Sun Moon Star: 30 Days - 30 Problems

Mathematics

SUN MOON STAR Academic Centre

SUN MOON STAR Academic Centre

လ

Mathematics 30 Days - 30 Problems Chapter 1 to 7 & 10, 11, 12 1

Mathematics

SUN MOON STAR Academic Centre

SUN MOON STAR Academic Centre လ ၀၉၄၂၁၇၄၇၃၉၆ ၀၉၉၅၆၁၉၂၁၉၄ လ လ

လ 2

Mathematics

SUN MOON STAR Academic Centre

1. 2.

3. 4.

Day 1

2

A function f is defined by f(x) = x – 4x + 6. Find the possible values of x which are unchanged by the mapping. Let R be the set of real numbers and a binary operation ʘ on R be defined by x ʘ y = xy – x + y for x, y R. Find the values of (2 ʘ 1) ʘ 3 and 2 ʘ (1 ʘ 3). Is the binary operation associative? Prove your answer. If x – 2 and x + 3 are factors of the polynomial f(x) = px3 + x2 – 13 + pq, find the values of p and q. The coefficient of x3 in the expansion of (1 + )n is 7, find the value of n.

5. 6. 7.

Use a sketch graph to obtain the solution set of x2. If the A.M between x and y is 4, calculate the numerical value for which x3 + y3 + 24xy. In an A.P, 40, 37, 34, … ,find the sum to the first 12 terms and find the sum from 13th term to 24th term.

8.

If A = (

9. 10. 11. 12. 13.

) and A2 + A– 1 = mB, where m is a real number, find the value of

), B = (

m. Find the solution set of the system of the equations 5x + 6y = 25 and 3x + 4y = 17 by matrix method; the variables are on the set of real numbers. A coin is tossed 4 times. Draw a tree diagram and list the possible outcomes. Find the probability that the number of heads is more than the number of tails. ⃗⃗ . ⃗ and ⃗⃗ are non-parallel and non-zero vectors, such that ⃗ + t ( ⃗⃗ + 2 ⃗) = 2 + + s( ⃗ Find t. ( ( Show that = are two angles in different quadrants such that find ( without using the table

14. Calculate

and

.

15. Show that the point (0,π) lies on the curve x2 cos2 y = sin y. Then find the equations of tangent and normal to the curve at the point (0,π). ***** S. M. S ac *****

Day 2 1. 2.

3. 4. 5. 6. 7.

Let R be the set of real numbers and Q be the set of positive real numbers. If g : Q R be –1 given by g(x) = log10 x, find g (2). Let J+ be the set of all positive integers. An operation ʘ on J+ is given by x ʘ y = (2x +y), for all positives integers x and y. Prove that ʘ is a binary operation on J+ and calculate (2 ʘ 3)ʘ4. Is the binary operation commutative? Given that the expression x2 – 5x + 7 leaves the same remainder whether divided by x – b or x– c, where b c, show that b + c = 5. If the coefficient of x2 in the expansion of (2x + k)6 is equal to the coefficient of x5 in the expansion of (2 + kx)8, find k. Find the solution set in R for the inequation (1 + 2x)3 + (1 – 2x)3 – 22. Find the smallest positive number in an A. P. 179, 173, 167, …. An A.P contains seven terms, the sum of three terms in the middle is 39 and the sum of the last three terms is 57. Find the series.

8.

Let A = (

), B = (

) and C = (

). Prove that A (B + C) = AB + AC. What

9.

is the name of this law? Use the matrix method to find the solution set of the system of the linear equations, 3y–2x=1 and x + 2y = 10.

3

Mathematics

SUN MOON STAR Academic Centre

10. A coin is tossed four times. Head or tail is recorded each time. By drawing three diagram, find the probability of getting exactly one tail, getting at least one tail, getting no tail. 11. If the vector ⃗ and ⃗⃗ are non-collinear, find the value of x, for which, the vectors ⃗ = (x – 2) ⃗ + ⃗⃗ and ⃗ = (2x + 1) ⃗ – ⃗⃗ are collinear. 12. Prove that cosec x – sin x= cos x cot x. 13. If sin θ = a, where θ is an acute angle, express in terms of a. 14. Calculate the gradient of the curve y = at the point (2,4). 15. Find the coordinates of the points on the curves x2 – y2 = 3xy – 39 at which the tangents are (i) parallel (ii) perpendicular to the line x + y = 1. ***** S. M. S ac *****

Day 3 1. 2. 3. 4. 5. 6. 7.

Let a function f be defined by f(x) = 7 – , x 0. Find the value of x for which f– 1 is undefined. A binary operation ʘ on R is defined by x ʘ y = x + y + 10xy. Show that the binary operation is commutative. Find the values of b such that (1 ʘ b) ʘ b = 485. If x + p is a common factor of x3 – x2 – 7x – 2 and x3 + 3x2 – 4, find the possible values of p. Find the coefficient of x–10 in the expansion of (1 – )8. Find the solution set in R of the inequation (3x – 5)2 – 2 0. Let S4 be the sum to the first four terms of an A.P and S* be the sum of the next four terms. If S* – S4 = 48, find the common difference of that A.P. An A.P contains thirteen terms. If the sum to first four terms is 32 and the sum of the last four terms is 176, find the middle term of that A.P.

8.

Given that X = (

) and Y = (

9.

Find the inverse of the matrix (

), find out whether or not (X + Y) (X – Y) = X2 – Y2. ). Hence determine the coordinates of the point of

intersection of the lines 9x – 2y – 13 = 0 and 2x + 3y + 4 = 0. 10. A family has four children. Draw a tree diagram to list all possible outcomes. If each outcome is equally likely to occur, find the probability that the last two children are girls. Find also the probability that exactly two children are boys. 11. The positive vectors relative to the origin O of the points L and M are ( ) and ( ) respectively. Find the unit vector parallel to ⃗⃗⃗⃗⃗⃗⃗. 12. Prove that 13. Given that tan 2A = cos 2A and sin 2A. 14.

( (

and that ∠A is acute, find without using the tables, the values of

, where a is constant, find a.

15. Find the equations of normal line to the curve y = x 2 – 5x + 6 at the points where the curve cuts the X- axis. ***** S. M. S ac *****

Day 4 1.

2.

Let f: R R and g: R R be f(x) = px + 5 and g(x) = qx – 3, where p 0, q 0. If g ◦ f : R R is the identity function on R, find the value of p. Then prove also that p is the reciprocal of q. An operation ʘ is defined by a ʘ b = a2 – 3ab + 2b2. Find (–2 ʘ 1) ʘ 4. Find p in the equation (p ʘ 3) – (5 ʘ p) = 3p – 17. 4

Mathematics

SUN MOON STAR Academic Centre 3. 4. 5. 6. 7. 8.

If n is an integer, find the remainder when 5x2n + 1 + 10 x2n – 3x2n – 1 + 5 is divided by x + 1. The first three terms in the expansion of (a + b)n, in ascending power of b, are denoted by p, q and r respectively. Show that = ( .

If y = x2 – 4x, find x when y = 0, and also find y when x = 2. Use a sketch graph to obtain the solution set of x2 – 4x 0. For a certain A.P Sn = (3n – 17), calculate u3 and u5 and un. The sum of the first four terms of an A.P is 38. The sum of their squares is 406. Find the third term and the fourth term. If A = (

), I is the unit matrix of order 2 and A2 – 10A + kI = 0, find the value of k.

Show also that (A – 7I) (A – 3I) = 0. 9.

