Nmtc Question Bank (3)

NMTC QUESTION BANK PRIMARY GROUP (STAGE –1)

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NUMBER SYSTEM

EXERCISE-1 \

OBJECTIVE

1.

How many 2 digit numbers greater than 10 are there, which are divisible by 2 and 5 but not by 4 or 25 ? [AMTI 2004] (A) 3 (B) 12 (C) 5 (D) 2

2.

4ab5 is a four digit number divisible by 55 where a, b are unknown digits. Then b – a is [AMTI 2004] (A) 1 (B) 4 (C) 5 (D) 0

3.

The sum of reciprocals of all divisors of 6 is (A) 1 (B) 2

(C) less than 2

[AMTI 2004] (D) Greater than 2

4.

The number of pairs of two digit square numbers, the sum or difference of which are also square numbers is [AMTI 2004] (A) 0 (B) 1 (C) 2 (D) 3

5.

The number of 3 digit even numbers that can be written using the digits 0, 3, 6 withoout repetition is [AMTI 2004] (A) 6 (B) 3 (C) 4 (D) 2

6.

When 1000 single digit non-zero numbers are added, the unit place is 5. The maximum carry over is this case is [AMTI 2004] (A) 495 (B) 895 (C) 899 (D) 995

7.

The number of 2 digit numbers of the form aa, with the same digit ‘a’ having exactly four divisor is [AMTI 2005] (A) 2 (B) 4 (C) 6 (D) 8

8.

Number of prime numbers dividing 2005 is (A) 4 (B) 3

[AMTI 2005] (C) 2

(D) 1

9.

A 2005 digit number has all its digits the same. If this number is divided by 11, then the remainder is [AMTI 2005] (A) 0 (B) 10 (C) either 0 or 10 (D) neither 0 nor 10

10.

Given that a, b, c and d are natural numbers and that a = bcd, b = cda, d = abc then (a + b + c + d)2 is [AMTI 2005] (A) 16 (B) 8 (C) 2 (D) 1

11.

The number of 3 digit number which end is 7 and are divisible by 11 is (A) 2 (B) 4 (C) 6

[AMTI 2006] (D) 8

12.

How many positive integers less than 100 can be written as the sum of 9 consecutive positive integers? [AMTI 2006] (A) 11 (B) 9 (C) 7 (D) 5

13.

In a six digit number 5 digits are prime numbers. The sum of all the digits is 24. The 2nd, 3rd and 5th digit are identical and the others are distinct digits. The number is divisible by 4. The last digit of the number is [AMTI 2006] (A) 2 or 4 (B) 4 or 6 (C) 4 or 6 or 8 (D) 2 or 6 or 8

14.

How many positive integers less than 100 can b written as the sum of 9 consecutive positive integers ? [AMTI 2006] (A) 11 (B) 9 (C) 7 (D) 5

15.

The product of two integers is 27. 33. 55. 73. Then the sum of the two numbers may be divisible by [AMTI 2006] (A) 16 (B) 9 (C) 25 (D) 49

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16.

Let a, b, c, d be two positive integers where a + b + c = 53, b + c + d = 51, c + d + a = 57 and d + a + b = 58. Then the greatest and the smallest number among a, b, c, d are respectively. [AMTI 2006] (A) b and d (B) a and c (C) c and a (D) d and b

17.

Number of prime numbers less than 100 whose sum of digits is 2 is (A) 1 (B) 2 (C) 3

(D) 4

All the divisors of 128 are arranged is ascending order the sixth divisor is (A) 16 (B) 64 (C) 32

(D) 128

18.

19.

Consider the fractions (A) 10

[AMTI 2006] [AMTI 2006]

1 2 3 9 10 , , ,....... , . The number of these fractions which are irreducible is 10 9 8 2 1 [AMTI 2006] (B) 8 (C) 6 (D) 4

20.

The natural numbers are written as below following some rule 1, 3, 6, 10....., then tenth number is [AMTI 2007] (A) 55 (B) 62 (C) 105 (D) 35

21.

20072007  100001 is a (A) Three digit number (B) Four digit number

22.

It is given that 5 (A) 10

3 1  b – 19 , then a + b : a 2 (B) 12

[AMTI 2007] (D) Six digit number

(C) Eight digit number

[AMTI 2007] (C) 9

(D) 15 a

b

b

a

23.

a and b are 2 primes of the form p and p + 1, and m = a + b ; N = a + b then [AMTI 2007] (A) M and N are composite (B) M is a prime bu N is composite (C) M and N are primes (D) M is composite, N is prime

24.

The value of

1

[AMTI 2007]

1

2 3

1 4

(A) 25.

77 60

The value of (A)

8 7

1 5 (B)

68 157

(C)

2007 2008

(D) None of these

1 1 1 1 1 1 1 + + + + + + is 1 2 23 34 45 56 67 78 (B)

7 8

(C) 1

[AMTI 2007] (D)

1 1569

26.

a,b,c are any three of the first four prime numbers. n = a2bc. The biggest and the smallest value of n are respectively. [AMTI 2007] (A) (1035, 50) (B) (735, 60) (C) (525, 50) (D) (735, 50)

27.

A number when divided by 899 gives a remainder 63. What remainder will be obtained by dividing the same number by 29 ? [AMTI 2007] (A) 5 (B) 18 (C) 19 (D) 21

28.

Which fraction is between (A)

29.

1 3

1 1 and ? 4 5

(B)

1 20

[AMTI 2008] (C)

9 40

(D)

1 6

How many whole numbers less than 100 satisfy all the following conditions ? [AMTI 2008] (1) If divided by 3, the remainder is 1 (2) If divided by 5, the remainder is 1 (3) If divided by 7, the remainder is 0 (A) 0 (B) 1 (C) 91 (D) 15

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30.

A 2009 digit number is multiplied by 54. The last two digits of the product are 6 and 8 in this order. If the same number is multiplied by 46, the last two digits are [AMTI 2009] (A) 0,9 (B) 8,6 (C) 3,2 (D) cannot be found

31.

The digits of a three digits number are 3,7 and x in that order and 37x = 33 + 73 + x3. The value of x is [AMTI 2009] (A) 1 or 2 (B) 0 or 2 (C) 1 or 0 (D) 0;1 or 2

32.

The largest of the four numbers given below is (A) 3.1416

(B) 3.1 416

[AMTI 2009] (C) 314.16

(D) 3.1416

33.

When a number n is divided by 10,000, the quotient is 1 and the remainder is 2011. The quotient and remainder when n is divided by 2011 are respectively. [AMTI 2011] (A) 4,1936 (B) 5,1956 (C) 490 (D) 590

34.

The sum of all four digit numbers formed by using all the four digits of the numbers 2011 (including) is [AMTI 2011] (A) 10877 (B) 12666 (C) 10888 (D) 12888

35.

The difference between the biggest and smallest 3 digit number each of which has difference digit is [AMTI 2011] (A) 885 (B) 785 (C) 587 (D) 588

36.

a, b, c are 3 natural numbers such that a < b < c and a + b + c = 6. The value of C is [AMTI 2011] (A) 1 (B) 2 (C) 3 (D) 1 or 2 or 3

37.

A says : “I am a 6– digit number and all middle digits are made of zeros”. B says to A : “I am your successor. My digit is the tens place is the same as your starting digit.” The value of the whole number A is [AMTI 2012] (A) 100009 (B) 100008 (C) 100007 (D) 200009

38.

The number of 3 digit numbers that are divisible by 2 but not divisible by 4 is [AMTI 2012] (A) 200 (B) 225 (C) 250 (D) 450

39.

a,b where a > b are natural numbers each less than 10 such that (a2 – b2) is a prime number. The number of such pairs (a, b) is [AMTI 2012] (A) 5 (B) 6 (C) 7 (D) 8

40.

When 26 is divided by a positive integer n, the remainder is 2. The sum of all the possible values of n is [AMTI 2014] (A) 57 (B) 60 (C) 45 (D) 74

41.

Mahadevan used his calculator (which he rarely uses) to multiply a number by 2. But by mistake he multiplied by 20. To obtain the correct result he must [AMTI 2014] (A) divided by 20 (B) divided by 40 (C) multiply by 10 (D) multiply by 0.1

42.

a 4273b is a six digit number in which a and b are digits. This number is divisible by 72. Then [AMTI 2014] (A) b – 2a = 0 (B) a – 2b = 0 (C) 2a – b = 4 (D) a + b = 13

43.

P and Q are natural numbers. If 25  P  18 = Q  15. The smallest values of P + Q is. [AMTI 2014] (A) 61 (B) 21 (C) 41 (D) 31

44.

The thousands digits in the multiplication 111111  11111 is (A) 1 (B) 2 (C) 3

[AMTI 2014] (D) 4

45.

A 3-digit number is divisible by 35. The greatest such number has in its tens place the digit [AMTI 2015] (A) 4 (B) 7 (C) 9 (D) 8

46.

When 22015 is completely calculated the units place of the number obtained is [AMTI 2015] (A) 2

47.

The L.C.M of 4 (A) 62

(B) 4

(C) 8

(D) 6

1 1 , 3 and 10 is 2 2 (B) 18

[AMTI 2015] (C) 63

(D) 64

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48.

The average of 10 consecutive odd numbers is 120. What is the average of the 5 smallest numbers among them [AMTI 2015] (A) 100 (B) 105 (C) 110 (D) 115

49.

abc is a three digit number where a,b,c are the digits. How many are there such that a × b × c = 12. [AMTI 2015] (A) 12 (B) 6 (C) 4 (D) 15

50.

Slok is a primary school child. He calculated the numebr of Sundays occurring in 45 consecutive days. He was very happy that he got the maximum Sundays. This maximum number is [AMTI 2015] (A) 6 (B) 7 (C) 8 (D) 15

51.

Which one is true for the following set of five natural numbers [AMTI 2015] 24, 25, 26, 27, 28 (A) When 3, 4, 5, 6 and 7 are added respectively to the numbers, we get a set of 5 primes. (B) When 5 is added to 24, 6 is subtracted from 25, 7 is added to 26, 8 is subtraced from 27 and 9 is added to 28, we get a set of 5 primes. (C) All the 5 numbers are composite. (D) When 1 is added to all we get a set of primes.

EXERCISE-2 \

SUBJECTIVE

1.

a234 is a four digit number which is divisible by 18 then a is ____________

[AMTI 2009]

2.

a, b, c are squares of three consecutive integers and (b – a) = 87 then c is ________. [AMTI 2009]

3.

The number of natural numbers (a, b) satisfying the relation 7 + a + b = 10 is _____________ . [AMTI 2009]

4.

A boy divided a certain number by 75 instead of 72 and got both quotient and remainder to be 72. What should be the quotient and remainder if it is divided by 72 _________ [AMTI 2009]

5.

Find the total number of digits in the number 1234 ... 2007 2008 2009.

5.

The sum of the digits of a two digit number is subtracted from the number. The units digit of the result is 6. The number of two digit numbers having this property is ___________ . [AMTI 2010]

6.

The sum of all natural less than 45 which are not divisible by 3 is _____________ [AMTI 2014]

7.

75 is written as the sum of 10 consecutive natural numbers. The maximum of the numebrs is __________. [AMTI 2015]

8.

The number of 4–digit numbers of different digits greater then 2000 which contains the digits of 2015 is [AMTI 2015] ___________.

[AMTI 2009]

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SEQUENCE

EXERCISE-1 OBJECTIVE 1.

Look at the following dot diagram

[AMTI 2004]

The pattren continues. The value of 1 + 3 + 5 + .... up to 100 terms is the number of dots shown in the (A) 100th diagram and the number of dots present in it is 1000 (B) 1000th diagram and the number of dots present in it is 10000 (C) 100th diagram and the number of dots present in it is 10,000 (D) 1000th diagram and the number of dots present in it is 1000 2.

In the sequence 1, 22, 333, ... 10101010101010101010, 11111111111111111111,.... the sum of the digits in the 200th term is [AMTI 2004] (A) 200 (B) 400 (C) 600 (D) 40000

3.

In the sequence of numbers 1, 2, 11, 22, 111, 222,.... the sum of the digits in the 99th term is [AMTI 2004] (A) 999 (B) 1998 (C) 500 (D) 1000

4.

If !5 = 4 + 6 – 5, !12 = 11 + 13 – 12 and !23 = 22 + 24 – 23, then what is the value of !40 + !41 + !42 + !43 + !44 + ...... + !49 + !50 ? [AMTI 2004] (A) 505 (B) 495 (C) 455 (D) 465

5.

The 25th term in the sequence (1,2), (2,3), (3,5), (4,7), (5,11), (6,13),..... is [AMTI 2005] (A) (25, 47) (B) (25, 49) (C) (25, 37) (D) (25, 97)

6.

The total number of dots in the first 100 rows is

(A) 550

(B) 560

[AMTI 2005]

(C) 5500

(D) 10000

7.

Nine bus stops are equally spaced along a bus route. The distance from the first to the third is 600m, How far is it from the first stop to the last ? [AMTI 2005] (A) 800m (B) 1600m (C) 1800m (D) 2400m

8.

What is the first digit of the smallest number whose sum of the digits is 2007 ? (A) 9 (B) 8 (C) 3 (D) 2

9.

S = 1 + 22 + 333 + 4444 + 55555 + 666666 + 7777777 [AMTI 2007] The digits of the number S are added. If you get a double digit number, again add the digits and continue to get a single digit number. The final single digit is (A) 9 (B) 5 (C) 8 (D) 1

10.

The value of

1

[AMTI 2007]

1

2 3

1 4

(A)

77 60

[AMTI 2006]

1 5 (B)

68 157

(C)

2007 2008

(D) none of these

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11.

3 1 a . If a is a term the next term is . Then the 2007th term is 8 1 a [AMTI 2007] 8 3 (B) (C) (D) 1 3 8

The first term of a sequence is

(A)

5 11

12.

8 boxes all of different sizes, are placed in a row ; 2008 books are distributed in such a way that each box receives 2 books more than its next immediate smaller box. How many books does the largest box receive ? [AMTI 2008] (A) 258 (B) 244 (C) 236 (D) 264

13.

n, a are natural numbers each greater than 1. If a + a + ... + a = 2010, and there are n terms on the left [AMTI 2009] hand side, then the number of ordered pairs (a, n) is (A) 7 (B) 8 (C) 14 (D) 16

14.

The natural numbers are written in the following form

[AMTI 2009]

First row 1 Second row 234 Third row 98765 Fourth row 10 11 12 12 14 15 16 ........ ....................... The number 2010 is in the (A) 45th row as the 72nd number from the right. (C) 45th row as the 16th number from the left

(B) 44th row as the 73nd number from the right (D) 44th row as the 17th number from the left

15.

In the sequence 1,4,3,6,5,8,7,10, ... we have t2n – 1 = 2n–1 and t2n = t2n – 1+ 3 [(ie) every odd term is that odd number and the next even term is 3 more than the previous odd term]. If tm = 2010, then m is equal to [AMTI 2009] (A) 1005 (B) 1004 (C) 2008 (D) 2010

16.

The 2009th letter of the word sequence MATHTALENT MATHTALENT MATHTALENT ... is [AMTI 2009] (A) A (B) H (C) L (D) N

17.

Given two adition problems a = 1 + 12 + 123 + .....+ 123456789 b = 987654321 + 87654321 + .... + 21 + 1 The digits in the thousandth place of a and b are respectively (A) 4 and 6 (B) 1 and 6 (C) 4 and 4

18.

[AMTI 2010]

(D) 1 and 4

The integers greater than 1 are arranged in 5 columns as follows. Column

Column

Column

Column

Column

(1)

(2)

(3)

(4)

(5)

2

3

4

5

8

7

6

10

11

12

16

15

14

Row 1 ® Row 2

9

Row 3 ® Row 4

17

[AMTI 2011]

13

In the odd numbered rows, the integers appear in the last 4 columns are increasing form left to right. In the even numbered rows, the integers appear in the first four columns are increasing from right to left. In which colum will the number 2012 appear ? (A) fourth (B) second (C) first (D) fifth

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19.

In the following sequence 11, 88, 16, 80, 21, 72, ..,...,...,... the blanks are two digit numbers. No [AMTI 2014]

number in the blank ends with (A) 1 20.

(B) 4

(C) 6

(D) 7

Laxman starts counting backwards from 100 by 7’s. He begins 100, 93, 86, ... which number will not come in his countdown ? (A) 65

21.

The value of

(A)

23 4

[AMTI 2014]

(B) 30

(C) 23

(D) 15

50 50 50 50    .......... ......... is 72 90 110 9900 (B)

32 7

(C)

[AMTI 2015]

1 2015

(D)

55 27

EXERCISE-2 SUBJECTIVE 1.

In a sequence, the first term t1 = 6, t2 = a + 3, t3 = 42 tn+2 = 3tn+1 – 2tn for n = 1, 2, .... ; then every term of the sequence is a multiple of ____________

[AMTI 2009]

2.

In the sequence 1, 2, 3, 4, 5, ...., 100000000 the percentage of square numbers is ______[AMTI 2009]

3.

Archie shoots two arrows at the target, for example, in the diagram her score is 5. If both the arrows hit the target then the number of different scores is __________.

[AMTI 2009]

1 2 3 6

4.

n is a two digit number. P(n) is the product of the digits of n and S(n) is the sum of the digits of n. If n = p(n) +S(n) then the units digit of n is ________

5.

[AMTI 2011]

A sequence of numbers 1, 2, 3, .......... follows the rule that every number from the 4th term is the sum of the previous three numbers. The tenth number in the sequence is _________.

[AMTI 2015]

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GEOMETRY

EXERCISE-1 OBJECTIVE 1.

In the figure shown B,C,D lie on the same line. mECD = 90°. m[2CDE] = mCED A = mBAC 2x° = mABC

[AMTI 2004]

? C 90°

B 2x°

x°

D

2x°

The value of mACD is

E (A) 120° 2.

(B) 150°

(C) 210°

(D) 240°

ABC is an isosceles triangle with mA = 20° and AB = AC D and E are points on AB and AC such that AD = AE. I is the midpoint of the segment DE. [AMTI 2004] If BD = ID, then the angle of IBC are

A 20° I

D

(A) 110°, 35°, 35° 3.

E C (C) 80°, 50°, 50°

B (B) 100°, 40°, 40°

(D) 90°, 45°, 45°

A,B,C and D are points on a line. E is a point outside this line. Given that AE = BE = AB = BC and CE = CD, we find that the measure of DEA is [AMTI 2004] E D C B A

(A) 90° 4. 5.

(B) 105°

(C) 120°

There are four points A,B,C,D on a straight line. The distance between A and B is 3 cms. C and D are both twice as far from A as from B. Then the distance between C and D is. [AMTI 2004] (A) 1 cm (B) 2 cms (C) 3 cms (D) 4 cms How big is the angle x ? [AMTI 2004] 36º

y

z x

(A) 30° 6.

(D) 150°

(B) 36°

80º

(C) 44°

L1

L2

In an isosceles acute angled triangle one angle is 50°. I. The other two angles are 65° and 65° II. The other two angles are 50° and 80° Then which one of the above ststements can be true ? (A) I only (B) II only (C) I and II both

(D) 64° [AMTI 2004]

(D) either I or II but not both

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7.

Three cubes of metal whose edges are 3cm. 4cm and 5cm are melted and formed into a single cube. If there is no wastage in the process, the edge of the new cube is [AMTI 2006] (A) 12 cm (B) 7 cm (C) 9 cm (D) 6 cm

8.

In the adjoining figure the parallel lines are marked by arrow lines. The value of the angle x is [AMTI 2007]

80°

70°

90°

x (A) 60° 9.

(B) 70°

(D) 50°

In the adjoining figure AG = 6 cm. AB = BC = CD = DE = EF = FG = 1 cm. Semicircles are drawn as in the diagram. The total length of the path A to G along the smaller smicircles is x. The length of the path (B to F) along the bigger semicircle is y. Then [AMTI 2007]

A

(A) x < y 10.

(C) 80°

C

B

E

D

(B) x = y

F

G

(C) x > y

(D) x + y = 6

In the adjoning figure x =

[AMTI 2007]

60° 130° G 50°

x D (A) 40°

B

(B) 30°

(C) 20°

(D) 10°

11.

Mahabir drove 5 km west, then 7 km south, then 4 km west, then 18 km north and then 9 km east. How far is he now from his starting place ? [AMTI 2008] (A) 11 km (B) 4 km (C) 14 km (D) 9 km

12.

Triangles ABC and DBC have a common base BC, AB = BC = AC = CD. ACD = 90°. Then ABD is [AMTI 2009] A

D B

13.

C

(A) 15° (B) 25° (C) 35° (D) 45° Three straight lines intersect in one point. Two of the angles are given. The value of the angle marked x is. [AMTI 2009] B C

110º

105º A

F (A) 35°

(B) 25°

D O

x

E (C) 45°

(D) 50°

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15.

16.

