen from pack of playing cards. What is the probability that it is either red or a picture card? Answer: 8/13

Section 1.17

1. Two fair dice are thrown. What is the probability that the sum of the dice is divisible by 5? Answer: 7/36 2. The captain of a baseball team assigns himself to the 4th place in the batting order. In how many ways can he assign the remaining places to his eight teammates if just three men are eligible for the first position? Answer: 15120 3. How many permutations can made out of the letters of the word ENGINEERING? Answer: 277,200 4. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that both are white. Answer: 3/28 5. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that all are of the same color. Answer: 13/28 6. A bag contains 4 red balls, 3 green balls, and 5 blue balls. The probability of not getting a red ball in the first draw is: Answer: 2/3 7. The face of a coin is either head or tail. If three coins are tossed, what is the probability of getting three tails or three heads? Answer: 1/4 8. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that one ball is white and the other is red. Answer: 15/28 9. A committee of 6 teachers is to be formed from 5 male teachers and 8 female teachers. If the committee is selected at random, what is the probability that it has an equal number of male and female teachers? Answer: 140/429 10. A recent survey reported that 60 percent of the students at a certain university are girls and 65 % of girls of this school play basketball. If a student at this school were selected at random, what is the probability that the student is a girl who plays basketball? Answer: 0.39

11. If 15 people can win prices in a estate lottery (assuming that there are no ties). How many ways can these 15 people win first, second,, third, fourth and fifth prizes? Answer: 360,360 12. A semiconductor company will hire 7 men and 4 women. In how many ways can the company choose from 9 men and 6 women who qualified for the position. Answer: 540 13. What is the probability of obtaining either four or five heads if a fair coin is tossed 10 times? Answer: 231/512 14. How many different signals each consisting of 6 flags hung in a vertical line can be formed from 4 identical red flags and 2 identical blue flags? Answer: 15 15. How many 3 digit numbers can be formed from the digits 2,4,6,8 and 9 if repetitions are allowed? Answer: 125 16. An ECE class of 40 students took examinations in Electronics and Communications. If 30 passed in Electronics, 36 passed in Communication and 2 failed in both subjects, how many students passed in both subjects? Answer: 28 17. A provincial chapter of IECEP held a lottery to raise funds for their organization, with P20,000 top prize and with 3,000 tickets printed and sold. What is the mathematical expectation of a member if she bought 30 tickets? Answer: 200 18. Rhazel draws 3 candies in succession from a jar containing 5 Kopiko candies, 6 Maxx¡¯s candies and 7 Stork candies. Find the probability of drawing these candies in order: Kopiko, Maxx, Stork. Answer: 35/816 19. How many line segments can be formed with 10 distinct points, no two of which are collinear? Answer: 45 20. In a dice game, one fair is used. The player wins P20.00 if he rolls either 1 or 6. He losses P10.00 if he turns up any other face. What is the expected winning for one roll of the die? Answer: P0.00

Section 1.18 1. At a meeting, after everyone had shaken hands with everyone else, it was found that 66 handshakes were exchanged. How many were at the meeting? Answer: 12 2. A pair of dice is tossed 10 times. What is the probability that no 7s 0r 11s appear as the sum of the sides facing up? Answer: 0.08 3. There are 3 copies each of 4 different books. In how many different ways can they be arranged on a shelf? Answer: 369,600 4. With the throw of two dice, what is the probability that the sum will be a prime number? Answer: 5/12 5. If there are 15 horses in a horse race, how many possibilities are there for the win, place, and show (first, second, and third) positions if all orders of finish are possible? Answer: 2730 6. A certain protocol uses 10 bits per message block. If the number of 1¡¯s per block is limited to exactly four, how many different message blocks can be generated? Answer: 210 7. Xyler flipped a coin 10 times. Each flip comes up either heads or tails. How many possible outcomes contain at most three tails? Answer: 176 8. How many permutations of the letters ABCDEFG contain the string BGCF? Answer: 24 9. In the ECE board examinations, the probability that an examinee pass in each subject is 0.8. What is the probability that he will pass in at least 2 subjects? Answer: 0.896 10. What is the probability of getting a 9 exactly once in 3 throws with a pair of dice? Answer: 0.263

11. How many numbers between 3000 and 5000 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 if repetition is not allowed? Answer: 240

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12. In how many ways can 10 different magazines be divide among A, B, and C so that A gets 5 magazines, B 3 magazines and C 2 magazines? Answer: 2,520

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13. A club has 25 members, 4 of whom are ECE¡¯s. In how many ways can a committee of 3 be formed so as to include at least one ECE? Answer: 970

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14. Droodles Charity Sweepstakes sold one hundred tickets, numbered 1, 2, 3, ¡-., 100 to 100 different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to the Bahamas). How many ways are there to award the prizes if the person holding ticket 13 does not win a prize? Answer: 90345024

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15. NMDA plans to enforce speed limits by using radar traps at 4 different locations within the expressway. The radar traps at each of the locations R1, R2, R3, and R4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket? Answer: 0.27

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16. NMDA plans to enforce speed limits by using radar traps at 4 different locations within the expressway. The radar traps at each of the locations R1, R2, R3, and R4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket as he passed through the radar trap located at R2? Answer: 1/9

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17. The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30minute period are 0.03, 0.18, 0.24, 0.28, 0.10, and 0.17, respectively. Find the probability that in this 30-minute period, at most 4 cars receive gas. Answer: 0.83

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18. What is the probability that a poker hand contains full house, that is, three of one kind and two of another kind? Answer: 6/4165 19. Find the probability that a hand of five cards in poker contains four cards of a kind. Answer: 1/4165 20. To avoid arrest, a syndicate places 7 narcotic tablets in a bottle containing 10 vitamin pills that are similar in appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of narcotics? Answer: 14/17

SECTION 1.19 1. The probability that a person, living in a certain town, owns a cat is estimated to be 0.3. Find the probability that the ninth person randomly interviewed in that town is the fourth one to own a cat. Answer: 0.076 Details 2. How many bit strings of length 10 contain at least three 1s and at least three 0s? Answer: 912

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3. In a certain high-school entrance examination, an examinee may select 5 problems from a set of 20 questions. In how many ways can she make her choice? Answer: 15504 Details 4. Determine the probability of getting exactly 2 tails when a coin is tossed four times. Answer: 3/8

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5. A bag contains 7 pink balls and 9 beige balls. If 2 balls are drawn in succession without replacement, find the probability that the two balls drawn are both pink. Answer: 7/40 Details 6. A certain 20 ¨C item examination has only two choices per question. What is the probability of getting exactly 15 correct answers? Answer: 969/65536

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7. What is the probability that a five-card poker hand contains the two of diamonds, the three of spades, the six of hearts, the ten of clubs, and the king of hearts? Answer: 1/52C5

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8. What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different kinds and a fifth card of a third kind)? Answer: 198/4165

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9. Dong's Pizza Company uses taste testing and statistical analysis of the data prior to marketing any new product. Consider a study involving four types of crusts (thin, thin with garlic and oregano, thin with bits of cheese and thick). Dong's is also studying four sauces, (standard, a new sauce with more garlic, a new sauce with fresh basil, and a new sauce with pandan). What is the probability that a judge will get a plain thick crust with a pandan sauce for his first taste test? Answer: 1/16 Details 10. The probability that an examinee passes the board examination is 0.8. Find the probability that the examinee will pass the examination on the fourth try. Answer: 4/625

11. If the odds against event E are 3:8, find the probability of success. Answer: 0.72

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12. The odds that A can solve a given problem are 5 to 7, and the odds that B can solve it are 3 to 6. Find the probability that either A or B can solve the problem. Answer: 3/4

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13. A marksman hits 75% of all his targets. What is the probability that he will hit exactly 4 of his next ten shot? Answer: 0.01622

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14. In a poker game consisting of 5 cards, what is the probability of holding 2 aces and 2 Queens? Answer: 33/54145

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15. Dennis Rodman sinks 50% of all his attempts. What is the probability that he will make exactly 3 of his next 10 attempts? Answer: 15/128 16. If seven coins are tossed simultaneously, find the probability that there will be at least six tails. Answer: 1/16 Details 17. A bag contains 3 white and 5 black balls. If two balls are drawn in succession without replacement, what is the probability that both balls are black? Answer: 5/14 Details 18. Five fair coins were tossed simultaneously. What is the probability of getting three heads and two tails? Answer: 5/16 Details 19. There are 4 white balls and 6 red balls in a sack. If the balls are taken out successively (the first ball is not replaced), what is the probability that the balls drawn are of different colors. Answer: 8/15 Details 20. A first bag contains 5 white balls and 10 black balls. The experiment consists of selecting a bag and then drawing a ball from the selected bag, find the probability of drawing a white ball from the first bag. Answer: 1/6

SECTION 1.20 1. A group of 4 people entered an opera house after the lights head dimmed. They are shown to the correct group of 4 seats by the usher. Each person holds a number stub. What is the probability that each is in the correct seat according to the numbers on seat and stub? Answer: 1/24

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2. There are 15 balls in a box: 8 balls are green, 4 are blue and 3 are white. Then 1 green and 1 blue balls are taken from the box and put away. What is the probability that a blue ball is selected at random from the box? Answer: 3/13

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3. A bag contains 4 white balls and 3 black balls. Another bag contains 3 white balls and 5 black balls. If one ball is drawn from each bag, determine the probability that the balls drawn will be 1 white and 1 black. Answer: 29/56

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4. An um contains 4 black balls and 6 white balls. What is the probability of getting one black ball and white ball in two consecutive draws from the urn? Answer: 0.53

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5. A bus is guaranteed to arrive at the bus stop between 8:00 am and 8:15 am. It is no more likely to arrive at any time than any other time in this interval (i.e. the probability density function is uniform). What is the probability that the bus will arrive at 8:05 am? Answer: 0

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6. A bus is guaranteed to arrive at the bus stop between 8:00 am and 8:15 am. It is no more likely to arrive at any time than any other time in this interval (i.e. the probability density function is uniform). What is the probability that the bus will arrive at 8:05 am? Answer: 1/3

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7. If 10 coins are to be flipped and the first 5 all come up heads, what is the probability that exactly 3 more heads will be flipped? Answer: 0.3125 Details 8. An organization has 26 members. How many ways are there to choose a chairman, vice chairman, secretary and treasurer of the organization, where no person can hold more than one office? Answer: 358800 Details 9. The probability for the ECE board examinees from a certain school to pass the subject in mathematics is 3/7 and for the subject of Communication is 5/7. If none of those examinees fail both subjects and there are four examinees who passed both subjects, find the number of examinees from that school who took the examinations. Answer: 28

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10 The dartboard has nine numbered blocks. Each block measuring 20cm x 20 cm. The number on each block is the score earned when a dart hits that block. A dart, which hits the unnumbered portion of the dartboard, gets a score of zero. Assuming all the darts hit the dartboard and with two darts, what is the probability of getting a total score of 11? Answer: 0.0128

11. The dartboard has nine numbered blocks. Each block measuring 20cm x 20 cm. The number on each block is the score earned when a dart hits that block. A dart, which hits the unnumbered portion of the dartboard, gets a score of zero. Assuming all the darts hit the dartboard, what is the probability of getting a score of zero with one dart? Answer: 0.64

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12. The dartboard has nine numbered blocks. Each block measuring 20cm x 20 cm. The number on each block is the score earned when a dart hits that block. A dart, which hits the unnumbered portion of the dartboard, gets a score of zero. Assuming all the darts hit the dartboard, what is the probability of getting a score of seven with one dart? Answer: 0.04

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13. An urn contains 4 white balls and 3 black balls. Another urn contains 3 white balls and 5 black balls. If one ball is drawn from each urn, determine the probability that the balls drawn will be 1 white and 1 black. Answer: 29/56

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14. An organization has 25 members, 5 of whom are ECE¡¯s. In how many ways can a committee of 3 be formed so as to include at least one ECE? Answer: 1160 15. In how many ways can a group of 6 people be seated on a row of 6 seats if a certain 2 refuse to sit next to each other? Answer: 480 ways

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16. What is the probability of drawing 6 white balls from a jar containing 9 white, 4 red, and 3 blue balls? Answer: 0.0105 Details 17. Ten books consisting of 5 mathematics books, 3 physics books, and 2 chemistry books are placed in a bookcase at random. What is the probability that the same books are all together? Answer: 1/420

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18. It is known that 3 out of 10 television sets are defective. If 2 television sets are selected at random from the 10, what is the probability that 1 of them is defective? Answer: 7/15

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19. A worker with a bunch of 10 keys is to open a locked but only one key can open. What is the probability that he will succeed in 2 trials? Answer: 0.1

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20. A bag contains 3 white balls and 5 red balls. If 2 balls are drawn in succession without returning the first ball drawn, what is the probability that the balls drawn are both red? Answer: 0.357

SECTION 1.21 1. If there are 10 distinct items taken 4 at a time, how many arrangements will there be? Answer: 5040 Details 2. In how many ways can 9 books be arranged on a shelf so that 5 of the books are always together? Answer: 14400 Details 3. How many ways can 3 men and 4 women be seated on a bench if the women are to together at all times. Answer: 576

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4. The probability that a man can marry his widowed sister is: Answer: 0

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5. How many permutation can be made out of the letters of the word ENGINEERING? Answer: 277,200

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6. In a guessing type of exam with 4 choices, what is the probability that you will get at least 7 of the 10 items? Answer: 0.0035 7. In a certain electronic factory, the ratio of the number of male to female workers is 2: 3. If one hundred new female workers are hired, the number of female workers will increase to 65% of the total number of workers. Find the original number of workers in the factory. Answer: 700 workers

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8. A card is drawn from a deck of 52 cards. Find the probability of drawing a king or a red. Answer: 0.5385 Details 9. What is the probability of obtaining at least 4 heads when a coin is tossed 5 times? Answer: 0.1875 10. Three copies of ECE books, 4 copies of EE books and 2 copies of ME books are covered with covers of different colors of each kind of book. In how many different ways can they be placed on a shelf? Answer: 1,260 ways

11. An urn contains 7 yellow balls and 10 green balls. Another urn contains 5 yellow balls and 3 green balls. If one ball is drawn from each urn, determine the probability that both are yellow. Answer: 35/136

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12. Find the probability of getting exactly 12 out of 30 questions on true or false questions. Answer: 0.08

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13. In how many ways can 9 books be arranged on a shelf so that 5 of the books are always together? Answer: 14400 Details 14. A number of ECE examinees decided to have an excursion after their licensure exam. They hire a boat for P 56. But for some reasons two of them were not able to join so that the share of the remaining in the group increased by P 3.20. How many are in the original group? Answer: 7

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5. An inexperience statistical clerk submitted the following statistics to his manager on the average rate of production of radios in an assembly line, 15 workers produce 3 radios in 2 hours. How many workers are employed in the assembly line working 40 hours per weekly production is 480 radios. Answer: 120

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16. A semiconductor company will hire 7 men and 4 women. IN how many ways can the company choose from 9 men and 6 women who is qualified for the position. Answer: 540

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17. Donaire made total bouts of 12 in 3 years. In how many ways can he end his record with 7 wins, 3 losses and 2 draws? Answer: 7920 Details 18. In how many ways can a party of 6 people be seated in a row of 6 if a certain 2 insist on sitting next to each other? Answer: 240 ways

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19. The probability that A will pass is •0†6. The probability that B will pass is 5/6. What is the probability that either A or B will pass is: Answer: 0.96

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20. In an examination, 42% failed in Algebra and 52% failed in English. If 17% students failed in both subjects, find the percentage of those students who passed in both subjects. Answer: 23%

SECTION 1.22 1. Three copies of ECE books, 4 copies of EE books and 2 copies of ME books are covered with covers of different colors of each kind of book. In how many different ways can they be placed on a shelf. Answer: 1260 Details 2. From a box containing 10 black balls, 7 blue balls and 5 white balls one ball is drawn at random. Determine the probability that it is blue or white. Answer: 6/11 Details 3. 6 books with black covers and 4 books with red covers are placed on a shelf. How many color arrangements are possible? Answer: 210

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4. In how many ways can 6 vacancies be filled with either 3 or 4 men if there are 9 men and 7 women applicants? Answer: 5,586 Details 5. How many permutation can be made out of the letters of the word ENGINEERING? Answer: 277,200

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6. If seven coins are tossed simultaneously, find the probability that they will just have three heads. Answer: 35/128

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7. A UN forces for Bosnia uses a type of missile that hits the target with a probability of 0.3. How many missiles should be fired so that there is at least an 80% probability of hitting the target?. Answer: 5

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8. In an election, one of the two candidates got 42% of the total votes and still lost by 3,680 votes. Find the total number of votes. Answer: 23,000 9. The value of probability of any outcome will never be equal to or exceed. Answer: 1

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10. If two events A and B are mutually exclusive events and the probability that A or B will happen is: Answer: Pa + Pb

1. Find the probability of getting exactly 12 out of 30 questions on true or false questions. Answer: 0.08

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2. In how many ways can 9 books be arranged on a shelf so that 5 of the books are always together? Answer: 14400 Details 3. Four electronics and three communication books are placed in a shelf. What is the probability that the Communications book will be together? Answer: 0.1428 Details 4. A door password consists of 2 digits from 0 to 9 and 2 letters of the English alphabet. How many different distinct passwords are possible if repetitions are not allowed? Answer: 58500 Details 5. Manny Pacquiao made total bouts of 12 in 3 years. In how many ways can he end his record with 7 wins, 3 losses and 2 draws? Answer: 7920 Details 6. From the digits 0, 1, 2, 3, 4, 5, 6 ,7, how many 3-digit odd numbers that are distinct can be formed? Answer: 144