Find the inverse of the matrix (

), where 2a

b and use it to solve the simultaneous

equation ax + by = 2a2 and x + 2y = b in terms of a and b. 10. How many 3-digit numerals can you form from 2, 5, 6 and 0 without repeating any digit? Find the probability of an odd number and find the probability of a numeral which is a multiple of 5. 11. Find the map of the point (2,0) which rotates through an angle of 90 about 0 in clockwise direction. 12. Prove the identity = tan . 13. Without the use of table, evaluate tan (x + y + z) given that tan x = , tan y = and tan z = . 14. When a marble is moving in a groove, the distance s cm from one end at time t sec is given by s = 5t – t2. Find the speed of the marble at t = 2 sec. 15. Calculate the gradient of the curve and find the coordinate of the point at which √ the gradient is 0. ***** S. M. S ac *****

Day 5 A function f: R R is defined by f(x) = evaluate the image of 3 under f.

2.

A binary operation ʘ is defined on the set of natural numbers by x ʘ y =

3. 4. 5. 6. 7. 8.

,x

1. If f – 1 (– 1) = 6, find the value of a and

1.

. Solve the

equation (a ʘ 2a) + (2a ʘ a) = 8a – 2. Find the value of n for which the division of x2n – 7xn + 9 by x – 3 gives a remainder of 27. Find, in ascending power of x, the first three terms of (1 + kx)5 (1 – 4x). If the coefficient of x is 16, find the value of k and the coefficient of x2. Find the solution set in R for the inequation (x + 3) (x + 1) 24. The sum of the first five consecutive terms of an A.P is 110. Find the middle term. The sum of four consecutive numbers in an A.P is 24. The product of the second and third numbers exceeds that of the first and last by 18. Find the numbers. Let A = (

), find p and q such that A2 = pA + qI, where I is the unit matrix of order 2.

9.

Try to solve x + y = 4 and 3x + 3y = 12 by matrices. Explain with the aid of a Cartesian diagram why you failed. 10. Box A contains 4 pieces of paper numbered 1, 2, 3 and 4. Box B contains 2 pieces numbered 1 and 2. One piece of paper is chosen at random from each box. Draw a tree diagram to list all possible outcomes of the experiment. Find the probability that the product of the two numbers chosen is at least 4. Find also the probability that the sum of the two chosen numbers is equal to their product. 11. If P is a point inside a parallelogram ABCD, prove that ⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗.

5

Mathematics

SUN MOON STAR Academic Centre 12. Prove that 13.

(

(

. Without using the tables calculate the

values of . 14. Find the rate of change of the function f(x) = √ at x = 8. √ 2 15. Find the coordinates of the points on the curve x – y2 = xy – 5 at which the tangents are parallel to the line x + y = 1. ***** S. M. S ac *****

Day 6 1.

5. 6. 7.

The function f is defined, for x R, by f: x , x 2. –1 –1 Use the formula of f to find f (5). A binary operation ʘ on N is defined by x ʘ y = the remainder when xy is divided 5. Is the binary operation commutative? Find [(2ʘ3)ʘ4] + [2ʘ(3ʘ4)]. Is the binary operation associative? The remainder when b (b – c)(b + c) is divided by b–2c is 6. Find the value of c. If the (r + 1)th term in the expansion of (2x – )9, where x doesn't equal to 0, is the term independent of x, then find the value of r. Find the solution set in R for the inequation x2 + 4x 0. In an A.P, the 6th term is 22 and the 10th term is 34. Find nth term. Find the sum of all three-digit natural numbers which are divisible by 4.

8.

Given that A = (

2.

3. 4.

), B =(

) and C = (

), find the values of a, b, c and d when

B + AC = 4A. Try to solve 9x + 6y = 4 and 6x + 4y = 2 by matrices. Explain with the aid of a Cartesian diagram why you failed. 10. Draw a tree diagram to list all possible two-digit numerals which can be formed by using the digits 2, 3, 5 and 6 without repeating any digit. If one of these numerals is chosen at random, find the probability that it is divisible by 13. Find also the probability that it is either a prime number or a perfect square. 11. OPQR is a parallelogram and OR is produced to S such that OS = 3OR. If Y is a point on OQ such that OQ = 4YQ, show that Y lies on PS. 12. Show that sec 2x 9.

13. Find the exact value of 4

.

14. Differentiate f(x) = √ with respect to x from the first principle. 15. Find the stationary points of the curve y = x2 (x – 2) and determine their nature. ***** S. M. S ac *****

Day 7 1. 2. 3. 4. 5.

A function f from A to A, where A is the set of positive integers, is given by f(x) = the sum of all positive divisors of x. Find the value of k, if f(15) = 3k + 6. The binary operation ʘ on R is defined by x ʘ y = ax2 + bx + cy, for all numbers x and y. If 1ʘ1 = 4, 2 ʘ 1 = 5 and 1 ʘ 2 = – 1, then find the values of a, b and c. Given that expression 2x3 + ax2 + bx + c leaves the same remainder when divided by x – 2 or by x + 1, prove that a + b = – 6. Write down the third and the fourth terms in the expansion of (a + bx)n. If these terms are equal, show that 3a = (n – 2) bx. Find the solution set in R for the inequation 12 – 2x2 5x.

6

Mathematics

SUN MOON STAR Academic Centre

7.

If S5 is the sum of first 5 terms of A.P., S* is the sum of the next 5 terms and S* – S5 = 75. Find the common difference of the series. Find the sum of all two-digit natural numbers which are divisible by 5.

8.

Show that A = (

6.

) satisfies A2 + I = 2A cos θ, where I is the unit matrix of

order 2. 9.

Solve the matrix equation (

)X=(

Hence find x and y if X = (

). ).

10. Construct a table of possible outcomes for the rolling of two dice. Find the probability that the sum of the scores on the two dice is a prime number. Find also the probability that the product of the scores on the two dice is divisible by 6 or 9. 11. The median AD of ∆ABC is produced to K so that ⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗. If ⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗. Prove that BKCG is a parallelogram. 12. Prove that 1 √

13. Show that

14. Given f(x) = (x + ) (x – ). Show that f '(x) = . 15. Find the equation of the normal line to the curve y = x 2 – 3x + 2 given that the gradient of the normal is . ***** S. M. S ac *****

Day 8 1. 2.

3. 4. 5. 6. 7. 8.

2

Functions f and g are defined by f(x) = 4x – 1 and g(x) = . –1 Find the value of x if (g ◦ f) (x) = 5. Let J be the set of positive integers. Show that the operation ʘ defined by a ʘ b = ab + a + b for a, b J, is a binary operation on J. Find the values of 2 ʘ 4 and 4ʘ2. Is the binary operation commutative? Why? Find all real roots of 1 – x + x2 + x3 = 0. In the expansion of (1 + x)a + (1 + x)b, the coefficient of x and x2 are equal for all positive integers a and b, prove that 3(a + b) = a2 + b2. Find the solution set in R for the inequation (1 + x) (6 – x) – 8. Insert 2 A.M. between 12 and 96. Show that there are 18 integers which are multiple of 17 and which lie between 200 and 500. Find the sum of all these integers. Given that D = (

) and that D2 + 2 D– 1 – kI = 0, where I is the unit matrix of order 2, find

the value of k. 9. If ps qr, find the 2 x 2 matrix X such that ( )X= ( ). Find also X– 1, if it exists. 10. Two dice are rolled. Find the probability of an outcome in which the score on the second die is greater than on the first and also that the total score on the dice is a prime. 11. In ∆ABC, ⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗⃗. Prove that 2 ⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗ = 6 ⃗⃗⃗⃗⃗⃗. 12. Prove the identity (

(

) √

13. Find the exact value of 14. Calculate

and

√

.

15. Find the equation of the normal line to the curve y = x2 – 3x + 2 at the point where x = 3. ***** S. M. S ac ***** 7

Mathematics

SUN MOON STAR Academic Centre

Day 9

3. 4. 5. 6. 7.