ABCD is a quadrilateral AB = AD, BC = CD. [AMTI 2010] BAD = BDC = 20°. The measure of the angles ABC, BCD and CDA are respectively. (A) 100°, 140°, 100° (B) 20°, 140°, 100° (C) 100°, 100°, 20° (D) 140°, 100°. 100° 1 In the diagram ABCD is a quadrilateral. ABC = 150°, DAB = ABC and BCD = 60°. Then ADP 3 [AMTI 2010] and APD are respectively. A R

17.

(A) 100° and 30° (B) 110° and 20° (C) 80° and 40° (D) 120° and 10° ABC is a triangl in which BAC = 60°, ACB = 80°, ADE is the angle bisector of BAC. Then the triangle BDE is [AMTI 2010] A

B

18.

P

C

D

80° C

60° D

E (A) Isosceles (B) Equilateral (C) Right angled The points A,B,C and D are marked on a line l as shown in the figure l D C B A  AB   is equal to AC = 12 cm, BD = 17 cm, AD = 22 cm. Then   CD 

(D) Scalene [AMTI 2011]

5 1 7 (B) (C) 2 (D) 7 2 5 ABCD is a rectangle in which AB = 20 cm, BC = 10 cm. An equilateral triangle ABE is drawn here and M is the midpoint of BE. Then CMB s equal to [AMTI 2011] (A)

19.

B

A

M D

C

E

20.

(A) 70° (B) 75° In the adjoining figure the value of x is

130°

E x B 140°

(A) 110°

(B) 130°

(C) 65°

(D) 90° [AMTI 2011]

A

150° D

(C) 120°

(D) 125°

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21.

Aruna has a piece of cloth measuring 128 cm by 72 cm. She wants to cut it into pieces of squares. [AMTI 2014] The greatest possible size of the square that she can cut is (A) 6 cm by 6 cm (B) 8 cm by 8 cm (C) 9 cm by 9 cm (D) 12 cm by 12 cm

22.

A rectangle has length 9 times its width. The ratio of its perimeter to the perimeter of the square of same area is [AMTI 2015] (A) 5 : 4 (B) 6 : 5 (C) 5 : 3 (D) 7 : 5

23.

ABCD is a square of side 1cm. O is the point of intersection of the diagonals. P is the midpoint of OB. [AMTI 2015] Then the length of AP2 (in cm) is (A)

24.

3 8

(B)

3 4

(C)

3 5

(D)

5 8

In the figure given below, the distance between two adjacent dots horizontally or vertically is 1 unit. A is the area of the shaded region in figure (1), B is the area of the Shaded region in figure (2). Then A : B is [AMTI 2015]

Figure (1)

(A) 4 : 3

Figure (2)

(B) 5 : 1

(C) 9 : 4

(D) 6 : 1

EXERCISE-2 SUBJECTIVE 1.

In the adjoining figure the arrowed lines are parallel. The value of x is ___________. [AMTI 2014]

x

108°

2.

28°

In the adjoining figure, the area of each circle is 4p square units. The area of the square in the same square units is ___________. [AMTI 2014]

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3.

The maximum number of rectangles with different perimeter and an area of 216 cm2. if the length and breadth of each rectangle is a multiple of 3 is _________ [AMTI 2014]

4.

A rectangle of dimensions 3cm by 8 cm is cut along the dotted line shown. The cut piece is then joined with the remaining piece to form a right angled triangle. The perimeter of this triangle is [AMTI 2014] _______cm.

4

3

5.

In the adjoining figure. ABC is a triangle. AD is perpendicular to CB produced, BE is parallel to CF. FH bisector CFG. The value of x + y is __________. [AMTI 2015]

A

30° y = 45

15° 48° 40°

C

B

D

x F x

H

G

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AREA & PERIMETER

EXERCISE-1 \

OBJECTIVE

1.

O is the cente of the regular pentagon. What part of the whole pentagon is the shaded region ? [AMTI 2006]

O

(A) 10% 2.

3.

(B) 20%

(C) 25%

(D) 30%

Bases of four equilateral triangles form a square. Indise the square four circles of radius 5 units are drawn as in the figure. The perimeter of the four cornered star is [AMTI 2006]

(A) 100 (B) 120 (C) 160 (D) 200 Four circles of equal radii are centred at the four vertices of a square. These 4 circles touch a fifth circle of equal radius placed inside the square. The ratio of the shaded area of the circles to the unshaded area [AMTI 2006] of the circles is

(A) 1 : 3

(B) 2 : 3

(C) 3 : 4

(D) 2 : 5

4.

Which one of the following is incorrect (A) Doubling the length of a rectangle doubles the area (B) Doubling the altitude of a triangle doubles the area (C) Doubling a given quantity may make its area lesser than the original (D) Doubling the radius of a circle doubles the area.

5.

A plank is placed on a tiled floor. What fraction of the floor is not covered by the plank ?[AMTI 2008]

(A) 6.

1 4

(B)

3 8

(C)

17 64

[AMTI 2008]

(D)

5 8

Tulsi bought a square-shaped carpet. It the perimeter of the study room is 16m, what was the area of the [AMTI 2008] original carpet ? 2 2 2 (A) 256 m (B) 64 m (C) 32 m (D) 16 m2

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7.

ABC is an equilateral triangle with side length 9 cm. P, Q, R, S trisect respectively the sides AC and AB. PR2Q, ST2R are equilateral triangles. Again P1, Q1 ; Q2, P2 ; S1, R1 ; S2 , R2 trisect the respective sides. All the small triangles with vertices R1 R2, R3 , T1, T2, T3 are equilateral. The total perimeter of the figure is [AMTI 2008] A

T1 T2 T3

S2

B1 S 1 B2

R

R1 S

P

R2 P1 Q1 Q2 R3 P2 Q

B (A) 40 8.

(D) 64

(B) 42 cm2

(C) 16 cm2

(D) 8 cm2

In figure A, five squares with sides 1cm, 2cm 3cm 4cm and 5cm are arranged int he ascending order. In figure B theyare arranged as shown. By how much does the perimeter of the figure B exceed that of figure A [AMTI 2009]

(A) 0 cm 10.

C (C) 44

4 identical circles, each of radius 2cm are drawn and then a few arcs are erased to obtain the design [AMTI 2008] shown here. What is the area of the design ?

(A) 4 cm2 9.

(B) 37

(B) 4cm

(C) 10cm

(D) 14cm

In the adjoining diagram the number of white squares yet to be shaded, so that the number of the shaded [AMTI 2009] squares equals half the number of white squares is

(A) 5

(B) 10

(C) 6

(D) 8

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11.

In the figure each of the triangles and the square has the same perimeter. The perimeter of the whole figure is 3 times the perimeter of the square.

(A) 24 12.

(B) 28

[AMTI 2009]

(C) 15

(D) 30

In the adjoing rangoli design the distance between any two adjacent dots is 1 unit. In the diagram we find the triangle ABC is equilateral. The number of smallest equilateral triangles thus formed by joining the dots suitably is

[AMTI 2010]

(A) 24 13.

(B) 28

(C) 15

(D) 30

A thin rectangular strip of paoer is 2011 cms long. It is divided into four rectangular strips of different sizes as in the figure.

[AMTI 2011]

A, B, C, D are the centres of the rectangles (1), (2), (3) and (4) respectively. Then (AB + CD) is equal to.

(A)

14.

2011 cms 3

(B)

2011 cms 2

(C)

2011 cms 4

(D)

2(2011) cms 3

Three trays have been arranged according to their weights in increasing order as follows.

(1) Where the symbols of the tray

(2)

(3)

are the three digits of numbers showing each of the weights. The position lies

[AMTI 2011]

(A) between (1) and (2)

(B) between (2) and (3)

(C) before tray (1)

(D) after tray (3)

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15.

ABCD is a rectangle and is divided into two regions P and Q by the broken Zig Zag line as shown.

Then (A) The perimeter of the region P is greater than the perimeter of the region Q. (B) Area of region P is equal to the area of the region Q. (C) Perimeter of region P is equal to the perimeter of region Q. (D) Area of the region P is greater than the area of the region Q. 17.

[AMTI 2011]

In the figure below pieces of squared sheets are shown Each small square is 1 square unit.

(1)

(2)

(3)

(4)

Two of them can be joined together without overlapping to form a rectangle. The area of this rectangle in [AMTI 2011] square units si (A) 18 (B) 19 (C) 16 (D) 17 18.

The base of a triangle is twice as long as a side of a square. Their areas are equal. Then the ratio of the [AMTI 2012] altitude of the triangle to this base to the side of the square is (A)

1 4

(B)

1 2

(C) 1

(D) 2

EXERCISE-2 SUBJECTIVE 1.

The maximum number of points of intersection of a circle and a triangle is m. The maximum number of [AMTI 2011] points of intersection of two triangles is n. Then the value of (m + n) is ______.

2.

In the figure, the radius of each of the smallest circles is

1 of the radius of the biggest circle. The radius 12 of each of the middle sized circles is three times the radius of the smallest circle. The area of the shaded portion is _______ times the area of the biggest circle. [AMTI 2011]

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3.

In the figure, (1), (2), (3) and (4) are squares. The perimeter of the squares (1) and (2) are respectively 20 and 24 units. The area of the entire figure is ______ . [AMTI 2011]

4.

An insect crawls from A to B along a square lamina which is divided by lines as shown into 16 equal squares. The insect always travels diagonally from one corner of a square to the other corner. While going it never visits the same corner of any square. If one diagonal of a smallest square is taken as 1 units, the maximum length of the path travelled by the insect is ________ . [AMTI 2012]

B

5.

A In the figure ABCD and CEFG are squares of sides 6cm and 2cm respectively. The area of the shaded [AMTI 2012] portion (in cm2) is ______ .

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MISCELLANEOUS

EXERCISE-1 \

OBJECTIVE

1.

In the adjoining figure, the number of triangles formed is

[AMTI 2004]

A F B (A) 6

E

O

C

D

(B) 7

(C) 10

(D) 16

2.

If distinct numbers are replaced for distinct letters in the following subtraction. [AMTI 2004] FOUR –ONE _______ TWO _______ Then the value of F and T are given by (A) F = 1, T = 9 (B) F = 1, T = 8 (C) F = 1 and T is any single digit other than 1 (D) F and T cannot be determined.

3.

In the figure given BCFE, DFEA are square, BC = 5 units, HE = 1 unit, the length and breadth of the rectangle ABCD are [AMTI 2004]

F

D

C

5 unit G

H 1unit

A (A) 8 units and 5 units 4.

B

E

(B) 5 units and 10 units (C) 5 units and 7 units

(D) 9 units and 5 units

How many squares are there altogether in this diagram ?

E F A (A) 8

(B) 9

[AMTI 2004]

D C

K J N G H B L N (C) 10

(D) 11

5.

Each letter stands for a different digit. Which letter has the lowest value ? AM + 4 _____ TIC (A) A (B) M (C) T (D) I

[AMTI 2004]

6.

In a month three Tuesdays were on even dates. Then, 21st of the month is a [AMTI 2006] (A) Sunday (B) Monday (C) Wednesday (D) Saturday

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7.

An isosceles triangle has equal sides 7 units long and the length of the third side is an integer. The [AMTI 2006] number of such triangles is (A) 13 (B) 11 (C) 9 (D) 7

8.

If five lines are drawn in the plane the maximum number of regions into which the plane is divided is [AMTI 2006] (A) 10 (B) 12 (C) 14 (D) 16

9.

The diameter of the circle in the picture is 10 cms. The perimeter of the region marked in thick line in cms [AMTI 2006] is

(A) 12 10.

(B) 16

(C) 20

In the adjoining figure we have 16 dots equally spaced in 4  4 grid. From A to B one has to go. He can go up or down, left or right from one dot t othe other. In the figure the length of the path is 7. The maximum [AMTI 2007] length of the path is

A (A) 12 11. 12.

(C) 15

[AMTI 2007] (D) 1

Each small square in the figure has a side length 1 cm. An ant travels from P to R. If it moves only along the lines, in how many ways can it reach R, using only the shortest route ? [AMTI 2007]

R

S

Q

P

(B) 3

(C) 4

(D) 6

In the adjoining diagram A is your house, B is your friend’s house and C is your school. There are 3 different routes from A to B and 4 different routes from B to C. You are starting from your house and after picking your friend go to school. The number of different routes in which you can do this is [AMTI 2007]

(A) 7

C

B

(B) 3

(C) 12

(D) 15

A square is divided into four identical rectangles. The perimeter of each one of these rectangles is 20 cm. [AMTI 2007] What is the perimeter of the square ?

(A) 80 cm 15.

(D) 16

The number of rectangles with integer sides and with perimeter 16 cm is (A) 8 (B) 4 (C) 3

A

14.

B

(B) 13

(A) 1 13.

(D) 24

(B) 32 cm

(C) 40 cm

(D) 50 cm

6 cubes are glued together as shown in the figure and then dipped in paint. The cubes are then separated. [AMTI 2007] How many faces (sides) of the cubes are not painted in total ?

(A) 5

(B) 6

(C) 4

(D) 8

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16.

There are 2009 equispaced points on a circle. The number of diameters determined by these points (the extremities of any diameter should be two of these 2009 points) are [AMTI 2009] (A) 0 (B) 2008 (B) 1004 (D) 2009

17.

One hundred flowers were kept in 4 baskets. After some time, 4,5,3 and 8 flowers were taken out from the first, second, third and fourth baskets respectively. Now all the 4 baskets have the same number of flowers. The number of flowers in the fourth basket at the beginning was [AMTI 2009] (A) 24 (B) 25 (C) 23 (D) 28

18.

To open a safe, some three digit code needs to be used. It is known that only three digits 0,1,2, exist in this code. The sum of the digits used in the code should be 2. The number of ways this code can beset is [AMTI 2009] (A) 3 (B) 6 (C) 9 (D) 12

19.

The number of ways of labeling the ray using the points shown in the figure is

[AMTI 2009]

• • • • •

D C B H O (A) 6 20.

(B) 4

(C) 1

(D) 10

ABC are single digits in this multiplication B could be AB 7 ___ BCA (A) 7

(B) 1

(C) 2

[AMTI 2012]

(D) 4

21.

Samrud, saket, slok, Vishwa and Arish have different amount of money in Rupees, each an odd number which is less than 50 Rs. The largest possible sum of theses amounts (in Rupees) is [AMTI 2014] (A) 229 (B) 220 (C) 250 (D) 225

22.

The sum of the present ages of 5 brothers is 120 years. How many years ago the sum was 80 years ? [AMTI 2014] (A) 6 (B) 7 (C) 8 (D) 9

23.

Two numbers are respectively 26% and 5% more than a third number. What percent is the first of the second ? [AMTI 2015] (A) 80 (B) 120 (C) 90 (D) 75 The average age of 24 students and the class teacher is 16 years. If the age of the class teacher is excluded, the average age reduces by 1 year. The age of the class teacher (in year) is [AMTI 2015] (A) 40 (B) 45 (C) 50 (D) 55

24.

25.

Mahadevan told his granddaughter, "I am 66 years old , of course not counting the Sundays". The correct age of Mahadevan is [AMTI 2015] (A) 77 (B) 78 (C) 79 (D) 81

26.

The ratio of the money with Samrud and Saket is 7 : 15 and that with saket and Vishwa is 7 : 16. If [AMTI 2015] Samrud has Rs. 490, the amount of money Vishwa has (A) 2000 (B) 4900 (C) 2400 (D) 2015

27.

Jingle has six times as much money as Bingle. Dingle has twice as much money as Bingle. pingle has six times as much many as Dingle. Pingle has _________ many times as much money as Jingle. (A) 1 (B) 2 (C) 3 (D) 4 [AMTI 2015]

EXERCISE-2 \

SUBJECTIVE

1.

In a piece of paer there are some number calculations. A drop of ink made a stain covering a number or an arithmetic symbol. The calculation looked like 121 – 2•3 – 41 + 123 = 0. The symbol or number under the stain is _______ [AMTI 2009]

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2.

There are eleven squares in the diagram, with number 7 in the first square and 6 in the 9th square. The sum of the numbers in any three consecutive squares is 21. The number in the second square is _______

7

[AMTI 2009]

6

11

22

33

11

55

11 11

33

3. 44

55

55

44

44

[AMTI 2009]

These are 5 boxes, each box containing numbe cards as shown. Cards are removed from each box so that at the end each box contains only one card and different boxes contain different cards. The card remaining in Box (1) is ____________ 4.

In the square array of 9 first natural numbrs the sums of the numbers in the two diagonals are each equal to 15. Then the first 36 natural numbers are represented as a square array then the sums of the numbers in the two diagonals each is equal to _________ . [AMTI 2009]

1 2 3 4 5 6 7 8 9 5.

The number of ways in which 100 can be written as the sum of two prime numbers is _________________ [AMTI 2009]

6.

In the adjoining figure ABC and DEF are equilateral triangle AB = BF = BC = CE = AC. AD and EF cut at O and are perpendicular to each other. The number of right angled triangles formed in the figure is _____ [AMTI 2011] A

F

B

O

C

E

D 7.

If the previous month is July, the the month 21 month from now is __________.

[AMTI 2014]

8.

Candles A and B are lit together. Candle A lasts 11 hours and candle B lasts 7 hours. After 3 hours the two candles have equal lengths remaining. The ratio of the original length of candle A t ocandle B is __________. [AMTI 2014]

9.

A, B, C are three toys. A is 50% costlier than C and B is 25% constlier than C. Then A is.[AMTI 2014]

10.

In a box there are green, red and blue beads. The number of beads which are not green is 9. The number of beads which are not red is 8 and the number of beads which are not blue is 7. Total number of beads in the box is ______. [AMTI 2015]

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ANSWER KEY NUMBER SYSTEM EXERC ISE-1

1. 8. 15. 22. 29. 36. 43. 50.

A C C A B C D B

2. 9. 16. 23. 30. 37. 44. 51.

A D B C C A D C

3. 10. 17. 24. 31. 38. 45.

B A B B C B D

4. 11. 18. 25. 32. 39. 46.

C D C B B B C

5. 12. 19. 26. 33. 40. 47.

B C A B B A C

6. 13. 20. 27. 34. 41. 48.

C D A A D D D

7. 14. 21. 28. 35. 42. 49.

B C B C A C D

2. 8.

2025 12

3.

2

4.

76, 0

5.

6929

5.

10

6.

675

6. 13. 20.

D B D

7. 14. 21.

D C A

D A B

7. 15. 22.

D A C

EXERC ISE-2

1. 7.

9 12

SEQUENCE EXERC ISE-1

1. 8. 15.

C A C

2. 9. 16.

B B D

3. 10. 17.

C B A

4. 11. 18.

B C A

5. 12. 19.

D A D

2.

0.01

3.

9

4.

9

5.

230

EXERC ISE-2

1.

6

GEOMETRY EXERC ISE-1

1. 8. 16. 23.

A A A D

2. 9. 17. 24.

 C D C

26.

144p 

3. 10. 18.

B C B

4. 11. 19.

D A B

5. 12. 20.

D D C

6. 13. 21.

27.

4

28.

20

29.

110

6. 13.

D B

7. 14.

B B

A B D

EXERC ISE-2

25.

136

AREA & PERIMETER EXERC ISE-1

1. 8. 15.

D C C

2. 9. 17.

C B A

3. 10. 18.

B A C

4. 11.

CD D

5. 12.

B D

2.

53/48

3.

471

4.

8

5.

30

EXERC ISE-2

1.

12

MISCELLANEOUS EXERC ISE-1

1. 8. 15. 22.

D D A C

2. 9. 16. 23.

A C A B

3. 10. 17. 24.

D C D A

4. 11. 18. 25.

D C B A

5. 12. 19. 26.

D D B C

6. 13. 20. 27.

A C C B

7. 14. 21.

2. 10.

8 12

3.

22

4.

111

5.

6

6.

10

7.

EXERC ISE-2

1. 8.

0 1:2

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may

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NMTC QUESTION BANK SUB JUNIOR GROUP (STAGE –1)

VIBRANT ACADEMY (India) Private Limited A-14(A), Road No.1, Indraprastha Industrial Area, Kota-324005 (Raj.) Tel.:(0744) 2423406, 2428666, 2428664, 2425407 Fax: 2423405 Email: [email protected]

Website : www.vibrantacademy.com

NUMBER SYSTEM \

EXERCISE–I OBJECTIVE

1.

The number of non negative integers which are less than1000 and end with only one zero is [AMTI 2004] (A) 90 (B) 99 (C) 91 (D) 100

2.

A transport company’s vans each carry a maximum load of 12 tonnes. 24 sealed boxes weighing 5 tonnes have to be transported to a factory. The number of van loads needed to do this is [AMTI 2004] (A) 9 (B) 10 (C) 11 (D) 12

3.

The digits of the year 2000 add up to 2. In how many years has this happened since the year 1 till this [AMTI 2004] year 2004? (A) 3 (B) 6 (C) 9 (D) 10

4.

A certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35. The number is [AMTI 2004] (A) 105 (B) 210 (C) 420 (D) 525

5.