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7. In a family of 5 children, what is the probability that the first three are boys? Answer: 1/8

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8. In how many ways you can invite 1 or more of your 5 friends in a certain show? Answer: 31

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9. During a certain war, Philippines troop released 4 bombs to USA each of which has a probability of hitting the enemy of 0.25. If 4 bombs are released simultaneously, what is the probability of hitting the Americans? Answer: 0.6835 Details 10. Oscar and Des work independently in troubleshooting a circuit. The probability that that Oscar can fix it is 0.6 while the probability that Des can fix it is 0.5. What is the probability that the circuit will be fixed? Answer: 0.8

1. If the probability that an event will happen exactly three times in five trials is equal to the probability that it will happen exactly two times in six trials, find the probability that it will happen in one trial. Answer: 0.451 Details 2. There are three candidates Nelson, Cynthia, Bogs for mayor of a certain town. If the odds are 7:5 that Nelson will win and those of Cynthia are 1:3, what is the probability that Bogs will win? Answer: 1/6

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3. In a certain electronic factory, the ratio of the number of male to female workers is 2:3. If 100 new female workers are hired, the number of female workers will increase to 65% of the total number of workers. Find the original number of female workers in the factory. Answer: 420

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4. A person draws 3 balls in succession from a box containing 5 red balls, 6 yellow balls and 7 green balls. Find the probability of drawing the balls in the order red, yellow and green. Answer: 0.04289

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5. In the ECE board examinations, the probability that an examinee will pass in each subjects is 0.8. What is the probability that he will pass in at least two subjects? Answer: 0.896 Details 6. In how many ways can 6 distinct books be arranged in bookshelf? Answer: 720

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7. In how many ways can 3 marines and 4 army soldiers be seated on a bench if the marines must be seated together? Answer: 720 ways

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8. Donaire made total bouts of 12 in 3 years. In how many ways can he end his record with 7 wins, 3 losses and 2 draws? Answer: 7920 Details 9. The probability that A will pass is •0†6. The probability that B will pass is 5/6. What is the probability that either A or B will pass is: Answer: 0.96

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10. From a box containing 10 black balls, 7 blue balls and 5 white balls one ball is drawn at random. Determine the probability that it is blue or white. Answer: 6/11

SECTION 1.23 1. A group of 7 friends are having lunch together. Each person eats at least 3/4 of a pizza. What is the smallest number of whole pizzas needed for lunch? Answer: 6

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2. z varies directly as x and inversely as y2. If x = 1 and y = 2, then z = 2. Find z when x = 3 and y = 4. Answer: 1.5

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3. Find the third proportional to 4 and 12. Answer: 36

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4. For the remainder of the division of x3 - 2x2 + 3kx + 18 by x - 6 to be equal to zero, k must be equal to. Answer: -9

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5. In what ratio must a peanut costing P240.00 per kg. Be mixed with a peanut costing P340.00 per kg so that the profit of 20% is made by selling the mixture at 360.00 per kg? Answer: 2:3

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6. The sum of two numbers is 21 and their product is 108. Find the sum of their reciprocals. Answer: 7/36 Details 7. Two friends A and B are respectively 5 and 8 years old. In how many years will the ratio of their ages be 3:4? Answer: 4

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8. If a two digit number has x for its unit¡¯s digit and y for its ten¡¯s digit, represent the number. Answer: 10y + x 9. The denominator of a certain fraction is three more than twice the numerator. If 7 is added to both terms of the fraction, the resulting fraction is 3/5. Find the original fraction. Answer: 5/13 Details 10. Given that w varies directly as the product of x and y and inversely as the square of z and that w=4 when x=2, y=6, and z=3. Find w when x=1, y=4 and z=2. Answer: 3

1. If x varies directly as y and inversely as z, and x=14 when y=7 and z=2, find x, when z=4 and y=16. Answer: 16

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2. The electrical resistance of a cable varies directly as its length and inversely as the square of its diameter. If a cable 600 meters long and 25 mm in diameter has a resistance of 0.1 ohm, find the length of the cable 75 mm in diameter with resistance of 1/6 ohm. Answer: 9000 m Details 3. The electrical resistance offered by an electric wire varies directly as the length and inversely as the square of the diameter of the wire. Compare the electrical resistance offered by two pieces of wire of the same material, one being 100 m long and 5 mm diameter, and the other is 50 m long and 3 mm in diameter. Answer: R1= 0.72 R2 4. The time required for an elevator to lift a weight varies directly with the weight and the distance through which it is to be lifted and inversely as the power of the motors. If it takes 20 seconds for a 5-hp motor to lift 50 lbs. through 40 feet, what weight can an 80-hp motor lift through a distance of 40 feet within 30 seconds? Answer: 1200 lbs.

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5. The time required by an elevator to lift a weight, vary directly with the weight and the distance through which it is to be lifted and inversely as the power of the motor. If it takes 30 seconds for a 10-hp motor to lift 100lbs through 50 feet, what size of motor is requires to lift 800 lbs. in 40 seconds through a distance of 40 feet? Answer: 48 hp

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6. The vibration frequency of a string varies as the square root of the tension and inversely as the product of the length and diameter. Of the testing. If the testing is 3ft long and 0.03 inch diameter vibrates at 720 times per second under 90 pounds tension, at what frequency will a 2ft, 0.025 inch string vibrate under 50 pounds tension. Answer: 966

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7. The volume of a hemisphere varies directly as the cube of its radius. The volume of the hemisphere with 2.54cm, radius is 20.75 cm3. What is the volume of a sphere with 3.25cm.radius of the same kind of material? Answer: 86.92

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8. The enrollment at college A and college B both grew up by 8% from 1980 to 1985. If the enrollment in college A grew up by 800 and the enrollment in college B grew up by 840, the enrollment at college B was how much greater than the enrollment in college A in 1985? Answer: 540

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9. A boat takes 2/3 as much time to travel downstream from C to D, as to return, If the rate of the river¡¯s current is 8 kph, what is the speed of the boat in still water? Answer: 40

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10. If Juan is 10% taller than Pedro and Pedro is 10% taller than Maria, then Juan is taller than Maria by how many percent? Answer: 21%

SECTION 1.24 1. The population of the Philippines doubled in the last 30 years from 1967 to 1997. Assuming that the rate of population rate increase will remain the same in what year wills the population triple? Answer: 2014

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2. Missy goes to a bake shop to buy some pastries for resale at Book Latte. She spends half her money for Revel Bars, and one-third of what remains for Triple Chocolate Brownies. She spends 150 for other pastries and still has 200 left from the amount she originally had. How much money did she have at the start? Answer: 1050

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3. Arcadio Corporation’s gross margin is 45% of sales. Operating expenses such sales and administration are 15% of sales. Arcadio is in 40% tax bracket. What percent of sales is their profit after taxes? Answer: 18%

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4. A boat propelled to move at 25 mi/hr in still water, travels 4.2 miles upstream in the same time that it can travel 5.8 miles downstream. Find the speed of the stream. Answer: 4

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5. The velocity of an airplane in still air is 125 kph. The velocity of the wind due east is 25 kph. If the plane travels east and returns back to its base again in 4 hours, at what distance does the plane travel due east? Answer: 240 km. Details 6. It takes a boat 3 times to travel upstream against a river current that it takes the same boat to travel downstream. If the speed of the boat is 40 kph, what is the speed of the current? Answer: 20 mph Details 7. Pedro runs with a speed of 20 kph. Five minutes later, Mario starts running to catch Pedro in 20 minutes. Find the velocity of Mario. Answer: 25 kph Details 8. A man rows downstream at the rate of 5 mph and upstream at the rate of 2 mph. How far downstream should he go if he is to return 7/4 hour after leaving? Answer: 2.5 mi

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9. A ship propelled to move at 25 mi/ hr in still water, travels 4.2 miles upstream in the same time that it can travel 5.8 miles downstream. Find the speed of the stream. Answer: 4

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10. If x varies directly as y and inversely as z, and when and , find x when z=4 and y=16. Answer: 16

1. The second of the four numbers is three less than the first, the third is four more than the first and the fourth is two more than the third. Find the fourth number if their sum is 35? Answer: 13

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2. A jogger starts a course at a steady rate of 8 kph. Five minutes later, a second jogger starts the same course at 10 kph. How long will it take the second jogger to catch the first? Answer: 20 min Details 3. An airplane flying the wing, took 2 hours to travel 1000 km and 2.5 Hours in flying back. What was the wind velocity in kph? Answer: 50

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4. Two planes leave Manila for a southern city, a distance of 900km. Plane A travels as a ground speed of 90 kph faster than the plane B. Plane A arrived in their destination 2 hours and 15 min ahead a plane B. What is the ground speed of plane A? Answer: 240 kph Details 5. On a certain trip, Ben drive 231 km in exactly the same time as Allan drive 308 km. If Allan¡¯s rate exceeded that of Ben by 13 kph, determine the rate Allan. Answer: 52 kph Details 6. The electric power which a transmission line can transmit is proportional to the product of its design voltage and current capacity, and inversely to the transmission distance. A 115-kilovolt line rate at 100 amperes can transmit 150 megawatts over 150 km. how much power, in megawatts can a 230 kilovolt line rate at 150 amperes transmit over 100 km? Answer: 675

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7. Find the ratio of an infinite geometric progression if the sum is 2 and the first term is 1/2. Answer: 3/4

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8. A hiker walks from his car to a distant lake and back again. He walks on smooth terrain for 2 hours until he reaches a 5-mile-long, rocky trail. His pace along the 5-mile-long trail is 2 mph. If he walks steadily with no stops, how long will it take the hiker to complete the entire trip from his car to the lake and back again? Answer: 9 hours Details 9. White flour and whole wheat flour are mixed together in a ratio of 5 parts white flour to 1 part whole wheat flour. How many pounds of white flour are in 48 pounds of this mixture? Answer: 40 pounds

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10. Timmy can sell 20 glasses of lemonade for 10 cents per glass. If he raises the price to 25 cents per glass, Timmy estimates he can sell 7 glasses. If so, how much more money will Timmy make by charging 25 cents instead of 10 cents per glass? Answer: -$0.25

1. A four-pound mixture of raisins and nuts is 2/3 raisins. How many pounds of nuts are there? Answer: 1.3 pounds

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2. To make 20 biscuits, Juanita uses 5 cups of flour to 1 cup of milk. If she uses 3 cups of milk, how many cups of flour will she use? Answer: 15

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3. When 2,000 pounds of paper are recycled or reused, 17 trees are saved. How many trees are saved if 5,000 pounds of paper is recycled? Answer: 42.5

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4. On a certain map, 1 cm = 12 km actual distance. If two places are 96 km apart, what is their distance on map? Answer: 8 cm Details 5. A person types 360 words in 4 minutes. How much time does he take to type 900 words? Answer: 10

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6. Two numbers are in the ratio 3 : 4. If the sum of numbers is 63, find the numbers. Answer: 27, 36 Details 7. If 2A = 3B = 4C, find A : B : C. Answer: 6:4:3 Details 8. What must be added to each term of the ratio 2 : 3, so that it may become equal to 4 : 5. Answer: 2

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9. The length of the ribbon was originally 30 cm. It was reduced in the ratio 5 : 3. What is its length now? Answer: 18

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10. Find the ratio of an infinite geometric progression if the sum is 2 and the first term is 1/2. Answer: 3/4

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SECTION 1.25 1. Mary was four times as old as Lea ten years ago. If she is now twice as old as Lea, how old is Mary. Answer: 30

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2. Jose’s rate of doing work three times as fast as Bong. On given day Jose and Bong work together for 4 hours then Bong was called away and Jose finishes the rest of the job in 2 hours. How long would it take Bong to do the complete job alone? Answer: 22 hrs. Details 3. A solution is made of water and pure acid. If 75% of the solution is water, how many litters of pure acid are in 20 liters of this solution? Answer: 5

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4. Three men A, B, and C can do a piece of work in t hours working together. Working alone, A can do the work in 6 hours more, B in 1 hour more, and C in twice the time if all working together. How long would it take to finish the work if all working together? Answer: 40 mins.

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5. Rukia has nickels, dimes, and quarters amounting to $1.85. If he has twice as many dimes as quarters, and the number of nickels is two less than twice the number of dimes, how many quarters does he have? Answer: 3

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6. How many liters of a 25% acid solution must be added to 80 liters of a 40% acid solution to have a solution that is 30% acid? Answer: 160L Details 7. What time after 2 o¡¯clock will the hands of the clock extend in opposite directions for the first time? Answer: 2:43.64

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8. Seven carpenters and 5 masons earn a total of 2,300 per day. At the same rate of pay 3 carpenters and 8 masons earn 2,040. What are the wages per day of the carpenter and a mason? Answer: 200 & 180

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9. A man and a boy can do 15 days a piece of work which would be done by 7 men and 9 boys in 2 days. How long would it take one man do it alone? Answer: 20 days

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10. Ana is 5 years older than Beth. In 5 years, the product of their ages is 1.5 times the product of their present ages. How old is Beth now? Answer: 20

1. An audience of 540 people is seated in rows having the same number of persons in each row. If 3 more persons seat in each row, it would require 2 rows less to seat the audience. How many persons were in each row originally? Answer: 27

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2. Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is now. How old is Ana? (ECE November 1995) Answer: 18

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3. One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4 hours. A drainpipe can empty the full content of the tank in 20 hours. With all the three pipes open, how long will it take to fill the tank? Answer: 2.5 hours

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4. A swimming pool is filled through its inlet pipe and then emptied through its outlet pipe in a total of 8 hours. If water enters through its inlet and simultaneously allowed to leave through its outlet, the pool is filled in 7 •0†5 hours. Find how long will it take to fill the pool with the outlet closed. Answer: 3

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5. An alloy of silver and gold weighs 15 oz. in air and 14 oz. in water. Assuming that silver losses 1/10 of its weight in water and gold losses 1/18 of its weight, how many oz. at each metal are in the alloy? Answer: Silver = 3.75 oz.; Gold = 11.25 oz.

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6. A 100-kilogram salt solution originally 4% by weight. Salt in water is boiled to reduce water content until the concentration is 5% by weight salt. How much water is evaporated? Answer: 20

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7. A father is now 41 and his son 9. After how many years will his age be just triple his son¡¯s age? Answer: 7

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8. Your father told you ¡°I was your age now when you were born¡±. If you are 21 years old today, how old is your father? Answer: 42

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9. Mark has nickels, dimes, and quarters amounting to $1.85. If he has twice as many dimes as quarters, and the number of nickels is two less than twice the number of dimes, how many quarters does he have? Answer: 3

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10. A US dollar equals 0.716 European euros and a Japanese yen equals 0.00776 European euros. How many US dollars equal a Japanese yen? Answer: 0.011

SECTION 1.26 1. Two times the father’s age is 8 more than six times his son’s age. Ten years ago, the sum of their ages was 44. The age of the son is: Answer: 15

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2. The ages of the mother and her daughter are 45 and 5 years, respectively. How many years will the mother be three times as old as her daughter? Answer: 15

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3. How many minutes after 10:00 o’clock will the hands of the clock be opposite of the other for the first time? Answer: 21.81 Details 4. A man travels in a motorized banca at rate of 12 kph from his barrio to the poblacion and come back to his barrio at the rate of 10 kph. If his total time of travel back and forth is 3 hours and 10 minutes, the distance from the barrio to the poblacion is: Answer: 17.27 km

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5. Nonoy left Pikit to drive to Davao at 6:15 PM and arrived at 11:45 PM averaged 30 mph and stopped 1 hour for dinner, how far is Davao from Pikit. Answer: 135

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6. From the time 6:15 PM to the time 7:45 PM of the same day, the minute hand of a standard clock describes an arc of: Answer: 540

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7. What time between the hours of 12:00 noon and 1:00 pm would the hour hand and the minute hand of a continuously driven clock be in straight line? Answer: 12:33 pm

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8. A job could be done by eleven workers in 15 days. Five workers started the job. They were reinforced with four more workers at the beginning of the 6th day. Find the total number of days it took them to finish the job. Answer: 20.56 Details 9. Delia can finish a job in 8 hours. Daisy can do it in 5 hours. If Delia worked for 3 hours and then Daisy was asked to help her finish it, how long will Daisy have to work with Delia to finish the job? Answer: 1.923 hours

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10. Three persons can do a piece of work alone in 3 hours, 4 hours and 6 hours respectively. What fraction of the job can they finish in one hour working together? Answer: 3/4

1. At what time after 12:00 noon will the hour hand and the minute hand of a clock first form a n angle of 120? Answer: 12:21.818

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2. Mr. Brown can wash his car in 15 minutes, while his son John takes twice as long as the same job. If they work together, how many minutes can they do the washing? Answer: 10

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3. A 40-gram alloy containing 35% gold is to be melted with a 20-gram alloy containing 50% gold. How much percentage of gold is the resulting alloy? Answer: 40%

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4. Two thousand kilogram of steel containing 8% of nickel is to be made by mixing steel containing 14% nickel with another steel containing 6% nickel. How much of the steel containing 14% nickel is needed? Answer: 500kg Details 5. A father is three times as old as his son. Four years ago, he was four times as old as his son was at that time. How old is his son? Answer: 12 years