Function f is defined by f: x 2x + 1. If f(kx) – 9f(x) + 8 = 0, for all values of x. Find the value of k. Let A = {x│0 x 360 }. Let the function t : A R be given by t(x) = sin x – cos x. (a) Find t(30 ), t(60 , t(90 ) to three significant figures. (b) If t( ) = 0.6, find such that 0 x 90 . Find the value of a for which (1 – 2a)x2 + 5ax + (a – 1)(a – 8) is divisible by x – 2 but not x – 1. Find the value of (√ + 1)6 – (√ – 1)6. Find the solution set in R for the inequation x2 . Given that sin2 x, cos2 x and 5 cos2 x – 3 sin2 x are in A.P., find the value of sin2 x. Find the sum of all integers between 50 and 400 which ends in 3.

8.

If A = (

9.

Given P = (

1. 2.

), prove that A2 + 2 A– 1 = 3I. ), Q = (

) and R = (

) , write down the inverse of the matrix P and use

it to find the matrix X in the matrix equation PX + Q = R. 10. Three coins are tossed simultaneously. Make a table to determine all possible outcomes. Find the probability of getting at least one head and find the probability of getting at most one head. How many wound you expect to obtain exactly one head in 800 trails? 11. In ∆PQR, X and Y are points on the sides PQ and PR respectively such that PX:XQ = PY:YR = 3:2. Prove by a vector method that XQRY is a trapezium. 12. Prove the identity 13. If 14. Calculate

and

(

.

15. Find the equation of the tangent line to the curve x2 + xy + y = 5 at the point where the curve cuts the line x = 1. ***** S. M. S ac *****

Day 10 1. 2. 3.

Functions f: R R and g: R R are given by f(x) = , x – 1 and g(x) = 2x – 1. –1 If (g◦f ) (x) = 3, find the value of x. A function f : R R is defined by f(x) = x + 1. Find the function g : R R in each the following. (a) (g ◦ f) (x) = x2 + 5x + 5. (b) (f ◦ g) (x) = x2 + 5x + 5. If (kx + 1) is a common factor of the polynomials 2x2 + 7x + 3 and 2x2 – 5x – 3, find the value of k and hence find also the remainder when 2x3 + x2 – 18x – 9 is divided by x + k.

4. 5. 6. 7.

Find the middle term of (1 – )14. Find the solution set in R for the inequation (2x + 1)2 4 (2x + 1). How many terms of the A.P. – 7, – 2, 3, 8, … add up to 155? In an arithmetic progression 44, 40, 36, … (a) find the sum to first 12 terms (b) find the sum from 13th term to 25th term.

8.

Given that A = (

9.

Solve the equation for 2 x 2 matrix X, 2(

) and B = (

), and find (A' + B– 1) (B – 2A). )+(

)X=(

).

10. Ten cards, bearing the letters P, R, O, P, O, R, T, I, O, N are placed in a box. Three cards are drawn out at random without replacement. Calculate the probabilities that the three cards bear the letters P, O, T in that order and in any order.

8

Mathematics

SUN MOON STAR Academic Centre

⃗⃗⃗⃗⃗ 11. PQRS is a square and K, L, M and N are midpoints of the sides ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ respectively. ⃗⃗⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ interest at O. Express 6 ⃗⃗⃗⃗⃗⃗⃗ + 7 ⃗⃗⃗⃗⃗⃗ + 8 ⃗⃗⃗⃗⃗⃗⃗ + 9 ⃗⃗⃗⃗⃗⃗⃗ in terms of a single vector ⃗⃗⃗⃗⃗⃗. 12. Prove the identity 13. If 14. Find

show that √

15. The curve with the equation y = x2 – 10x + 21 cuts the Y- axis at A and the X- axis at B and C. D is a point (2,5). Find the gradients of the curve at A, B and C and also find the equation of tangent at D. ***** S. M. S ac *****

Day 11

1. 2. 3.

Functions f and g are defined by f(x) = 4x – 3 and g(x) = 2x + 1. Find (f ◦ g) (x) and f– 1(x) in simplified forms. Show also that (f ◦ g)– 1 (x) = g – 1 (f – 1 (x)). Let f : R R and g :R R are defined by f(x) = 3x – 1 and g(x) = x + 7. Find (f – 1◦g) (x) and (g – 1 ◦f) (x). What are the values of (f – 1◦g) (3) and (g – 1◦f)(2)? The expression (polynomial) 2x2 + 5x – 3 leaves a remainder of 2p2 – 3p when divided by 2x – p. Find the value of p.

4. 5. 6. 7.

Find the two middle terms of (3a – )9. Find the solution set in R for the inequation 2x (x + 2) (x + 1) (x +3). How many terms of the A.P. 3 , 5 , 7 , … should be taken so that the sum is 296? A semicircle is divided into n sectors such that the angles of the sectors form an arithmetic progression. If the smallest angle is 5 and the largest angle25 , calculate n.

8.

Show that (AB)– 1 = B – 1A – 1 for A = (

9.

Given that A = (

) and B = (

).

), write down the matrix A– 1 and use it to solve the

) and B = (

equation AX = B – A. 10. Eleven cards bearing the letters M, A, T, H, E, M, A, T, I, C, S are placed in a box. Two cards are drawn at random without replacement. Find the probability that the two cards bear the letters A, C in that order. Find also the probability that the two cards are of the same letter. 11. In the quadrilateral OABC, D is the midpoint of BC and G is a point on AD such that AG:GD ⃗⃗ ⃗⃗⃗⃗⃗⃗ = 2:3. If ⃗⃗⃗⃗⃗⃗ = ⃗, ⃗⃗⃗⃗⃗⃗ ⃗ express ⃗⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ in terms of ⃗ ⃗⃗ ⃗ 12. Prove that 13. In ΔABC, if find a. 14. Calculate

and

√

√

√

√

.

15. Find the stationary points on the curves y = 27 + 12x + 3x2 – 2x3 and determine the nature of these points. ***** S. M. S ac *****

Day 12 1. 2.

3.

The functions f and g are defined by f(x) = 3x – 2 and g(x) = 2x – 3. Find the inverse functions of f and g. Show also that (f ◦ g)– 1 (x) = (g – 1 ◦ f – 1 )(x). Let R be the set of real numbers. A binary operation ʘ is defined by ʘ : R R. (x,y) x ʘ y = x2 + y2. (a) Evaluate [(2 ʘ 3) ʘ 4] + [2 ʘ (3 ʘ 4)]. (b) Show that (x ʘ y) ʘ x = x ʘ (y ʘ x). When f(x) = (x – 1)3 +6(px + 4)2 is divided by x + 2 , the remainder is –3. Find the value of p.

9

Mathematics

SUN MOON STAR Academic Centre

5. 6. 7.

Obtain the first four terms in the expansion of (1 + 2x)9. Use this expansion to find an approximate value of (1.02)9. Find the solution set in R for the inequation x x2 – 12. If un = 2 un – 1 + 5 and u4 = 11, deduce to u1. Find the sum of all multiples of 7 between 400 and 500.

8.

Given that A = (

4.

) and B = (

), write down the inverse matrix of A and find the

matrix P and Q such that PA = 2I and AQ = 2B. 9.

Let A = (

). Solve for 2x2 matrix X such that AX = 2A' + 5B – 1.