Three people each think of a number which is the product of two different primes. The product of three [AMTI 2004] numbers which are thought of is (A) 120 (B) 12100 (C) 240 (D) 3000

6.

 1 The last digit in the finite decimal representation of the number   5 (A) 2

(B) 4

2004

(C) 6

is

[AMTI 2004] (D) 8

7.

A four digit number of the form abaa (a’s and b’s are the digits of the four digit number) is divisible by 33. The number of such four digit numbers is [AMTI 2004] (A) 36 (B) 6 (C) 3 (D) 1

8.

The sum of the digits of the number 10n – 1 is 3798. The value of n is (A) 431 (B) 673 (C) 422

9.

Using all the digits 1 to 9 only once, how many nine digit prime numbers can you write ? (A) 1

10.

[AMTI 2004] (D) 501

(B) None

(C) 9

[AMTI 2005] (D) More than 100

How many times does the numeral 2 appear in a book having page numbers 1 to 250 ? [AMTI 2005] (A) 81

11.

12.

13.

(B) 106

(C) 55

(D) 76

Look at the additions on the left. Here A,B,C are different digits. If C is placed in the units column the [AMTI 2005] total is 45. If C is placed in the tens column the sum is 99. Then the value of A is AB A B C C ____ _____ 4 5 9 9 ____ _____ (A) 4 (B) 3 (C) 9 (D) 6 You have five pieces of 6 cm rods and 4 pieces of 7cm rods. Using some of all of them, which one of [AMTI 2005] the following lengths you cannot measure ? (A) 30 (B) 29 (C) 31 (D) 33 How many two digits numbers divide 109 with a remainder of 4 ? (A) 2 (B) 4 (C) 3

[AMTI 2005] (D) None

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14.

The highest power of 2 that divides the sum of the numbers 4 + 44 + 444 + .... + 444 .......4 is 100 times

[AMTI 2005] (A) 2

(B) 3

(C) 4

(D) 5

15.

a and b are square numbers ; the l.c.m of a and 140 is 560 and the l.c.m of b and 140 is 700. Then the l.c.m of a and b is [AMTI 2005] (A) 400 (B) 1600 (C) 2500 (D) 4900

16.

a, b, c, d are natural numbers such that a = bc, b = cd, c = da and d = ab. Then (a + b) (b + c) (c + d) [AMTI 2005] (d + a) is equal to (A) (a + b + c + d)2 (B) (a + b)2 + (c + d)2 (C) (a + d)2 + (b + c)2 (D) (a + c)2 + (b + d)2

17.

How many 4 digits numbers with middle digits 97 are divisible by 45 ? (A) 0 (B) 2 (C) 4

[AMTI 2005] (D) 1

18.

The products of three consecutive odd numbers is 357 627. What is their sum ? (A) 213 (B) 243 (C) 153 (D) 209

[AMTI 2005]

19.

The number of integers whose square is a factor of 2000 is (A) 3 (B) 6 (C) 10

[AMTI 2005] (D) 12

20.

Let n be a 3 digit number such that n = sum of the squares of the digits of n. The number of such n is [AMTI 2006] (A) 0 (B) 1 (C) 2 (D) More than 2

21.

The number of prime numbers less than 1 lakh, whose digital sum is 2 (digital sum of a number is the [AMTI 2006] sum of its digits) (A) 5 (B) 4 (C) 3 (D) none of these

22.

The least number of numbers to be deleted from the set {1, 2, 3, ...... 13, 14, 15} so that the product of the remaining numbers is a perfect square is [AMTI 2006] (A) 1 (B) 2 (C) 3 (D) 4

23.

The largest positive integer n for which n200 < 6300 is (A) 12 (B) 13 (C) 17

[AMTI 2007] (D) 14

24.

The sum of two numbers is 1215 and their GCD is 81. How many pairs of such number are possible ? [AMTI 2007] (A) 2 (B) 4 (C) 6 (D) 8

25.

The number of digits in 815540 (when written in base 10 from) is (A) 42 (B) 45 (C) 55

(D) 2007

The number of two digit numbers whose digit sum is divisible by 6 is (A) 13 (B) 8 (C) 7

(D) 22

26.

[AMTI 2007] [AMTI 2007]

27.

A number when divided by 899 gives a remainder 63 what is the remainde when number is divided by 29 ? [AMTI 2007] (A) 5 (B) 28 (C) 16 (D) 12

28.

The number A7389B where A,B are digits is divisible by 72, then (A,B) is (A) (6,3) (B) (3,6) (C) (4,7)

29.

[AMTI 2007] (D) (7,5)

Given the alphametic where each letter represents a different digit, the value of C in the least value for [AMTI 2007] ABCD is.

ABCD –BCDA

[AMTI 2007]

1512 (A) 1

(B) 2

(C) 6

(D) 7

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[4]

30.

If 36a4 = a6 , then a3 is equal to (A)

31.

1 6 a 6

[AMTI 2006]

(B) 6a4

(C)

The quotient of 100100 and 5050 is (A) 5050 (B) 50100

1 2 a 6

(D) 6a2 [AMTI 2006]

50

50

(C) 200

(D) 400

32.

A number is formed by writing the first 10 primes in the increasing order. half of the digits are now crossed out, so that the number formed by the remaining digits without changing the order, is as large [AMTI 2006] as possible. The second digit from the left of the new number is (A) 2 (B) 3 (C) 5 (D) 7

33.

Let n be the number of integers less than 10,000 which are divisible by all integers from 2 to 10. Then [AMTI 2006] (A) n = 0 (B) 1  n < 5 (C) 5 < n < 10 (D) 10 < n <15

34.

Let the length of a positive integer n be defined as the number of prime factors of n, counting repetitions. Length of 36 is 4 as 36 = 2.2.3.3 Length of 64 is 6 as 64 = 26. The number of numbers less than [AMTI 2006] 100 with maximum length is (A) 5 (B) 4 (C) 3 (D) 2

35.

The number of primes less than 100 which have 7 as the unit digit is (A) 6 (B) 7 (C) 8

[AMTI 2007] (D) 9

36.

A six digit number is formed by repeating a three digit number twice (like 245245). Such numbers are always divisible by [AMTI 2007] (A) 1001 (B) 25 (C) 101 (D) 111

37.

Twenty four children are seated equally spaced around a circle and numbered from 1 to 24. What is [AMTI 2007] the number of the child who sits diametrically opposite to the child number 10. (A) 21 (B) 23 (C) 22 (D) 20

38.

If n = 1010–1, the number of digits in n3 is (A) 30 (B) 28

39.

[AMTI 2008] (C) 32

(D) 27

Let m be the number of perfect squares in 1,2,3 .....1000000. Let ‘n’ be the number of perfect cubes in 1,2,3.........1000000. Then the value of (A) 0.01

(B) 0.1

n is m

[AMTI 2008] (C) 10

(D) 100

40.

The difference of the squares of 2 consecutive natural number is 2008 (A) Only one such pair exists (B) Infinitely many such pairs exist (C) No such pair exists (D) Exactly two pairs exist.

41.

The number 1612 is obtained from the number 84 by raising the smaller number to the power n. Then n [AMTI 2008] is (A) 3

(B) 4

(C)

16 3

(D)

[AMTI 2008]

8 3

42.

The tens place of two three digit numbers is 8 and both the numbers are divisible by 4. Then the difference between the biggest and the smallest such numbers is [AMTI 2008] (A) 888 (B) 808 (C) 708 (D) 788

43.

The number of digits swhen 2008 is written in the decimal from is (A) 2008 (B) 1004 (C) 74

44.

45.

The unit digit of the number 222008 is (A) 6 (B) 2

[AMTI 2008] (D) 19 [AMTI 2008]

(C) 4

(D) 8 a

b

If a,b are natural numbers such that a + b = 2008, then (–1) + (–1) is (A) 1 (B) –1 (C) 2

[AMTI 2008] (D) 2 or –2

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46.

The number 111........1111 is a 2009 digit number. It is multiplied by 2009. The third digit from the left of [AMTI 2009] the product is (A) 1 (B) 2 (C) 3 (D) 9

47.

The remainder when the number (2  3  4  2007  2008  2009) – 2008 is divided by 2009 is (A) 0 (B) 2008 (C) 1

48.

[AMTI 2009] (D) 2007

In the adjoining addition each letter represents a digit (identical letters represent the same digit, [AMTI 2009] different letters represent different digits) The value of A is

AA +AA CAB (A) 6 49.

(B) 7

(C) 8

(D) 9

n    , m, n positive integers with n < 100. The value of (m + n) is [AMTI 2011] Given 63.63 = m  21  100  

(A) 21

(B) 24

(C) 104

(D) 101

50.

n is the natural number and (n + 2) (n + 4) is odd then the biggest power of 2 that divides (n + 1) (n + 3) for any n is [AMTI 2011] (A) 1 (B) 2 (C) 3 (D) 4

51.

n is a negative integer. The expression having the greatest value is (A) – 2n2 + 2n (B) – 2n2 – 2n (C) 2n2 + 2n

[AMTI 2011] (D) 2n2 – 2n

52.

A number is called a palindrome if it reads the same forward or backward. e.g. 13531 is palindrome. The difference between the the biggest 10 digit palindrome and the smallest 9 digit palindrome is [AMTI 2011] (A) 976666666 (B) 9888888888 (C) 9899999998 (D) 9777777777

53.

A computer is printing a list of the seventh powers of all natural numbers, that is the sequence 17, 27, 37,...... The number of terms (or numbers) between 521 and 428 are [AMTI 2011] (A) 12 (B) 130 (C) 14 (D) 150

54.

A natural number n has exactly two divisors and (n + 1) has three divisors. The number of divisors of [AMTI 2011] (n + 2) is (A) 2 (B) 3 (C) 4 (D) Depends on the value of n

55.

p q + = r where p,q,r are positive integers and p = q. The biggest value of r less than 100 is 9 10 [AMTI 2011] (A) 19 (B) 57 (C) 76 (D) 95

56.

The positive integers p, q. p – q and p + q are all prime numbers. The sum of all these numbers is [AMTI 2011] (A) divisible by 3 (B) divisible by 5 (C) divisible by 7 (D) prime

57.

100th term of the sequence 1,3,3,3,5,5,5,5,5,7,7,7,7,7,7,7......... is (A) 15 (B) 13 (C) 17

[AMTI 2011] (D) 19

58.

Using the digits 2 and 7, and addition or subtraction operations only, the number 2010 is written. The maximum number of 7 that can be used, so that the total numbers used is a minimum is [AMTI 2010] (A) 284 (B) 286 (C) 288 (D) 290

59.

In the addition problem shown, different letters represent different digits. If the carry over from adding [AMTI 2010] the unit digit is 2, then (A + I) cannot be

(A) 2

(B) 4

(C) 7

(D) 9

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60.

a3 + b3 = p1 × p2, a3 – b3 = p3 , where p1, p2, p3 are prime numbers, a > b, 2  a, b  5. The following statements are also given : (1) p1 + p2 + p3 is a prime number (2) p3 – (p1 + p2) is a prime number. (3) p1 ~ p2 is a prime number. Then the values of a and b are respectively. [AMTI 2010] (A) (3, 4) (B) (4, 3) (C) (3, 2) (D) (2, 3)

61.

p is a prime number greater than 3. When p2 is divided by 12 the remainder is (A) Always an odd number greater than 2 (B) always 1 (C) 1 or 11 (D) always an even number

62.

Five positive integers are written around a circle so that no two or three adjacent number have a sum divisible by 3. In this collection the number of numbers divisible by 3 is . [AMTI 2011] (A) 0 (B) 1 (C) 2 (D) 3

63.

In the grid shown in the diagram A and B erase 4 number each from the table. The sum of the numbers erased by them are in the ratio 10 : 11 After erasing the numbers, the number left out on the table is [AMTI 2011]

[AMTI 2010]

13 14 15 20 21 22 27 28 29

(A) 21

(B) 14

(C) 22 2

64.

2

 p  1  p  1  –   is p is an odd prime. Then  2    2  (A) A fraction less than p. (C) A natural number not equal to p

65.

(D) 13 [AMTI 2012]

(B) A fraction greater than p. (D) A natural number equal to p.

The last two digits of 32012, when represented in decimal notation, will be (A) 81 (B) 01 (C) 41

[AMTI 2012] (D) 21

66.

There exists positive integers x, y such that both the expressions (3x + 2y) and (4x – 3y) are exactly [AMTI 2012] divisible by (A) 11 (B) 7 (C) 23 (D) 17

67.

Let n be the smallest non prime integer greater than 1 with no prime factors less than 10 then [AMTI 2012] (A) 100 < n  110 (B) 110  n  120 (C) 120 < n  130 (D) 130 < n  140

68.

a, b, c, d, e are five integers such that a + b = b + c = c + d = d + e = 2012, a + b + c + d + e = 5024. [AMTI 2012] Then th value of (d – a) is (A) 8 (B) 12 (C) 10 (D) 4

69.

Five two digit numbers (none of the digits is zero) add up to 100. If each digit is replaced by its 9 complement, then the sum of these five new numbers is. [AMTI 2012] (A) 295 (B) 195 (C) 380 (D) 395

70.

In the adjoining incomplete magic square the sum of all numbers in any row or column or diagonals is [AMTI 2013] a constant value. The value of x is 17 23

5 6

10 11

(A) 18

(B) 23

12

20

x

21 9

(C) 22

(D) 16

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71.

The sum of three different prime numbers is 40. What is the difference between the two biggest ones among them? [AMTI 2013] (A) 8 (B) 12 (C) 20 (D) 24

72.

Peter has written down four natural numbers. If he chooses three of this numbers at a time and adds up each triple, he obtains totals of 186, 206, 215 and 194. The largest number Peter has written is [AMTI 2013] (A) 93 (B) 103 (C) 81 (D) 73

73.

The natual numbers from 1 to 20 are listed below in such a way that the sum of each adjacent pair is a prime number. [AMTI 2013] 20, A, 16, 15, 4, B, 12, C, 10, 7, 6 D, 2, 17, 14, 9, 8, 5, 18, E. The number D is (A) 1 (B) 3 (C) 11 (D) 13

74.

How many ordered pairs of natural numbers (a, b) satisfy a + 2b = 100 (A) 33 (B) 49 (C) 50

75.

The value of (A)

3 210

 5

6 7

  5

6 7

 5

(B) 210

6 (C) 4

7

 5

[AMTI 2013] (D) 99



6  7 is

210

[AMTI 2013]

(D) 104

76.

The least number which, when divided by 52 leaves a remainder 33, when divided by 78 leaves 59 as [AMTI 2013] remainder and when divided by 117 leaves 98 as remainder is (A) 553 (B) 293 (C) 468 (D) 449

77.

The sum of five distinct positive integers is 90. What can be the second largest number of the five at most ? [AMTI 2014] (A) 82 (B) 43 (C) 34 (D) 73

78.

The number of digits when (999999999999)2 is expanded is (A) 26 (B) 24 (C) 32

[AMTI 2014] (D) 16

79.

Samrud wrote 4 different natural numbers. He chose three numbers at a time and added them each time. He got the sums as 115, 153, 169, 181. The largest of the numbers Samrud first wrote is [AMTI 2014] (A) 37 (B) 48 (C) 57 (D) 91

80.

Saket wrote a two digit number. He added 5 to the tens digit and subtracted 3 from the units digit of [AMTI 2014] the number. The resulting number is twice the original number. The original number is. (A) 47 (B) 74 (C) 37 (D) 73

81.

Five consecutive natural numbers cannot add up to (A) 225 (B) 222 (C) 220

82.

[AMTI 2014] (D) 200

In the adjoining figure the different numbers denote the area of the corresponding rectangle in which the number is there. The value of x is [AMTI 2014]

2014 1007 x (A) 3014 83. 84.

(B) 1125

(C) 2139

What is the remainder when 287 + 3 is divided by 7 (A) 2 (B) 3 (C) 4

(D) 250 [AMTI 2014] (D) 5

Three different integers have a sum 1 and product 36. Then (A) Certainly all of them are positive (B) Only one is negative (C) Exactly two of them are negative

85.

125

[AMTI 2015]

(D) All the three are negative

22015  22015  .........  22015 The value of  divided by 22015 is

[AMTI 2015]

256 terms

(A) 256

(B) 273

(C) 22015

(D) 2015

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86.

* is an operation defined as a*b =

ab  ba where a,b are natural numbers. ab

(Ex : a = 155, b = 60 then a * b =

15560  60155 ). If a = 2015, b = 5 then a b lies between 155  60

(A) 40 and 50

(B) 50 and 60

(C) 30 and 40

[AMTI 2015]

(D) 53 and 54

2

87.

n is a natural number. The number of possible remainders of n when divided by 7 is (A) 2 (B) 3 (C) 4 (D) 5

88.

The units digit of a 4 digit number (5 + 1) (52 + 1) (53 + 1). ........... (52015 + 1) is [AMTI 2015] (A) 9 (B) 8 (C) 6 (D) 4 If the product of the digits of a 4-digits numbers is 75, the sum of the digits of the number is [AMTI 2015] (A) 12 (B) 13 (C) 14 (D) 15

89.

90.

The hypotenuse ‘c’ and one side ‘a’ of a right triangle are consecutive integers. The square of the third [AMTI 2015] side is (A) c – a

91.

[AMTI 2015]

The fraction

(A)

(B) ca

(C) c + a

(D)

c a

2121212121 210 when reduced to its simplest form is 1121212121 211

73 70

(B)

37 7

(C)

70 37

[AMTI 2015]

(D)

70 13

EXERCISE–II SUBJECTIVE a

b

1.

If 60 = 3 and 60 = 5 then the value of 12

2.

The value of x satisfying the equation

1 a b 2(1b )

is equal to _________

1

=

1

1

3 is ________ 4

[AMTI 2007]

[AMTI 2010]

1

1

1

1 x

3.

p is a prime number and p = a2 – 1. The number of divisors of a + p is ________

4.

The biggest value of

5.

The number of integers n for which

6.

The unit digit of the number 32011 is _________

7.

A bar code is form using 25 black and cirtain white bars. White and black bars are alternete. The first and the last are black bars. some of the black bars are thin and others are white. The number of white bars is 15 more than the thin black bars. [AMTI 2011] The number of thick black bars is ______

10a (a  N) is never greater than ______ 10  a n is the square of an integer is ________ 20  n

[AMTI 2010] [AMTI 2010]

[AMTI 2011] [AMTI 2011]

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[9]

8.

By drawing 10 lines of which 4 are horizontal and 6 are vertical crossing each other as in the figure, one can get 15 cells. With the same 10 lines of which 3 are vertical and 7 horizontal we get 12 cells. The maximum number of cells that could be got by drawing 20 lines. (some horizontal and some vertical) is _________ [AMTI 2011]

9.

Three digit numbers are written with different digits. The sum of the biggest and the smallest of such [AMTI 2009] numbers is __________

10.

A certain number n is divisible by 21, 28 and 49. The smallest possible value of n is ________ [AMTI 2009]

11.

Rite reads 41 pages every day. She started reading a book containing 2010 pages. The number of pages [AMTI 2009] she reads on the last day to finish the book is ________

12.

The sum of two natural numbers is 484. There HCF is 11. The number of such possible pairs is _____ [AMTI 2012]

13.

p is the difference a real number and its reciprocal. q is the difference between the square of the same real number and the square of the reciprocal. Then the value of p4 + q2 + 4p2 is _________ [AMTI 2012]

14.

a,b,c,d are five integers such that a + b = b + c = c + d = d + e = 2012 and a + b + c + d + e = 5024. Then the value of (d – a) _______ [AMTI 2012]

15.

Ab, CD, EF, GH and IJ are five 2-digit numbers such that each letter stands for a different digit. The largest possible sum of these five numbers is _________ [AMTI 2012]

16.

Two consecutive natural numbers are respectively divisible by 4 and 7. The sum of their respective quotient is 8. Then the sum between the numbers is _______ [AMTI 2012]

17.

The number of prime numbers p for which p + 2 and p + 5 are also prime number is _______ [AMTI 2012]

18.

A natural number less than 100 has remainder 2 when divided by 3, remainder 3 when divided by 4 and remainder 4 when divided by 5. The remainder when the number is divided by 7 is __________. [AMTI 2013]

19.

The units digit of (2013)2013 is ______________.

20.

The number of naturla numbers ‘n’ for which

[AMTI 2013]

(n  2) (n  2) is a natural number is _________. (n 3)

[AMTI 2013] 21.

The product fo two natural numbers ‘a’ and ‘b’ divides 48. a,b are not relatively prime to each other. The number of pairs(a,b) where 1 < a + b < 48 (a –b) is ___________. [AMTI 2013]

22.

The ratio of a two digit number and the sum of its digits is 4 : 1. IF the digit in the units place is 3 more than the digit in the tens place, then the number is ___________. [AMTI 2013]

23.

If

24.

The number of three digit numbers of the form ab5 which are divisible by 9 is ________.