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6. A production supervisor submitted the following report on the average role of production of printed circuit boards (PCB) in an assembly line, 1.5 workers produce 12 PCB’s in 2 hours. How many workers are employed in an assembly line working 40 hours each week with a weekly production of 8000 PCB’s? Answer: 50

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7. Pedro can paint a fence 50% faster than Juan and 20% faster that Pilar and together they can paint a given fence in 4 hours. How long will it take Pedro to paint the same fence if he had to work alone? Answer: 10 hrs Details 8. What time after 2 o’clock will the hands of the clock extend in opposite directions for the first time? Answer: 2:43.64

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9. How many liters of a 25% acid solution must be added to 80 liters of a 40% acid solution to have a solution that is 30% acid? Answer: 160 L Details 10. Given two numbers such that the difference of twice the first and the second number is 12. If the sum of the first and the second number is 36, find the number. Answer: 16, 20

SECTION 1.27 1. A carpenter and his helper together can repair a house in 10 days. It takes the helper 5 days longer than the carpenter to do the repair when each works alone. How many days would it take the helper to do the repair if he is to work alone? Answer: 22.8 days

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2. Two workers A and B together can complete a job in 7 days. A works twice as fast as B. How many days would it take B to do the job working alone? Answer: 21days 3. If soldering lead contains 63% silver, ______ grams of soldering lead can be made from 520 grams of silver. Answer: 825.4 Details 4. Pipes A and B can fill an empty tank in 6 and 3 hours respectively. Drain C can empty a full tank in 24 hours. How long will an empty tank be filled if pipes A and B with drain C open? Answer: 2.182 hours

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5. It takes pump (A) 4 hours to empty a swimming pool. It takes pump (B) 6 hours to empty the same swimming pool. If the two pumps are started together, at what time will the two pumps have emptied 50% of the water in the swimming pool? Answer: 1 hour 12 minutes

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6. If Chicago is 10% taller than Ishida and Ishida is 10% taller than Chad, then Ichigo is taller than Chad by how many percent? Answer: 21%

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7. If thrice the smaller number exceeds the larger by 12. Find the larger number if the two numbers are consecutive odd integers. Answer: 9 8. An investor has P100,000, part of which he invested at 12% interest and the rest at 18%. He received a total annual interest of P15,300. How much did he invest at 18% interest rate? Answer: 55,000 9. An object travels at fifteen feet per minute. How many feet does it travel in 24 minutes and 40 seconds? Answer: 370

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10. The sum of the digits of a two digit number is 11. If the digits are reversed, the resulting number is seven more than twice the original number. What is the original number? Answer: 38

1. Crew 1 can finish the installation of an antenna tower in 200 hours while crew 2 can finish the same job in 300 hours. How long will it take both crews to finish the same job working together? Answer: 120 hours

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2. The time required for two examinees to solve the same problems differs by two minutes. Together they can solve 32 problems in one hour. How long will it take for the slower problem solver to solve the problem? Answer: 5min Details 3. Wendy was four times as old as Lily ten years ago. If she is now twice as old as Lily, how old is Wendy? Answer: 30

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4. In how many minutes after 2 o¡¯clock will the hands of the clock extend in opposite directions for the first time? Answer: 2: 43 min & 38 sec

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5. Peter’s age 13 years ago was 1/3 of his age 7 years hence. How old is Peter? Answer: 23

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6. A man is 41 years old and in seven years he will be four times as old as his son is at that time. How old is his son now? Answer: 5

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7. A pump can pump out a tank in 11 hours. Another pump can pump out the same tank in 20 hours. How long it will take both pumps together to pump out the tank? Answer: 7 hours

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8. The sum of the reciprocals of two numbers is 11. Three times the reciprocal of one of the numbers is three more than twice the reciprocal of the other number. Find the numbers. Answer: 1/5 and 1/6

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9. One number if five less than the other number. If their sum is 135, what are the numbers? Answer: 65&70 Details 10. In a two-digit number, the unit¡¯s digit is 3 greater than the ten’s digit. Find the number if it is 4 times as large as the sum of its digits. Answer: 36

SECTION 1.28 1. Find two consecutive even integers such that the square of the larger is 44 greater than the square of the smaller integer. Answer: 10&12 Details 2. The product of three consecutive integers is 9240. Find the third integer. Answer: 22

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3. The sum of the digits of the three-digit number is 14. The hundreds digit being 4 times the units digit. If 594 is subtracted from the number, the order of the digits will be reversed. Find the number. Answer: 842

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4. The sum of two numbers is 21, and one number is twice the other. Find the numbers. Answer: 7 and 14

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5. Ten less than four times a certain number is 14. Determine the number. Answer: 6

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6. Kim sold a watch for P3500.00 at a loss of 30% on the cost price. Find the corresponding loss or gain if he sold it for P5050.00. Answer: 1% gain

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7. A certain XEROX copier produces 13 copies every 10 seconds. If the machine operates without interruption, how many copies will it produce in an hour? Answer: 4680 Details 8. At a certain printing plant, each of the machines prints 6 newspapers every second. If all machines work together but independently without interruption, how many minutes will it take to print the entire 18000 newspapers? ( Hint: let x = number of machines) Answer: 50/x Details 9. A 10-meter tape is 5 mm short. What is the correct length in meters? Answer: 9.995 m

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10. A man rows downstream at the rate of 5mph and upstream at the rate of 2mph. How far downstream should he go if he is to return in 7/4 hours after leaving? Answer: 2.5 mi

1. The distance passed over by a certain pendulum bob in succeeding swings form the geometric progression 16,12,9,... feet respectively. Calculate the total distance traversed by the bob before coming to rest. Answer: 64 ft Details 2. A man has P100,000, part of which he invested at 12% interest and the rest at 18%. He received a total annual interest of P15,300. How much did he invest at 18% interest rate? Answer: 55,000 Details 3. A man who is on diet losses 24 lb. in 3 months, 16lb. in the next 3 months and so on for a long time. What is the maximum total weight loss? Answer: 72

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4. Ten less than four times a certain number is 14. Determine the number. Answer: 6

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5. In a club of 40 executives, 33 likes to smoke Marlboro and 20 like to smoke Philip Moris. How many like both? Answer: 13

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6. The tens digit of a two-digit number is 1 less than twice the unit¡¯s digit. They differ by 4. Find the number. Answer: 95

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7. A certain two-digit numbers is 1 less than five times the sum of its digits. If 9 were added to the number, its digits would be reversed. Find the number. Answer: 34

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8. The length of a rectangle is 3 times its width. If the width of the rectangle is 5 inches, what is the rectangle's area, in square inches? Answer: 75

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9. The sum of three succeeding odd integers is 75. The largest integer is: Answer: 27

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10. Two times the father¡¯s age is 8 more than six times his son¡¯s age. Ten years ago the sum of their ages is 44. The age of the son is: Answer: 15

1. Minnette’s age 13 years ago was 1/3 of her age 7 years hence. How old is Minnette? Answer: 23

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2. Of the 316 people watching a movie, there are 78 more children than women and 56 more women than men. The number of men in the movie house is: Answer: 42

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3. The sum of the reciprocal of two numbers is 11. Three times the reciprocal of one of the numbers is three more than twice the reciprocal of the other number. Find the numbers? Answer: 1/5 & 1/6

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4. In a racing contest, there are 240 cars which will have fuel provision that will last for 15 hours. Assuming a constant hourly consumption for each car, how long will the fuel provision last if 8 cars withdraw from the race every hour after the first. Answer: 25 hours

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5. A number is divided into two parts such that when the greater part is divided by the smaller part, the quotient is 3, and the remainder is 5. Find the smaller number if the sum of the two numbers is 37. Answer: 8

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6. A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work? Answer: 18.9

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7. Eight men can dig 150 ft of trench in 7hrs. Three men can backfill 100ft of the trench in 4hrs. The time it will take 10 men to dig and fill 200 ft of trench is: Answer: 9.867hrs.

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8. How many minutes after 3:00 will the minute hand of the clock overtakes the hour hand? Answer: 16-4/11 minutes

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9. It is now between 3 and 4 o¡¯clock and in twenty minutes the minute hand will be as much as the hour-hand as it is now behind it. What is the time now? Answer: 3:06.06

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10. The product if two numbers is 1400. If three (3) is subtracted from each number, their product becomes 1175. Find the bigger number. Answer: 50

SECTION 1.29 1. Find the mean, median and mode respectively of the following numbers: 13, 13, 14, 12, 11, 10, 9, 11, 8, 11, 5, and 15 Answer: 11, 11, 11

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2. The standard deviation of the numbers 1, 4, &7 is: Answer: 3

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3. The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint. 3.4

2.5

4.8

2.9

3.6

2.8

3.3

5.6

3.7

2.8

4.4

4.0

5.2

3.0

4.8

Calculate the sample mean for this data. Answer: 3.787 Details 4. A random sample of 20 pieces of cotton fiber is taken and the absorbency on each piece was measured. The following are the absorbency values: 18.71

21.41

20.72 21.81

19.29

22.43

20.17 23.71

19.44

20.50 18.92

20.33

23.00

22.85 19.25

21.77

22.11

19.77 18.04

21.12

Calculate the median for the above sample values. Answer: 20.61 Details 5. Given the data set: 1.7, 2.2, 3.9, 3.11 and 14.7 Determine the sample mean and median respectively. Answer: 5.12, 3.9

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6. A sample size of 10 is taken with results given by: 7.07, 7.00, 7.10, 6.97, 7.00, 7.03, 7.01, 6.98, 7.08, 7.01. Calculate the sample variance. Answer: 0.0019

7. The following data represent the length of life in years, measured to the nearest tenth of 39 similar fuel pumps: 2.0 3.0 0.3 3.3 1.3 0.4 0.2 6.0 5.5 6.5 0.2 2.3 1.5 4.0 5.9 1.8 4.7 0.7 4.5 0.3 1.5 0.5 2.5 5.0 1.0 6.0 5.6 6.0 1.2 0.2 Compute for the sample range. Answer: 6.3

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8. Compute the mean value of the numbers from the tabulated values shown. (Numbers)

(Probability)

5.3

0.38

1.8

0.49

-2.6

0.13

Answer: 2.558 Details 9. Find the mode of the given set of numbers: 5,6,6,7,7,7,8,8,9,9,9,10,10,13 Answer: 7 and 9 10. Find the range of the set of number 9,3,8,8,6,5,11 and 15. Answer: 12

1. What is the variance of the following data {1, 2, 3, 4, and 5}? Answer: 2.5

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2. Given the following samples of distributed population {100, 200, 300, 350, 375, 375, 450}, what is the median? Answer: 350

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3. Find the standard deviation of the following -14, -4, -10, -12, 5, -1. Answer: 6.658 Details 4. In a lotto, there are pingpong balls numbered from 1 to 45. Six balls are drawn 1 at a time, to determine the winning combination in any order. Determine the probability that you will win the jackpot prize. Answer: 1/8145060

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5. There are 5 pocket holes at the periphery of a round horizontal platform. In how many ways can 5 balls of different colors be placed with 1 ball in each pocket. Answer: 24

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6. In statistics, a pictorial description of the probability concepts of independent and dependent events is called Answer: Venn diagram Details 7. What is the standard deviation of 2, 4, 6, 8 and 10? Answer: 2.83

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8. In a club of 40 executives, 33 like to smoke Marlboro and 20 like to smoke Philip Morris. How many like both? Answer: 13

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9. A class of 40 students took examination in Electronics and Communications. If 30 passed in Electronics, 36 passed in Communications and 2 failed in both subjects, how many students passed in both subjects? Answer: 28

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10. In a class of 40 students, 27 like Calculus and 25 like Chemistry. How many like both Calculus and Chemistry ? Answer: none of these are correct

SECTION 1.30 1. Find the mean, median, mode and range of this data set: 19, 18, 21, 16, 15, 17, 20, 18 Answer: 18,18,18,6

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2. On his first three quizzes, Patrick earned a 15, 18, and 16. (A perfect score would have been 20 points.) What does he need to earn on the next quiz to have a mean score of at least 17? Answer: 19

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3. Find the median of the set of numbers - 1,2,3,4,5,6,7,8,9 and 10. Answer: 5.5

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4. Find the median of the set of numbers - 21, 3, 7, 17, 19, 31, 46, 20 and 43. Answer: 20

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5. Find the median of the set of numbers - 100, 200, 450, 29, 1029, 300 and 2001. Answer: 300

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6. The following represents age distribution of students in an elementary class. Find the mode of the values - 7, 9, 10, 13, 11, 7, 9, 19, 12, 11, 9, 7, 9, 10, 11. Answer: 9

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7. Find the mode from these test results ¨C 90, 80, 77, 86, 90, 91, 77, 66, 69, 65, 43, 65, 75, 43, 90. Answer: 90

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8. Find the mode from these test results - 17, 19, 18, 17, 18, 19, 11, 17, 16, 19, 15, 15, 15, 17, 13, 11. Answer: 17

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9. Find the mean of these set of numbers - 100, 1050, 320, 600 and 150. Answer: 444

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10. The following numbers represent the ages of people on a bus ¨C 3, 6, 27, 13, 6, 8, 12, 20, 5, 10. Calculate their mean of their ages. Answer: 11

1. These numbers are taken from the number of people that attended a particular church every Friday for 7 weeks - 62, 18, 39, 13, 16, 37, 25. Find the mean. Answer: 30

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2. Find the mean of the set of ages in the table below:

Answer: 12.4

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3. The heights (in cm) of students of a class is given to be 163, 158, 167, 174, 148. Find the variance. Answer: 76.4

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4. There are 500 staffs in a company. The hours used in leisure per week by 5 of the employees is 5, 2, 8, 10, 7. Find the sample variance. Answer: 9.3

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5. Scores on a history test have average of 80 with standard deviation of 6. What is the z-score for a student who earned a 75 on the test? Answer: -0.833

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6. The weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with standard deviation of .1 ounce. What is the z-score corresponding to a weight of 8.17 ounces? Answer: 1.7

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7. Books in the library are found to have average length of 350 pages with standard deviation of 100 pages. What is the z-score corresponding to a book of length 80 pages? Answer: -2.7

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8. The temperature is recorded at 60 airports in a region. The average temperature is 67 degrees Fahrenheit with standard deviation of 5 degrees. What is the z-score for a temperature of 68 degrees? Answer: 0.2

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9. A group of friends compares what they received while trick or treating. They find that the average number of pieces of candy received is 43, with standard deviation of 2. What is the z-score corresponding to 20 pieces of candy? Answer: -11.5

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10. The mean growth of the thickness of trees in a forest is found to be .5 cm/year with a standard deviation of .1cm/year. What is the z-score corresponding to 1 cm/year? Answer: 5

1. Find the skewness of the following data: 500, 1000, 1000, 1500, 1500, 1500, 2000, 2000, 2000, 2000, 2500, 2500, 2500, 3000, 3000, 3500 Answer: 0

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2. You are given these numbers: 1, 2, 3, 4, 5, 9, 23, 32, and 69. Find the skewness. Answer: 1.53

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3. Calculate the skewness of given data; x: 2,3,4,5 ? Answer: 0

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4. Calculate the skewness of following data; x: 5, 10, 15 Answer: 0

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5. If the mean is greater than the mode, the distribution is ___. Answer: Positively skewed

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6. If the mean is less than the mode, the distribution is ___. Answer: Negatively skewed

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7. The following represents age distribution of students in an elementary class. Find the mode of the values - 7, 9, 10, 13, 11, 7, 9, 19, 12, 11, 9, 7, 9, 10, 11. Answer: 9

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8. Find the mode from these test results - 90, 80, 77, 86, 90, 91, 77, 66, 69, 65, 43, 65, 75, 43, 90. Answer: 90

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9. The following numbers represent the ages of people on a bus ¨C 3, 6, 27, 13, 6, 8, 12, 20, 5, 10. Calculate their mean of their ages. Answer: 11. Details 10. There are 5 pocket holes at the periphery of a round horizontal platform. In how many ways can 5 balls of different colors be placed with 1 ball in each pocket. Answer: 24

1. The positive values of x so that 4x, 5x + 4, 3x2 – 1will be in arithmetic progression is: Answer: 3 2. There are line (9) arithmetic means between 11 and 51. The sum of the progression is: Answer: 341 3. Find the quadratic mean of the numbers 8,9,9,13. Answer: 9.94 4 .Find the 100th term of the sequence 1.01, 1.00, 0.99… Answer: 0.02 5. The geometric mean of a and b is: Answer: √ab 6. The fourth term of a geometric progression is 189 and the sixth term is 1701, the 8th term is: Answer: 15309 7. The geometric mean and the harmonic mean of two numbers are 12 and 36/5 respectively. What are the numbers? Answer: 36 & 4 8. If x, 4x+8, 30x +24 are in geometrical progression, find the common ratio. Answer: 6 9. The 1st, 4th, 8th terms of an A.P. are themselves geometric progression (G.P.). What is the common ratio of the G.P.? Answer: 4/3 10. What value of x makes the three terms x, x/(x + 1) and 3x/[(x + 1)(x + 2)] those of a geometric sequence? Answer: -1/2