) and B = (

10. Eleven cards bearing the letters E, X, A, M, I, N, A, T, I, O, N are placed in a box. Three cards are drawn out at random without replacement. Calculate the probability that the three cards bear the letters A, I, M in that order and in any order. ⃗⃗ ⃗⃗. OP is drawn ⃗⃗⃗⃗⃗⃗ 11. In a quadrilateral OLNM, OM//LN, where ⃗⃗⃗⃗⃗⃗ ⃗ ⃗⃗⃗⃗⃗⃗⃗ parallel to MN to meet the diagonal ML at P. If LP = LM, find the value of k. 12. Prove that 13. Find the largest angle of ΔABC with a = 4, b = 7, c = 8. 14. Find

[(

)

(

) ]

. 3

15. Determine the turning points on the curve y = x – 4x2 – 3x + 18. State whether each of these points is a maximum or a minimum. ***** S. M. S ac *****

Day 13 1.

Let f: x a + bx, a, b R, be a function from R into R such that f(2b) = b and (f◦f)(b) = ab. If f is not a constant function, then find the formula for f.

2.

The binary operation on R is defined by x ʘ y = – xy, for all real numbers x and y. Show that the operation is commutative, and the possible values of a such that a ʘ y = a + 2. When f(x) = (x + 3)3 (x – 1) – px + 6 is divided by x + 3, the remainder is 28. Find the value of p and hence show that x – 1 is a factor of f(x). Find the first four terms in the expansion of (1 + x2)8. Use your result to estimate the value of (1.01)8. Find the solution set in R for the inequation (2 – x)2 – 16 0. The sum to first n terms of a series Sn = 3n + 4n2. Show that it is an A.P. and find u10. In an A.P whose first term is –27, the tenth term is equal to the sum of the first 9 terms. Calculate the common difference and the twentieth term.

3. 4. 5. 6. 7. 8.

Given that A = (

)

and B = (

), find A–

1

and use it to solve the equation

XA=3B+2A. 9.

Solve the matrix equation ( X=(

) X = (

). Hence find x and y if

).

10. Out of 13 applications for a job, there are 5 women and 8 men. It is desired to select 2 persons for the job. Find the probability that at least one of selected person will be a woman. 11. The points A and B have position vectors ⃗ and ⃗⃗ respectively, relative to an origin O. The point P divides the line segment OA in the ratio 1:3 and the point R divides the line segment AB in the ratio 1:2. Given that PRBQ is a parallelogram, find the position vector of Q. 12. Show that 13. In ΔABC, a:b:c = 1:3:√ find ∠C. 14. Differentiate cos2 √ and ( with respect to x.

10

Mathematics

SUN MOON STAR Academic Centre

15. Find the stationary point of the curve y = 3 – (2x – 1)4 and determine its nature. ***** S. M. S ac *****

Day 14 1. 2. 3. 4. 5. 6. 7. 8.

Functions f and g are defined by f(x) = ax + b, where a and b are and constants, g(x) = , –1 x – 1. Given that f(2) = g (2) and (f ◦ g) (–3) = – 9. Calculate the values of a and b. A binary operation ʘ on R is defined by x ʘ y = yx + 2xyyx – xy. Evaluate (2 ʘ1) ʘ1. The expression (polynomial) 6x2 – 2x + 3 leaves a remainder of 3 when divided by x – p. Determine the values of p. When (1 – x) (1 + ax)6 is expanded as far as the term in x2, the result is 1 + bx2. Find the value of a and b. Find the solution set in R for the inequation 4(2x – 3)2 x2. The 9th term of an A.P. is 499 and 499th term is 9. Find the term which is equal to zero. For a certain A.P, Sn = (3n – 17). Calculate S1, S2, S3 and S4. Hence find the first four terms of the correspondence sequence and a formula for the nth term. The matrices A and B are given by A = (

) and B = (

). Find the matrices P and

Q such that P = 2A + B2 and AQ + BQ = I. 9.

Find the inverse of the matrix (

) and use it to solve the following system of equations,

y – x = 1 and x + y = 3. 10. A box contains 12 discs of which 3 are white, 4 are red and 5 are blue. Two discs are to be drawn at random, in succession, each being replaced after its colour has been noted. Find the probability that both the two discs out are blue. Find also the probability that exactly one of two discs drawn out is blue. 11. The position vectors of points A, B and C relative to an origin O are ⃗⃗ ⃗ ⃗ ⃗ ⃗⃗ ⃗ ⃗ ⃗ respectively. Show that A, B and C are collinear and AB = BC + AC. 12. Show that 13. Find if a = 12, b = 5 and c = 13. 14. Find if x + sin y = cos (xy) and if √ 15. Find the x- coordinate, for 0

x

, of the stationary point on the curve y = ***** S. M. S ac *****

√

Day 15

7.

For the function f(x) = , find f – 1 and verify that f ◦ f – 1 and f – 1◦f both equal I. Let J+ be the set of all positives integers. Is the function ʘ defined by x ʘ y = x + 2y a binary operation on J+? If it is a binary operation, solve the equation (k ʘ 5) – (3 ʘ k) = 2k + 13. When the polynomial x3–3x2+kx+7 is divided by x+3, the remainder is 1. Find the value of p. Given that (p – )6 = r – 96 x + sx2 + ... , find p, r and s. The functions f and g are defined by f : x 2x2 + 4x + 5, x R and g : x x + 4, x R. Find the set of values of x for which f(x) g(x). The eighth term of an A.P is 150 and the fifty-third term is –30. Determine the number of terms whose sum is equal to zero. If 2x – 14, x – 4 and x are three consecutive terms of an A.P, find the value of x.

8.

Given that A = (

1. 2. 3. 4. 5. 6.

), B = (

) and C = (

result to find the matrix such that BXA = C.

11

). Find A– 1 and B– 1 and use the

Mathematics

SUN MOON STAR Academic Centre 9. 10.

11.

12. 13.

Find the inverse of the matrix (

) and use it to find the solution set of the system of

equations, 7x + 4y = 16 and 2y + 3x = 6. A box contains 2 black, 4 white, 3 red balls. One ball is drawn at random from the box, and kept aside. From the remaining balls in the box, another ball is drawn at random and kept besides the first. This process is repeated till all the balls are drawn from the box. Find the probability that the balls drawn are in the sequence of 2 black, 4 white and 3 red. The vector ⃗⃗⃗⃗⃗⃗ has a magnitude of 39 units and has the opposite direction as ( ). The vector ⃗⃗⃗⃗⃗⃗⃗ has a magnitude of 25 units and has the opposite directions as ( ). Express ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗⃗ as column vectors and find the unit vector in the direction of ⃗⃗⃗⃗⃗⃗ . ( Prove that ( Solve ΔABC if BC = 20, AC = 18 and

( 14. Find if y = . Find also if 15. Find the stationary points of the curve y = x4 – 4x3 and determine the nature of each. ***** S. M. S ac *****

Day 16 1.

3

7.

The function f : x ax + bx + 30. Then the values x = 2 and x = 3 which are unchanged by the mapping. Find the values of a and b. Let ʘ be the binary operation on R defined by a ʘ b = a2 + b2 for all a,b R. Show that (a ʘ b) ʘ a = a ʘ (b ʘ a). Solve the equation 4 ʘ (x ʘ 2) = 185. If x3 + ax2 – 8bx + 5 and 2x3 – bx2 + 4ax – 18 have a common factor x – 2. Find the values of a and b. Expand (2 + )6 in ascending powers of x up to the term in x 3. Hence, find an approximate value of (1.9975)6. Find the solution set of the following inequation and illustrate it on the numbered line, x2 + 9 0 by graphical method. The first term of an A.P is 3, its nth term is 23. If the sum of the first n terms of that A.P is 143, find n. If the sum of the first n terms of a series is Sn = (n – 1)(n + 1), find the rth term of the series.

8.

Let AB = (

9.

Find the inverse of the matrix (

2. 3. 4. 5. 6.

) + I where B = (

). Find 2 x 2 matrix A.