[AMTI 2013]

25.

The smallest multiple of 9 with no odd digits is _________.

[AMTI 2014]

26.

Arish found the value of 319 to be 11a2261467. he found all the digits correctly except the digit denoted by a. The value of ‘a’ is __________ [AMTI 2014]

10

2013

 25

 10 2

2013

 25

 = 2

10



n

then the value of n is ________.

[AMTI 2013]

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27.

The number of numbers in the list 1,2,3,4, ......... 2015 which are perfect squares and also perfect cubes is ____________. [AMTI 2015]

28.

Using the digits of the number 2015, four digit numbers of different digits are formed. The number of such numbers greater than 2000 and less than 6000 is _________ [AMTI 2015]

29.

The remainder when 20150020150002015002015 is divided by 3 is ____________.

30.

If

p =1+ q 2

1

[AMTI 2015]

where p, q have no common factors then p + q = ___________

1 1

3

4

1 5 [AMTI 2015]

31.

If

p =1+ q

5

where p, q have no common factors, then p + q = ___________

4

1 1

3 1

1 2 [AMTI 2015]

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PATTERN DETECTION EXERCISE–I OBJECTIVE 1.

If a* = a + 1 and *a = a – 1, then 1* – *1 + 2* – *2 + 3* – *3 + ..... + 1000* – *1000 is equal to [AMTI 2004] (A) 1000

2.

(C) 2000

(D) – 2000

The tenth term of the sequence (2,5), (3,7), (5,11), (7,13) (11,17),.... is (A) (22,29)

3.

(B) – 1000

(B) (19,13)

(C) (20,14)

[AMTI 2004] (D) (29,37)

Sum of all integers less than 100 which leave a remainder 1 when divided by 3 and leave a remainder 2 [AMTI 2006]

when divided by 4 is (A) 416 4.

(B) 1717

(C) 1250

(D) 1314

A three digit number with digits A,B,C in that order is divisible by 9. A is an odd digit and C is an even digit. [AMTI 2006]

B and C are non zero. The number of such three digit number is (A) 4 5.

(B) 8

(C) 16

(D) 20

A sequence of numbers T1, T2, T3,.....Tn ..... is constructed using the rule that the nth term Tn has n ‘n’ for example the 25th term T25 = 25252525....25, where 25 is repeated 25 times. The term containing the same number of 2s as T226 is (A) T622

6.

[AMTI 2006] (B) T262

(C) T254

(D) T452

The digit 1 is attached to the right of a 3 digit number making it a 4 digit number which is 7777 more than [AMTI 2006]

the given number. The sum of the digits of the number is (A) 23 7.

(B) 18

(C) 17

(D) 16

Given a sequence of two digit numbers grouped in brakets as follows : (10), (11, 20), (12, 21, 30), (13, 23, 31, 40)......(89, 98), (99). The digital sum of the numbers in the bracket [AMTI 2010]

having maximum numbers is. (A) 9 8.

(D) 18

(B) 38

(C) 37

[AMTI 2010] (D) 36

The 2015th letter of the sequence ABCDEDCBABCDEDCBA.... is (A) A

10.

(C) 9 or 10

n = 1 + 11 + 111 + .......+ 1111111111. The digital sum of n is (A) 39

9.

(B) 10

(B) B

(C) C

[AMTI 2015] (D) E

In the sum 3 + 33 + 333 + 3333 + ........ 2015 terms the number formed by taking the last four digits in that [AMTI 2015]

order is (A) 6365

(B) 6255

(C) 6465

(D) 6565

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EXERCISE–II SUBJECTIVE 1.

The natural numbers are filled as per the table shown where represents a square not filled. We find in the middle squares the numbers 2, *, 5, *, 8, , .....

1 3

2 *

* 7 9 *

5 6 8 * * 10 11 12

13 14 ... ...

[AMTI 2009]

* 4

* ...

The number found in the 2009th middle square is _________ 2.

Let a sequence of numbers be denoted as t1, t2, t3....., where t1 = 1 and tn = tn–1 + n (n is a natural number). Find t2t3,t4,t10,t2011.

[AMTI 2011]

3.

In the sequence 1,1,1,2,1,3,1,4,1,5...... The 2014th term is _________

[AMTI 2014]

4.

The value of 1 – 2 + 3 – 4 + 5 – ........... + 2015 is ____________.

[AMTI 2015]

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ALGEBRA EXERCISE–I OBJECTIVE 1.

If a + 2a + 3a + ..... + 1000a = 2b + 4b + 6b + .....+ 2000b = 3c + 6c + 9c + .....+ 3000c then a : b : c is as [AMTI 2004] (A) 1 : 2 : 3 (B) 3 : 2 : 1 (C) 2 : 3 : 6 (D) 6 : 3 : 2

2.

a and b are two natural numbers with a + b = 8. If a  b and a2 + b2 has minimum value then a and b are given by. [AMTI 2004] (A) 7,1 (B) 6,2 (C) 4,4 (D) 5,2

3.

In a magic square, each row and each column and both main diagonals have the same total. The number [AMTI 2004] that should replace x in this partially completed magic square is

13 5

15

x

4.

5.

(A) More information needed (B) 9 (C) 10 (D) 12 The product of Hari’s age in years on his last birthday and his age now in complete months is 1800. Hari’s age on his last birthday was [AMTI 2004] (A) 9 (B) 10 (C) 12 (D) 15 The largest positive integer which cannot be written in the form 5m + 7n where nm and n are positive integers is. [AMTI 2004] (A) 25 (B) 35 (C) > 100 (D) > 350

6.

Ram is 7 years younger than Ravi. IN four years time, Ram will be half of Ravi’s age. The sum of their ages now is [AMTI 2004] (A) 13 (B) 15 (C) 17 (D) 19

7.

Five books and 2 pencils cost Rs. 79 but 2 books and 5 pencils cost Rs. 40. What is the total cost of 1 [AMTI 2005] book and 1 pencil. (A) Rs. 39 (B) Rs. 19.5 (C) Rs. 17 (D) Rs. 23.8

8.

If a2 + a + 1 = 0 then a2 + (A) Positive intger tive intger

1 a2

is a

[AMTI 2005] (B) Positive Fraction which is not an integer(C) Nega(D) Negative Fraction which is not an integer

9.

Which of the following can never be a commnon factors of 287 + x and 378 + x where x can be any natural [AMTI 2005] number ? (A) 26 (B) 13 (C) 91 (D) 7

10.

You join a job. Your pay for the first day is Rs. 5 Each day after that your pay will be twice as much as it was the day before. Your pay on the tenth will be [AMTI 2005] (A) Rs. 100 (B) Rs. 250 (C) Rs. 5120 (D) Rs. 2560

11.

A quiz has 20 questions with seven points awarded for each correct answer, two points deducted for each wrong answer and zero for each question omitted. Ram scores 87 points. How many questions did he [AMTI 2005] omit ? (A) 2 (B) 5 (C) 7 (D) 9

12.

The weight of a dog is 8 kg plus one third its weight. The weight of the dog is (A) 11 kg (B) 12 kg (C) 14 kg (D) 15 kg

[AMTI 2006]

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[14]

13.

If x,y,z,a,b,c are real and none of these quantities is zero xy = a, xz = b and yz = c then x2 + y2 + z2 = [AMTI 2007] (A) a2 + b2 + c2

(B)

a2  b2  c 2 abc

(C)

abc a2  b2  c 2

(D)

(ab )2  (bc )2  (ca ) abc

2

14.

a,b,c are three non zero digits. The sum of all the numbers formed by these three digits is always divisible [AMTI 2008] by (A) a + b + c (B) 222 (C) 9(a + b +c) (A) Only (A) is true (B) Only (A) and (B) are true (C) only (B) and (C) are true (D) Only (A) and (C) are true

15.

a,b,c,d are real such that a – 2005 = b + 2006 = c – 2007 = d + 2008. The greatest among a, b,c,d is [AMTI 2007] (A) a (B) b (C) c (D) d

16.

A student got x marks in a test. The student who got the first mark gets 48 more than this student who got x marks. If the total marks of both the students is 110, the highest mark secured is [AMTI 2007] (A) 83 (B) 92 (C) 79 (D) 100

17.

A boy on being asked what

16 16 of a fraction was made the mistake of dividing the fraction by , and got 17 17

an answer which exceeded the correct answer by

(A)

18.

(B)

It is given that

(A) 19.

64 85

65 84

33 . The correct answer is 340 (C)

32 17

[AMTI 2007]

(D) None of these

x 4 = , which one of the following is incorrect y 5

xy 9 = y 5

(B)

y  2x 13 = x 4

(C)

x2  y2 41 = xy 20

[AMTI 2007]

(D)

2x 2  y 2 9 = xy 20

If a,b,c,d are positive integers such that a = bcd, b = cda, c = dab and d = abc, then the value of

(a  b  c  d)4 is (ab  bc  cd  da)2 (A) 4 20.

[AMTI 2007]

(B) 16

(C) 1

(D)

1 2

In a school, there are 5 times as many boys as girls, and 6 times as many girls as teachers. If b,g,t represent the boys, girls and teachers respectively the total number of boys, girls and teachers in the [AMTI 2007] school is (A) 37 b

(B)

37 b 30

(C) 30 g

(D) 37g

21.

Consider the equation x2 + y2 = 2007. Given x is a real number and y is a natural number, The number of [AMTI 2007] solutions of the equation is (A) 0 (B) 2006 (C) 88 (D) 44

22.

The number of ordered triples (a, b, c) 1  a,b,c  9 such that ac = b2 – 1 is (A) 9 (B) 7 (C) 14 (D) 18

23.

x is 1

1 6

th

of 3

(A) 2x = y

rd 3 1 1 and y is 2 of 2 . Then 4 6 3

(B) y < x

[AMTI 2008]

[AMTI 2008] (C) x < y

(D) x = y

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24.

The digital sum of a number is the sum of the digits of the number. A three digit number divisible by 9 has the following property. The digital sum of the quotient of this number divided by 9 is 9 less then the digital [AMTI 2009] sum of the original number. The number of such three digit numbers is. (A) 2 (B) 3 (C) 4 (D) 5

25.

A three digit number ab7 = a3 + b3 + 73. Then a is (A) 6 (B) 4 (C) 7

26.

27.

[AMTI 2010] (D) 8

1 1 1 + = where a,b are natural numbers. a b 13

[AMTI 2010]

(1) a = b = 26 (2) a = 13, b = 13  14 (3) a = 14, b = 13  14 of these statements the correct statements are (A) (1) and (2) (B) (1) and (3) (C) (2) and (3)

(D) (1) (2) and (3)

If A + B = C, B + C = D, D + A = E then A + B + C is (A) E (B) D + E (C) E – D

(D) B – D + C

[AMTI 2010]

28.

Among the participants of this screening test, some of them together got the correct answers for all the problems, not all of them got more than 8 problems correct. The maximum and minimum number of [AMTI 2010] problems solved by the three together are (A) 25, 25 (B) 24, 26 (C) 25, 27 (D) 25, 28

29.

a is real number such that a3 + 4a – 8 = 0. Then the value of a7 + 64a2 is (A) 128 (B) 164 (C) 256

[AMTI 2010] (D) 180

30.

Aruna and baskar wrote the three digit number 888 and Aruna changed two of its digit and wrote the biggest three digit number divisible by 8. Bhaskar too changed two of the digits of 888 and wrote the [AMTI 2011] smallest three number divisible by 8. The difference of the new number obtained is (A) 848 (B) 856 (C) 864 (D) 872

31.

a,b,c are positive reals such that a(b + c) = 32, b(c + a) = 65, and c(a + b) = 77. Then abc = [AMTI 2011] (A) 100 (B) 110 (C) 220 (D) 130

32.

If x +

1 1 = – 1, then the value of x3 – 3 is x x

(A) 0

(B) 1

[AMTI 2012] (C) – 1

(D) 2

33.

The number of natural numbers a (< 100) such that (a3 - a2) is a square of a natural number is [AMTI 2012] (A) 7 (B) 8 (C) 9 (D) 10

34.

If 3a + 1 = 2b – 1 = 5c + 3 = 7d + 1 = 15, then the value of (3a – b + 5c – 9d) is (A) 0 (B) 1 (C) 2 (D) 3

35.

If a + b + c = 0 where a, b, c are non zero real numbers, then the value of

a

2

  2



 bc  b 2  ca c 2  ab is

(A) 1

36.



[AMTI 2012]

(B) abc

[AMTI 2013] (C) a2 + b2 + c2

(D) 0

There are four non-zero numbers x,y,z and u. If x = y –z, y = z–u, z = u –x, then the value of is equal to (A) 1

x y z u    y z u x [AMTI 2013]

(B)

1 2

(C) 0

(D) –

1 2

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37.

If Mahadevan gets 71 in his next examination, his average will be 83. If he gets 99, his average will be 87. How many exams Mahadevan has already taken ? [AMTI 2014] (A) 3 (B) 4 (C) 5 (D) 6

38.

If 3a + 3b = 756, 7a + 2c = 375 and 5a + 3 = 128, then the value of a + b + c is (A) 12 (B) 14 (C) 18 (D) 20

[AMTI 2015]

EXERCISE–II SUBJECTIVE 1.

2. 3.

Anu, babu and Chitra have 51 balls all together. Babu gives 7 balls to Chitra and Chitra gives 5 balls to Anu and Anu gives 4 balls to Babu. If all the three have finally equal number of balls. Then the number of balls Anu had at the start is _________ [AMTI 2008]

1 2 3 n + 2 + 3 + ......n , on complete simplification has the denominator ___________ 2 3 4 n 1 [AMTI 2010] a = 1 + 3 + 5 + 7 + .......+ 2009 [AMTI 2010] b = 2 + 4 + 6 + 8 + ......+ 2010 then the value of (a – b)2 is _________ 1

4.

The number of terms in the expansion (a + b + c)3 is ________

5.

After the school final exams are over, all students in a class exchange their photographs. Each student gives his photograph to each of the remaining students and gets the photograph of all his friends. There [AMTI 2010] are totally 870 exchanges to photos. The numbers of students in the class is ____.

6.

a,b,c are the digits of a 9 digit number abcabcabc. The quotient when this number is divided by 1001001 [AMTI 2011] __________

7.

If a2 – b2 = 2011 where a, b are integers, then the most negative value of (a + b) is ________ [AMTI 2011] x x and y are real numbers such that xy = x + y = (x  0), (y  0). Then the numaral value of (x – y) is y [AMTI 2011] ______

8.

[AMTI 2010]

9.

a,b,c are three positive numbers. The second number is greater than the first by the amount the third number is greater than the second. The product of the two smaller numbers is 85 and that of the two bigger numbers if 115. Then the value of (2012a – 1006c) is __________ [AMTI 2012]

10.

x=

11.

[AMTI 2012] The number of multiples of 9 less than 2012 and having sum of the digits as 18 is _______ [AMTI 2012]

y x a2 y and y = the value of x(y + 2) + + when a = 2012 is ___________ y 1 y 2 x

12.

A class contains three girls and four boys. Every saturday, five students go on a picnic, a different group is sent each wek. During the picnic, each person (boy or girl) is given a cake by the accompanying teacher. After all possible groups of five have gone once ; the total number of cakes received by the girls during the picnic is ________ [AMTI 2012]

14.

There are three persons Samrud, Saket and Vishwa. Samrud is twice the age of Saket and Saket is twice the age of VIshwa. Their total ages will be trebled in 28 years. The present age of Samrud is ___________. [AMTI 2013]

15.

Five year ago, the average age of A, B, C and D is 45 years. E joins them now. The average age of all the [AMTI 2014] five now is 49 years. The present age of E is __________.

16.

The value of x which satisfies the equation

5 5

6 6

= 1 is _______

[AMTI 2014]

5 6x

17.

The age of a father is three times that of his son 10 years after the father age is twice that of his son. The father age will be 60 after __________ years. [AMTI 2014]

18.

The number of (x, y, z) which satisfies the equations xy = 6, yz = 15, zx = 10 simultaneouly is ______ [AMTI 2014]

19.

If

a b a b

=

p a 2  ab  b 2 1 . The value of 2 2 in the form q is _______. 2 a  ab  b

[AMTI 2014]

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GEOMETRY EXERCISE–I OBJECTIVE 1.

In the adjoining figure, ABC is right angled at A ; mB = mC. The bisectors of angles B and C meet at I. Then m BIC is [AMTI 2004] A (A) 135° I C (B) 115° (C) 100°

B

(D) 90° 2.

The number of isosceles triangles in which one angle is 4 times another angle is (A) 2 (B) 1 (C) Infinitely many (D) 4

3.

Nine dots are arranged such that they are equally spaced horizontally and vertically as in the figure. The number of triangles which are not right angled triangle that can be formed with the above dots as vertices is [AMTI 2004]

(A) 18

(B) 21

(C) 40

[AMTI 2004]

(D) 32

4.

In the triangle ABC, D is a point on the line segment BC such that AD = BD = CD. The measure of angle BAC is [AMTI 2004] (A) 60° (B) 75° (C) 90° (D) 120°

5.

If all the diagonals of a regular hexagon are drawn, the number of points of intersection, not counting the corners of the hexagon is [AMTI 2004] (A) 6 (B) 13 (C) 7 (D) 12

6.

The area of the shaded region in the diagram is

(A) 9

(B) 3 2

[AMTI 2004]

(B) 18

(D) 6 3 – 3 2

7.

ABCD is a square and an equilateral triangle CDE is drawn inward. Then (A) E lies on AB and AEB is a straight angle (B) E lies is the interior of the square and AEB = 150° (C) E lies is the interior of the square and AEB = 120° (D) E lies outside the square and AEB = 150°

[AMTI 2005]

8.

In the adjacent figure BA and BC are produced to meet CD and AD produced in E and F. Then AED + CFD is [AMTI 2006] B 50° A 90° C E

D

F

(A) 80°

(B) 50°

(C) 40°

(D) 160°

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[18]

9.

In a triangle ABC, BE is the angle bisector of ABC, where E is lies on AC. EF is the angle bisector of BEC, where F lies on BC. Also EF = EC. Then, [AMTI 2006] A

E

(A) AB = BC 10.

C

F

B

(C) ABC=ACB= 72° (D) BAC = 72°

(B) AC = BC

In an isosceles triangle ABC, AC = BC, BAC is bisected by AD where D lies on BC. It is found that AD = AB then ACB equals. [AMTI 2006] C

D

B

A

(A) 72° 11.

(B) 54°

(C) 36°

(D) None of these

ABC is right angled triangle with BAC = 90°. AH is drawn perpendicular to BC where H lies on BC. If AB = 60 and AC = 80, then BH = [AMTI 2006] A

80 60

B

(A) 36 12.

(B) 32

C

H

(C) 24

(D) 30

In a plane 3 line and a circle are given. If points of intersection of two lines or that of a line with the circle are counted, the maximum numbers of points of intersection possible in this. (A) 12

13.

(B) 9

(C) 6

[AMTI 2006]

(D) 5

In an isosceles acute angled triangle one angle is 50°. I The other two angles are 65° and 65°. II The other two angles are 50° and 80° III. The other two angles can be anything. Then which of the following statemnents [AMTI 2006]

is true. (A) I only 14.

(B) II only

(C) I and II only

(D) III only

In an isosceles triangle the equal sides are 7 units each and the length of the base is an integer. From [AMTI 2006]

these a triangle with the greatest perimeter is selected. Its perimeter is (A) 23

(B) 25

(C) 27

(D) 29

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15.

Two rectangles ABCD and DBEF are as shown in the figure. The area of rectangle DBEF in square units is [AMTI 2006] F D 3 A

(A) 10 16.

C 4

E

B

(B) 12

(C) 14

(D) 15

In the adjoing figure A = 60°, C = 50°. BDG = 30°,, GEF = 20°. Then

[AMTI 2007]

A F 60° 20°

G

E 50°

30° D

(A) EG = 2FG 17.

18.

B

C

(B) EG > FG

(C) EG = FG

(D) EG < FG

The lengths of the altitudes of a triangle are in the ratio 1 : 2 : 3 then

[AMTI 2007]

(A) one angle of the triangle must be 60°

(B) the triangle is a right angled triangle

(C) the triangle is an obtuse angled triangle

(D) such a triangle does not exist

In a triangle if one angle is greater than the sum of the other two angles then the triangle is [AMTI 2007] (A) Acute angled

19.

(B) Right angled

(C) Obtuse angled

(D) Equilateral

Three identical rectangles are overlapping as in the diagram. The length and breadth of each rectangle are respectively 2007 cm and 10 cm. The area of each of the shaded square portion is 16 cm2. [AMTI 2007] 2007 cm 10 cm

The perimeter of the outer boundary of the figure in cm is (A) 10070 (B) 12070 (C) 14070 20.

(D) 11070

OP, OQ, OR are rays OS bisects ROP, POQ = 62°, ROQ = 102° Then SOQ =

[AMTI 2007]

S R

Q

O

(A) 20° 21.