Section 2 1. Determine the sum of the first 4 terms of the sequence whose general term is given by 3n -2. Answer: 112 2. Solve for x in the following equation: x + 3x + 5x + … + 49x = 625. Answer: 1 3. Determine the 5th term of the sequence whose sum of n terms is given by 2n+3 – 5. Answer: 128 4. Determine x so that 2x + 1, x2 + x + 1, 3x2 – 3x + 3 are consecutive terms of an arithmetic progression. Answer: 2 5. Find the ratio of an infinite geometric series of the sum is 2 and the first term is ½. Answer: ¾ 6. The sum of a geometric series is follows: S = 1.0 + 1.1 + 1.21 + 1.1331 + … up to 50 th term, Answer: 1163.91 7. The sum of the three numbers in AP is 33, if the sum of their squares is 461, find the numbers. Answer: 4, 11, 18 8. Three numbers are in AP such that the sum of the first and third is 12 and the product of the first and second is 24. Find the largest of three numbers. Answers: 8 9. The arithmetic mean of 6 numbers is 17. If two numbers are needed to the progression, the new set of number will have an arithmetic mean of 19. What are two numbers if their difference is 4? Answer: 23, 27 10. The first term of the arithmetic progression (AP) is 6 and the 10th term is 3 times the second number. What is the common difference? Answer: 2

1. The motion of a particle through a certain medium is such that it moves two thirds as far each second as in the preceding second. If it moves 6m of the first second, how far will it move before coming to rest? Answer: 18 2. A man who is on diet losses 24 lb. in 3 months, 16 lb. is the next 3 months and so on for a long time. What is the maximum total weight loss? Answer: 72 3. The first term of an arithmetic progression (AP) is 6 and the 10th term is 3 times the second number. What is the common difference. Answer: 2 4. The 10th term of the series a, a-b, a-2b, … is: Answer: 1-9b 5. The arithmetic mean of 80 numbers is 55. If two numbers namely 250 and 850 are removed what is the arithmetic mean of the remaining numbers? Answer: 42.31 6. The sum of five arithmetic means between 34 and 42 is: Answer: 190 7. There are 6 geometric means between 4 and 8748. Find the sum if all terms. Answer: 13120 8. Determine x so that x, 2x+7, 10x-7 will form a geometric progression. Answer: 7 9. The sum of all numbers between 0 and 10000 which is exactly divisible by 77 is: answer: 645645 10. There are seven arithmetic means between 3 and 35. Find the sum of all terms. Answer: 171

1. The sum of the progression 5, 8, 11, 14 … is 1025. How many terms are there? Answer: 25 2. The arithmetic mean of two numbers is 5, and their harmonic mean is 24/5. Find the numbers. Answer: 4 & 6 3. Find the sum of the numbers divisible by 6 which lie between 75 and 190 Answer: 2508 4. How many term of the progression 4, 7, 10, 13, … must be taken so that the sum will be 69 Answer: 6 5. If the third of a GP is 20 and the 6th term is 160, what is the first term? Answer: 5 6. Find the 15th term of the progression 1/4, 1/7, 1/10, … Answer: 1/46 7. The fourth term of the geometric progression is 189 and the sixth term is 1701, the 8th term is: Answer: 15309 8. The sum of the three in AP is 33, if the sum of their squares is 461, find the numbers. Answer: 4, 11, 18 9. The motion of a particle through a certain medium is such that it moves two thirds as far each second as in the preceding second. If it moves 6m of the first second, how far will it move before coming to rest. 10. There are line (9) arithmetic means between 11 and 51. The sum of progression is: Answer: 341

Section 3 1. If n is a positive integer such that n!/(n – 2)! = 342, find n. Answer: 19 2. A critical number c of a function is a number in the domain of f such that: Answer: f’(c) = 0 or f’ (c) is undefined 3. What is the range of the function y=x=2/3? Answer: y>0 4. How many integers are there in the solution set of |x – 2| ≤ 5? Answer: 11 5. How many positive integers are there in the solution set of x/x – 2 > 5? Answer: 0 6. What is the domain of the function f(x) = 3x3-7? Answer: X ≥ 1.33 7. The number of integers that satisfy the inequality x2 + 48 < 16x is: Answer: 7 8. Given: A = {2,6,10}. Which of the following is equal to the given set A? Answer: All are correct 9. Let V be the set of letters in the English alphabet. The |V| is ______. Answer: 26 10. Given A = {11,12,13} and B = {11,13,15}. Find A-B. Answer: {12}

1. If set Y is the set of all positive integers less than 10 and set B is a subset of set Y, set B may be ______. Answer: {6,7,8} 2. Given A = {1,2} and B = {a, b}. Find AxB. Answer: {(1,a), (1,b), (2,a), (2,b)} 3. Given A = {1,2,3} and B = {a, b, c. Find the cardinality of AxB. Answer: 9 4. Given A = {1,2,3} and B = {1,3,5}. Find AB. Answer: {1,2,3,5} 5. Given A = {1,2,3} and B = {1,3,5}. Find AB. Answer: {1,3} 6. Given A = {a,b,c}, B = {a,c,e}, and C = {e,f,g}. Which of the given sets are disjoint? Answer: A & C 7. Given A = {11,12,13} and B = {11,13,15}. Find A –B. Answer: {12} 8. Given A = {31,32,33} and B = {31,33,35}. Find B\As Answer: {35} 9. Let A be the set of positive integers than 10 with a universal set of all positive integers. Find A. Answer: {1,2,3,4,5,6,7,8,9,10} 10. Let A = {0,2,4,6,8,10}, B = {0,1,2,3,4,5,6} and C = {4,5,6,7,8,9,10}. What is (A B) C? Answer: {0,2,4,5,6,7,8,9,10}

Section 4 1. Let A={4, 6, 7, 8, 9}, B={6, 7, 8, 25}. Determine A XOR B. Answer: {4, 9, 25} 2. Let A = {a, b, c}, B = {b, c, d} and C = {b, c, e}. Find (A ∪ B) ∩ (A ∪ C). Answer: {a, b, c} 3. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find A – B. Answer: {1, 2, 4, 5} 4. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find B – A. Answer: {0, 6} 5. Find the power set of A = {a, b}, where a and b are distinct elements. Answer: {Ø, {a}, {b}, {a, b}} 6. Let A = {41, 42, 43, 44} and B = {43, 44, 45, 46}. Find the symmetric difference of A and B. Answer: {41, 42, 45, 46} 7. Let A={0,2,4,6,8,10}, B={0,1,2,3,4,5,6} and C={4,5,6,7,8,9,10}. What is A ∩ B∩ C? Answer: {4,6} 8. Find the floor and ceiling functions, respectively, of 7/8. Answer: 0, 1 9.If Z is the set of odd positive integers less than 10, then, the cardinality of Z is: Answer: 5 10.If a and b are both even numbers, which of the following could be and odd integer? Answer: (a + 1)/(b + 1)

1.Given the sets: A={a,c,e}; B={c,e,a} and C={a,c,c,c,e}. Which of the following statements is false? Answer: A≠C 2. Given the sets: A={3,4,5,6,7} and B={4,5,6}. Which of the following statements is false? Answer: A⊆ B 3. Given Set A={-4,-5,1,3,13}, if we have the relationship C ⊂ A. What should set C be? Answer: {-4, 1, 13} 4. Find the cardinality of the power set of {8, 13, 21, 57}. Answer: 16 5. Which of the following is a null set? Answer: {x|x is an integer such that x^2 = 2} 6.How many elements does the set Ƿ(Ƿ(Ø)) has/have? Answer: 2 7. Given |G| is 9 and |H| is 3. All of the given are possible values for the cardinality of the union of set G and set H, except __. Answer: 7 8. Given |P| is 12 and |Q| is 7. All of the given are possible values for the cardinality of the intersection. Answer: 9 9. Let f1 and f2 be functions from R to R such that f1(x) = x2 and f2(x) = x – x2. What is the function f1f2. Answer: x3- x4 10. Let {an} be a sequence that satisfies the recurrence relation an = an-1 – an-2 for n=2,3,4… and suppose that a0 = 3 and a1 = 5. What is a2? Answer: 2

SECTION 5 1

List the member of Set W, where W = {x | x is the square of an integer and x <30}.

Answer: {0, 1, 4, 9, 16, 25} 2

Find the cardinality of A^2 if A = {0, 1, 3}.

Answer: 9 3

Find the truth set of the given predicate: P(x) = x^2 < 3. The domain is the set of integers.

Answer: {-1, 0, 1} 4

Let f = 0.5 + g, where g is the ceiling function of 3/2. Determine the floor function of f.

Answer: 2 5. Let f = 0.5 x g, where g is the floor function of 5/2. Find the value of the floor function of f. Answer: 1 6. Let A = {g, h, i, j, k} and B = {11, 12, 13, 14} with f(g) = 12, f(h) = 11, f(i) = 14, f(j) = 11 and f(k) = 14. The image of the subset S = {h, i, j} is ________. Answer: {11, 14} 7. Let X = {1, 2, 3} and Y = {a, b, c, d}. Given a function H from X to Y. Which of the following shows a one-to-one function? Answer: H(1) = c, H(2) = a, H(3) = d 8. Let X = {1, 2, 3, 4} and Y = {a, b, c}. Given a function H from X to Y. Which of the following shows an onto function? Answer: H(1) = c, H(2) = b, H(3) = a, H(4) = a 9. Let f be the function from {a, b, c, d} to {1, 2, 3, 4}. Which of the following definitions will make f bijective? Answer: f(a) = 4, f(b) = 1, f(c) =3, f(d) = 2 10. A function is bijective if it is both one-to-one and onto. Answer: One-to-one and onto

1. Let f be the function from {p, q, r} to {11, 12, 13} such that f(p) = 12, f(q) = 13 and f(r) = 11. Find the inverse of f. Answer: f-1(11) = r, f-1(12) = p, f-1(13) = q 2. Find the inverse of the given function described as follows: Let f: Z → Z such that f(x) = x + 1. Answer: f-1(y) = y – 1 3. Let Set A = (-1, 0] and Set B = [0 ,1). Find the complement of Set A if the universal set is the set R of all real numbers. Answer: ( -∞ , -1 ] ∪ ( 0, ∞ ) 4. An engineering system has two components. Let us define the following events: A = first component is good; A’ = first component is defective; B = second component is good; B’ = second component is defective Describe the following events in terms of A, A’, B, and B’: (a) at least one component is good (b) one is good and one is defective. Answer: A∪B, AB’∪A’B 5. Under which conditions is xy/x-y negative? Answer: x < y < 0 6. The set O of odd positive integers less than 10 can be expressed by _____________. Answer: {1, 3, 5, 7, 9} 7. What is the Cartesian product of A = {1, 2} and B = {a, b}? Answer: {(1, a), (2, a), (1, b), (2, b)} 8. What is the cardinality of the set of odd positive integers less than 10? Answer: 5 9. Which of the following two sets are equal? Answer: A = {1, 2, 3} and B = {2, 1, 3} 10. What is the Cardinality of the Power set of the set {0, 1, 2}. Answer: 8

SECTION 6 1. The complement of the set A is _____________. Answer: U – A 2. The bit string for the set {2, 4, 6, 8, 10} (with universal set of natural numbers less than or equal to 10) is ____________________. Answer: 0101010101 3. Let Ai = {i, i+1, i+2,…}. Then set {n, n+1, n+2, n+3,…} is the _________ of the set Ai. Answer: Intersection 4. The bit strings for the sets are 1111100000 and 1010101010. The union of these sets is ___________. Answer: 1111101010 5. The set difference of the set A with null set is __________. Answer: A 6. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then number of elements in A U B is: Answer: 4 7. Let the set A is {1, 2, 3} and B is { 2, 3, 4}. Then number of elements in A ∩ B is: Answer: 2 8. Let the set A is {1, 2, 3} and B is {2, 3, 4}. Then the set A – B is: Answer: {1} 9. In which of the following sets A- B is equal to B – A. Answer: A={1, 2, 3}, B ={2, 3, 1} 10. Let A be set of all prime numbers, B be the set of all even prime numbers, C be the set of all odd prime numbers, then which of the following is true? Answer: All of the mentioned

1 If A has 4 elements B has 8 elements then the minimum and maximum number of elements in A U B are respectively. Answer: 8, 12 2

If A is {{Φ}, {Φ, {Φ}}, then the power set of A has how many element?

Answer: 4 3

Which sets are not empty?

Answer: { x: x is a prime number less than 5 and is odd} 4

The shaded area of figure is best described by,

Answer: A ∩ B 5

The shaded area of figure is best described by,

Answer: A U B – B 6

If n(A)=20 and n(B)=30 and n(A U B) = 40 then n(A ∩ B) is:

Answer: 10 7

The shaded area of figure is best described by,

Answer: Venn Diagram

Answer: B – (A ∩ B) – (C ∩ B) 8. Let A : All badminton player are good sportsperson. B: All person who plays cricket are good sportsperson. Let X denotes set of all badminton players, Y of all cricket players, Z of all good sportsperson. Then which of the following statements is correct? Answer: Z contains both X and Y 9 If n(A)=10 , n(B)=30,n(C)=50 and if set A,B,C are pairwise disjoint then which of the following is correct? Answer: All of the mentioned 10

In the given figure the if n(A)=20,n(U)=50,n(C)=10 and n(A∩B)=5 then n(B)=?

Answer: 35

SECTION 7 1

The shaded area of figure is best described by,

Answer: A U B – (A ∩ B) 2

Let C and D be two sets then which of the following statements are true?

1) C U D = D U C

2) C ∩ D = D ∩ C Answer: Both of these statements 3

If set C is {1, 2, 3, 4} and C – D = Φ then set D can be:

Answer: {1, 2, 3, 4} 4

Let C and D be two sets then C – D is equivalent to:

Answer: C ∩ D’ 5

For two sets C and D the set (C – D) ∩ D will be:

Answer: Φ 6

Which of the following statement regarding sets is false.

Answer: (A U B)’ =A’ U B’ 7

Let C = {1,2,3,4} and D = {1, 2, 3, 4} then which of the following hold not true in this case.

Answer: C ∩ D = C – D 8

If C’ U (D ∩ E’) is equivalent to:

Answer: (C ∩ ( D’ U E))’

9 Let Universal set U is {1, 2, 3, 4, 5, 6, 7, 8} ,(Complement of A) A’ is {2, 5, 6, 7}, A ∩ B is {1, 3, 4} then the set B’ will surely have of which of the element. Answer: 8 10

Let a set be A then A ∩ φ and A U φ are respectively.

Answer: φ, A

1 If in sets A, B, C, the set B ∩ C consists of 8 elements, set A ∩ B consists of 7 elements and set C ∩ A consists of 7 elements then the minimum element in set A U B U C will be: Answer: 4,845 2

Let set A ={1, 2} and C be {3, 4} then A X B (Cartesian product of set A and B) is:

Answer: {(1, 3) , (2, 4), (1, 4) , (2, 3) } 3

If set A has 4 elements and B has 3 elements then set n(A X B) is:

Answer: 12 4

If set A has 3 elements then number of elements in A X A X A are:

Answer: 27 5

Which of the following statements regarding sets is false?

Answer: A X B = B X A 6

If n(A X B) = n(B X A) = 36 then which of the following may hold true?

Answer: n(A)=6, n(b)=6 7

Let the sets be A, B, C, D then (A ∩ B) X (C ∩ D) is equivalent to:

Answer: (A X C) ∩ (B X D) 8 B) is:

If set A and B have 3 and 4 elements respectively then the number of subsets of set (A X

Answer: 4096 9

If set A X B=B X A then which of the following sets may satisfy.

Answer: A={1, 2} , B={2, 1} 10

If a set contains 3 elements then the number of subsets is:

Answer: 8

Section 8 1 At a meeting, after everyone had shaken hands with everyone else, it was found that 66 handshakes were exchanged. How many were at the meeting? Answer: Union set 2

If a set is empty then number of subsets will be:

Answer: 1 3

If the number of subsets of a set are 4 then the number of elements in that sets are:

Answer: 2 4

Let a set be A={1, 2, 3} then the number of subsets containing two elements will be:

Answer: 3 5

Let the set be A= {a , b, c, {a,b}} then which of the following is false.

Answer: {a} Є A 6 3, is:

If A={1, 2, 3, 4} ,then the number of the subsets of A that contain the element 2 but not

Answer: 4 7 Let A(1), A(2), A(3), A(100) be 100 sets such that number of elements in A(i)=i+1 and A(1) is subset of A(2), A(2)is subset of A(3), A(99) is subset of A(100). The the number of elements in union of the all the sets are: n(A(1) U A(2) U A(3) U A(100)): Answer: 101 8 A function is said to be ______________ if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. Answer: One-to-one 9

The value of ⌊1/2. ⌊5/2⌋ ⌋ is ______________.

Answer: 1 10

Which of the following function f: Z X Z → Z is not onto?

Answer: f(a, b) = |b|

1 The domain of the function that assign to each pair of integers the maximum of these two integers is ___________. Answer: Z+ X Z 2 Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + 1 and g(x) = 3x + 4. Then the composition of f and g is ____________. Answer: 6x + 9 3

The inverse of function f(x) = x3 + 2 is ____________.

Answer: f -1 (y) = (y – 2) 1/3 4

The g -1({0}) for the function g(x)= ⌊x⌋ is ___________.

Answer: {x | 0 ≤ x ≤ 1} 5

What is domain of function f(x)= x1/2 ?

Answer: [0, ∞) 6

What is range of function f(x) = x-1 which is defined everywhere on its domain?