) and use it to solve the following system of equations,

3y + 4x + 7 = 0 and 14x + 12y + 32= 0. 10. A bag contains 12 balls: three red, three blue, three green and three yellow. Three balls are drawn from the bag in succession, without replacement. What is the probability that the first is red, the second is green or blue, and the third yellow? 11. The position vectors of three points A, B and C, relative to an origin O, are ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ respectively. The midpoint of AB is M and the point N is such that ⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ Find ⃗⃗⃗⃗⃗⃗⃗⃗ in terms of ⃗ ⃗ 12. Given that A = B + C, prove that 13. Solve the triangle ABC if ∠A = 64 14. Find the value of a and b for which

∠B = 50 ( )

(

.

15. Find the minimum value of the sum of a positive number and its reciprocal. ***** S. M. S ac *****

12

Mathematics

SUN MOON STAR Academic Centre

Day 17 1.

A function f is defined by f(x) = for all x 1, where k is a constant. If f – 1(7) = 4, find the value of k. If g(x) = 2x + 3, find the formula of f – 1 ◦ (g ◦ f) in simplified form.

2.

An operation ʘ on R is defined by x ʘ y = + 2xy. Show that ʘ is commutative. Find the values of p such that p ʘ 3 = p + 10. If 2x is a factor of x3 – 4x2 + 5x + a2 + 3a where a is a constant then find the value of a. If (1 – 3x)7 = a0 + a1x + a2x2 + a3x3 + ... + a7x7, find a0, a1 and a2. Show that a1 + a2 + a3 + ... + a7 = – 129. Find the solution set of the following inequation and illustrate it on the numbered line, x2+4 0 by algebraic method. If the first, second and last terms of an A.P are a, b and 2a respectively, then show that its sum is ( .

3. 4. 5. 6. 7.

If 18, x, 8, y, z is an A.P, find x + y + z.

8.

Given that M = (

9. 10.

11.

12. 13.

), I is a unit matrix of order 2 and M2 – 9M + (4k +2) I = 0, find the

value of the number of k. Find also the inverse of M. Find the solution set of the system of equation 3x – 7y = 35, x + y = 5, by matrix method; the variables are on the set of real numbers. In a car park, there were 4 white cars and x black cars. One car is chosen at random. Given that the probability that it will be black is , calculate the value of x. Using your value of x, find the probability that the first two cars that will leave the car park will be the same colour. The position vectors of A and B relation to an origin O are ⃗ and 4 ⃗⃗ respectively. The point ⃗⃗⃗⃗⃗⃗ and the point E on ⃗⃗⃗⃗⃗⃗ is such that⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ . The line D on ⃗⃗⃗⃗⃗⃗ is such that ⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗, express ⃗⃗⃗⃗⃗⃗ in two segments OE and BD intersect at point X. If ⃗⃗⃗⃗⃗⃗ different forms and hence find the value of k and m. If show that A and B are two points on opposite banks of a river. From A, a line AC=275 ft is laid off and ∠CAB=125 ∠ are measured. Find AB.

14. If y =

, find the rate of change of y with respect to x at x = 2.

Find also if x3 – 4xz + z2 = 14. 15. If a piece of string, 200 ft long, is made to enclose a rectangle, show that the enclose area is the greatest when the rectangle is a square. ***** S. M. S ac *****

Day 18 1. 2. 3. 4. 5. 6. 7.

Let f : R R and g : R R are defined by f(x) = kx – 1, where k is a constant and g(x) = x + 12. Find the value of k for which (g◦f) (2) = (f◦g) (2). The operation ʘ on the set N of natural numbers is defined by x ʘ y = xy. Find the value of a such that 2 ʘ a = (2 ʘ 3) ʘ 4. Find also b such that 2 ʘ (3 ʘ b) = 512. Find the factors of 2x3 + x2 – 13x + 6. Use the Binomial Theorem to estimate the value of (1.99)5, correct to four decimal places. Find the solution set of the following inequation and illustrate it on the numbered line, x2 + 7 0 by graphical method. If S1, S2, S3 are the sum of n, 2n, 3n terms of an A.P., then show that 3 (S2 –S1) = S3. The eight term of an A.P. is 150 and the fifty-third term is – 30. Determine the number of terms whose sum is zero.

13

Mathematics

SUN MOON STAR Academic Centre 8.

Given that A = (

) and det A = 7, find the value of a. If I is unit matrix of order 2,

2

verify that A – 7A + 7I = 0. 9. By using matrix method, find the solution set of the system of equations 6x + 3y = 15 2y – 3x = – 18. 10. The classes of students are comprised as follows: Class A 4 girls 6 boys Class B 4 girls 5 boys Class C 6 girls 8 boys One student is selected at random from each class. Calculate the probability that from each class. Calculate the probability that the three selected students are all girls. 11. A, B and C are points with position vectors ⃗ ⃗ ( ⃗ ⃗ and ⃗ ⃗ respectively. Find ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗ . Given that B lies on AC, find the value of . 12. Given that

( (

13. Two runners start from the same point at 6: 00 A.M., one of them heading north at 6 m.p.h and other heading N 65˚ E at 8 m.p.h. What is the distance between them at 9:00 that morning? 14. Differentiated y = x2 – 5x + 4 with respective to x from the first principle. 15. Given that the volume if a solid cylinder of radius r cm is 250π cm3, find the value of r for which the total surface area of the solid is minimum. ***** S. M. S ac *****

Day 19

1. 2. 3.

If the function f is defined by f(x) = x + 3, find the function g such that (g◦f)(x) = 2x2 + 3. If a ʘ b = a2 – 3ab + 2b2, find (– 2 ʘ 1) ʘ 4. Find p if (p ʘ 3) – (5 ʘ p) = 3p – 17. The expressions x3 – 7x + 6 and x3 – x2 – 4x + 24 have the same remainder when divided by x+p. Find the values of p.

4.

Find the coefficient of

5.

7.

Find the solution set of the inequation and illustrate it on the numbered line, 2x3 – x2 + 5 0 by algebraic method. If k is a positive integer, show that the sum of the A.P. 3k + 2, 3k + 5, 3k + 8, … , 3k + 44 is divisible by 5. For a certain A.P. Sn = (3n – 17). Find the first three terms of the corresponding sequence.

8.

Given that A = (

6.

2

in the expansion of (

) and B = (

–

)10.

), find the value of k for which the determinant of AB

is – 20. Hence find the inverse matrix of B. 9. By using matrix method, find the solution set of the system of equations 5x + 2y = 11 4x – 3y = 18. 10. Three groups of children consist of 3 boys and 1 girl, 2 boys and 2 girls, 1 boy and 3 girls respectively. If a child is chosen at random from each group, find the probability that 1 boy and 2 girls are chosen. 11. Point A and B have position vectors ( ) ( ) respectively, relative to an origin O. Given that C with position vector ( ) lies on AB produced, calculate the value of k and the value of | ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ |. 12. Given that

( (

evaluate tan y.

13. A and B are two points on one bank of the river, distant from one another 649 yds. C is on the other bank and the measures of the angles are respectively 48˚32’ and 75˚25’. Find the width of the river. 14

Mathematics

SUN MOON STAR Academic Centre

14. Find the derivatives of the function and (3 – x2)3 √ . 15. A rectangular field is surrounded by a fence on three of its sides and a straight hedge on the fourth side. If the area of the field is to be 11250 square metres, find the smallest possible length of the fence. ***** S. M. S ac *****

Day 20 1. 2.

7.