(B) 15°

P

(C) 25°

(D) 10°

The perimeter of a right angled triangle is 144 cm and the hypotenuse is 65 cm. Its area in square cm is [AMTI 2007] (A) 506 (B) 508 (C) 1440 (D) 504

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22.

Three circles C1, C2, C3 with radii r1, r2, r3 where r1 < r2 < r3 are placed as in the figure. r3 L C3

r2 M C2

[AMTI 2007]

r1 C1

Then r2 = (A) 23.

r3  r1

(B)

r3  r1

r32  r12

(C)

(D)

r3r1

In the adjoining figure l1 and l2 are parallel lines. t is a transversal which cuts l1, l2 at A,B respectively. The angles at A, B (refer figure) are trisected. The measure of the angles ACB and ADB are respectively. [AMTI 2008] t x x x

A C

D B

(A) 60° , 120°

l1

y y

l2

y

(B) 120°, 60°

(C) x – y, x + y

(D) 2(x – y), 2(x + y)

24.

ABC is an isosceles triangle with AB = AC = 2008 cm. ADC is drawn as an equilateral triangle on AC outsie ABC. AD is parallel to BC. The bisector of D meets AB in E, say. Then BE is equal to [AMTI 2008] (A) 1004 cm (B) 2008 cm (C) 0 (D) 502 cm

25.

In the adjoning figure ABC, DEF are equilaeral triangles. AB = 8 cm ad DE = 3 cm. Then the possible value of AE + BD + CF is [AMTI 2008] A

E F

D

C

B

(A) 6.9 cm 26.

(B) 7.1 cm

(C) 5.2 cm

(D) 8.3 cm

On line segment AB = 2008 cm a square and a regular hexagon are drawn as shown in the diagram. The [AMTI 2008] distance between their centres P, Q, in cm is

A Q

R

P

B (A) 1004 3 27.

(B) 1004 ( 3 +1)

(C) 2008 3

(D) 1004 ( 3 – 1)

The measure of one of the angle of a right triangle is five times that of a second. angle. Then the possibility of the second largest angle is [AMTI 2008] (A) 72° (B) 75° (C) 72° or 75° (D) none of these

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28.

A rectangular sheet of paper is folded so that the corners A, B go to A’ B’ as in the figure. Then ZXY is

(A) an acute angle (C) a right angle 29.

(B) an obtuse angle [AMTI 2008] (D) a variable angle depending on the point X

ABCD is a square and BCE is an equilateral triangle constructed externally. The measure of AED is. [AMTI 2009]

B

A

E D (A) 30° 30.

C

(B) 20°

(C) 35°

(D) 15°

In the figure AB = AC = CD and BAC = 32° then BAD is

[AMTI 2009]

A

B (A) 37° 31.

(B) 64°

D

C (C) 69°

(D) 74°

A square and a triangle overlap as shown here. The shaded area is two third the area of the square and 40% of the area of the triangle. The ratio of the area of the square to the area of the triangle is

(A) 5 : 3

(B) 3 : 5

(C) 4 : 15

(D) 7 : 15

[AMTI 2009]

32.

Two regular polygons have the number of sides in the ratio 3 : 2 and the interior angles in the ratio [AMTI 2010] 10 : 9 in that order. The number of sides of the polygons are respectively. (A) 6, 4 (B) 9, 6 (C) 12, 8 (D) 15, 10

33.

In the adjoining rangoli design each of the four sided figures is a rhombus and the distance between any two dots is 1 unit. The total area of the design is [AMTI 2010]

(A) 36 3

(B) 9 3

(C) 24 3

(D) 18 3

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34.

In the adjoining figure, ABCD is a square of side 4 units. Semicircles are drawn outside the squares with diameter 2 units as shown. The area of the shaded portion in square units is [AMTI 2010]

(A) 8 35.

A

B

D

C

(B) 16

(C) 16 – 2

(D) 8 - 

In the figure, PQRS is a rectangle of area 2011 square units. K, L, M, N are the mid points of the respective sides. O is the mid point of MN. [AMTI 2011] The area of the triangle OKL is equal to (in square units) N

S K

M

P

(A)

2011 5

(B)

R O

Q

L

2(2011) 5

(C)

2011 4

(D)

3(2011) 8

36.

Two squares 17 cm  17 cm overlap to form a rectangle 17 cm  30 cm. The area of the overlapping region [AMTI 2011] is (A) 289 (B) 68 (C) 510 (D) 85

37.

A point P inside a rectangle ABCD is jointed to the angular points. Then [AMTI 2012] (A) Sum of the areas of two of the triangles so formed is equal to the sum of the other two. (B) The sum of the area of the triangles so formed is a whole number whatever may be the dimensions of the rectangle (C) The sum of the areas of a pair of opposite triangles is greater than half the area of the rectangle. (D) None of these

38.

In the adjoning figure ABCD and BGFE are rhombus. AB = 10 cm, GF = 3 cm. GE meets DC at H. A = 60° [AMTI 2012] The perimeter ABEHD is ( in cm) D

H

C

E

A

(A) 47 39.

B

G

(C) 39

(D) 33

PQRS is a parallelogram. MP and NP divide SPQ into three equal parts (MPQ > NPQ) and MQ and NQ divide RQP into 3 equal parts (MQP > NQP). If k(PNQ) = (PMQ) then k = [AMTI 2012] (A)

40.

(B) 40

F

1 2

(B) 1

(C)

3 2

(D)

1 3

The perimeter of an isosceles right angled triangle is 2012. Its area is (A) 2012(3 –

2)

(B) (1006)2(3– 2 )

(C) (2012)2

[AMTI 2012] (D) (1006)2

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41.

In the adjoining figure, all the three semicircles have equal radius of 1 unit. The area of the shaded portion [AMTI 2013] is

3 +1 (D) 4 2 The radius of a circle is increased by 4 units and the ratio of the areas of the original and the increased [AMTI 2013] circle is 4 : 9. The radius of the original circle is (A) 6 (B) 4 (C) 12 (D) 8 (A)  + 2

42.

(B) 5

(C)

43.

The length of two sides of an isosceles triangle are 5 units and 16 units. The perimeter of the triangle (in [AMTI 2014] the same unit) is (A) 26 (B) 37 (C) 26 or 37 (D) none of these

44.

ABCD is a rectangle. P is the mid-point of DC and Q is a point on AB such that AQ = of the area of ABCD to AQPD.

1 AB. What is ratio 3 [AMTI 2014]

1 3 2 5 (B) (C) (D) 2 4 7 12 Three equal square are kept as in the diagram. C, D being the mid point of the respective sides of the lower squares. If AB = 100 cm, Area of each square is (in cm2) [AMTI 2014] (A)

45.

B

D

C

A

(A) 1200 46.

(B) 1500

(C) 900

(D) 1600

In the adjoining figure, AB, CD EF and GH are straight lines passing through a single point. The value of x + y + z + v = [AMTI 2014] A

37°

C

E x G

v

y 85° 55°

z

H

F B

D

(A) 155° 47.

(B) 164°

(C) 174°

(D) 148°

In the adjoining diagram AB = AD DCB = 23°. The measure of DBC is.

[AMTI 2014]

A 44° D

B

(A) 55°

(B) 58°

23°

(C) 56°

C

(D) 45°

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48.

In the adjoining figure ABCD is a square. BCE is an equilateral triangle. The measure of BEA is [AMTI 2014] (A) 15° (B) 20° (C) 18° (D) 16°

49.

The ratio of the angles of a quadrilateral are 7 : 9 : 10 : 10. Then (A) One angle of the quadrilateral is greater than 120° (B) Only one angle of the quadrilateral is 90° (C) The sum of some two angles of the quadrilateral is 100° (D) There are exactly two right angles as interior angles.

50.

ABCD is a square of area 64 cm2. The centre square has are 16cm2. The remaining are four congruent [AMTI 2015] rectangles. The ratio of the length to breadth of a rectangle is A

[AMTI 2015]

B

16

D

(A) 2

(B) 3

C

(C) 4

(D) 5

EXERCISE–2 SUBJECTIVE 1.

ABC is a right angled triangle with B = 90°. BDEF is square. BE is perpendicular to AC. The measure of DEC is ________ [AMTI 2010]

2.

There are 2009 equispaced points on a circle. The number of isosceles triangles with 2 of their vertices at [AMTI 2009] these points and the third vertex at the centre of the circle is _________

3.

In the diagram APB and DPC have a common vertex P. XPY is the common bisector of the angles APB and DPC. If APB = 150°, CPD = 40° then BPD is ________. [AMTI 2009]

4.

Two circles of different radii touch externally. APQB is the line of diameter as shown in the diagram. AD, BC and DRC are tangents so that the area of the rectangle ABCD is 16 cm2. The area of the triangle PQR [AMTI 2009] is ________ cm2

A

P

O

B

C R Two of the altitudes of the scalene triangle ABC have lngths 4 and 12. If the length of the third altitude is also an integer, then its greatest value can be _______ [AMTI 2009] D

5. 6.

From a point within an equilateral triangle perpendiculars are drawn to the three sides and are 5, 7 and 9 cms in length. The perimeter of the triangle is _______ cm. [AMTI 2010]

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7.

ABCD is a rectangle rotated clockwise about A by 90° as shown. The rotation takes B to B’, C to C’, D to [AMTI 2010] D’ AB = 6 cm, BC’ = 10 cm. The breadth of the rectangle ABCD is ______

D'

A

B

C

D 8.

B' C' AB is a line segment 2000 cm long. The following design of semicircles is drawn on AB, with AP = 5 cm and repeating the design. The area enclosed by the semicircular designs from A to B is _____ [AMTI 2010] A

B

9.

The perimeter of a right angled triangle is 132. The sum of the square of all its sides is 6050. The sum of [AMTI 2011] the legs of the triangle is _______.

10.

All sides of the convex pentagon ABCDE are equal in length. A = B = 90° . Then E is equal to __ [AMTI 2011]

11.

The angles of a polygon are in the ratio 2 : 4 : 5 : 6 : 6 : 7. The difference between the greatest and least [AMTI 2011] angle of the polygon is _________

12.

ABCD is a trapezium with AB and CD parallel. If AB = 16 cm BC = 17 cm, CD = 8 cm, DA = 15 cm then [AMTI 2012] the area of the trapezium (in cm2) is _________

13.

A square is inscribed in another square each of whose 4 vertices lies on each side of the square. The area

25 times the area of the bigger one. Then the ratio with which each vertex of the 49 [AMTI 2012] smaller square divides the side of the bigger suqrae is ___________

of the smaller square is

14.

PSR is an isosceles triangle in which PS = PR. SP is produced to O such that PO = SP. Then SRO is [AMTI 2012] equal to ___________

15.

CAB is an angle whose measure is 70°. ACFG and ABDE are squares drawn outside the angle. The [AMTI 2012] diagonal FA meets BE at H. Then the measure of the EAH is ________

16.

AB and AC are two straight line segments enclosing an angle 70º . Squares ABDE and ACFG are drawn outside the angle BAC. The diagonal FA is produced to meet the diagonal EB in H. then EAH = _________. [AMTI 2013] In the pentagon ABCDE, D = 2B and the angles A, C, E are each equal to half the sum of the angles [AMTI 2013] B and D. The maximum interior angle of the pentagon is _________.

17.

18.

A triangle whose sides are integers has a perimeter 8. The area of the triangle is _________. [AMTI 2013]

19.

ABC and ADC are isosceles triangles with AB=AC = AD. BAC = 40º, CAD = 70º. The value of [AMTI 2013] BCD + BDC = ___________.

20.

The sum of the square of the length of the three sides of a right triangle is 800. The length of the [AMTI 2014] hypotenuse is ____________.

21.

In the adjoining figure ABCD is a rectangle. E is the midpoint of AD. F is the midpoint of EC. The area of [AMTI 2014] the rectangle ABCD is 120 cm2. A

E

D

F B

C

The area of the triangle BDF is ___________ cm2

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22.

In the adjoining figure, ABCD is a rectangle. AD = 2, AB = 1, AE is the arc of the circle centred D. The length BE is equal to ______________ [AMTI 2015] B

A

E

D

C

COMMERCIAL MATHS EXERCISE-1 OBJECTIVE 1.

Some donkeys and some chickens totally 20 in all were seen in a garden. I counted 64 legs in all. How many donekeys were here [AMTI 2005] (A) 12 (B) 10 (C) 14 (D) 17

2.

Nine numbers are written in ascending order. The middle number is also the average of the nine numbers. The average of the 5 larger numbers is 68 and the average of the 5 smaller numbers is 44. The sum of all the numbers is [AMTI 2006] (A) 540 (B) 450 (C) 504 (D) 501

3.

In a box there are some coins and rings which are made of either gold or silver. 60% of the objects are coins, 40% of the rings are gold and 30% of the coins are silver. The precentage of gold articles is [AMTI 2006] (A) 24% (B) 16% (C) 42% (D) 58%

4.

If the average of 20 different positive integers is 20 then the greatest possible number among these 20 numbers can be [AMTI 2006] (A) 210 (B) 200 (C) 190 (D) 180

5.

20% of 50% is what percent of 25% of 40% ? (A) 80% (B) 60%

6.

7.

[AMTI 2007] (C) 65%

(D) 100%

x,y,z are three sums of money such that y is the simple interest on x and z is the simple interest on y for the same time and rate then [AMTI 2007] (A) x2 = yz (B) y2 = zx (C) z2 = xy (D) None of these

a A man gets   b

th

th

b of Rs. 100 and   of Rs. 100 again. he gives away Rs. 200. Then the man a [AMTI 2007]

(A) loses by the transaction (B) will not lose by the transaction (C) loss or gain depends on whether a > b or a < b respectively (D) loss or gain depends on whether a < b or a > b respectively. 8.

On the sale of 10 m cloth, a gain equal to the selling price of 2m cloth was obtained. The gain is [AMTI 2007] (A) 20% (B) 10% (C) 25% (D) 40%

9.

The value of a commodity is increased by x% first and again increased by x%. The total increase is [AMTI 2008] (A) 2x%

(B) 4x%

 x 2   % 2 x  (C)  100  

(D) x2%

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10.

The average of ten different positive integers is 10. The smallest is 5. The biggest of these numbers can be [AMTI 2009] (A) 55 (B) 49 (C) 25 (D) 19

11.

The percentage of natural numbers from 10 to 99 both inclusive which are the product of consecutive [AMTI 2010] natural numbers is

7 7 (B) 7 (C) 10 (D) 9 9 9 Aruna, Bhanu and Rita have some amount of money. The ratio of the money of Aruna to that of Bhanu is 7 : 15 and the ratio of the money of Bhanu and Rita is 7 : 16. If Aruna has Rs. 490, the amounts of money Rita has is (in Rupees) [AMTI 2013] (A) 1500 (B) 1600 (C) 2400 (D) 3600 A sum of money is divided between two persons in the ratio 3 : 5. If the share of one person is Rs. 2000 [AMTI 2013] more than that of the other, then the sum of money is (in rupees) (A) 6000 (B) 8000 (C) 10,000 (D) 12,000 (A) 9

12.

13.

14.

Two numbers are respectively 20% and 50% more than a third number. What percent the first number is with respect to the second number? [AMTI 2013] (A) 70% (B) 30% (C) 80% (D) 60%

15.

There are some toys. One third of them are sold at a profit of 15%, one fourth of the total are sold at a profit of 20% and the rest for 24% profit. The total profit is Rs. 3200. The total price to the toys is (in [AMTI 2013] rupees) (A) 32000 (B) 64000 (C) 16000 (D) 48000

16.

If 150% of a certain number is 300, then 30% of the number is (A) 50 (B) 60 (C) 70

[AMTI 2014] (D) 65

17.

The ratio of two natural numbers is 7 : 9. If each number is decreased by 2, the ratio becomes 3 : 4. The sum of the two numbers is [AMTI 2015] (A) 23 (B) 32 (C) 48 (D) 12

18.

The speed of two runner are respectively 15 km/hr and 16 km/hr. To cover a distance d km one takes 16 [AMTI 2015] minutes more that the other. Then d = (in kilometres) (A) 32 (B) 48 (C) 64 (D) 128

19.

a% of the quantity P is added to P. To the increased quantity b% of the increased quantity is added. C% [AMTI 2015] of the result is added to the result and the final quantity is Q. Then P is (A)

Q  100  100  100 (a  b  c )

(B)

Q 100a  100b  100c

(C)

Q  100  100  100 (100  a)  (100  b)  (100  c )

(D)

Q  100  100  100 (100  a)  (100  b)  (100  c )

20.

A student has to score 30% marks to get through in an examination. If he gets 30 marks and fails by 30 [AMTI 2015] marks the maximum marks set for the examination is (A) 90 (B) 200 (C) 250 (D) 125

21.

a,b,c,d are real numbers such that 1015  a  2015, 3015  b  4015, 5015  c  6015 and 7015  d  8015. The maximum value of

cd is ab

[AMTI 2015]

1403 1402 1401 (B) (C) (D) 2015 403 403 403 A black and white photograph is 70% black and 30% white. It is enlarged three times. The percentage of [AMTI 2015] white in the enlargment is (A)

22.

(A) 90%

(B) 66

2 % 3

(C) 33

1 % 2

(D) 30%

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EXERCISE-II SUBJECTIVE 1.

When a barrel is 40% empty it contains 80 litres more than when it is 20% full. The capacity of the barrel (in litres) is ______. [AMTI 2011]

2.

There is a famine in a place. But there is sufficient food for 400 people for 31 days. After 28 days 280 of them left the place. Assuming that all of them consume the same amount of food, the number of [AMTI 2013] days for which the rest of the food would last for the remaining people is __________.

3.

The daily wages of two person are in the ratio 3 : 5. They work in a place and the employer is satisfied with their work and gives Rs. 20 more to each. Then the ratio of their wages comes to 13 : 21. The [AMTI 2014] sum of the original wages of the two persons is ________

4.

Samrud got an average mark 85 in his first 8 tests and an average 81 for the first 9 tests. His mark in the 9th test is _________ [AMTI 2015]

MESLLANIOUS EXERCISE–I OBJECTIVE 1. 2.

The image of INMO when reflected in a mirror is (A) I WO (B) OMNI

[AMTI 2004] (C) INWO

(D) OWNI

In the given addition sum, H, E, O are different digits. The value of the letter O is

[AMTI 2005]

HE HE HE HE OH (A) 1 3.

(B) 8

(C) 9

(D) 4

A cube of edge 4 cms is painted externally. It is then cut into one cm cubes. How many of these do not have red paint on any face ? [AMTI 2005]

(A) 4

(B) 8

(C) 56

(D) 16

4.

How often the hands (hour hand and minute hand) of a clock are in a straight line every day ? [AMTI 2007] (A) 2 (B) 4 (C) 16 (D) 44

5.

A leap year started on a monday. The days appearing 53 times are (A) Sunday and Monday (B) Monday and Tuesday (C) Tuesday and Wednesday (D) Saturday and Sunday

[AMTI 2008]

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6.

In a certain month in the new academic year, Sunday are on even dates in that month. Which day occurs on three even dates of the following month. [AMTI 2009] (A) Thursday or Friday (B) Wednesday or Thursday (C) Tuesday or Wednesday (D) Monday or Sunday

7.

In a football league, a particular team played 50 games in a season. The team never lost three games consecutively and never won four games concecutively, in that season. If N is the number of games the [AMTI 2012] team won in that season, then N satisfies (A) 25  N  30 (B) 25  N  36 (C) 16  N  38 (D) 16  N  30

8.

The years of 20th century and 21 century are of 4 digits. The number of year which are divisible by the product of the four digits of the year is [AMTI 2012] (A) 7 (B) 8 (C) 9 (D) none of these

9.

There are four types of dolls called Dingle (D), Pingle (P), Jingle(J) and Mungle (M) .All toys of same category have same weight. No two toys of different category have some weight. They balance as [AMTI 2015] shown. How many Jingles will balance one Mungle ? D D

(A) 2

P P P

D

(B) 3

M

P

(C) 4

M

(D) 5

EXERCISE–II SUBJECTIVE 1.

Observe the above diagram formed by unit squares. Not including the middle square hole, the number [AMTI 2009] of unit squares needed to build the tenth pattern is ________

2.

Aruna, Bhavani and Christina wear a saree of different colours (blue, yellow or green). it is known that, if Aruna wears blue then Bhavani wears green, if Aruna wears yellow Christina wears green and if Bhavani does not wear yellow, Christina wears blue. The saree Aruna is wearing has the colour _____. [AMTI 2012]

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ANSWER KEY NUMBER SYSTEM EXERC ISE-1

1. 8. 15. 22. 29. 36. 43. 50. 57. 64. 71. 78. 85.

C C A C C A D C B D D B A

2. 9. 16. 23. 30. 37. 44. 51. 58. 65. 72. 79. 86.

D B A D D C A D B C C D C

3. 10. 17. 24. 31. 38. 45. 52. 59. 66. 73. 80. 87.