Answer: (-∞, ∞) 7

If f(x) = 2x then range of the function is:

Answer: (0, ∞) 8

If f(x) = x2 + 4 then range of f(x) is given by:

Answer: [4, ∞) 9

An injection is a function which is:

Answer: one-one 10. what is domain of function f(x)=x^-1 for it to be defined everywhere on domain? Answer: (-∞, ∞)-{0}

Section 9 1

A mapping f : X -> Y is one one if:

Answer: If f(x1) = f(x2) then x1 = x2 for all x1, x2 in X. 2 A function is defined by mapping f : A -> B such that A contains m elements and B contains n elements and m ≤ n then number of one one functions are: Answer: nCm x m! 3 A function is defined by mapping f: A -> B such that A contains m elements and B contains n elements and m>n then number of one one functions are: Answer: 0 4 Set A has 3 elements and set B has 4 elements then number of injections defined from A to B are? Answer: 24 5 A function is defined by mapping f : A -> B such that A contains m elements and B contains n elements and 1≤n≤m then number of onto functions are: Answer: r=1∑r=nnCr (-1)n-r rm 6 A function is defined by mapping f: A->B such that A contains m elements and B contains n elements and m > n then number of bijections are: Answer: 0 7

A floor function map a real number to:

Answer: greatest previous integer 8

A ceil function map a real number to:

Answer: smallest following integer 9

A function f(x) is defined as f(x) = x – [x], where [.] represents GIF then:

Answer: f(x) will be fractional part of x 10

Floor(2.4) + Ceil(2.9) is equal to:

Answer: 5

1 If x, and y are positive numbers both are less than one, then maximum value of floor(x + y) is? Answer: 1 2 is?

If x, and y are positive numbers both are less than one, then maximum value of ceil(x + y)

Answer: 2 3

If X = Floor(X) = Ceil(X) then:

Answer: X is a Integer 4

A function f(x) is defined from A to B then f -1 is defined:

Answer: from B to A 5

If f is a function defined from R to R , is given by f(x) = 3x – 5 then f –1(x) is given by:

Answer: (x+5)/3 6

If f is a function defined from R to R , is given by f(x) = x2 then f –1(x) is given by:

Answer: does not exist since it is not a bijection 7

The cardinality of the set A = {1, 2, 3, 4, 6} is:

Answer: 5 8

If A is a subset of B:

Answer: Cardinality of B is greater than A 9

If A is a subset of B and B is a subset of C, then cardinaity of A U B U C is equal to:

Answer: Cardinality of C 10

If A is any statement, then which of the following is a tautology?

Answer: A ∨ ¬A

Section 10 1

If A is any statement, then which of the following is not a contradiction?

Answer: A ∨ F 2 A compound proposition that is neither a tautology nor a contradiction is called a ___________. Answer: Contingency 3

¬ (A ∨ q) ∧ (A ∧ q) is a ___________.

Answer: Contradiction 4

(A ∨ ¬A) ∨ (q ∨ T) is a __________.

Answer: Tautology 5

A → (A ∨ q) is a __________.

Answer: Tautology 6

The contrapositive of p → q is the proposition:

Answer: ¬q → ¬p 7

The inverse of p → q is the proposition:

Answer: ¬p → ¬q 8

The converse of p → q is the proposition:

Answer: q → p 9 What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?” Answer: If the home team wins, then it is not drizzling 10 What is the converse of the conditional statement “If it ices today, I will play ice hockey tomorrow. Answer: “I will play ice hockey tomorrow only if it ices today.”

1 What are the contrapositive of the conditional statement “I come to class whenever there is going to be a test. Answer: “If I do not come to class, then there will not be a test.” 2 What are the inverse of the conditional statement “ A positive integer is a composite only if it has divisors other than 1 and itself.” Answer: “If a positive integer is not composite, then it has no divisors other than 1 and itself.” 3 What are the converse of the conditional statement “When Raj stay up late, it is necessary that Raj sleep until noon.” Answer: “If Raj sleep until noon, then Raj stay up late.” 4 What are the contrapositive of the conditional statement “Medha will find a decent job when she labour hard.”? Answer: “If Medha will not find a decent job, then she not labour hard.” 5 What are the inverse of the conditional statement “If you make your notes, it will be a convenient in exams.” Answer: “If you do not make notes, then it will not be a convenient in exams.” 6 The compound propositions p and q are called logically equivalent if ________ is a tautology. Answer: p ↔ q 7

p → q is logically equivalent to:

Answer: ¬p ∨ q 8

p ∨ q is logically equivalent to:

Answer: ¬p → q 9

¬ (p ↔ q) is logically equivalent to:

Answer: p↔¬q 10 What is the contrapositive of the conditional statement? “The home team misses whenever it is drizzling?” Answer: If the home team wins, then it is not drizzling

1

p ∧ q is logically equivalent to:

Answer: ¬ (p → ¬q) 2

Which of the following statement is correct?

Answer: All of mentioned 3

p ↔ q is logically equivalent to:

Answer: (p → q) ∧ (q → p) 4

(p → q) ∧ (p → r) is logically equivalent to:

Answer: p → (q ∧ r) 5

(p → r) ∨ (q → r) is logically equivalent to:

Answer: (p ∧ q) → r 6

¬ (p ↔ q) is logically equivalent to:

Answer: p ↔ ¬q 7

Let P (x) denote the statement “x >7.” Which of these have truth value true?

Answer: P (9) 8 The statement,” Every comedian is funny” where C(x) is “x is a comedian” and F (x) is “x is funny” and the domain consists of all people. Answer: ∀x(C(x) → F (x)) 9 The statement, “At least one of your friends is perfect”. Let P (x) be “x is perfect” and let F (x) be “x is your friend” and let the domain be all people. Answer: ∃x (F (x) ∧ P (x)) 10 Let domain of m includes all students , P (m) be the statement “m spends more than 2 hours in playing polo”. Express ∀m ¬P (m) quantification in English. Answer: No student spends more than 2 hours in playing polo

Section 11 1

Factor the expression x2 + 6x + 8 as completely as possible.

Answer: (x+4)(x+2) 2

For all real numbers x, the minimum value of 1 + 2cos(4x) is:

Answer: -1 3

Find the value of x which will satisfy the equation √(x-2) / √x = 1.

Answer: 4 4

If x = 2.0001, which of the following expressions has the largest value?

Answer: 2/(x - 2) 5

Solve: √(2x-5)-√(x-2)=2.

Answer: 27 6

If 16 is 4 more than 4x, find x-1.

Answer: 2 7

Give the factors of: a2-x2

Answer: (a+x)(a-x) 8

(a-b)3 is equivalent to which of the following?

Answer: a3-3a2b+3ab2-b3 9

Reduce the following complex fraction into simple functions.

Answer: (a-2)/(a-4) 10

Reduce the following complex fraction into simple fractions.

Answer: -x+y/x+y

1

Find the value of x which will satisfy the equation √x - 2/ √x =1.

Answer: 4 2

Solve for a in the equation: a = 64x4y.

Answer: 43x+y 3

Simplify: 3x - 3x-1 - 3x-2.

Answer: 5×3x-2 4

Which of the following is true?

Answer: 55+55+55+55+55=56 5

Simplify:∛2x4 -∛16x4 +2∛54x4.

Answer: 5∛2x4 6

Solve for x: 3x5x+1 = 6x+2.

Answer: 2.1544 7

Factor the expression x3 + 8.

Answer: (x+2)(x2-2x+4) 8

Factor the expression (x4 – y4) as completely as possible.

Answer: (x2+y2)(x+y)(x-y) 9

Simplify:

Answer: a-6b7 10

Solve for x: 37x+1 = 6561.

Answer: 1

Section 12 1

If 3a = 7b, then 3a2/7b2 =?

Answer: 7/3 2

If x to the ¾ power equals 8, then x equals:

Answer: 16 3

If 33y = 1, what is the value of y/33?

Answer: 0 4

Find the value of x that will satisfy the following expression: √(x-2)=-√x+2.

Answer: x = 9/4 5

Which of the following is equivalent to:

Answer: Figure 4

6

Simplify the following:

7a+2 – 8(7a+1) + 5(7a) + 49(7a-2). Answer: -7a 7

Solve for x:

Answer: 16/25 & 0

8

Factor the expression: 16 – 10x + x2.

Answer: (x-8)(x-2) 9

Which of the following is not an identity?

Answer: 2(x-1)+3(x+1) = 5x+4 10

Solve for x:

Answer: Any value

1

Resolve (x+2)/(x2-7x+12) into partial fraction.

Answer: 6/(x-4)-5/(x-3) 2 If the angles of the triangle are 2x, x + 15, and 2x + 15, find the smallest of the angle in mills. Answer: 800 mils 3

If 1/x + 1/y = 3 and 2/x – 1/y = 1. Then x is equal to:

Answer: 3/4 4

If 3x = 4y then (3x2)/(4y2 ) is equal to:

Answer: 4/3 5

What is the smallest value of x that satisfies the equation: x(x + 4) = -3.

Answer: -3 6

If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y = ?

Answer: 6 7

If a = 3, then 2 / (1/7 + 1/a) = ?

Answer: 21 / 5 8

If -3/(a - 3) = 3/(a + 2), then a = ?

Answer: ½ 9

Solve the z if the equation is 4 x 10-5 = z.

Answer: 0.00004 10

If 3a = 7b, then 3a2/7b2 =?

Answer: 7/3

Section 13 1

If (x+3) : 10=(3x-2): 8, find (2x-1).

Answer: 3 2

The linear distance between -4 and 17 on the number line is:

Answer: 20 3

For what value of k the equation below has no value of x: 2x + 3 = x - 2kx – 5.

Answer: -0.5 4

Find the value of A for which the equation A(2x+3)-(x-4) = 3x+10 is an identity.

Answer: 2 5

Solve for x if 8y = 3x - 11.

Answer: (8y + 11)/3 6

If x + 4y = 5 and 5x + 6y = 7, then 3x + 5y = ?

Answer: 6 7

If f(x) = x2 + x + 1, then f(x) – f(x-1)=.

Answer: 2x 8

Solve the simultaneous equations: 3x – y = 6; 9x – y = 12.

Answer: x = 1; y = -3 9

Solve algebraically: 4x2 + 7y2 = 32 11y2 – 3x2 = 41.

Answer: y = 4, x = ±1 and y = -4, x = ±1 10

Find k so that the expression kx2 – 3kx + 9 is a perfect square.

Answer: 4

1

Which of the following quadratic equations will have two real and distinct roots?

Answer: 6x2-61x +143 = 0 2

The roots of the equation 2x2 – 3x + 20 = 0 are.

Answer: complex and unequal 3

Which of the following quadratic trinomial is NOT factorable?

Answer: 6x2- 52x – 60 4

The value of k for which the roots of 8x2+8kx+3k+2 = 0 are real and equal is:

Answer: 2 5

The degree of the polynomial f(x, y, z) = 7x3y2-4xz5+2x2y is:

Answer: 3 6

The least common multiple of (x-2)2 and x2+x-6 is:

Answer: (x-2)2(x+3) 7

Given f(x) = (x – 4) (x + 3) + 4, when f(x) is divided by x – k, the remainder is k. Find k.

Answer: 4 8

The equation y = (x-1)/ (x+2) is not defined at x =?

Answer: -2 9

If the expression x3 + 2hx - 2 is equal to 6 when x = -2, what is the value of h?

Answer: -4 10 From the equation 12x3 – 8x2 + kx + 18 = 0, find the value of k if one root is the negative of the other. Answer: -27

Section 14 1

Simplify:

Answer: y5/2 /x 2

Simplify:

Answer: 1/x2y7z3 3

Simplify:

Answer: α-6b7

4

Simplify:

Answer: 1/x3y 5

4xy – 4x2 –y2 is equal to:

Answer: –(2x-y)2 6

Find the quotient of 3x5 – 4x3 + 2x2 + 36x + 48 divided by x3 – 2x2 + 6.

Answer: 3x2+ 6x + 8 7

The quotient of (x5 +32) by (x+2) is:

Answer: x4– 2x3+ 4x2– 8x + 16 8

Find the sum and product of roots of the equation x3 + 2x2 – 23x – 60 = 0.

Answer: -2, 60 9 = 0.

Find the equation whose roots are two times the roots of the equation x3 – 6x2 + 11x – 6

Answer: x3– 12x2+ 44x – 48 = 0 10

If one root of 9x^2 – 6x + k = 0 exceed the other by 2, find the value of k.

Answer: -8

1

If ax3 + bx2 + cx + d is divided by x - 2, then the reminder is equal to.

Answer: 8a + 4b + 2c + d 2 For the remainder of the division of x3 - 2x2 + 3kx + 18 by x - 6 to be equal to zero, k must be equal to: Answer: -9 3

If one root of 9x^2-6x+k=0 exceeds the other by 2, find the value of k.

Answer: -8 4

Find k in the equation 4x2 + kx + 1 = 0 so that it will only have one real root.

Answer: 4 5

Given f(x) = (x+3)(x-4) + 4 when divided by (x-k), the remainder is k. Find k.

Answer: 4 6

The polynomial x3 + 4x2 -3x + 8 is divided by x-5. What is the remainder?

Answer: 218 7

In the quadratic equation Ax2 + Bx + C = 0, the product of the roots is:

Answer: C/A 8

In the equation 3x2 + 4x + (2h – 5) = 0, find h if the product of the roots is 4.

Answer: 17/2 9 Find the value of k in the quadratic equation (2k + 2) x2 + (4 – 4k) x + k – 2 = 0 so that the roots are reciprocal of each other. Answer: -4 10

If one root of 9x^2 – 6x + k = 0 exceed the other by 2, find the value of k.

Answer: -8

Section 15 1

If (x -3) is a factor of the polynomial x4 – 4x3 – 7x2 + kx + 24, what is the value of k?

Answer: 22 2

Find the sum of the roots 5x2 -10x + 2 = 0.

Answer: 2 3

Which of the following quadratic equations will have two real and distinct roots?

Answer: 6x2-61x +143 = 0 4

The roots of the equation 6x2 – 61x + 143 = 0 are:

Answer: real and distinct 5

Find the value of k that will make x2 – 28x +k have equal roots.

Answer: 196 6

If x:y:z =4: - 3:2 and 2x+4y-3z = 20, find the value of x.

Answer: -8 7

The roots of the equation 2x2 – 3x + 20 = 0 are:

Answer: complex and unequal 8

If x3+3x2+(5+k)x+2-k is divided by x+1 and the remainder is 3, then the value of k is:

Answer: -2 9

Which of the following quadratic trinomial is NOT factorable?

Answer: 6x2- 52x – 60 10

The value of k for which the roots of 8x2+8kx+3k+2 = 0 are real and equal is:

Answer: 2 1

The degree of the polynomial f(x, y, z) = 7x3y2-4xz5+2x2y is:

Answer: 3 2

The least common multiple of (x-2)2 and x2+x-6 is:

Answer: (x-2)2(x+3) 3

In the equation x^2 + x = 0, one root is equal to:

Answer: none of these choices

4

The roots of x^2 + x + 1 = 0 are:

Answer: none of these choices 5

(a - b)^3 = ?

Answer: a^3 - 3a^2b + 3ab^2 - b^3 6

The equation whose roots are the reciprocals of the roots of 2x2 - 3x - 5 = 0 is:

Answer: 5x2+ 3x - 2 = 0 7 The maximum possible number of positive roots that the equation x4 + 2x3 – 3x2 +bx –5 = 0 can have if b represent a positive real number is: Answer: 3 8

The roots of a quadratic equation are 1/3 and ¼. What is the equation?

Answer: 12x2– 7x + 1 = 0 9

The equation whose roots are the reciprocals of the roots of the equation 2x2 -3x - 5 = 0.

Answer: 5x2+3x–2=0 10

The remainder when 2x4-kx-15x2-3x-2 is divided by (x-3) is 4. What is the value of k?

Answer: 4

1

If x3+3x2+(5+k)x+2-k is divided by x+1 and the remainder is 3, then the value of k is____.

Answer: -2 2

Solve: 5x3 – 5x2 – 10x = 0.

Answer: 0,-1,2 3

Solve: 25y2 – 3 = 0.

Answer: +/- √3/5 4

Solve for the quadratic equation below:x2– 6x +1=0.

Answer: 3 ± √8 5

Solve for the equation below: t5-9t3= 0.

Answer: 0, 3, -3 6

Solve for the quadratic equation below:

(z-2)2 -36 =0; Answer: -4, 8 7

Solve for x in: y= 10/(3-7x).

Answer: 3y-10/7y 8

Find the sum and product of roots of the equation x3 + 2x2 – 23x – 60 = 0.

Answer: -2, 60 9

In the equation 3x2 + 4x + (2h – 5) = 0, find h if the product of the roots is 4.

Answer: 17/2 10

If (x -3) is a factor of the polynomial x4 – 4x3 – 7x2 + kx + 24, what is the value of k?