Let the function f(x) = , x 2 and g(x) = , x 3. Find the formula for f◦g. 2 The operation ʘ is defined by x ʘ y = x + xy – 3y2. x,y R. If 4 ʘ x = 17, find the possible values of x. Find also (2 ʘ 1) ʘ 3. Given that the remainder when x3 – x2 + ax is divided by x + a is twice the remainder when it is divided by x – 2a, find the values of a. Given that (1 + ax)n = 1 – 12x + 63x2 + ... , find a and n. Find the solution set of the inequation 2x2 – 3 and illustrate it on numbered line. The sum of n terms of two A.P.’s are in the ratio of 13 – 7n:3n + 1; prove that their first terms are as 3:2 and their second terms are as – 4:5. The fourth and sixth terms of an A.P. are x and y respectively. Show that the 10th term is 3y – 2x.

8.

Given that A = (

3. 4. 5. 6.

9. 10. 11.

12. 13.

), det A = – 5, find c. Hence verify that A2 – 4A – 5I = 0, when I is the unit

matrix of order 2. Use the matrix method to find the solution set of the system of equations: 3x – 7y = 44 and 8y + 2x + 34 = 0. Out of the 20 applicants for a job, there are 8 women and 12 men. It is desired to select 3 persons for this job. Find the probability that at least one person of the selected person will be a woman. The vector ⃗⃗⃗⃗⃗⃗ has a magnitude of 39 units and has the same direction as ( ). The vector ⃗⃗⃗⃗⃗⃗⃗ has a ⃗⃗⃗⃗⃗⃗⃗ as column vectors and magnitude of 25 units and has the same direction as ( ). Express ⃗⃗⃗⃗⃗⃗ find the unit vector in the direction of ⃗⃗⃗⃗⃗⃗ . Given that tan α = p and tan (α + β) = q and express tan β in terms of p and q. Calculate the value of tan (α + β) when p=1 and q=0.5. A man walking due to west along a level road observes a tower in a direction N 47˚ W. After walking 135 m, he observes it in the direction N 38˚ W. How far is the tower from the road?

14. Given that xy = sin x, prove that 15. Two positive numbers x and y vary in such a way that xy = 18. Another number z is defined by z = 2x + y. Find the values of x and y for which z has a stationary value and show that this value of z is a minimum. ***** S. M. S ac *****

Day 21 1. 2. 3. 4. 5.

The functions f and g are defined by f(x) = 3x + 1 and g(x) = , x – 1, find the composite function f◦g and hence find the (f◦g)(2). Let R be the set of real numbers. Is the function ʘ defined by a ʘ b = a2 – 2ab + 3b2 for all a, b R, a binary operation? Is ʘ commutative? Why? Find what value p must have in order that x + 2 may be a factor of 2x 3+3x2+px– 6. Find the other factors. When (x + y)8 is expanded in descending powers of x, the third and fourth terms have equal values when x = p and y = q where p and q are positive numbers and their sum is 1. Find the value of p. Find the solution set in R of the inequation (1 + 2x)3 + (1 – 2x)3 – 22.

15

Mathematics

SUN MOON STAR Academic Centre 6.

The sum of the first n terms of an A.P 3, 5 , 8, … is equal to the (2n)th term of an A.P 16 , 28 ,

7.

40 , …., calculate n. The sum of the first five consecutive terms of an A.P. is 110. Find the middle term.

8.

Solve the matrix equation for 2x2 matrix X, 2 (

9.

Find the inverse of the matrix A = (

)+(

)X=(

).

) and use it to find the solution set of the system of

equations x + 3y = 7 and 5y – 2x = – 3. 10. If the probabilities that students P and Q will pass an examination are and respectively, find the probability that both P and Q will pass the examination. Find also the probability that at most one of P and Q will fail the examination. 11. OABC is a parallelogram with the vertex O at the origin and the vertices A and C at (4,6) and (8,2) respectively. If P and Q are midpoints of OA and BC respectively, show that OPBQ is a parallelogram by using vector method. √

( 12. Find all values x between 0˚ and 360˚ which satisfy the equation 13. To approximate the distance between two points A and B on opposite sides of a swamp, a surveyor selects a point C and measures it to be 215 meters from A and 310 meters from B. Then they measure the angle ACB, which turns out to be 49˚. What is the distance from A to B. 14. If y = (3 + 4x)e– 2x , then prove that . 2 15. If the area of a rectangle is 49 cm , show that the perimeter is the smallest when the rectangle is a square and find the smallest perimeter. ***** S. M. S ac *****

Day 22

7.

A function h is defined by h(x) = , x 3. Show that h(3 + p) + h(3 – p) = 2, where p is the positive and find the positive number q such that h(q) = q – 1. Given (3a – b) ʘ (a + 3b) = a2 – 3ab + 4b2, evaluate 4 ʘ 8. The expression ax3 – x2 + bx –1 leaves the remainder of – 33 and 77 when divided by x + 2 and x –3 respectively. Find the values of a and b, and the remainder when divided by x – 1. In the binomial expansion of (a + b)n, the coefficient of fourth and thirteenth terms are equal to each other. Find n. Find the solution set of the inequation x2 by algebraic method and illustrate it on numbered line. x2, (8x + 1) and (7x + 2), where x 0, are the 2nd, 4th and 6th terms respectively of an A.P. Find the value of x, the common difference and the first term. Three consecutive terms of an arithmetic series have sum 21 and product 315. Find the numbers.

8.

Find the 2x2 matrix in the equation: X (

9.

Find the inverse of the matrix A = (

1. 2. 3. 4. 5. 6.

)=3(

)

2(

).

) and use it to solve the system 7x + 8y = 10 and 5x + 6y =

7. 10. Two independent events, A and B each have two possible outcomes success or failure. The probability of success in B is half the probability of success in A. If the probability of both A and B resulting in failures is , calculate the probability that the outcome of event B is success. ⃗⃗⃗⃗⃗⃗⃗ 11. In ∆XYZ, L and M are midpoints of YZ and XZ respectively. Prove that ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ 12. Solve the equation in 0 √ 13. A man standing at a point P, sees two trees X and Y which are respectively 250 m and 310 m away from him. If ∠XPY=120˚, how far apart are the two trees.

16

Mathematics

SUN MOON STAR Academic Centre

14. If y = A cos (ln x) + B sin (ln x), where A and B are constants; show that x2y'' + xy' + y = 0. 15. Find two positive numbers whose sum is 20 and whose product is as large as possible. ***** S. M. S ac *****

Day 23

1.

The function f(x) = ax3 + bx +30. Then the values x = 2 and x = 3 which are unchanged by the mapping. Find the values of a and b.

2.

Let R be the set of real numbers and binary operation ʘ on R be defined by x ʘ y = – 2xy for x, y R. Find the value of 3 ʘ 2 and (3 ʘ 2) ʘ 16. If a and b are two real numbers such that a ʘ b = 8, find the relation between a and b. The polynomial ax3 + bx2 – 5x + 2 is exactly divisible by x2 – 3x – 4. Find the values of a and b. What is the remainder when it is divided by x + 2? If the 21st and 22nd terms in the expansion of (1 – x)44 are equal, find x. Find the solution set of the inequation 12 + x – x2 0, by graphical method. In an A.P., the 13th term is 25 and the sum of the first 11 terms is 121. Find the first term and the common difference of that A.P. Show also that the sum of the first n terms of that A.P. is n2. If k is a positive integer, show that the sum of the A.P. 3k + 2, 3k + 5, 3k + 8, …, 3k + 44 is divisible by 5.

3. 4. 5. 6.

7.

8. If ps qr, find the matrix X such that ( )X = ( ). Find also X – 1 if it exists. 9. Find the solution set of the system of equations 3x + 2y = 7 and 5x – y = 3 by matrix method. 10. A, B, C on shot each a target. The probability that A will hit the target is , and the probability that B will hit the target is and the probability that C will hit the target is . If they fire together, calculate the probability that (i) all three shots hit the target and (ii) C’s shot only hits the target. 11. ABCDEF is a regular hexagon. If G is the common point of intersection of the diagonals, prove by vector method that AB//ED and AD = 2BC. 12. Find the value of without using the table. 13. To find the distance from the house at A to the house at B, a surveyor measure the angle BAC to be 40˚, then walk off a distance of 100 ft to C, and measures the angle ACB to be 65˚. What is the distance from A to B? 14. If y cos x = ex, show that . 15. A rectangular field is surrounded by a fence on three of its sides and a straight hedge on the fourth side. If the length of the fence is 320 metres, find the maximum area of the field enclosed. ***** S. M. S ac *****

Day 24 1.