D B B B C A D C D D C A C

4. 11. 18. 25. 32. 39. 46. 53. 60. 67. 74. 81. 88.

A B A A D B C B C C B B C

5. 12. 19. 26. 33. 40. 47. 54. 61. 68. 75. 82. 89.

C B B A B C C A B B D D C

6. 13. 20. 27. 34. 41. 48. 55. 62. 69. 76. 83. 90.

C C A A D B D D C D D C C

7. 14. 21. 28. 35. 42. 49. 56. 63. 70. 77. 84. 91.

C B C B A B B D A B C C C

2. 9. 16. 23. 30.

–2 1089 41 4030 381

3. 10. 17. 24. 31.

2 588 0 10 29

4. 11. 18. 25.

10 1 3 288

5. 12. 19. 26.

3 10 3 6

6. 13. 20. 27.

7 2q2 3 3

7. 16 14. 12 21. 8 28. 12

5.

D

6.

B

7.

C

3.

1007

4.

1008

7. 14. 21. 28. 35.

C B C D D

EXERC ISE-2

1. 8. 15. 22. 29.

2 81 360 36 2

PATTERN DETECTION EXERC ISE-1

1. 8.

C C

2. 9.

D C

3. 10.

A A

4.

2.

3, 6, 10,........, 2023066

D

EXERC ISE-2

1.

*

ALGE BR A EXERC ISE-1

1. 8. 15. 22. 29. 36.

D C C D A B

2. 9. 16. 23. 30. 37.

C A C C B D

3. 10. 17. 24. 31. 38.

D D A D B B

4. 11. 18. 25. 32.

C B C B A

5. 12. 19. 26. 33.

B B B B C

6. 13. 20. 27. 34.

A D B A A

EXERC ISE-2

1.

16

2.

1

3.

10052

4.

10

5.

30

6.

100a + 10b + c

7.

–2011

8.

3 2

9.

5533

10.

2012

11.

109

12.

45

14.

24

15.

45

16.

1

17.

30

18.

1

19.

91 73

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GEOMETRY EXERC ISE-1

1. 8. 15. 22. 29. 36. 43. 50.

A C B D A B B B

2. 9. 16. 23. 30. 37. 44.

A C C A C A D

3. 10. 17. 24. 31. 38. 45.

D C D C B D D

4. 11. 18. 25. 32. 39. 46.

C A C D C A C

5. 12. 19. 26. 33. 40. 47.

4.

B B B B B B D

6. 13. 20. 27. 34. 41. 48.

A C A C C C A

7. 14. 21. 28. 35. 42. 49.

B C D C C D B

4

5.

5

6. 126/ 3

EXERC ISE-2

1.

45°

2.

2017036

3.

125

7.

2

8.

1250 

9.

77 units 10.

150°

11.

120°

12. 180

13.

4 3

14.

90°

15.

25°

16.

25°

17.

144

18. 2 2

19.

145

20.

20

21.

15

22.

2– 3

D C C

6. 13. 20.

B B B

7. 14. 21.

B C A

B

6.

C

7.

C

COMMERCIAL MATHS EXERC ISE-1

1. 8. 15. 22.

A C C C

2. 9. 16.

C C B

3. 10. 17.

D D B

4. 11. 18.

A B C

5. 12. 19.

2.

10

3.

640

4.

49

EXERC ISE-2

1.

200 litres

MISCELLANEOUS EXERC ISE-1

1. 8.

A D

2. 9.

C D

2.

Green

3.

B

4.

D

5.

EXERC ISE-2

1.

92

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NMTC QUESTION BANK JUNIOR GROUP (STAGE –1)

VIBRANT ACADEMY (India) Private Limited A-14(A), Road No.1, Indraprastha Industrial Area, Kota-324005 (Raj.) Tel.:(0744) 2423406, 2428666, 2428664, 2425407 Fax: 2423405 Email: [email protected]

Website : www.vibrantacademy.com

NUMBER SYSTEM EXERCISE –I O B J E C T IV E

1.

2.

The number of two digit numbers divisible by the product of the digit is : (A) 5 (B) 8 (C) 14

[AMTI 2004] (D) 33

In this addition each letter represents a different digit. Which is the absent digit ?

ABCD +BCD G H IJK (A) 1

[AMTI 2004] (B) 3

(C) 4

(D) 5

3.

A number with 8 digits is a multiple of 73 and also a multiple of 137. The second digit from the left equals 7. Then the 6th digit from the left equals : [AMTI 2004] (A) 1 (B) 7 (C) 9 (D) can be any digit

4.

Left n be the least positive integer such that 1260n is the cube of a natural number. Then n satisfies : [AMTI 2004] (A) 1 < n < 50 (B) 50 < n < 100 (C) 100 < n < 1000 (D) 1000 < n < 10000

5.

If (43)x in base x number system is equal to (34)y in base y number system the possible value for x + y is [AMTI 2004] : (A) 16 (B) 14 (C) 12 (D) 10

6.

In each of the following 2003 fractions the sum of the numerator and denominator equals 2004 : 2 3 2003   1  2003 , 2002 , 2001 ,...., 1  . The number of fractions < 1 which are irreducible (no common factor  

between numerator and denominator) is (A) 664 (B) 332 7.

For how many integers n is (A) 3

8.

[AMTI 2004] (C) 1002

(D) 1001

9 – (n  2)2 a real number ?

(B) 5

(C) 7

[AMTI 2004] (D) infinitely many

Let [x] denote the greatest integer less than or equal to x, what is the value of [ 1 ] + [ 2 ] + [ 3 ] + ..... + [ 2004 ] ? (A) 58850 (B) 59730

[AMTI 2004] (C) 59950

(D) 56718

9.

How many solutions are there for (a,b) if 7ab 73 is a five digit number divisible by 99 ? (A) 3 (B) 2 (C) 0 (D) 1

10.

The number 10790 –7690 is divisible by : (A) 61 (B) 62

(C) 64

[AMTI 2004]

[AMTI 2004] (D) none of these

11.

(22, 48), (61, 76), (29, 34) are some pairs of distinct two digit numbers whose product ends with digit 6. [AMTI 2005] How many such pairs are there ? (A) 477 (B) 315 (C) 549 (D) 405

12.

The digits 1,2,3,4 are used to generate 256 different 4-digit numbers. The sum of the 256 numbers is: [AMTI 2005] (A) 71440 (B) 711040 (C) 704110 (D) 741040

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13.

Number with two digits or more in which the digits reading from left to right occur in strictly increasing order are called as “sorted numbers”. For example 125, 14 and 239 are sorted numbers while 255, 74 and 198 are not. Suppose that a complete list of sorted numbers is prepared and written in increasing order, the 100th number on this list is : [AMTI 2005] (A) 389 (B) 356 (C) 345 (D) 258

14.

The last two digit of (2006)2005 (A) 16 (B) 36

[AMTI 2005] (C) 76

(D) 96

15.

The number of prime numbers less than 100 which can be expressed as the sum of the squares of two natural numbers is : [AMTI 2005] (A) 11 (B) 12 (C) 16 (D) 20

16.

If

97 =w+ 19

1 1 x y

(A) 16

where w,x,y are integers then w + x + y equals :

(B) 17

(C) 18

[AMTI 2005]

(D) 19

17.

How many of these expressions x3 + y4 , x4 + y3, y3 + y3 and x4 – y4 are positive for all possible numbers x and y for which x > y ? [AMTI 2005] (A) 1 (B) 0 (C) 2 (D) 3

18.

The number of ordered pair of digits (A, B) such that A 3640548981270644 B is divisible by 99 is : [AMTI 2005] (A) 3 (B) 2 (C) 1 (D) zero

19.

2

3

Let A = a + a + a + ..... + a (A) positive integer

2006

and a = –1, then

(B) negative integer

A 3 – a2 A2 – a

is a :

(C) negative fraction

[AMTI 2006] (D) positive fraction

20.

How many possible values can one by multiplying 2 different numbers from the set [AMTI 2006] {4,8,9,16,27,32,64,81,243} ? (A) 12 (B) 24 (C) 32 (D) 48

21.

Define a * b = 2a + 2b – ab. For example 4 * 3 = 2 × 4 + 2 × 3 – 4  3 = 2. If 3 * x = 2 * x, then x is : [AMTI 2006] (A) 2 (B) 1 (C) 0 (D) No such x exists

22.

If ABCD = (ABC) × (BD), where A, B, C and D are digit not necessarily distinct, then the value of C is: [AMTI 2006] (A) 0 (B) 1 (C) any digit (D) any nonzero digit

23.

Define a * b = a + b + 1. and if a * 5 = b * 4 = 11 then a * b is (A) 10 (B) 11 (C) 12

[AMTI 2006] (D) 13

24.

a, b and c are three natural numbers. Exactly two of them are odd. Then : (A) a + b + c + ab + bc + ca is odd (B) ab + bc + ca is even (D) a2 + bc is even (C) a3bc is odd

[AMTI 2006]

25.

Given x + 3y = 100 where x and y are positive integers. The number of pairs satisfying the above equation [AMTI 2006] is : (A) 100 (B) 97 (C) 34 (D) 33

26.

A three digit number with digits A,B,C in that order is divisible by 9. A is an odd digit and C is an even digit. [AMTI 2006] B and C are non zero. The number of such three digit numbers is : (A) 20 (B) 16 (C) 8 (D) 4

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27.

The number of prime numbers less than 1 million whose digital sum is 2 is : [AMTI 2006] (A) 5 (B) 4 (C) 3 (D) none of these

28.

The value of

36 24 24 36

of on simplification is :

[AMTI 2006]

1

 3 2 (A)   2

(B)

3 212

(C)

32 23

 3  (D)    32 

12

29.

In Eldorado, the base in the number system is b, unlike our decimal system where 235 means 2. 102 + 3. 10 + 5. The place values in Eldorado are powers of b and the currency is in dinars. Ram gave a 1000 dinar bill for a bicycle costing 440 dinars and receives 340 dinars as change. The base b is : [AMTI 2006] (A) 12 (B) 8 (C) 7 (D) none of these

30.

A certain number leaves a remainder 4 when divided by 6. The remainder when the number is divided by [AMTI 2006] 9 is : (A) 1 or 4 (B) 4 or 7 (C) 1 or 7 (D) 1,4 or 7

31.

Given the alphmetic where each letter represents a different digit the number of distinct solution is :

ABCD –B C D A 1 51 2 (A) 0

[AMTI 2006] (B) 1

(C) 6

(D) 7

32.

The least number of numbers to be deleted from the set {1,2,3,..... , 13,14,15} so that the product of the remaining number is perfect square is : [AMTI 2006] (A) 4 (B) 3 (C) 2 (D) 1

33.

If (43)x in base x number system is equal to (34)y in base-y number system then the value for x and y are [AMTI 2006] respectively : (A) 3,4 (B) 4,3 (C) 7,9 (D) 9,7

34.

An eight digit number is a multiple of 73 and 137. If the second digit from left is 7, what is the 6th digit from [AMTI 2006] the left of the number ? (A) 7 (B) 9 (C) 5 (D) 3

35.

Two natural numbers differ by 41. The bigger number is greater than 30 times the smaller number plus 10. [AMTI 2007] The smaller number is : (A) 11 (B) 2007 (C) 71 (D) 1

36.

1   1   1  1     is : The value of 1 – 2  1 – 2  1 – 2  ...... 1 – 2   3   4  2007 2   

(A)

37.

2008 2007

39.

1004 2007

The number of natural numbers n for which (A) 8

38.

(B)

(B) 2

The units digit of 32007 × 72008 × 132009 is : (A) 3 (B) 1

(C)

2007 2008

[AMTI 2007]

(D) 1

15n2  8n  6 is a natural number is : n (C) 3 (D) 4

[AMTI 2007]

[AMTI 2007] (C) 9

(D) 7

A two digit number is increased by 20% when its digits are reversed. Then the sum of the digits of the [AMTI 2007] number is : (A) 2 (B) 7 (C) 9 (D) 8

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40.

The units digit of (1+9+92+93+.....+92007) is : (A) 1 (B) 9

[AMTI 2007] (C) 0

(D) 8

41.

A boy multiplies 423 by a certain number and obtains 65589 as his answer. If both the fives are wrong but the other digits are correct, the right answer is : [AMTI 2007] (A) 63489 (B) 60489 (C) 62689 (D) 60689

42.

The ratio between a two digit number and the sum of the digits of that number is a : b. If the digit in the units place is n more than the digit in the tens place then the number in the tens place is : [AMTI 2007] (A)

43.

n(a – b ) 11b – 2a

(B)

n(a  b ) 11b – 2a

(C)

2n(a – b ) 11b – 2a

(D)

(a – b) n(11b – 2n)

The digits 1,2,3,4,5 and 6 are written in a spiral like fashion begining from the central marked cell. Which digit is written on the cell placed exactly 100 cells above the marked cell ? [AMTI 2008]

(A) 1

(B) 3

(C) 5

(D) 6

44.

The first digit from the left of a four digit number is equal to the number of zeroes in the number. The second digit is equal to the number of digits 1, the third digit is equal to the number of digits 2 and the [AMTI 2008] fourth digit is equal to the number of digits 3. How many numbers have this property ? (A) 0 (B) 2 (C) 3 (D) 4

45.

Let A be the least number such that 10A is a perfect square and 35 A is perfect cube. Then the number of [AMTI 2008] positive divisors of A is : (A) 72 (B) 64 (C) 45 (D) 80

46.

What is the least possible value of the expression 2008–BHA–SK–ARA if it known that each alphabet represents a different non zero digit ? [AMTI 2008] (A) 106 (B) 108 (C) 1580 (D) none of these

47.

A positive integer ‘n’ is such that its only digits are 3’s. ‘n’ is exactly divisible by 383. When by 1000, what is the remainder ? (A) 351 (B) 781

48.

n is divided 383

[AMTI 2008] (C) 651

The number of 2 digit numbers having exactly 6 factors is : (A) 4 (B) 20 (C) 16

(D) 931 [AMTI 2008] (D) 12

49.

A teacher wrote the numbers 1,2,3, .....2008 on the black board; one of the students erases m of these numbers. Among the numbers remaining on the board, not a single one is the product of any two of the [AMTI 2008] others. If the smallest possible value of m is x then (A) x < 50 (B) 50  x < 200 (C) 200  x < 500 (D) 500  x < 1000

50.

The number of two digit numbers that increase by 75% when their digits are reversed is : [AMTI 2008] (A) 3

(B) 4

(C) 5

(D) 7

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51.

Arun thought of a possible integer. Balu multiplied this by either 5 or 6. Chitra added 5 or 6 to Balu’s number. Devi subtracted either 5 or 6 from the number got by chitra and the final result was. 73. Then the [AMTI 2008] numbers obtained by arun, Balu and Chitra are respectively. (A) 12, 72, 78, (B) 12,60,66 (C) 12,72,77 (D) 12,60,65

52.

a,b are positive integers, If log9 a = log12 b = log16 (a + b) then the value of (A)

4 3

(B)

1 3 2

(C)

8 5

b is a (D)

[AMTI 2008]

1 5 2

53.

A positive integer m is triangular if m = 1 + 2 + 3 + ..... + n for some integer n > 0. If m is triangular, then (8m + 1 ) is : [AMTI 2009] (A) also triangular (B) a perfect square (C) a perfect cube (D) none of the above three

54.

The sum of the digit of a number ‘n’ is 30. Which one of the following cannot be the sum of the digits of the number n + 3 ? [AMTI 2009] (A) 6 (B) 15 (C) 21 (D) 24

55.

A numerical puzzle is constructed with positive integers greater than 1 in each cell as shown in the figure. In the adjacent grid with 6 cells which number cannot be filled in the botton most cell a b axb

(A) 90 56.

58.

59.

ab – cd ab  cd

61.

(C) 153

(B)

bc – ad c d

(C)

ab – cd cd

The number of solutions of the equation x log10 x = 100x is : (A) 0 (B) 1 (C) 2

(D) 210

(D)

ab – cd b–c

[AMTI 2010] (D) 3

Given a and b are integers the expression (a2+a+2011) (2b+1) is : (A) Odd for exactly 2010 values of a.b. (B) Odd for all values of a,b. (C) Even for exactly one value of a and two values of b. (D) Odd for exactly for one value of a and one value of b.

[AMTI 2010]

2 The equation log2x   (log2x)2 + (log2x)4 = 1 has x

(A) A root less than 1. (C) Two irrational roots 60.

(B) 144

The number which when subtracted from the terms of ratio a : b makes it equal to c : d is : [AMTI 2010] (A)

57.

[AMTI 2009]

[AMTI 2010] (B) Has only one root greater than 1 (D) No real

How many distinct rational numbers (a,b,c,d) are there with a.log10 2 + b.log10 3 + c.log10 5 + d.log10 7 = 2011. (A) 0 (B) 1 (C) 5

[AMTI 2011] (D) 2011

The number of positive integral values of n for which (n3 – 8n2 + 20n – 13) is a prime number is : [AMTI 2011] (A) 2 (B) 1 (C) 3 (D) 4

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62.

The number of positive integers ‘n’ for which 3n–4, 4n–5 and 5n – 3 are all primes is : [AMTI 2011] (A) 1

(B) 2

(C) 3

(D) infinite

63.

The number of digits in the sum 100 + 1002 + 1003 + ..... 1002011 is : (A) 4023 (B) 4022 (C) 4024

64.

For how many positive integrals values of x  100 is (3x – x2) divisible by 5 ? (A) 16 (B) 20 (C) 24 (D) 36

65.

a, b are positive integers such that (A) the sum of their square is S (B) the sum of their cubes is C times the sum of the numbers itself. (C) S – C = 28. The numbers of such pairs (a, b) is (A) 1 (B) 2 (C) 3

[AMTI 2011] (D) none of these [AMTI 2012]

[AMTI 2013] (D) 6

66.

The number of numbers of the form 30a0b03 where a, b are digits which are divisible by 13 is [AMTI 2013] (A) 5 (B) 6 (C) 7 (D) 0

67.

The number of ordered triples (x, y, z) such that x, y, z are primes and xy + 1 = z is [AMTI 2013] (A) 0 (B) 1 (C) 2 (D) infinitely many

68.

a, b, c are digits of a 3-digit number such that 64a + 8b + c = 403, then the value of a + b + c + 2013 is [AMTI 2013] (A) 2024 (B) 2025 (C) 2034 (D) 2035

69.

What is the sum of the digits of (9999999999)3 (A) 99 (B) 108

[AMTI 2013] (C) 180

(D) 199

70.

The number of three digits number which are divisible by 3 and have the additional property that the sum of their digits is 4 times their middle digit is [AMTI 2013] (A) 7 (B) 4 (C) 11 (D) 10

71.

Nine numbers are written in asscending order. The middle number is the average of the nine numbers. The average of the five largest number is 68 and the average of the five smallest numbers is 44. The sum of all numbers is [AMTI 2013] (A) 560 (B) 504 (C) 112 (D) 122

72.

N is a five digit number. 1 is written after the 5 digit of N to make it a six digit number, which is three times the same number with 1 written before N. (If N = 23456 it means 234561 and 123456) . Then the middle digit of the number N is [AMTI 2014] (A) 2 (B) 4 (C) 6 (D) 8

73.

The five digit number a679b is a multiple of 72. Then the value of a + b is (A) 3 (B) 5 (C) 6

[AMTI 2014] (D) 7

74.

The number of two digit numbers having the property that when they are divided by the sum of their digits, the quotient is 7 without remainder is [AMTI 2014] (A) 0 (B) 1 (C) 3 (D) 4

75.

The sum of all values of integers n for which (A) 0

76.

(B) 7

n2  9 is also an integer is n 1 (C) 8

The number of values of x which satisfy the equation 5 2.x 8 x –1 = 500 is : (A) 1 (B) 2 (C) 3

[AMTI 2014] (D) 9 [AMTI 2015] (D) 0

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77.

A number when divided by 899 gives a remainder 63. The remainder when this number is divided by 29 is: (A) 6 (B) 7 (C) 8 (D) 5 [AMTI 2015]

78.

The number of natural number pairs (x, y) in which x > y and (A) 1

79.

(B) 2

(C) 3 8 x  27 x

The number of real x which satisfies the equal by

12 x  18 x

(A) 2

(C) 4

(B) 3

5 6  = is : x y

[AMTI 2015] (D) 4

=

7 is : 6

[AMTI 2015]

(D) 0

EXERCISE – II SUBJECTIVE

1.

The number of 100 digit number having the sum of their digits equal to 3 is ________.

2.

All the numbers from 1 to 2009 are written one after the other. The number of digits in this number is _____ [AMTI 2009]

3.

The sum of the digits of the number (100020–20) expressed in decimal notation is ____

4.

a,b,c,d and e are positive reals. If a – 2c + e = 0, b – 2c + d = 0, c – 2d + e = 0 then the set consisting [AMTI 2009] of the largest and smallest numbers among a,b,c,d,e is _______ .

5.