Answer: 22

Section 1.16 1. In how many ways can a party of 6 people be seated on a row of 6 seats if a certain 2 refuse to sit next to each other? Answer: 480 ways 2. A number between 1 and 10000 is randomly selected. What is the probability that it will be divisible by 4 and 5? Answer: 0.05 3. In how many ways can 9 different books be arranged on a shelf so that 3 of the books are never all together? Answer: 332,640 4. How many 4-digits even numbers can be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if each digit is to be used only once in each number? Answer: 2,520 5. A guy has 8 flowers of different variety. In how many ways can he select 2 or more flowers to form a bouquet? Answer: 247 6. At a conference, after everyone had shaken hands with everyone else, it was found that 45 handshakes were exchanged. How many were at the conference? Answer: 10 7. When two dice are thrown, what is the probability that the sum of the two faces shown is 6? Answer: 5/36 8. Two dice are tossed. What is the probability that the sum of the two dice is greater than 3? Answer: 11/12 9. How many ways can 3 men and 4 women be seated on a bench if the women to be together? Answer: 576 10. In how many ways can a caravan of 9 covered wagons from India be arranged in a circle? Answer: 40320

11. Two balls are drawn one at a time from a basket containing 4 black balls and 5 white balls. If the first ball is returned before the second ball is drawn, find the probability that both balls are black. Answer: 0.198

12.In how many ways can 3 acacias, 4 narras, and 2 mahoganies be arranged along a property line if one does not distinguish among trees of the same kind? Answer: 1260 13. Allen picked 10 differently colored beads. How many different bracelets can he make if he is going to use only 6 of the 10 beads per bracelet? Answer: 12600 14. Suppose a die is rolled three times. What is the probability that all three rolls are the same? Answer: 1/36 15. Ricky and George each throw dice. If Ricky gets a sum of four what is the probability that George will get less than of four? Answer: 1/12 16. One box contains four cards numbered 1, 3,5,and 6. Another box contains three cards numbered 2, 4, and 7. One card is drawn from each bag. Find the probability that the sum is even. Answer: 5/12 17. The probability of getting credit in an examination is 1/3. If three students are selected at random, what is the probability that at least one of them got a credit? Answer: 19/27 18. In how many ways can a hostess select six luncheon guests from 10 women if she is to avoid having particular two of them together at the luncheon? Answer: 140 19. One letter is taken from each of the words PARALLEL and LEVEL at random. What is the probability of getting the same letter? Answer: 1/5 20. A card is chosen from pack of playing cards. What is the probability that it is either red or a picture card? Answer: 8/13

Section 1.17

1. Two fair dice are thrown. What is the probability that the sum of the dice is divisible by 5? Answer: 7/36 2. The captain of a baseball team assigns himself to the 4th place in the batting order. In how many ways can he assign the remaining places to his eight teammates if just three men are eligible for the first position? Answer: 15120 3. How many permutations can made out of the letters of the word ENGINEERING? Answer: 277,200 4. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that both are white. Answer: 3/28 5. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that all are of the same color. Answer: 13/28 6. A bag contains 4 red balls, 3 green balls, and 5 blue balls. The probability of not getting a red ball in the first draw is: Answer: 2/3 7. The face of a coin is either head or tail. If three coins are tossed, what is the probability of getting three tails or three heads? Answer: 1/4 8. A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that one ball is white and the other is red. Answer: 15/28 9. A committee of 6 teachers is to be formed from 5 male teachers and 8 female teachers. If the committee is selected at random, what is the probability that it has an equal number of male and female teachers? Answer: 140/429 10. A recent survey reported that 60 percent of the students at a certain university are girls and 65 % of girls of this school play basketball. If a student at this school were selected at random, what is the probability that the student is a girl who plays basketball? Answer: 0.39

11. If 15 people can win prices in a estate lottery (assuming that there are no ties). How many ways can these 15 people win first, second,, third, fourth and fifth prizes? Answer: 360,360 12. A semiconductor company will hire 7 men and 4 women. In how many ways can the company choose from 9 men and 6 women who qualified for the position. Answer: 540 13. What is the probability of obtaining either four or five heads if a fair coin is tossed 10 times? Answer: 231/512 14. How many different signals each consisting of 6 flags hung in a vertical line can be formed from 4 identical red flags and 2 identical blue flags? Answer: 15 15. How many 3 digit numbers can be formed from the digits 2,4,6,8 and 9 if repetitions are allowed? Answer: 125 16. An ECE class of 40 students took examinations in Electronics and Communications. If 30 passed in Electronics, 36 passed in Communication and 2 failed in both subjects, how many students passed in both subjects? Answer: 28 17. A provincial chapter of IECEP held a lottery to raise funds for their organization, with P20,000 top prize and with 3,000 tickets printed and sold. What is the mathematical expectation of a member if she bought 30 tickets? Answer: 200 18. Rhazel draws 3 candies in succession from a jar containing 5 Kopiko candies, 6 Maxx¡¯s candies and 7 Stork candies. Find the probability of drawing these candies in order: Kopiko, Maxx, Stork. Answer: 35/816 19. How many line segments can be formed with 10 distinct points, no two of which are collinear? Answer: 45 20. In a dice game, one fair is used. The player wins P20.00 if he rolls either 1 or 6. He losses P10.00 if he turns up any other face. What is the expected winning for one roll of the die? Answer: P0.00

Section 1.18 1. At a meeting, after everyone had shaken hands with everyone else, it was found that 66 handshakes were exchanged. How many were at the meeting? Answer: 12 2. A pair of dice is tossed 10 times. What is the probability that no 7s 0r 11s appear as the sum of the sides facing up? Answer: 0.08 3. There are 3 copies each of 4 different books. In how many different ways can they be arranged on a shelf? Answer: 369,600 4. With the throw of two dice, what is the probability that the sum will be a prime number? Answer: 5/12 5. If there are 15 horses in a horse race, how many possibilities are there for the win, place, and show (first, second, and third) positions if all orders of finish are possible? Answer: 2730 6. A certain protocol uses 10 bits per message block. If the number of 1¡¯s per block is limited to exactly four, how many different message blocks can be generated? Answer: 210 7. Xyler flipped a coin 10 times. Each flip comes up either heads or tails. How many possible outcomes contain at most three tails? Answer: 176 8. How many permutations of the letters ABCDEFG contain the string BGCF? Answer: 24 9. In the ECE board examinations, the probability that an examinee pass in each subject is 0.8. What is the probability that he will pass in at least 2 subjects? Answer: 0.896 10. What is the probability of getting a 9 exactly once in 3 throws with a pair of dice? Answer: 0.263

11. How many numbers between 3000 and 5000 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 if repetition is not allowed? Answer: 240

Details

12. In how many ways can 10 different magazines be divide among A, B, and C so that A gets 5 magazines, B 3 magazines and C 2 magazines? Answer: 2,520

Details

13. A club has 25 members, 4 of whom are ECE¡¯s. In how many ways can a committee of 3 be formed so as to include at least one ECE? Answer: 970

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14. Droodles Charity Sweepstakes sold one hundred tickets, numbered 1, 2, 3, ¡-., 100 to 100 different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to the Bahamas). How many ways are there to award the prizes if the person holding ticket 13 does not win a prize? Answer: 90345024

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15. NMDA plans to enforce speed limits by using radar traps at 4 different locations within the expressway. The radar traps at each of the locations R1, R2, R3, and R4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket? Answer: 0.27

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16. NMDA plans to enforce speed limits by using radar traps at 4 different locations within the expressway. The radar traps at each of the locations R1, R2, R3, and R4 are operated 40%, 30%, 20%, and 30% of the time, and if a person who is speeding on his way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that he will receive a speeding ticket as he passed through the radar trap located at R2? Answer: 1/9

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17. The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30minute period are 0.03, 0.18, 0.24, 0.28, 0.10, and 0.17, respectively. Find the probability that in this 30-minute period, at most 4 cars receive gas. Answer: 0.83

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18. What is the probability that a poker hand contains full house, that is, three of one kind and two of another kind? Answer: 6/4165 19. Find the probability that a hand of five cards in poker contains four cards of a kind. Answer: 1/4165 20. To avoid arrest, a syndicate places 7 narcotic tablets in a bottle containing 10 vitamin pills that are similar in appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of narcotics? Answer: 14/17

SECTION 1.19 1. The probability that a person, living in a certain town, owns a cat is estimated to be 0.3. Find the probability that the ninth person randomly interviewed in that town is the fourth one to own a cat. Answer: 0.076 Details 2. How many bit strings of length 10 contain at least three 1s and at least three 0s? Answer: 912

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3. In a certain high-school entrance examination, an examinee may select 5 problems from a set of 20 questions. In how many ways can she make her choice? Answer: 15504 Details 4. Determine the probability of getting exactly 2 tails when a coin is tossed four times. Answer: 3/8

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5. A bag contains 7 pink balls and 9 beige balls. If 2 balls are drawn in succession without replacement, find the probability that the two balls drawn are both pink. Answer: 7/40 Details 6. A certain 20 ¨C item examination has only two choices per question. What is the probability of getting exactly 15 correct answers? Answer: 969/65536

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7. What is the probability that a five-card poker hand contains the two of diamonds, the three of spades, the six of hearts, the ten of clubs, and the king of hearts? Answer: 1/52C5

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8. What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different kinds and a fifth card of a third kind)? Answer: 198/4165

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9. Dong's Pizza Company uses taste testing and statistical analysis of the data prior to marketing any new product. Consider a study involving four types of crusts (thin, thin with garlic and oregano, thin with bits of cheese and thick). Dong's is also studying four sauces, (standard, a new sauce with more garlic, a new sauce with fresh basil, and a new sauce with pandan). What is the probability that a judge will get a plain thick crust with a pandan sauce for his first taste test? Answer: 1/16 Details 10. The probability that an examinee passes the board examination is 0.8. Find the probability that the examinee will pass the examination on the fourth try. Answer: 4/625

11. If the odds against event E are 3:8, find the probability of success. Answer: 0.72

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12. The odds that A can solve a given problem are 5 to 7, and the odds that B can solve it are 3 to 6. Find the probability that either A or B can solve the problem. Answer: 3/4

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13. A marksman hits 75% of all his targets. What is the probability that he will hit exactly 4 of his next ten shot? Answer: 0.01622

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14. In a poker game consisting of 5 cards, what is the probability of holding 2 aces and 2 Queens? Answer: 33/54145

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15. Dennis Rodman sinks 50% of all his attempts. What is the probability that he will make exactly 3 of his next 10 attempts? Answer: 15/128 16. If seven coins are tossed simultaneously, find the probability that there will be at least six tails. Answer: 1/16 Details 17. A bag contains 3 white and 5 black balls. If two balls are drawn in succession without replacement, what is the probability that both balls are black? Answer: 5/14 Details 18. Five fair coins were tossed simultaneously. What is the probability of getting three heads and two tails? Answer: 5/16 Details 19. There are 4 white balls and 6 red balls in a sack. If the balls are taken out successively (the first ball is not replaced), what is the probability that the balls drawn are of different colors. Answer: 8/15 Details 20. A first bag contains 5 white balls and 10 black balls. The experiment consists of selecting a bag and then drawing a ball from the selected bag, find the probability of drawing a white ball from the first bag. Answer: 1/6

SECTION 1.20 1. A group of 4 people entered an opera house after the lights head dimmed. They are shown to the correct group of 4 seats by the usher. Each person holds a number stub. What is the probability that each is in the correct seat according to the numbers on seat and stub? Answer: 1/24

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2. There are 15 balls in a box: 8 balls are green, 4 are blue and 3 are white. Then 1 green and 1 blue balls are taken from the box and put away. What is the probability that a blue ball is selected at random from the box? Answer: 3/13

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3. A bag contains 4 white balls and 3 black balls. Another bag contains 3 white balls and 5 black balls. If one ball is drawn from each bag, determine the probability that the balls drawn will be 1 white and 1 black. Answer: 29/56

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4. An um contains 4 black balls and 6 white balls. What is the probability of getting one black ball and white ball in two consecutive draws from the urn? Answer: 0.53

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5. A bus is guaranteed to arrive at the bus stop between 8:00 am and 8:15 am. It is no more likely to arrive at any time than any other time in this interval (i.e. the probability density function is uniform). What is the probability that the bus will arrive at 8:05 am? Answer: 0

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6. A bus is guaranteed to arrive at the bus stop between 8:00 am and 8:15 am. It is no more likely to arrive at any time than any other time in this interval (i.e. the probability density function is uniform). What is the probability that the bus will arrive at 8:05 am? Answer: 1/3

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7. If 10 coins are to be flipped and the first 5 all come up heads, what is the probability that exactly 3 more heads will be flipped? Answer: 0.3125 Details 8. An organization has 26 members. How many ways are there to choose a chairman, vice chairman, secretary and treasurer of the organization, where no person can hold more than one office? Answer: 358800 Details 9. The probability for the ECE board examinees from a certain school to pass the subject in mathematics is 3/7 and for the subject of Communication is 5/7. If none of those examinees fail both subjects and there are four examinees who passed both subjects, find the number of examinees from that school who took the examinations. Answer: 28

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10 The dartboard has nine numbered blocks. Each block measuring 20cm x 20 cm. The number on each block is the score earned when a dart hits that block. A dart, which hits the unnumbered portion of the dartboard, gets a score of zero. Assuming all the darts hit the dartboard and with two darts, what is the probability of getting a total score of 11? Answer: 0.0128

11. The dartboard has nine numbered blocks. Each block measuring 20cm x 20 cm. The number on each block is the score earned when a dart hits that block. A dart, which hits the unnumbered portion of the dartboard, gets a score of zero. Assuming all the darts hit the dartboard, what is the probability of getting a score of zero with one dart? Answer: 0.64

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12. The dartboard has nine numbered blocks. Each block measuring 20cm x 20 cm. The number on each block is the score earned when a dart hits that block. A dart, which hits the unnumbered portion of the dartboard, gets a score of zero. Assuming all the darts hit the dartboard, what is the probability of getting a score of seven with one dart? Answer: 0.04

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13. An urn contains 4 white balls and 3 black balls. Another urn contains 3 white balls and 5 black balls. If one ball is drawn from each urn, determine the probability that the balls drawn will be 1 white and 1 black. Answer: 29/56

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14. An organization has 25 members, 5 of whom are ECE¡¯s. In how many ways can a committee of 3 be formed so as to include at least one ECE? Answer: 1160 15. In how many ways can a group of 6 people be seated on a row of 6 seats if a certain 2 refuse to sit next to each other? Answer: 480 ways

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16. What is the probability of drawing 6 white balls from a jar containing 9 white, 4 red, and 3 blue balls? Answer: 0.0105 Details 17. Ten books consisting of 5 mathematics books, 3 physics books, and 2 chemistry books are placed in a bookcase at random. What is the probability that the same books are all together? Answer: 1/420

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18. It is known that 3 out of 10 television sets are defective. If 2 television sets are selected at random from the 10, what is the probability that 1 of them is defective? Answer: 7/15

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19. A worker with a bunch of 10 keys is to open a locked but only one key can open. What is the probability that he will succeed in 2 trials? Answer: 0.1

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20. A bag contains 3 white balls and 5 red balls. If 2 balls are drawn in succession without returning the first ball drawn, what is the probability that the balls drawn are both red? Answer: 0.357

SECTION 1.21 1. If there are 10 distinct items taken 4 at a time, how many arrangements will there be? Answer: 5040 Details 2. In how many ways can 9 books be arranged on a shelf so that 5 of the books are always together? Answer: 14400 Details 3. How many ways can 3 men and 4 women be seated on a bench if the women are to together at all times. Answer: 576

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4. The probability that a man can marry his widowed sister is: Answer: 0

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5. How many permutation can be made out of the letters of the word ENGINEERING? Answer: 277,200

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6. In a guessing type of exam with 4 choices, what is the probability that you will get at least 7 of the 10 items? Answer: 0.0035 7. In a certain electronic factory, the ratio of the number of male to female workers is 2: 3. If one hundred new female workers are hired, the number of female workers will increase to 65% of the total number of workers. Find the original number of workers in the factory. Answer: 700 workers

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8. A card is drawn from a deck of 52 cards. Find the probability of drawing a king or a red. Answer: 0.5385 Details 9. What is the probability of obtaining at least 4 heads when a coin is tossed 5 times? Answer: 0.1875 10. Three copies of ECE books, 4 copies of EE books and 2 copies of ME books are covered with covers of different colors of each kind of book. In how many different ways can they be placed on a shelf? Answer: 1,260 ways

11. An urn contains 7 yellow balls and 10 green balls. Another urn contains 5 yellow balls and 3 green balls. If one ball is drawn from each urn, determine the probability that both are yellow. Answer: 35/136

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12. Find the probability of getting exactly 12 out of 30 questions on true or false questions. Answer: 0.08

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13. In how many ways can 9 books be arranged on a shelf so that 5 of the books are always together? Answer: 14400 Details 14. A number of ECE examinees decided to have an excursion after their licensure exam. They hire a boat for P 56. But for some reasons two of them were not able to join so that the share of the remaining in the group increased by P 3.20. How many are in the original group? Answer: 7

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5. An inexperience statistical clerk submitted the following statistics to his manager on the average rate of production of radios in an assembly line, 15 workers produce 3 radios in 2 hours. How many workers are employed in the assembly line working 40 hours per weekly production is 480 radios. Answer: 120

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16. A semiconductor company will hire 7 men and 4 women. IN how many ways can the company choose from 9 men and 6 women who is qualified for the position. Answer: 540

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17. Donaire made total bouts of 12 in 3 years. In how many ways can he end his record with 7 wins, 3 losses and 2 draws? Answer: 7920 Details 18. In how many ways can a party of 6 people be seated in a row of 6 if a certain 2 insist on sitting next to each other? Answer: 240 ways

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19. The probability that A will pass is •0†6. The probability that B will pass is 5/6. What is the probability that either A or B will pass is: Answer: 0.96

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20. In an examination, 42% failed in Algebra and 52% failed in English. If 17% students failed in both subjects, find the percentage of those students who passed in both subjects. Answer: 23%

SECTION 1.22 1. Three copies of ECE books, 4 copies of EE books and 2 copies of ME books are covered with covers of different colors of each kind of book. In how many different ways can they be placed on a shelf. Answer: 1260 Details 2. From a box containing 10 black balls, 7 blue balls and 5 white balls one ball is drawn at random. Determine the probability that it is blue or white. Answer: 6/11 Details 3. 6 books with black covers and 4 books with red covers are placed on a shelf. How many color arrangements are possible? Answer: 210