4.

Functions f and g are defined by f(x) = 2x + 1 and g(x) = , x 3. Find the values of x for –1 which (f◦g )(x) = x – 4. Given that x ʘ y = x2 + xy + y2, x, y R, solve the equation (6 ʘ k) – (k ʘ 2) = 8 – 8k. Is ʘ commutative? Why? Given that x3 – 2x2 – 3x – 11 and x3 – x2 – 9 have the same remainder when divided by x + a, determine the values of a and the corresponding remainders. If, in the expansion of (3√ + )n, the ratio of the 7th term from the beginning to the 7th term

5.

from the end is equal to , find the value of n. Use the graphical method to find the solution set of (x + 2)2

2. 3.

√

17

2x + 7.

Mathematics

SUN MOON STAR Academic Centre 6. 7. 8.

An A.P is such that the 5th term is three times the 2nd term. Show that the sum of the first eight terms is four times the sum of the first four terms. An arithmetic progression has 22 terms. The sum of the odd terms is 253 and the sum of the even terms is 275. Find the last term. Find all possible matrices of the form X = (

) such that X2 = 16 I, where I is a unit matrix

of order 2. 9. Find the solution set of the system of equations –2x – y = 0 and x – 2y = 5 by matrix method. 10. A die is rolled 360 times. Find the expected frequency of a factor of 6 and the expected frequency of a prime number. If all the score obtain in these 360 trails are added together, what is the expected total score? 11. In the quadrilateral ABCD, M and N are the midpoints of AC and BD respectively. Prove that ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗ 12. Use the compound angle formula to find in surd form. 13. A ship leaves harbor on a course N 72˚ E and after travelling for 50 km, changes course to 108˚. After a further 106 km, find the distance of the ship from the harbor and its bearing from the harbor. 14. If y = 3x sin 3x + cos 3x, than prove that . 15. Find the two positive numbers whose product is 361 and whose sum is as small as possible. ***** S. M. S ac *****

Day 25 1. 2.

3. 4. 5. 6.

7.

Let f(x) = 2x – 1, g(x) = , x 1. Find the formula for (g◦f)– 1 and state the domain of (g◦f)– 1. A binary operation ʘ on R is defined by x ʘ y = x2 + y2, for all real numbers x and y. Show that the binary operation is commutative and find the value of 2 ʘ (3 ʘ 1). Solve the equation x ʘ 2√ = 3 ʘ 4. Given that f(x) = x3 + px2 – 2x + 4√ has a factor x + √ , find the values of p. Show that x – 2√ is also a factor and solve the equation f(x) = 0. If in the binomial expansion of (a – b)n , n 5, the sum of the 5th term and 6th term is zero, then show that . Find the solution set of the inequation (2 + x) (3 – 2x) 2x + 1. Which term of the progression 19 + 18 + 17 + … is the first negative term? What is the smallest number of terms which must be taken for their sum to be negative? Calculate this sum exactly. The number of terms in an A.P. is 40 and the last term is – 54. Given that the sum of the first 15 terms added to the sum of the first 30 terms is zero, calculate the sum of the progression.

8.

Find the matrix X of the form X = (

9.

Find the solution set of the system of equations 4x + 3y = 24 and 3x + 2y = 9 by matrix method. A spinner is equally likely to point to any one of the number 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. What is the probability of scoring a number divisible by 2? If the arrow is spun 1000 times, how many would you expect scoring a number not divisible by 2? OPQR is a parallelogram and OR is produced to S such that OS = 3 OR. If Y is a point on OQ such that OQ = 4YQ, show by means of a vector method that Y lies on PS. Find the values of without using the table. A ship sails 15 miles on a course of S 40˚ 10' W and then 21 miles on a course N 28˚20' W. Find the distance and direction of the last position from the first.

10.

11. 12. 13.

) such that X3 = (

18

).

Mathematics

SUN MOON STAR Academic Centre 14. If y = e2x cos 3x, prove that . 15. What is the smallest perimeter possible for a rectangle of area 16 ft2? ***** S. M. S ac *****

Day 26 1. 2. 3.

4. 5. 6. 7. 8.

A function f is defined by f(x) = , x 1, where k is a constant. If f – 1(7) = 4, find the value of k. If g(x) = 2x + 3, find the formula of f – 1◦(g◦f) in simplified form. A binary operation ʘ on R is defined by a ʘ b = a2 – 2b, for all a, b R. If 4 ʘ (2 ʘ k) = 20, find the value of (k ʘ 5) ʘ k. Given that f(x) = x2n – (p + 1)x2 + p, where n and p are positive integers, show that x – 1 is a factor of f(x), for all values of p. When p = 4, find the value of n for which x – 2 is a factor of f(x) and hence factorize f(x) completely. Find the coefficient of x4 in the expression of (x2 – 5x + 12) (x – )6. Find the solution set in R of the inequation (x + 2)2 3 (x + 8) and illustrate it on numbered line. A certain A.P has 25 terms. The last three terms are , and . Calculate the value of x and the sum of all terms of the progression. Find the sum of first 20 terms of the A.P. 2, 5, 8, … . Find also the sum of the terms between the 25th term and the 40th term of that A.P. Use the definition of inverse of matrix; find the inverse of (

).

Find the solution set of the system of equations 2x – 2y = 5 and 2x + 3y = 12 by matrix method. 10. A box contains 5 cards numbered as 2, 3, 4, 5 and 9. A card is chosen, and the number is recorded. Draw a tree diagram and tabulate possible outcomes. Find the probabilities that (a) getting two prime numbers (b) getting two odd numbers (c) getting a pair of numbers where the sum is a prime number. ⃗⃗⃗⃗⃗⃗ intersect at Y, prove by a 11. X is the midpoint of the side ⃗⃗⃗⃗⃗⃗ of the parallelogram PQRS. If ⃗⃗⃗⃗⃗⃗ vector method that SY = SQ. 9.

12. Find the six trigonometric ratios of . 13. A man travels 15 km in a direction N 80˚ E and then 5 km in a direction N 40˚ E. What is his final distance and bearing from his starting point? 14. Given that y = , show that 15. Find the approximate change in the volume of a sphere when its radius increases from 2 cm to 2.05 cm. ***** S. M. S ac *****

Day 27

1. 2. 3. 4. 5.

Let functions f and g are defined by f(x) = 3x – 1 and g(x) = x + 7. Find (f – 1◦g) (x) and (g– 1◦ f ) (x). What are the values of (f – 1◦g) (3) and (g – 1◦f) (2)? Let J+ be the set of positive integers and a binary operation ʘ be defined by a ʘ b = a (3a + b) for a, b J+. Find the values of 2 ʘ 1 and (2ʘ1)ʘ4. Find also the value of p if p ʘ (p + 1) = 39. When the polynomial y3 + 3y2 – 2y + p is divided by y + 2, the remainder is R. When the polynomial is divided by y – 2, the remainder is R. Find the value of R. In expansion of (3 + 4x)n, the coefficient of x4 and x5 are in the ratio 3 : 4. Find the value of n. Calculate the ratio of the coefficient of x5 and x6. Find the solution set in R of the inequation (2x – 1)2 – 25 0 and illustrate it on numbered line. 19

Mathematics

SUN MOON STAR Academic Centre 6.