[AMTI 2009]

[AMTI 2009]

a3 a5 a6 5 a2 a4 = + + + + , where 0  ai < i, i = 1,2,3,4,5,6. Then a2 + a3 + a4 + a5 + a5 + a6 6 3! 5! 6! 2! 4! is ______ [AMTI 2010]

6.

If a number n is divisible by 8 and 30, then the smallest number of divisors that n has is ______. [AMTI 2010]

7.

A two digit number is equal to the sum of the product of its digits and the sum of its digits. Then the units [AMTI 2010] place of the number is _____ .

8.

The number of perfect square divisors of the number 12! is _____ .

9.

The difference between the largest 6 digits number with no repeated digits and the smallest six digit number with no repeated digits is ______. [AMTI 2011]

10.

Three consecutive integers lying between 1000 and 9999, both inclusive, are such that the smallest is

[AMTI 2010]

divisible by 11 and the middle one by 9 and the largest by 7. The sum of the largest such four digit numbers is _____. 11.

[AMTI 2011]

While multiplying two numbers a and b Renu reverted the digits of a two digit number and obtained the product to be 391. Renu realized that she made a mistake as her correct answer must be even, The correct product is _____ .

12.

[AMTI 2011]

Let x and y be two distinct three digit positive integers such that their average is 600. Then the maximum

x value of y is _____ .

[AMTI 2011]

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13.

Let N = 101010...... 101 be a 2011 digit number with alternating 1’s and 0’s. The sum of the digits of the [AMTI 2011]

product of N with 2011 is _____ . 14.

A two digit number is 6 times the sum of digits. The number formed by interchanging the digits is k times [AMTI 2012]

the sum of the digits. Then value of k is _____ 15.

A two digit number is less than the sum of the squares of its digits by 11 and exceeds twice the product of its digits by 5. The two digit number is _____

[AMTI 2012]

16.

The value of x which satisfies the equation 52. 54.56.....52x = (0.04)–28 is ____

[AMTI 2012]

17.

When the number 333332 + 22222 is written as a single decimal number, the sum of its digits is [AMTI 2013]

___________ 18.

The number of three digits numbers such that the product of their digits is a prime number is

19.

The eight digits 6, 5, 5, 4, 4, 3, 2 and 1 are used to form two 3-digit numbers and one 2-digit number. The largest possible sum of three number is ____________

[AMTI 2013]

20.

The number of integers greater than 1 and less than 70 that can be written as ab (where b >1 is)____

21.

The six digit number that increases 6 times when its last three digits are carried to the beginning of the

[AMTI 2013]

22.

number without their order being changed is

[AMTI 2014]

The symbol [x] means the integral part of x

[AMTI 2014]

Ex : [2.3] = 2, [ 5 ] = 3 . Then the value of [ 1 ] +[ 2 ]+[ 3 ] .........+ [ 100 ] 23.

If a, b, c, d are positive integers such that a5 = b4, c3 = d2 and c – a = 19, then the numerical value of d – b is ________ (you can express in powers of numbers)

[AMTI 2014]

24.

n = 560560560560563. Saket divided n2 by 8. He will get a remainder ________.

[AMTI 2014]

25.

The least positive integer by which 396 be multiplied to make a perfect cube is _______. [AMTI 2014]

26.

Mahadeven was asked what is

16 16 of a certain fraction. By mistake he divided the fraction by and got 17 17

an answer, which exceeds the correct answer by 27.

33 . The correct answer is : 340

[AMTI 2015]

d is an integer greater than 1. When the numbers 1059, 1417 and 2312 are divided by ‘d’, the remainders in each case is the same (r). The value of (d-r) is _______ . 1 log5 3

[AMTI 2015]

1

+ 27 log36 is __________ . 9 + 3 log7 9

28.

The value of

29.

When 265 is divided by a two digit number the remainder is 5. The number of such two digit number is

81

__________ . 30.

[AMTI 2015]

[AMTI 2015]

A 5-digit integer is reversed. The difference of the original and the reversed number is not zero. The largest prime number that must be a factor of this difference is __________ .

[AMTI 2015]

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ALGEBRA EXERCISE – I OBJECTIVE 1.

Given the sequence a, ab, aab, aabb, aaabb,aaabbb, ..... upto 2004 terms, the total number of times a’s [AMTI 2004] and b’s are used from 1 to 2004 terms are : (A) 2004 a’s and 2003 b’s (B) 4008 a’s and b’s (C) 1002 × 1003 a’s and (1002)2 b’s (D) 10032 a’s and 1002 × 1003 b’s

2.

Given that (a–5)2 + (b–c)2 + (c–d)2 + (b + c + d–9)2 = 0 then (a + b + c) (b + c + d) is : (A) 0 (B) 11 (C) 20 (D) 99

[AMTI 2004]

3.

x4 – y4 = 15, x and y are positive integers. Then x4 + y4 is : (A) 17 (B) 31 (C) 32

[AMTI 2004]

4.

If a,b are positive real numbers and true ? (A) b = a2 +1

5.

If

a

a =a b

(D) 113

a a where a is a mixed fraction, which of the following is b b

[AMTI 2004] (B) a = b2 – 1

(C) a = b2 + 1

(D) b = a2 – 1

a b c p q r p 2 q2 r 2   = 1 and   = 0 then the value of 2  2  2 is : p q r a b c a b c

(A) 0

(B) –11

(C) 9

[AMTI 2004] (D) 1

6.

If the roots of the equation x2 – 2ax + a2 + a – 3 = 0 are real and less than 3 then (A) a < 2 (B) 2  a  3 (C) 3 < a  4 (D) a > 4

7.

If a function f(x) is defined such that 10f(x) =  ab   (A) f   1  ab 

8.

1 x where x < 1, then f(a) + f(b) is equal to [AMTI 2004] 1 x  ab   (C) f   1  ab 

(D) none of these

A sequence a0,a1,a2,a3.......,an ... is defined such that a0 = a1 = 1 and an+1 = (an–1an) +1 for n  1. Which of [AMTI 2004] the following is true ? (A) 4 a2004

9.

 ab   (B) f   1  ab 

[AMTI 2004]

(B) 3 a2003

(C) 5 | a2004

(D) 2 a2003

A cubic polynomial P is such that P(1) = 1, P(2) = 2, P(3) = 3 and P(4) = 5. Then P(6) is [AMTI 2004] (A) 7

10.

(C) 13

Which of the following is the best approximation to

(A)

11.

(B) 10

3 5

(B)

33 50

Given that (1–x) (1+x+x2 + x3 + x4) =

(A)

31 64

(B)

31 32

(D) 16

(23  1)(33  1).....(10003  1) (23  1)(33  1).....(10003  1)

(C)

333 500

(D)

[AMTI 2004]

3333 5000

31 and x is a rational number. Then 1 + x + x2 + x3 + x4 + x5 is : 32

(C)

63 64

(D)

63 32

[AMTI 2005]

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12.

Given

a + a2 + a3 + ..... =

5 6 25 66

a2 + a3 + a4 + ..... =

(A)

13.

b + b2 + b3 + ..... =

6 5

b2 + b3 + b4 + ..... =

36 , then a2 – b2 is : 55

–1 11

(B)

1 11

[AMTI 2005]

(C)

–1 30

(D)

11 30

Given the equation of the circle x2 + y2 = 100, the number of points (a,b) lying on the circle, where ‘a’ and [AMTI 2005] ‘b’ are both integers is : (A) 2

(B) 4

(C) 8

(D) 12

14.

The roots of the equation x5 – 40x4 + Px3 + Qx2 + Rx + S = 0 are in geometric progression. The sum of their reciprocals is 10. Then |S| is equal to : [AMTI 2005] (A) 16 (B) 32 (C) 4 (D) 1

15.

The number of solutions (x,y) where x and y are integers, satisfying 2x2 + 3y2 + 2x + 3y = 10 is : [AMTI 2006] (A) 0 (B) 2 (C) 4 (D) none of these

16.

The first two terms of a sequence are 0 and 1. The nth terms Tn = 2Tn–1 – Tn–2 , n  3. For example the third term T3 = 2T2 – T1 = 2 – 0 = 2. The sum of the first 2006 terms of this sequence is : [AMTI 2006] (A)

2006  2007 2

(B)

2005  2006 2

(C) 2006

(D) 2005

17.

Consider the following sequence : a1 = a2 = 1, ai = 1+ minimum {ai-1 , ai–2} for i > 2. Then a2006 = [AMTI 2006] (A) 1003 (B) 1002 (C) 1001 (D) none of these

18.

If (x,y) is a solution set of the system of equation xy = 8 and x2y + y2x + x + y = 54, then x2 + y2 = [AMTI 2007] (A) 62 (B) 46 (C) 20 (D) 100

19.

If a,b,c are real ; a  0, b  0, c  0 and a + b + c  0 and (c + a) = (A) 1

1 1 1 1   = then (a + b) (b + c) a b c abc

[AMTI 2007] (B) 3 abc

(C) 0

(D) abc

20.

xn+1 – xn – x + 1 is exactly divisible by (x–1)2 if n is : (A) an odd positive integer (B) an even positive integer (C) an odd prime (D) any positive integer

[AMTI 2007]

21.

The number of real solutions of the equations 1 + x + x2 + x3 = x4 + x5 is : (A) 1 (B) 5 (C) 4

[AMTI 2007] (D) 3

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22.

If a,b,c,d are positive integers such that a = bcd, b = cda, c = abd and d = abc, the value of (a+b+c+d)4 / (ab+bc+cd+da)2 is : [AMTI 2007] (A) 16

23.

(B) 1

(C) 34

(D)

44 32

a,b,c are three numbers satisfying the following conditions. (i) abc  0 (ii) a + b + c = abc (iii) (a + b) (b + c) (c + a)  0 and (iv)

ab bc ca + + = kabc, then k = 1 – ab 1 – bc 1 – ca

(A) 0

(B) 1

(C) –1

[AMTI 2007] (D) 2

24.

The sum of the fourth powers of the roots of the equation x3 – x2 – 2x + 2 = 0 is : (A) 1 (B) 5 (C) 9 (D) 13

25.

If tan and tan are the roots of the quadratic equation x2 – Ax + B = 0 and cot and cotare the roots of [AMTI 2007] x2 – Cx + D = 0 then CD = (A) AB

26.

(B)

1 A –B

(C)

A B

(D)

2

If x + y + z = 2007, xy + yz + zx = 4011 x  1; y  1; z  1 then the value of

[AMTI 2007]

A B

1 1 1 + 1– y + is: 1– x 1– z [AMTI 2007]

(A) 1

27.

(B) 2008

For the equation

1 2008

(D) 0

( x – a )( x – b ) ( x – c )( x – d) = x–a–b x–c–d

[AMTI 2007]

(A) x =

ab(c – d) – cd(a  b) is the only solution ab – cd

(B) x =

ab(c – d) – cd(a  b) is one of the solutions ab – cd

(C) x =

abc d is the only solution 4

(D) x =

(a  b) – (c  d) is a solution. 4

1 28.

(C)

The sum of

2 11 2

1 +

3 2 2 3

1 +

1

4 3 3 4

+ ..... +

25 24  24 25

is [AMTI 2008]

(A)

9 10

(B)

4 5

(C)

14 15

(D)

7 15

29.

For how many real values of a will x2 + 2ax + 2008 = 0 has two integer roots ? (A) 2 (B) 4 (C) 6 (D) 8

[AMTI 2008]

30.

The number of positive integer solutions of ab–24= 2a is ? (A) 0 (B) 2 (C) 8

[AMTI 2008]

31.

(D) 12

Here is Morse-Thue sequence. Start with 0 ; to each initial segment append its complement, as follows; 0, 01, 0110, 01101001... What will be the last 8 digits of the 2008th term of this sequence. [AMTI 2008] (A) 11110000 (B) 10010110 (C) 01101001 (D) 00001111

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32.

33.

P = 20082007 – 2008; Q = 20082 + 2009. The remainder when P is divided by Q is : [AMTI 2008] (A) 4031964 (B) 4032066 (C) 4158972 (D) 40682896  1   2   3   99  + f + f  +....+ f   equals: If f(x) + f(1–x) is equal to 10 for all real numbers x then f  100 100 100        100 

[AMTI 2008] (A) 490 34.

(B) 495

(C) 500

(D) 505

If x,y are positive real numbers satisfying the system of equations x2 + y xy = 336, y2 + x xy = 112, [AMTI 2008]

then x + y equals (A) 35.

(B)

448

224

(C) 20

The number of integer solutions of the equation 2x (4–x) = 2x + 4 is : (A) 3 (B) 4 (C) 0

(D) 40 [AMTI 2008] (D) infinite

36.

If a,b,c are positive integers such that a2 + 2b2–2ab = 169 and 2bc – c2 = 169 then a + b + c is : [AMTI 2008] (A) 0 (B) 169 (C) 13 (D) 39

37.

p and q are primes. It is given that the quadratic equation x2 –px + q = 0 has distinct real roots. Then (p + 2q) is : [AMTI 2008] (A) 7 (B) 19 (C) 9 (D) 13

38.

The number of real solutions of the equation (x + 1) (3x –2) = 1 is : (A) 0 (B) 1 (C) 2

39.

The number of integer values of a for which x2 + 3ax + 2009 = 0 has two integer roots is :

[AMTI 2009] (D) more than 2

[AMTI 2009] (A) 3 40.

(B) 4

1  x n1 n = 0,1,2, .... Then the value of x20111 is : xn

(A) 1

(C) x1

(D) x2

(B) 18

(C) 2010

[AMTI 2010] (D) 69

The remainder when the polynomial x + x3 + x9 + x27 + x81 + x 243 is divided by x2 –1 : (A) 6x

43.

(B) x0

[AMTI 2010]

If xy = 6 and x2y + y2x + x + y = 63, the value of x2 + y2 is : (A) 81

42.

(D) 8

A sequence of real numbers xn is defined recursively as follows. x0 , x1 are arbitrary positive real numbers and xn+2 =

41.

(C) 6

(B) 2x

(C) 3x

[AMTI 2010]

(D) 1

Consider the sequence 4,4,8,2,0,2,2,4,6,0,.... where the nth term is the units place of the sum of the previous two terms for n  3. If Sn is the sum to n terms of this sequence then the smallest ‘n’ for which Sn > 2010 is : (A) 253

44.

[AMTI 2010] (B) 502

(C) 503

(D) 504

Three teams of wood-cutters take part in a competition. The first and the third teams put together produced twice the amount cut by the second team. The second and the third team put together yielded a three-fold output as compared with the first team. Which of the teams won the competition? [AMTI 2011] (A) first team

(B) second team

(C) third team

(D) there is a tie

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45.

a and b are the roots of the quadratic equation x2 + x –

1 22

= 0 where x is the unknown and  is a real

parameter. The minimum value of a4 + b4 is :

[AMTI 2011]

2 (A) 2 2

(B)

(C)

1 2

(D) 2 + 2

2

46.

There are three natural numbers. The second is greater than the first by the amount the third is greater than the second. The product of the two smaller numbers is 85 and the product of the two larger numbers is 115. If the numbers are x, y, z with x< y < z then the value of (2x + y + 8z) is : [AMTI 2011] (A) 117 (B) 119 (C) 121 (D) 78

47.

If x > y > 0 and

xy = xy

(A) 5 48.

2 , the value of

x2  y2 is : xy

(B) 4

(C) 1

(D) 6

The value of ‘a’ for which the expression 1 1  1  1 1   1  24 a  2   4    8 4 8  a  a  1   a  a  1   a  4 a  1      

(A) 4048

49.

[AMTI 2012]

(B) 6036

[AMTI 2012] 1

– 2log4 a  2 takes the value 2012 is (C) 6037

1  a, b, c are real numbers and none of them zero and E =  a   a 

(D) 8047 2

2

1  1  1    . Then E is equal to  a    b    ab  a  b  ab  

50.

2

1 1     – +  b   +  ab  b ab    [AMTI 2012]

(A) 2012 when a = b = 2012

(B) 2012 when ab = 2012

(C) 4 for all real values of a and b

(D) 2012 for all real values of a and b

If a = 2012, b = –1005, c = – 1007, then the value of

a4 b4 c4 + + + 3abc is bc ca ab [AMTI 2012]

(A) 2012 51.

If one root of

(B) 1

a–x +

(A) (2000, 2012)

(C) 0

(D) (2012)3

bx =

[AMTI 2012] a  b is 2012, then a possible value of a,b is (B) (4024, 2012) (C) (1000, 1012) (D) (1012, 1000)

52.

If a = 2012, b = 2011, c = 2010 then the value of a2 + b2 + c2 – ab – bc – ca is : (A) 0 (B) 2012 (C) 3 (D) 4024

53.

x and y are real numbers such that 7x – 16y = 0 and 4x – 49 y = 0, then the value of (y – x) is [AMTI 2013] (A)

5 2

(B)

19 5

(C)

4115 2013

(D)

[AMTI 2012]

1569 784

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54.

The number of positive integeral values of (x, y) which satisfy the equation simultaneously is (A) 1

55.

x +

3

y = 4 ; x + y = 28

[AMTI 2013] (B) 2

(C) 0

The number of real solutions of the equation x + (A) 1

3

(B) 2

(D) 3

x 2  x 3  1 = 1 is (C) 3

[AMTI 2013] (D) 0

56.

For some natural number ‘n’, the sum of the fist ‘n’ natural numbers is 240 less than the sum of the first (n + 5) natural numbers. Then n itself is the sum of how many natural numbers starting with 1. [AMTI 2014] (A) 7 (B) 6 (C) 9 (D) 10

57.

The number of real solutions x of the question (A) 0

(B) 1

3

x – 1  3 x – 3  3 x – 5 = 0 is : (C) 2 (D) 3

a a b b

58.

When a = 2015 and b = 2016, value of (A) 0

59.

2 b a b

–

(C) (2015)2

(B) 1

ab : a–b (D)

[AMTI 2015]

2016

An arithmetical progression has positive terms. The ratio of the difference of the 4th and 8th term to 15th term is

4 and the square of the difference of the 4th and the 1st term is 225. Which term of the series is 15

[AMTI 2015]

2015 ? (A) 225

60.

 a  b (a – b)

+

[AMTI 2015]

(B) 404

(C) 403

The number of real solutions of the equation (A) 0

(B) 1

(D) 410

| x – 3 | – | x  1| = 1 is : 2 | x  1| (C) 2

[AMTI 2015] (D) 3

EXERCISE –II SUBJECTIVE 1.

The value of

[AMTI 2009]

1  2008 1  2009 1  2010 1  2011 .2013 is _______ 2.

Each term of a sequence is the sum of its preceding two terms from the third term onwards. The second [AMTI 2009] term of the sequence is –1 and the 10th term is 29. The first term is ____ .

3.

The number of pairs of positive integers (x,y) satisfying the equation x2 + y2 + 2xy – 2008x–2008y – 2009 [AMTI 2009] = 0 is _____ .

4.

The value of

5.

Both the roots of the quadratic equation x2 – 12x + K = 0 are prime numbers. The sum of all such values [AMTI 2010] of K is ______ .

6.

Let f(x) be a polynomial of degree 1. If f(10) – f(5). = 15, then f(20) – f(5) equals _____. ]

3

20  14 2 +

3

20 – 14 2 is _______ .

[AMTI 2010]

[AMTI 2010]

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7.

In the sequence a1, a2, ...., an the sum of any three consecutive terms is 40. If the third term is 10 and the [AMTI 2011] eighth term is 8 then the 1000th term is ____.

8.

f(x) is a quadratic polynomial with f(0) = 6, f(1) = 1 and f(2) = 0. Then f(3) = ____ .

9.

The value of

10.

If a,b,c,d, satisfy the equation a + 7b + 3c + 5d = 0, 8a + 4b + 6c + 2d = – 16, 2a + 6b + 4c + 8d =16 [AMTI 2012] 5a + 3b + 7c + d = –16 then the value of (a + d) (b + c ) = _____ .

11.

The combined age of a man and his wife is six times the combined ages of their children. Two years ago their united ages were ten times the combined ages of their children. Six years hence their combined age will be three times the combined age of the children. The number of children they have is ____. [AMTI 2012]

12.

The sum of the roots of the equation x 3 x 2 = ( x )x is ____ .

13.

The least value of the positive integer ‘n’ such that (n + 20) + (n + 21) + (n + 22) + .....+ (n + 100) is a [AMTI 2013] perfect square is ___________

14.

The number of real values (x, y) for which 2x+1 + 3y = 3y+2 – 2x is ___________

[AMTI 2013]

15.

If f(x) = ax + b and f(f(f(x))) = 27x + 26 then (a + b) = __________

[AMTI 2013]

16.

a  0, b  0 The number of real number pair (a, b) which satisfy the equation a4 + b4 = (a + b)4 is [AMTI 2013]

17.

m, n are natural numbers. The number of pairs (m, n) for which m2 + n2 + 2mn – 2013 m – 2013n – 2014 = 0 is __________ [AMTI 2013]

18.