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4. In how many ways can 6 vacancies be filled with either 3 or 4 men if there are 9 men and 7 women applicants? Answer: 5,586 Details 5. How many permutation can be made out of the letters of the word ENGINEERING? Answer: 277,200

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6. If seven coins are tossed simultaneously, find the probability that they will just have three heads. Answer: 35/128

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7. A UN forces for Bosnia uses a type of missile that hits the target with a probability of 0.3. How many missiles should be fired so that there is at least an 80% probability of hitting the target?. Answer: 5

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8. In an election, one of the two candidates got 42% of the total votes and still lost by 3,680 votes. Find the total number of votes. Answer: 23,000 9. The value of probability of any outcome will never be equal to or exceed. Answer: 1

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10. If two events A and B are mutually exclusive events and the probability that A or B will happen is: Answer: Pa + Pb

1. Find the probability of getting exactly 12 out of 30 questions on true or false questions. Answer: 0.08

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2. In how many ways can 9 books be arranged on a shelf so that 5 of the books are always together? Answer: 14400 Details 3. Four electronics and three communication books are placed in a shelf. What is the probability that the Communications book will be together? Answer: 0.1428 Details 4. A door password consists of 2 digits from 0 to 9 and 2 letters of the English alphabet. How many different distinct passwords are possible if repetitions are not allowed? Answer: 58500 Details 5. Manny Pacquiao made total bouts of 12 in 3 years. In how many ways can he end his record with 7 wins, 3 losses and 2 draws? Answer: 7920 Details 6. From the digits 0, 1, 2, 3, 4, 5, 6 ,7, how many 3-digit odd numbers that are distinct can be formed? Answer: 144

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7. In a family of 5 children, what is the probability that the first three are boys? Answer: 1/8

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8. In how many ways you can invite 1 or more of your 5 friends in a certain show? Answer: 31

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9. During a certain war, Philippines troop released 4 bombs to USA each of which has a probability of hitting the enemy of 0.25. If 4 bombs are released simultaneously, what is the probability of hitting the Americans? Answer: 0.6835 Details 10. Oscar and Des work independently in troubleshooting a circuit. The probability that that Oscar can fix it is 0.6 while the probability that Des can fix it is 0.5. What is the probability that the circuit will be fixed? Answer: 0.8

1. If the probability that an event will happen exactly three times in five trials is equal to the probability that it will happen exactly two times in six trials, find the probability that it will happen in one trial. Answer: 0.451 Details 2. There are three candidates Nelson, Cynthia, Bogs for mayor of a certain town. If the odds are 7:5 that Nelson will win and those of Cynthia are 1:3, what is the probability that Bogs will win? Answer: 1/6

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3. In a certain electronic factory, the ratio of the number of male to female workers is 2:3. If 100 new female workers are hired, the number of female workers will increase to 65% of the total number of workers. Find the original number of female workers in the factory. Answer: 420

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4. A person draws 3 balls in succession from a box containing 5 red balls, 6 yellow balls and 7 green balls. Find the probability of drawing the balls in the order red, yellow and green. Answer: 0.04289

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5. In the ECE board examinations, the probability that an examinee will pass in each subjects is 0.8. What is the probability that he will pass in at least two subjects? Answer: 0.896 Details 6. In how many ways can 6 distinct books be arranged in bookshelf? Answer: 720

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7. In how many ways can 3 marines and 4 army soldiers be seated on a bench if the marines must be seated together? Answer: 720 ways

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8. Donaire made total bouts of 12 in 3 years. In how many ways can he end his record with 7 wins, 3 losses and 2 draws? Answer: 7920 Details 9. The probability that A will pass is •0†6. The probability that B will pass is 5/6. What is the probability that either A or B will pass is: Answer: 0.96

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10. From a box containing 10 black balls, 7 blue balls and 5 white balls one ball is drawn at random. Determine the probability that it is blue or white. Answer: 6/11

SECTION 1.23 1. A group of 7 friends are having lunch together. Each person eats at least 3/4 of a pizza. What is the smallest number of whole pizzas needed for lunch? Answer: 6

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2. z varies directly as x and inversely as y2. If x = 1 and y = 2, then z = 2. Find z when x = 3 and y = 4. Answer: 1.5

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3. Find the third proportional to 4 and 12. Answer: 36

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4. For the remainder of the division of x3 - 2x2 + 3kx + 18 by x - 6 to be equal to zero, k must be equal to. Answer: -9

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5. In what ratio must a peanut costing P240.00 per kg. Be mixed with a peanut costing P340.00 per kg so that the profit of 20% is made by selling the mixture at 360.00 per kg? Answer: 2:3

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6. The sum of two numbers is 21 and their product is 108. Find the sum of their reciprocals. Answer: 7/36 Details 7. Two friends A and B are respectively 5 and 8 years old. In how many years will the ratio of their ages be 3:4? Answer: 4

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8. If a two digit number has x for its unit¡¯s digit and y for its ten¡¯s digit, represent the number. Answer: 10y + x 9. The denominator of a certain fraction is three more than twice the numerator. If 7 is added to both terms of the fraction, the resulting fraction is 3/5. Find the original fraction. Answer: 5/13 Details 10. Given that w varies directly as the product of x and y and inversely as the square of z and that w=4 when x=2, y=6, and z=3. Find w when x=1, y=4 and z=2. Answer: 3

1. If x varies directly as y and inversely as z, and x=14 when y=7 and z=2, find x, when z=4 and y=16. Answer: 16

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2. The electrical resistance of a cable varies directly as its length and inversely as the square of its diameter. If a cable 600 meters long and 25 mm in diameter has a resistance of 0.1 ohm, find the length of the cable 75 mm in diameter with resistance of 1/6 ohm. Answer: 9000 m Details 3. The electrical resistance offered by an electric wire varies directly as the length and inversely as the square of the diameter of the wire. Compare the electrical resistance offered by two pieces of wire of the same material, one being 100 m long and 5 mm diameter, and the other is 50 m long and 3 mm in diameter. Answer: R1= 0.72 R2 4. The time required for an elevator to lift a weight varies directly with the weight and the distance through which it is to be lifted and inversely as the power of the motors. If it takes 20 seconds for a 5-hp motor to lift 50 lbs. through 40 feet, what weight can an 80-hp motor lift through a distance of 40 feet within 30 seconds? Answer: 1200 lbs.

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5. The time required by an elevator to lift a weight, vary directly with the weight and the distance through which it is to be lifted and inversely as the power of the motor. If it takes 30 seconds for a 10-hp motor to lift 100lbs through 50 feet, what size of motor is requires to lift 800 lbs. in 40 seconds through a distance of 40 feet? Answer: 48 hp

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6. The vibration frequency of a string varies as the square root of the tension and inversely as the product of the length and diameter. Of the testing. If the testing is 3ft long and 0.03 inch diameter vibrates at 720 times per second under 90 pounds tension, at what frequency will a 2ft, 0.025 inch string vibrate under 50 pounds tension. Answer: 966

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7. The volume of a hemisphere varies directly as the cube of its radius. The volume of the hemisphere with 2.54cm, radius is 20.75 cm3. What is the volume of a sphere with 3.25cm.radius of the same kind of material? Answer: 86.92

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8. The enrollment at college A and college B both grew up by 8% from 1980 to 1985. If the enrollment in college A grew up by 800 and the enrollment in college B grew up by 840, the enrollment at college B was how much greater than the enrollment in college A in 1985? Answer: 540

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9. A boat takes 2/3 as much time to travel downstream from C to D, as to return, If the rate of the river¡¯s current is 8 kph, what is the speed of the boat in still water? Answer: 40

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10. If Juan is 10% taller than Pedro and Pedro is 10% taller than Maria, then Juan is taller than Maria by how many percent? Answer: 21%

SECTION 1.24 1. The population of the Philippines doubled in the last 30 years from 1967 to 1997. Assuming that the rate of population rate increase will remain the same in what year wills the population triple? Answer: 2014

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2. Missy goes to a bake shop to buy some pastries for resale at Book Latte. She spends half her money for Revel Bars, and one-third of what remains for Triple Chocolate Brownies. She spends 150 for other pastries and still has 200 left from the amount she originally had. How much money did she have at the start? Answer: 1050

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3. Arcadio Corporation’s gross margin is 45% of sales. Operating expenses such sales and administration are 15% of sales. Arcadio is in 40% tax bracket. What percent of sales is their profit after taxes? Answer: 18%

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4. A boat propelled to move at 25 mi/hr in still water, travels 4.2 miles upstream in the same time that it can travel 5.8 miles downstream. Find the speed of the stream. Answer: 4

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5. The velocity of an airplane in still air is 125 kph. The velocity of the wind due east is 25 kph. If the plane travels east and returns back to its base again in 4 hours, at what distance does the plane travel due east? Answer: 240 km. Details 6. It takes a boat 3 times to travel upstream against a river current that it takes the same boat to travel downstream. If the speed of the boat is 40 kph, what is the speed of the current? Answer: 20 mph Details 7. Pedro runs with a speed of 20 kph. Five minutes later, Mario starts running to catch Pedro in 20 minutes. Find the velocity of Mario. Answer: 25 kph Details 8. A man rows downstream at the rate of 5 mph and upstream at the rate of 2 mph. How far downstream should he go if he is to return 7/4 hour after leaving? Answer: 2.5 mi

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9. A ship propelled to move at 25 mi/ hr in still water, travels 4.2 miles upstream in the same time that it can travel 5.8 miles downstream. Find the speed of the stream. Answer: 4

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10. If x varies directly as y and inversely as z, and when and , find x when z=4 and y=16. Answer: 16

1. The second of the four numbers is three less than the first, the third is four more than the first and the fourth is two more than the third. Find the fourth number if their sum is 35? Answer: 13

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2. A jogger starts a course at a steady rate of 8 kph. Five minutes later, a second jogger starts the same course at 10 kph. How long will it take the second jogger to catch the first? Answer: 20 min Details 3. An airplane flying the wing, took 2 hours to travel 1000 km and 2.5 Hours in flying back. What was the wind velocity in kph? Answer: 50

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4. Two planes leave Manila for a southern city, a distance of 900km. Plane A travels as a ground speed of 90 kph faster than the plane B. Plane A arrived in their destination 2 hours and 15 min ahead a plane B. What is the ground speed of plane A? Answer: 240 kph Details 5. On a certain trip, Ben drive 231 km in exactly the same time as Allan drive 308 km. If Allan¡¯s rate exceeded that of Ben by 13 kph, determine the rate Allan. Answer: 52 kph Details 6. The electric power which a transmission line can transmit is proportional to the product of its design voltage and current capacity, and inversely to the transmission distance. A 115-kilovolt line rate at 100 amperes can transmit 150 megawatts over 150 km. how much power, in megawatts can a 230 kilovolt line rate at 150 amperes transmit over 100 km? Answer: 675

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7. Find the ratio of an infinite geometric progression if the sum is 2 and the first term is 1/2. Answer: 3/4

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8. A hiker walks from his car to a distant lake and back again. He walks on smooth terrain for 2 hours until he reaches a 5-mile-long, rocky trail. His pace along the 5-mile-long trail is 2 mph. If he walks steadily with no stops, how long will it take the hiker to complete the entire trip from his car to the lake and back again? Answer: 9 hours Details 9. White flour and whole wheat flour are mixed together in a ratio of 5 parts white flour to 1 part whole wheat flour. How many pounds of white flour are in 48 pounds of this mixture? Answer: 40 pounds

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10. Timmy can sell 20 glasses of lemonade for 10 cents per glass. If he raises the price to 25 cents per glass, Timmy estimates he can sell 7 glasses. If so, how much more money will Timmy make by charging 25 cents instead of 10 cents per glass? Answer: -$0.25

1. A four-pound mixture of raisins and nuts is 2/3 raisins. How many pounds of nuts are there? Answer: 1.3 pounds

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2. To make 20 biscuits, Juanita uses 5 cups of flour to 1 cup of milk. If she uses 3 cups of milk, how many cups of flour will she use? Answer: 15

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3. When 2,000 pounds of paper are recycled or reused, 17 trees are saved. How many trees are saved if 5,000 pounds of paper is recycled? Answer: 42.5

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4. On a certain map, 1 cm = 12 km actual distance. If two places are 96 km apart, what is their distance on map? Answer: 8 cm Details 5. A person types 360 words in 4 minutes. How much time does he take to type 900 words? Answer: 10

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6. Two numbers are in the ratio 3 : 4. If the sum of numbers is 63, find the numbers. Answer: 27, 36 Details 7. If 2A = 3B = 4C, find A : B : C. Answer: 6:4:3 Details 8. What must be added to each term of the ratio 2 : 3, so that it may become equal to 4 : 5. Answer: 2

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9. The length of the ribbon was originally 30 cm. It was reduced in the ratio 5 : 3. What is its length now? Answer: 18

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10. Find the ratio of an infinite geometric progression if the sum is 2 and the first term is 1/2. Answer: 3/4

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SECTION 1.25 1. Mary was four times as old as Lea ten years ago. If she is now twice as old as Lea, how old is Mary. Answer: 30

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2. Jose’s rate of doing work three times as fast as Bong. On given day Jose and Bong work together for 4 hours then Bong was called away and Jose finishes the rest of the job in 2 hours. How long would it take Bong to do the complete job alone? Answer: 22 hrs. Details 3. A solution is made of water and pure acid. If 75% of the solution is water, how many litters of pure acid are in 20 liters of this solution? Answer: 5

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4. Three men A, B, and C can do a piece of work in t hours working together. Working alone, A can do the work in 6 hours more, B in 1 hour more, and C in twice the time if all working together. How long would it take to finish the work if all working together? Answer: 40 mins.

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5. Rukia has nickels, dimes, and quarters amounting to $1.85. If he has twice as many dimes as quarters, and the number of nickels is two less than twice the number of dimes, how many quarters does he have? Answer: 3

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6. How many liters of a 25% acid solution must be added to 80 liters of a 40% acid solution to have a solution that is 30% acid? Answer: 160L Details 7. What time after 2 o¡¯clock will the hands of the clock extend in opposite directions for the first time? Answer: 2:43.64

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8. Seven carpenters and 5 masons earn a total of 2,300 per day. At the same rate of pay 3 carpenters and 8 masons earn 2,040. What are the wages per day of the carpenter and a mason? Answer: 200 & 180

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9. A man and a boy can do 15 days a piece of work which would be done by 7 men and 9 boys in 2 days. How long would it take one man do it alone? Answer: 20 days

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10. Ana is 5 years older than Beth. In 5 years, the product of their ages is 1.5 times the product of their present ages. How old is Beth now? Answer: 20

1. An audience of 540 people is seated in rows having the same number of persons in each row. If 3 more persons seat in each row, it would require 2 rows less to seat the audience. How many persons were in each row originally? Answer: 27

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2. Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is now. How old is Ana? (ECE November 1995) Answer: 18

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3. One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4 hours. A drainpipe can empty the full content of the tank in 20 hours. With all the three pipes open, how long will it take to fill the tank? Answer: 2.5 hours

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4. A swimming pool is filled through its inlet pipe and then emptied through its outlet pipe in a total of 8 hours. If water enters through its inlet and simultaneously allowed to leave through its outlet, the pool is filled in 7 •0†5 hours. Find how long will it take to fill the pool with the outlet closed. Answer: 3

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5. An alloy of silver and gold weighs 15 oz. in air and 14 oz. in water. Assuming that silver losses 1/10 of its weight in water and gold losses 1/18 of its weight, how many oz. at each metal are in the alloy? Answer: Silver = 3.75 oz.; Gold = 11.25 oz.