7.

If the sum of the first 6 terms of an A.P is 42 and the first term is 2, find the common difference. If the sum of the first 2n terms of that A.P exceeds the sum of the first n terms by 154, find the value of n. An A.P, with first term 8 and common different d, consists of 101 terms. Given that the sum of the last three terms is 3 times the sum of the first three terms, find the value of d.

8.

Use the definition of inverse of matrix; find the inverse of each of (

9.

Find the inverse of the matrix (

)

), and use it to solve the following system of equations;

y – 5x = 7 3x + 2y = 1. 10. A box contains 4 marbles of 2 blue, 1 red and 1 yellow. A marble is chosen, the colour is recorded, and the marble is not replaced. Then another marble is chosen and the colour is recorded. Draw a tree diagram to determine possible outcomes. Hence find the probabilities of (a) choosing 2 blue marbles and (b) choosing 2 different colours. 11. Find the matrix which will rotate 30 and then reflects in the line OY. What is the map of the point (1,0)? 12. Express as a single trigonometric ratio. 13. A ship is 13 km from a boat in a direction N 47˚ E and a lighthouse is 15 km from that boat in a direction S 25˚ E. Calculate the distance between the ship and the lighthouse. 14. If y = x2 + 2x + 3, show that (

(

15. Using the derivative of a suitable function, find an approximate value of √ ***** S. M. S ac *****

Day 28 1. 2.

3. 4.

5. 6. 7. 8.

The function f and g are defined by f(x) = 4x – 3 and g(x) = , x – 1. Find the inverse functions –1 of f and g. Find also the formula for (g◦f) . Let J+ be the set of all positive integers. An operation ʘ on J+ is given by x ʘ y = x (2x + y), for all positive integers x and y. Prove that ʘ is a binary operation on J+ and calculate (2 ʘ 3) ʘ 4. Is the binary operation commutative? If x3 + mx2 – x + 6 has x – 2 as a factor, and leaves a remainder n when divided by x – 3, find the values of m and n. Obtain the first four terms of the expansion of (1 + p)6 in ascending power of p. By writing p = x+x2, obtain the expansion of (1 + x + x2) as far as the term in x2. Hence find the value of (1.0101)6 to three decimal places. Find the solution set in R of the inequation (2x + 1)2 9 and illustrate it on numbered line. If T1, T2, T3 are the sum of n terms of three series in A.P., the first term of each being 1 and the respective common difference being 1, 2, 3, then show that T1 + T3 = 2T2. The sum of the first 4 terms of an A.P. is 16 and the sum of their squares is 84. Find the first four terms. Use the definition of inverse of matrix to find the inverse of (

9.

By using a matrix method, find the solution set of the system of equations; 3x – 5y = 6 3y + 2x = 23. 10. Three groups of people are comprise as follows. First group Second group Third group

4 women 3 women 3 women

3 men 3 men 4 men 20

).

Mathematics

SUN MOON STAR Academic Centre

11. 12. 13. 14. 15.

One person is selected at random from each group. Calculate the probability that the three selected people are all women. Find the matrix which will reflect in the line OY followed by a translation through 3 units horizontally and – 2 units vertically. What is the map of the point (4, – 1)? Express in terms of A town P is 25 miles away from the town Q in the direction N 35˚ E and a town R is 10 miles from Q in the direction N 42˚ W. Calculate the distance and bearing of P from R. Given that x2 – y2 = 5, show that x2y'' + xy' = y. If the radius of a circle increases from 4 cm to 4.04 cm, find the approximate increase in the area. ***** S. M. S ac *****

Day 29 1. 2. 3. 4. 5. 6. 7. 8.

Given that f(x) =

10.

11.

12. 13.

, f(8) = 1, f (– 2) = 2, show that 2

= 10. 2

A binary operation ʘ on R is defined by a ʘ b = a – 2ab + b . Show that ʘ is commutative. If (3 ʘ k) – (2k ʘ 1) = k – 28, find the values of k. The polynomials ax3 – 3x2 + 4 and 2x3 – 5x + a when divided by x – 2, leave the remainders p and q respectively. If p – 2p = 4, find the value of a. Find the independent term of x in the expansion of ( )9. Find the solution set in R of the inequation –4 – 3x2 0. The sum of the first six terms of an A.P is 55.5 and the sum of the next six terms is 145.5. Find the common difference of the A.P and the first terms. Show that there are 18 integers which are multiple of 17 and which lie between 200 and 500. Find the sum of all these integers. In the following matrices, show that the given matrices are the inverse of the other. A=(

9.

–1

) and B = (

).

Find the solution set of the system of equations 2x + 3y + 4 = 0 and – 5x + 4y + 13 = 0 by matrix method. A spinner is equally likely to point to any one of 1, 2, 3, 4, is spun once and a die is rolled. Make a table of order pairs (Spinner, Die). Hence, find (where E means even and O means odd.) (a) P(E,E) (b) P((E,O) or (O,E)) (c) P (total of 10) (d) P (product of 1) (e) P (total less than 6). Write down the matrix which rotates through an angle of 60 anticlockwise about the origin. Find the map of the point (4,2) under this rotation. Find also the point which is mapped to the point (2,0) by this rotation. Express cot 2x in terms of cot x. A, B and C are three cities, B is 20 miles from A in a direction N 47˚ E. C is 27 miles away from B in a direction N 65˚ W. Find the distance and direction of A from C.

14. If y = cos2 2x, prove that

By using this result show that, if z = sin2 2x, then

15. Determine the turning point on the curve y = 2x3 + 3x2 – 12x + 7 and state whether it is a maximum or a minimum. Then sketch the graph of the curve. ***** S. M. S ac *****

21

Mathematics

SUN MOON STAR Academic Centre

Day 30 1.

5. 6. 7.

Functions f and g are defined by f(x) = 3x + 2 and g(x) = , x 2. Find the formulae of f◦g –1 –1 and g . Solved the equation g (x) = x. An operation ʘ on the set of positive real numbers defined by a ʘ b = log2 a + log5 b. Find 16ʘ(2ʘ625). The expression x3 + ax2 + bx + c is divisible by both x and x – 3 and leaves a remainder of -40 when divided by x + 2. Find the value of a , b and c, and hence factorize the expression completely. Given that the coefficient of x2 in the expansion of (4 + kx) (2 – x)6 is zero, find the value of k. Find the solution set in R of the inequation (2x + 3)2 0. If 19, a – b, a + b, 85 is an A.P., find the values of a and b. Find the sum of all odd numbers between 100 and 150.

8.

A=(

2. 3.

4.

). Find A2, A3 and A4 and hence deduce a formula for An, where n is a positive

integer. Try to solve the system of equations x + y = 1 and 2x + 2y = 2. Explain with the aid of a Cartesian diagram why you failed. 10. The probabilities that the students A and B will pass an examination are and respectively. Find the probabilities that (a) both A and B pass the examination (b) exactly one of A and B passes the examination. 11. ABCDEFGH is a regular octagon. If ⃗⃗⃗⃗⃗⃗ ⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗ find vectors ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗ in terms of ⃗ and ⃗⃗. 9.

12. Given that

√

and that both A and B are in the same quadrant, calculate ( ( the values of 13. In ΔABC, if AB = x, BC = x + 2 and AC = x – 2 where x 4, prove that . Find ( √

the integral values of x for which A is obtuse. 14. Let f(x) = where k is a constant. Find f '(x) in terms of x and k. If f '(2) = 0.35, find the value of k. 15. Find the coordinates of the points on the curve x2 – y2 = xy – 5 at which the tangents are parallel to the line x + y = 1. ***** S. M. S ac *****

လ 22

Mathematics

SUN MOON STAR Academic Centre

23