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6. A 100-kilogram salt solution originally 4% by weight. Salt in water is boiled to reduce water content until the concentration is 5% by weight salt. How much water is evaporated? Answer: 20

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7. A father is now 41 and his son 9. After how many years will his age be just triple his son¡¯s age? Answer: 7

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8. Your father told you ¡°I was your age now when you were born¡±. If you are 21 years old today, how old is your father? Answer: 42

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9. Mark has nickels, dimes, and quarters amounting to $1.85. If he has twice as many dimes as quarters, and the number of nickels is two less than twice the number of dimes, how many quarters does he have? Answer: 3

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10. A US dollar equals 0.716 European euros and a Japanese yen equals 0.00776 European euros. How many US dollars equal a Japanese yen? Answer: 0.011

SECTION 1.26 1. Two times the father’s age is 8 more than six times his son’s age. Ten years ago, the sum of their ages was 44. The age of the son is: Answer: 15

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2. The ages of the mother and her daughter are 45 and 5 years, respectively. How many years will the mother be three times as old as her daughter? Answer: 15

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3. How many minutes after 10:00 o’clock will the hands of the clock be opposite of the other for the first time? Answer: 21.81 Details 4. A man travels in a motorized banca at rate of 12 kph from his barrio to the poblacion and come back to his barrio at the rate of 10 kph. If his total time of travel back and forth is 3 hours and 10 minutes, the distance from the barrio to the poblacion is: Answer: 17.27 km

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5. Nonoy left Pikit to drive to Davao at 6:15 PM and arrived at 11:45 PM averaged 30 mph and stopped 1 hour for dinner, how far is Davao from Pikit. Answer: 135

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6. From the time 6:15 PM to the time 7:45 PM of the same day, the minute hand of a standard clock describes an arc of: Answer: 540

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7. What time between the hours of 12:00 noon and 1:00 pm would the hour hand and the minute hand of a continuously driven clock be in straight line? Answer: 12:33 pm

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8. A job could be done by eleven workers in 15 days. Five workers started the job. They were reinforced with four more workers at the beginning of the 6th day. Find the total number of days it took them to finish the job. Answer: 20.56 Details 9. Delia can finish a job in 8 hours. Daisy can do it in 5 hours. If Delia worked for 3 hours and then Daisy was asked to help her finish it, how long will Daisy have to work with Delia to finish the job? Answer: 1.923 hours

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10. Three persons can do a piece of work alone in 3 hours, 4 hours and 6 hours respectively. What fraction of the job can they finish in one hour working together? Answer: 3/4

1. At what time after 12:00 noon will the hour hand and the minute hand of a clock first form a n angle of 120? Answer: 12:21.818

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2. Mr. Brown can wash his car in 15 minutes, while his son John takes twice as long as the same job. If they work together, how many minutes can they do the washing? Answer: 10

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3. A 40-gram alloy containing 35% gold is to be melted with a 20-gram alloy containing 50% gold. How much percentage of gold is the resulting alloy? Answer: 40%

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4. Two thousand kilogram of steel containing 8% of nickel is to be made by mixing steel containing 14% nickel with another steel containing 6% nickel. How much of the steel containing 14% nickel is needed? Answer: 500kg Details 5. A father is three times as old as his son. Four years ago, he was four times as old as his son was at that time. How old is his son? Answer: 12 years

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6. A production supervisor submitted the following report on the average role of production of printed circuit boards (PCB) in an assembly line, 1.5 workers produce 12 PCB’s in 2 hours. How many workers are employed in an assembly line working 40 hours each week with a weekly production of 8000 PCB’s? Answer: 50

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7. Pedro can paint a fence 50% faster than Juan and 20% faster that Pilar and together they can paint a given fence in 4 hours. How long will it take Pedro to paint the same fence if he had to work alone? Answer: 10 hrs Details 8. What time after 2 o’clock will the hands of the clock extend in opposite directions for the first time? Answer: 2:43.64

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9. How many liters of a 25% acid solution must be added to 80 liters of a 40% acid solution to have a solution that is 30% acid? Answer: 160 L Details 10. Given two numbers such that the difference of twice the first and the second number is 12. If the sum of the first and the second number is 36, find the number. Answer: 16, 20

SECTION 1.27 1. A carpenter and his helper together can repair a house in 10 days. It takes the helper 5 days longer than the carpenter to do the repair when each works alone. How many days would it take the helper to do the repair if he is to work alone? Answer: 22.8 days

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2. Two workers A and B together can complete a job in 7 days. A works twice as fast as B. How many days would it take B to do the job working alone? Answer: 21days 3. If soldering lead contains 63% silver, ______ grams of soldering lead can be made from 520 grams of silver. Answer: 825.4 Details 4. Pipes A and B can fill an empty tank in 6 and 3 hours respectively. Drain C can empty a full tank in 24 hours. How long will an empty tank be filled if pipes A and B with drain C open? Answer: 2.182 hours

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5. It takes pump (A) 4 hours to empty a swimming pool. It takes pump (B) 6 hours to empty the same swimming pool. If the two pumps are started together, at what time will the two pumps have emptied 50% of the water in the swimming pool? Answer: 1 hour 12 minutes

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6. If Chicago is 10% taller than Ishida and Ishida is 10% taller than Chad, then Ichigo is taller than Chad by how many percent? Answer: 21%

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7. If thrice the smaller number exceeds the larger by 12. Find the larger number if the two numbers are consecutive odd integers. Answer: 9 8. An investor has P100,000, part of which he invested at 12% interest and the rest at 18%. He received a total annual interest of P15,300. How much did he invest at 18% interest rate? Answer: 55,000 9. An object travels at fifteen feet per minute. How many feet does it travel in 24 minutes and 40 seconds? Answer: 370

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10. The sum of the digits of a two digit number is 11. If the digits are reversed, the resulting number is seven more than twice the original number. What is the original number? Answer: 38

1. Crew 1 can finish the installation of an antenna tower in 200 hours while crew 2 can finish the same job in 300 hours. How long will it take both crews to finish the same job working together? Answer: 120 hours

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2. The time required for two examinees to solve the same problems differs by two minutes. Together they can solve 32 problems in one hour. How long will it take for the slower problem solver to solve the problem? Answer: 5min Details 3. Wendy was four times as old as Lily ten years ago. If she is now twice as old as Lily, how old is Wendy? Answer: 30

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4. In how many minutes after 2 o¡¯clock will the hands of the clock extend in opposite directions for the first time? Answer: 2: 43 min & 38 sec

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5. Peter’s age 13 years ago was 1/3 of his age 7 years hence. How old is Peter? Answer: 23

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6. A man is 41 years old and in seven years he will be four times as old as his son is at that time. How old is his son now? Answer: 5

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7. A pump can pump out a tank in 11 hours. Another pump can pump out the same tank in 20 hours. How long it will take both pumps together to pump out the tank? Answer: 7 hours

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8. The sum of the reciprocals of two numbers is 11. Three times the reciprocal of one of the numbers is three more than twice the reciprocal of the other number. Find the numbers. Answer: 1/5 and 1/6

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9. One number if five less than the other number. If their sum is 135, what are the numbers? Answer: 65&70 Details 10. In a two-digit number, the unit¡¯s digit is 3 greater than the ten’s digit. Find the number if it is 4 times as large as the sum of its digits. Answer: 36

SECTION 1.28 1. Find two consecutive even integers such that the square of the larger is 44 greater than the square of the smaller integer. Answer: 10&12 Details 2. The product of three consecutive integers is 9240. Find the third integer. Answer: 22

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3. The sum of the digits of the three-digit number is 14. The hundreds digit being 4 times the units digit. If 594 is subtracted from the number, the order of the digits will be reversed. Find the number. Answer: 842

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4. The sum of two numbers is 21, and one number is twice the other. Find the numbers. Answer: 7 and 14

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5. Ten less than four times a certain number is 14. Determine the number. Answer: 6

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6. Kim sold a watch for P3500.00 at a loss of 30% on the cost price. Find the corresponding loss or gain if he sold it for P5050.00. Answer: 1% gain

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7. A certain XEROX copier produces 13 copies every 10 seconds. If the machine operates without interruption, how many copies will it produce in an hour? Answer: 4680 Details 8. At a certain printing plant, each of the machines prints 6 newspapers every second. If all machines work together but independently without interruption, how many minutes will it take to print the entire 18000 newspapers? ( Hint: let x = number of machines) Answer: 50/x Details 9. A 10-meter tape is 5 mm short. What is the correct length in meters? Answer: 9.995 m

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10. A man rows downstream at the rate of 5mph and upstream at the rate of 2mph. How far downstream should he go if he is to return in 7/4 hours after leaving? Answer: 2.5 mi

1. The distance passed over by a certain pendulum bob in succeeding swings form the geometric progression 16,12,9,... feet respectively. Calculate the total distance traversed by the bob before coming to rest. Answer: 64 ft Details 2. A man has P100,000, part of which he invested at 12% interest and the rest at 18%. He received a total annual interest of P15,300. How much did he invest at 18% interest rate? Answer: 55,000 Details 3. A man who is on diet losses 24 lb. in 3 months, 16lb. in the next 3 months and so on for a long time. What is the maximum total weight loss? Answer: 72

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4. Ten less than four times a certain number is 14. Determine the number. Answer: 6

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5. In a club of 40 executives, 33 likes to smoke Marlboro and 20 like to smoke Philip Moris. How many like both? Answer: 13

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6. The tens digit of a two-digit number is 1 less than twice the unit¡¯s digit. They differ by 4. Find the number. Answer: 95

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7. A certain two-digit numbers is 1 less than five times the sum of its digits. If 9 were added to the number, its digits would be reversed. Find the number. Answer: 34

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8. The length of a rectangle is 3 times its width. If the width of the rectangle is 5 inches, what is the rectangle's area, in square inches? Answer: 75

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9. The sum of three succeeding odd integers is 75. The largest integer is: Answer: 27

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10. Two times the father¡¯s age is 8 more than six times his son¡¯s age. Ten years ago the sum of their ages is 44. The age of the son is: Answer: 15

1. Minnette’s age 13 years ago was 1/3 of her age 7 years hence. How old is Minnette? Answer: 23

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2. Of the 316 people watching a movie, there are 78 more children than women and 56 more women than men. The number of men in the movie house is: Answer: 42

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3. The sum of the reciprocal of two numbers is 11. Three times the reciprocal of one of the numbers is three more than twice the reciprocal of the other number. Find the numbers? Answer: 1/5 & 1/6

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4. In a racing contest, there are 240 cars which will have fuel provision that will last for 15 hours. Assuming a constant hourly consumption for each car, how long will the fuel provision last if 8 cars withdraw from the race every hour after the first. Answer: 25 hours

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5. A number is divided into two parts such that when the greater part is divided by the smaller part, the quotient is 3, and the remainder is 5. Find the smaller number if the sum of the two numbers is 37. Answer: 8

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6. A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work? Answer: 18.9

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7. Eight men can dig 150 ft of trench in 7hrs. Three men can backfill 100ft of the trench in 4hrs. The time it will take 10 men to dig and fill 200 ft of trench is: Answer: 9.867hrs.

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8. How many minutes after 3:00 will the minute hand of the clock overtakes the hour hand? Answer: 16-4/11 minutes

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9. It is now between 3 and 4 o¡¯clock and in twenty minutes the minute hand will be as much as the hour-hand as it is now behind it. What is the time now? Answer: 3:06.06

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10. The product if two numbers is 1400. If three (3) is subtracted from each number, their product becomes 1175. Find the bigger number. Answer: 50

SECTION 1.29 1. Find the mean, median and mode respectively of the following numbers: 13, 13, 14, 12, 11, 10, 9, 11, 8, 11, 5, and 15 Answer: 11, 11, 11

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2. The standard deviation of the numbers 1, 4, &7 is: Answer: 3

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3. The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint. 3.4

2.5

4.8

2.9

3.6

2.8

3.3

5.6

3.7

2.8

4.4

4.0

5.2

3.0

4.8

Calculate the sample mean for this data. Answer: 3.787 Details 4. A random sample of 20 pieces of cotton fiber is taken and the absorbency on each piece was measured. The following are the absorbency values: 18.71

21.41

20.72 21.81

19.29

22.43

20.17 23.71

19.44

20.50 18.92

20.33

23.00

22.85 19.25

21.77

22.11

19.77 18.04

21.12

Calculate the median for the above sample values. Answer: 20.61 Details 5. Given the data set: 1.7, 2.2, 3.9, 3.11 and 14.7 Determine the sample mean and median respectively. Answer: 5.12, 3.9

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6. A sample size of 10 is taken with results given by: 7.07, 7.00, 7.10, 6.97, 7.00, 7.03, 7.01, 6.98, 7.08, 7.01. Calculate the sample variance. Answer: 0.0019

7. The following data represent the length of life in years, measured to the nearest tenth of 39 similar fuel pumps: 2.0 3.0 0.3 3.3 1.3 0.4 0.2 6.0 5.5 6.5 0.2 2.3 1.5 4.0 5.9 1.8 4.7 0.7 4.5 0.3 1.5 0.5 2.5 5.0 1.0 6.0 5.6 6.0 1.2 0.2 Compute for the sample range. Answer: 6.3

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8. Compute the mean value of the numbers from the tabulated values shown. (Numbers)

(Probability)

5.3

0.38

1.8

0.49

-2.6

0.13

Answer: 2.558 Details 9. Find the mode of the given set of numbers: 5,6,6,7,7,7,8,8,9,9,9,10,10,13 Answer: 7 and 9 10. Find the range of the set of number 9,3,8,8,6,5,11 and 15. Answer: 12

1. What is the variance of the following data {1, 2, 3, 4, and 5}? Answer: 2.5

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2. Given the following samples of distributed population {100, 200, 300, 350, 375, 375, 450}, what is the median? Answer: 350

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3. Find the standard deviation of the following -14, -4, -10, -12, 5, -1. Answer: 6.658 Details 4. In a lotto, there are pingpong balls numbered from 1 to 45. Six balls are drawn 1 at a time, to determine the winning combination in any order. Determine the probability that you will win the jackpot prize. Answer: 1/8145060

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5. There are 5 pocket holes at the periphery of a round horizontal platform. In how many ways can 5 balls of different colors be placed with 1 ball in each pocket. Answer: 24

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6. In statistics, a pictorial description of the probability concepts of independent and dependent events is called Answer: Venn diagram Details 7. What is the standard deviation of 2, 4, 6, 8 and 10? Answer: 2.83

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8. In a club of 40 executives, 33 like to smoke Marlboro and 20 like to smoke Philip Morris. How many like both? Answer: 13

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9. A class of 40 students took examination in Electronics and Communications. If 30 passed in Electronics, 36 passed in Communications and 2 failed in both subjects, how many students passed in both subjects? Answer: 28

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10. In a class of 40 students, 27 like Calculus and 25 like Chemistry. How many like both Calculus and Chemistry ? Answer: none of these are correct

SECTION 1.30 1. Find the mean, median, mode and range of this data set: 19, 18, 21, 16, 15, 17, 20, 18 Answer: 18,18,18,6

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2. On his first three quizzes, Patrick earned a 15, 18, and 16. (A perfect score would have been 20 points.) What does he need to earn on the next quiz to have a mean score of at least 17? Answer: 19

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3. Find the median of the set of numbers - 1,2,3,4,5,6,7,8,9 and 10. Answer: 5.5

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4. Find the median of the set of numbers - 21, 3, 7, 17, 19, 31, 46, 20 and 43. Answer: 20

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5. Find the median of the set of numbers - 100, 200, 450, 29, 1029, 300 and 2001. Answer: 300

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6. The following represents age distribution of students in an elementary class. Find the mode of the values - 7, 9, 10, 13, 11, 7, 9, 19, 12, 11, 9, 7, 9, 10, 11. Answer: 9

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7. Find the mode from these test results ¨C 90, 80, 77, 86, 90, 91, 77, 66, 69, 65, 43, 65, 75, 43, 90. Answer: 90

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8. Find the mode from these test results - 17, 19, 18, 17, 18, 19, 11, 17, 16, 19, 15, 15, 15, 17, 13, 11. Answer: 17

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9. Find the mean of these set of numbers - 100, 1050, 320, 600 and 150. Answer: 444

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10. The following numbers represent the ages of people on a bus ¨C 3, 6, 27, 13, 6, 8, 12, 20, 5, 10. Calculate their mean of their ages. Answer: 11

1. These numbers are taken from the number of people that attended a particular church every Friday for 7 weeks - 62, 18, 39, 13, 16, 37, 25. Find the mean. Answer: 30

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2. Find the mean of the set of ages in the table below:

Answer: 12.4

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3. The heights (in cm) of students of a class is given to be 163, 158, 167, 174, 148. Find the variance. Answer: 76.4

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4. There are 500 staffs in a company. The hours used in leisure per week by 5 of the employees is 5, 2, 8, 10, 7. Find the sample variance. Answer: 9.3

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5. Scores on a history test have average of 80 with standard deviation of 6. What is the z-score for a student who earned a 75 on the test? Answer: -0.833

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6. The weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with standard deviation of .1 ounce. What is the z-score corresponding to a weight of 8.17 ounces? Answer: 1.7

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7. Books in the library are found to have average length of 350 pages with standard deviation of 100 pages. What is the z-score corresponding to a book of length 80 pages? Answer: -2.7

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8. The temperature is recorded at 60 airports in a region. The average temperature is 67 degrees Fahrenheit with standard deviation of 5 degrees. What is the z-score for a temperature of 68 degrees? Answer: 0.2

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9. A group of friends compares what they received while trick or treating. They find that the average number of pieces of candy received is 43, with standard deviation of 2. What is the z-score corresponding to 20 pieces of candy? Answer: -11.5

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10. The mean growth of the thickness of trees in a forest is found to be .5 cm/year with a standard deviation of .1cm/year. What is the z-score corresponding to 1 cm/year? Answer: 5

1. Find the skewness of the following data: 500, 1000, 1000, 1500, 1500, 1500, 2000, 2000, 2000, 2000, 2500, 2500, 2500, 3000, 3000, 3500 Answer: 0

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2. You are given these numbers: 1, 2, 3, 4, 5, 9, 23, 32, and 69. Find the skewness. Answer: 1.53

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3. Calculate the skewness of given data; x: 2,3,4,5 ? Answer: 0

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4. Calculate the skewness of following data; x: 5, 10, 15 Answer: 0

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5. If the mean is greater than the mode, the distribution is ___. Answer: Positively skewed

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6. If the mean is less than the mode, the distribution is ___. Answer: Negatively skewed

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7. The following represents age distribution of students in an elementary class. Find the mode of the values - 7, 9, 10, 13, 11, 7, 9, 19, 12, 11, 9, 7, 9, 10, 11. Answer: 9

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8. Find the mode from these test results - 90, 80, 77, 86, 90, 91, 77, 66, 69, 65, 43, 65, 75, 43, 90. Answer: 90

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9. The following numbers represent the ages of people on a bus ¨C 3, 6, 27, 13, 6, 8, 12, 20, 5, 10. Calculate their mean of their ages. Answer: 11. Details 10. There are 5 pocket holes at the periphery of a round horizontal platform. In how many ways can 5 balls of different colors be placed with 1 ball in each pocket. Answer: 24