If a, b, c are real and a + b + c = 0 the value of a(b - c)3 + b(c – a)3 + a(a – b)3 is

19.

m, n are natural numbers. If (m – 8) (m – 10) = 2n, the number of pairs (m, n) is_____.

3

5  2 13 +

3

5 – 2 13 is = ______ .

[AMTI 2011] [AMTI 2012]

[AMTI 2012]

[AMTI 2013]

[AMTI 2014] 20.

 3x  x 3 1 x     If f(x) = log for – 1 < x < 1 and it is found that f  2 1 x   1  3x

   = Kf(x), then the value of K is ____.  [AMTI 2014]

3

1.2.4  2.4.8  ......  n.2n.4n is _________. 1.3.9  2.6.18  .........  n3n.9n

21.

The value of

22.

n is a natural number. It is given that (n + 20) + (n + 21) + ......+ (n + 100) is a perfect square. Then the [AMTI 2014] least value of n is __________.

23.

The value of 3 1 2 .

24.

1 1 2  1 1  (a  b)2      When a = 5 and b = 403, the value of  2 b 2 a  b  a b  ab a

25.

The sum of all even two digit numbers is __________ .

26.

In a G.P. of real numbers, the sum of the first two terms is 7. The sum of the first six terms is 91. The sum of the first four terms is __________ . [AMTI 2015]

27.

If x3 + 3xy2 = 14, y3 + 3yx2 = 13, x,y are real then value of x2 + y2 is __________ .

6

[AMTI 2014]

3 – 2 is __________ .

[AMTI 2015]

   

–1

is __________ . [AMTI 2015] [AMTI 2015]

[AMTI 2015]

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GEOMETRY EXERCISE – I OBJECTIVE 1.

The sum of all angles except one of a convex polygon is 2190° (where the angles are less than 180°) Then [AMTI 2004] the possible number of sides of the polygon is (A) 13 (B) 15 (C) 17 (D) 19

2.

In a right angled triangle with legs 4 and 8, the area of the largest square that can be inscribed in the [AMTI 2004] triangle is (A)

3.

8 3

(B)

4 3

(C)

16 9

(D)

16 9

Two circles with centres A and B and radius 2 touch each other externally at C . A third circle with centre [AMTI 2004] C and radius 2 meets the other two at D, E. Then area ABDE is

D

E

B

A C (A) 3 2 4.

(B) 6 2

(C) 3 3

(D) 6 3

In ABC, A = 90° and I is the incentre. The perpendicular distance of I from BC is to

8 . Then AI is equal [AMTI 2004]

A Q R I B (A)

8

C

P

(B) 3

(C) 12

(D) 4

5.

In an isosceles triangle, the centroid, the orthocentre, the incentre and the circumcentre are [AMTI 2004] (A) conincident (B) collinear (C) in the interior of the circumcircle (D) in the interior of the incircle

6.

A solid cuboid has edges of length a, b, c. What is the surface area ?

[AMTI 2004]

C

b a

(A) (a + b + c)2 – (a2 + b2 + c2) (C) 2(a2 + b2 + c2)

(B) abc (D) ab + bc + ca

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7.

A circle and a parabola are drawn on a piece of paper. The number of regions they divided the paper into is at most. [AMTI 2004] (A) 3 (B) 4 (C) 5 (D) 6

8.

Find the area in square centimeters of the shaded rectangle if all other shapes 1,2,3 in the large [AMTI 2005] 9 cm  5 cm rectangle are squares.

9 cm 3 2

5 cm

1 (A) 4 9.

(B) 3

(C) 2

(D) 1.5

In the diagram PS = PQ and QS = QR. If SPQ = 80° then QRS equals

[AMTI 2005]

P 80°

Q

R

S (A) 15° 10.

(B) 20°

(C) 25°

Three squares are joined together at the corners onto two vertical poles as shown aside. Find the value of [AMTI 2005] x when the angles are given as shown.

30°

75°

124°

(A) 39

x°

90°

90°

11.

(D) 30°

(B) 41

(C) 43

(D) 44

A cone is made from a circular sector, by joining the two radii and the ratio of the radius and slant height of the cone is as 1: 2. Then the angle of the sector from which the cone is made is [AMTI 2005]

l = 2r r (A) 90° 12.

(B) 35°

(C) 180°

(D) 70°

In the adjoining figure P, M, Q and R are collinear points are PM = MQ = MS. It is also given SR2 = PR.QR. Then [AMTI 2005]

S

P (A) QSR = MSP

M

(B) QSR = MSQ

Q

R (C) QSM = PSM

(D) SQR = SMP

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13.

In a triangle ABC, A = 30°, BC = 6 cm. Then the radius of the circum circle is (A) 3 cm (B) 6 cm (C) 8 cm (D) 4.5 cm

[AMTI 2005]

14.

AB is the diameter of the circle. Then QPB is

[AMTI 2005]

P A

B

40°

Q (A) 40° 15.

(B) 50°

(C) 60°

(D) 30°

PQRS is a common diameter of the three circles. The area of the middle circle is the average of the areas of the other two. [AMTI 2005] If PQ = 2, RS = 1 then the length of QR is P (A) 1 + 6 (B)

Q

6 –1 2

R

(C) 4 (D) 3 16.

S

Of the points (0,0), (2,0), (3,1), (2,1), (3,3), (4,3) and (2,4) at most how many can be on a circle ? [AMTI 2005] Y G

F

E D C B

X

O (A) 5

(B) 3

(C) 4

(D) 6

17.

In an n sided regular polygon the radius of the circum-circle is equal in length to the shortest diagonal. Then n is [AMTI 2006] (A) 6 (B) 8 (C) 12 (D) Such a polygon does not exist.

18.

All the six faces of a cube are extended in all directions. The number of regions into which the whole space is divided by these 6 planes is [AMTI 2006] (A) 9 (B) 16 (C) 24 (D) 27

19.

In a ABC, A = 30°, AC = 10 units and BC = 4 units. Which of the following statements is correct ? (A) B is acute (B) B is obtuse [AMTI 2006] (C) B is 90° (D) such a triangle does not exist

20.

BC is the diameter of a semicircle. The sides AB and AC of a triangle ABC meet the semicircle in P and Q respectively. PQ subtends 140° at the centre of the semi-circle. The A is [AMTI 2006] A

P

B

(A) 10°

(B) 20°

Q

C

(C) 30°

(D) 40°

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21.

In rectangle ABCD, C is trisected by CE and CF where E lies on AB and F on AD. If BE = 6 cm and AF = 2 cm, which of the following integers is nearest to the area of the rectangle ABCD in sq. cm ? [AMTI 2006] C

D

F 2 A

(A) 130 22.

E

B

6

(B) 150

(C) 170

(D) 190

In the adjacent figure BA and BC are produced to meet CD and AD produced in E and F. Then AED + CFD is [AMTI 2006] A

E

50°

B

D C F

(A) 80° 23.

(B) 50°

(C) 40°

(D) 160°

A hoop is resting vertically at stair case as shown in the diagram AB = 12 cm and BC = 8 cm. The radius of the hoop is [AMTI 2006]

C

A

(A) 13 24.

B

(B) 12 2

(C) 14

(D) 13 2

In a triangle ABC, P and Q are midpoints of AB and AC respectively. R is a point on BC such that BR = 2RC. Let S be the intersection of PQ and AR. If the area of triangle AQS is 1 sq. unit, then the area of trapezium PSRB is [AMTI 2006] A

Q

P

S

B

(A) 6

(B) 8

R

C

(C) 9

(D) 12

25.

A solid cube is chopped off at each of its 8 corners to create an equilateral triangle with 3 new corners. The 24 corners are now joined to each other by diagonals. How many of these diagonals completely lie in the interior of the cube ? [AMTI 2006] (A) 36 (B) 96 (C) 108 (D) 120

26.

In a polygon there are 6 right angles and the remaining angles are all equal to 200° each. The number of [AMTI 2007] sides of the polygen is (A) 15 (B) 12 (C) 9 (D) 23

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27.

PQRS is a rectangle of area 2000 cm2. Two lines parallel to the sides are drawn to cut the rectangle into four rectangle of areas A,B,C,D, Given A = 1000 cm2, B = 500 cm2, C = 300 cm2, D = 200 cm2 which of the P Q [AMTI 2007] following is possible ? (A) Such a division is not possible A B (B) Exactly two such divisions are possible L (C) At least two such divisions are possible C D R (D) In such a division the point L lies on one of the diagonals S

28.

In a rectangle the length is x units more than its breadth. If its length is increased by y units and its breadth is decreased by z units, the area is unaltered. The breadth of the rectangle is [AMTI 2007] (A)

29.

( x  y )z yz

(B)

z xy

(C)

( x  y )z yz

 ( x  y )z   (D)   yz 

Two circles touch internally at A. AOB is the diameter of the bigger circle of centre O. OPQ is a tangent [AMTI 2007] to the smaller circle. APL is a straight line. If BOL = 42° then LOQ =

L Q

P

42°

(A) 84° 30.

B

O

A

(B) 104°

(C) 90°

(D) 117°

In the adjoining figure AB = AC. DEF is an equilateral triangle . Then

[AMTI 2007]

A

D b

E c

a B

(A) a + b + c = 180° 31.

F

(B) a + b = 2c

C

(C) a =

bc 2

(D) a + c = 2b

In the adjoining figure the diameter of the circle is 2007 cm. Then x =

[AMTI 2007]

2007 cm

(A) 32.

2007 cm 2

(B)

2007 cm 4

(C) 2007 cm

(D)

2  2007 cm 3

The sides of a triangle are integers. The perimeter is 8. The area is (A)

8

(B) 8

(C) 5

[AMTI 2007] (D) 12

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33.

Let A denote the ratio of the volume of a cube to that of the sphere which will fit inside the cube. Let B be the ratio of the volume of a cube inscribed in a sphere to that of the sphere then [AMTI 2007] (A) A2 = 27 B2 (B) B2 = 27 A2 (C) A2 = 9B2 (D) B2 = 9A2

34.

Two semi circles are constructed as shown in the figure. The chord PQ of the greater circle touches the smaller circle and is parallel to the diameter of larger circle. If the length of PQ is 10 units, then what is the [AMTI 2008] area between the semi circles ?

Q

P

(A) 50

(B) 50

(C)

1 .25 2

(D)

1 .25 2

35.

ABC is a right angled triangle with A = 63° and B = 27°. ACFG, BCDE are square drawn external to the triangle ABC, such that AE intersects BC at H, BG intersects AC at K. The the size of angle measure DKH is [AMTI 2008] (A) 27° (B) 36° (C) 63° (D) 45°

36.

ABC is an equilateral . G is the centroid and AG = 2 cm S is the circumcircle of the triangle. The area [AMTI 2008] of the shaded portion (in cm2) is

A S G B

(A) (4– 3 3 )

(B)

16  9 3 4

C

(C) 3(4 – 3 3 )

(D)

1 (4 – 3 3 ) 3

37.

Two of the altitudes of a scalene triangle have lengths 4 and 12. If the length of the remaining altitude is [AMTI 2008] also an integer, then its maximum value is (A) 1 (B) 2 (C) 5 (D) 8

38.

Three faces of a rectangular parallelepiped have areas 48 cm2 18 cm2 and 24 cm2. Then the sum of the length, breadth and height of the parallelepiped (in cm) is [AMTI 2008] (A) 17 (B) 27 (C) 14 (D) 11

39.

ABCD is a square. With centres B and C and radius equal to the side of the square, circles are drawn to [AMTI 2008] cut one another at E inside the square. BDE is equal to (A) 22

40.

1 2



(B) 30°

(C) 15°

(D) 37°

AB is a fixed diameter of a circle. PQ is a chord whose length is equal to the radius of the circle, which [AMTI 2008] does not intersect the diameter AB.AP and BQ meet at R (A) ARB = 45° when PQ is parallel to AB (B) ARB = 60° for all positions of PQ (C) ARB = 60° only when PQ is parallel to AB (D) ARB is a variable angle depending on the position of PQ

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[23]

41.

Six equal circles are arranged as in the figure. The height of the triangle is 4. Then the radius of the circle is [AMTI 2009]

4

(A) ( 3 –1)

 3  1   (B)  2   

4 (C)

(D)

3 1

3

42.

The number of squares on a coordinate plane with one vertex at A(–2, 2) and at least one of the coorcinate [AMTI 2009] axes as axis of symmetry of the square is (A) 3 (B) 5 (C) 6 (D) 7

43.

In the adjacent figure AE = AF, then

[AMTI 2009] A y

E

F x b

a

c

B

(A) a + b + c = 180º

(B) a + c = b

C

(C) 2c + a = b

(D) 2a + c = b

44.

Let f(n) be the number of non-congruent integer sided triangles with perimeter n (f(3) = 1, f(4) = 0, f(5) = 1). Then f(10) is [AMTI 2009] (A) 2 (B) 3 (C) 4 (D) greater than 4

45.

The diagonal BD of a quadrilateral ABCD bisects the angle ABC is given that AC = BC, BDC = 80° ACB = 20°. BAD is [AMTI 2009] (A) 110° (B) 120° (C) 130° (D) 140°

46.

If two circles of radii 13 and 15 intersect and the length of the common chord is 24, then the distance [AMTI 2009] between their centres is (A) 10 (B) 12 (C) 14 (D) 16

47.

Given a triangle of area 80cm2, circles of radii 2cm are drawn with centres at the vertices of the triangle. What is the area of the triangle out side of these circles (in cm2) [AMTI 2009] (A)  – 70 (B)  – 76 (C) 78 –  (D) 80 – 2

48.

In the diagram, four semi circles of radius 1 cm each with their centres at the mid points of the sides of square are drawn. A circle is inscribed touching all the four semicircles. The radius of the inner circle is [AMTI 2009]

(A) 49.

2 1

(B)

 ( 2 – 1) 2

(C)

3 –1

(D) 2 2 – 2

Through D, the mid-point of the side BC of a triangle ABC, a straight line is drawn to meet AC at E and AB [AMTI 2010] produced at F so that AE = AF. Then the ratio BF : CE is : (A) 1 : 2 (B) 2 : 1 (C) 1 : 3 (D) None of these

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50.

In the bigger of two concentric circles two chord AB and AC are drawn to touch the smaller circle at D at E. Then BC is equal to : [AMTI 2010]

(A) 3DE

(B) 4DE

(C) 2DE

(D)

3 DE 2

51.

The internal bisector AE of the angle A of triangle ABC is : (A) not greater than the median through A for all triangles. (B) not greater than the median through A for only acute angled triangles. (C) Not greater than the median through A for only obtuse angled triangles. (D) not less than the median through A for all triangles.

[AMTI 2010]

52.

In the adjoining diagram ABC is an equilateral triangle and BCDE is a square. The side of the equilateral [AMTI 2010] triangles is 2010. The radius of the circle is :

A

(A) 2010 53.

C

E

D

(B) 4020

(C) 6030

(D) 8040

If p is the perpendicular drawn from the vertex of a regular tetrahedron to the opposite face and if each [AMTI 2010] edge is equal to 2 units, then p is (A) 8 3

54.

B

(B)

8 3 2

(C)

8 3 5

(D)

24 3

P is a point inside an equilateral triangle of side 2010 units. The sum of the lengths of the perpendiculars drawn from P to the sides is equal to : [AMTI 2010]

2010 (A) 2010

(B) 2010 3

(C) 1005 3

(D)

3

55.

ABCD is a rectangle in which AB = 8, AD = 9. E is on AD such that DE = 4. H is on BC such that BH = 6. EC and AH cut at G. GF is drawn perpendicular to AD produced. Then GF = [AMTI 2011] (A) 20 (B) 22 (C) 18 (D) 15

56.

The sides of the base of a rectangular parallelepiped are a and b. The diagonal of the parallelepiped is [AMTI 2011] inclined to the base plane at an angle . Then the lateral surface area of the solid is : (A) 2(a + b)

a 2  b 2 tan

(C) (a2 + b2) a  b tan

(B) (a + b)

a 2  b 2 tan

(D) 2(a2 + b2)

a  b tan

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57.

In the adjoining figure, O is the circum centre of the triangle ABC. The perpendicular bisector of AC meets AB at P and CB produced at Q. Then : [AMTI 2011]

(A) 2PQB = 3PBO

(B) 3PQB = 2PBO

(C) 4PQB = 5PBO

(D) None of these

58.

There are 15 radial spokes in a wheel, all equally inclined to one another. Then there are two spokes [AMTI 2011] which (A) lie along a diameter of the wheel (B) are perpendicular to each other (C) are inclined at an angle of 120° (D) include an angle less than 24°

59.

Two regular polygons of same number of sides have sides 40cm and 9cm in length. The lengths of the side of another regular polygon of the same number of sides and whose area is equal to the sum of the [AMTI 2012] areas of the given polygons is (in cm) : (A) 49 (B) 31 (C) 41 (D) 360

60.

In rectangle ABCD, AB = 2BC = 4cm E and F are midpoints of AB and CD respectively. ESD and ETC are arcs of circles centred at A and B respectively. If the perpendicular bisector line l of EF cuts the arcs at S and T as in the diagram, then ST is equal to (in cm) [AMTI 2012]

(A) (4 – 2 3 )

(B) (3 + 3 )

(C) (2 + 2 3 )

(D) (4 3 – 2)

61.

C1 and C2 and two non-intersecting circles whose radii are in the ratio 1 : 2. A third circle cuts the smaller circle at A and B and the bigger one at C and D. AB and CD intersect at P. The ratio of the lengths of the [AMTI 2012] tangents form P to the circles C1 and C2 is : (A) 1 : 4 (B) 1 : 8 (C) 1 : 1 (D) 1 : 2

62.

The angles of a triangle are in the ratio 2 : 3 : 7. The length of the smallest side is 2012cm. The radius of [AMTI 2012] the circum circle of the triangle (in cm) is : (A) 2013 (B) 2011 (C) 4024 (D) 2012

63.

ABC is triangle with AB = 13 cm, BC = 14cm, and CA = 15cm. AD and BE are altitudes from A and B to BC and AC respectively. H is the point of intersection of AD and BE. Then the ratio

HD = HB [AMTI 2012]

(A)

3 5

(B)

12 13

(C)

4 5

(D)

5 9

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64.

In the figure shown, BD = CD, BE = DE, AP = PD and DG||CF.

[AMTI 2012]

area of ADH Then area of ABC is equal to (A)

65.

1 5

(B)

2 9

(C)

3 13

AX and BX are two adjacent sides of a regular polygon. If ABX =

(D)

1 6

1 AXB, then the number of sides of 3

[AMTI 2012]

the polygon is : (A) 6

(B) 7

(C) 9

(D) 5

66.

A rectangular block of dimensions 16 × 10 × 8 units is painted. It is cut in to cubes of dimensions [AMTI 2012] 1 × 1 × 1. The number of cubes which are not painted at all is : (A) 945 (B) 672 (C) 812 (D) 796

67.

Two sides of a triangle are 10cm and 5cm in length and the length of the median to the third side is

1 cm. The area of the triangle is 6 x cm2. The value of x is 2 (A) 12 (B) 13 (C) 14 6

[AMTI 2013] (D) 15

68.

ABCD is a rectangle. Through C a variable line is drawn so as to cut AB at X and DA produced at Y. Then BX.DY is [AMTI 2013] (A) twice the area of the rectangle ABCD (B) equal to the area of the rectangle ABCD (C) a variable quantity which lies between the area of rectangle ABCD and twice the area of the rectangle ABCD (D) always a constant less than the area of rectangle ABCD

69.

In the adjoining figure O is the centre of the circle. ACOB is a square with A on the circle. Through B a line parallel to OA is drawn to cut the circle at D nearer to A. Then BOD = [AMTI 2013]

B

O

A

C

D

(A) 20°

(B) 18°

(C) 15°

(D) 22

1 ° 2

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70.

In the adjoining figure PQRS is a square of side 2 units PTR and QTS are quadrants of circles of radius 2 units. With SR as diameter a semicircle is drawn. A, B denote the areas of the portions shaded. Then (A – B) = [AMTI 2013] P

Q

B T A

R

S

(A) 71.

3 –4 2

(B)  –

1 3

(C)

3 1 – 2 4

(D) 

In the adjoining figure AB is a diameter of a circle. AB is produced to P such that BP = radius of the circle. PC is a tangent to the circle. The tangent at B and AC produced cut at E. Then CDE is [AMTI 2013] C

E D P

A

B

(A) isosceles with EC = ED (C) equilateral 72.

(B) isosceles with EC = CD (D) a scalene triangle

In the adjoining figure, PQ = 42 cm. QR is the tangent to the semicircle at Q. If the difference of the areas ofregions A,B is 357 cm 2, then the base QR of the right triangle PQR is (in cm) (take  = 22/7) [AMTI 2014]

P B

A A R

Q (A) 42 73.

(B) 48

(C) 52

(D) 50

The sides of a right angled triangle are all integrs. Two sides are primes that differ by 50. The smallest [AMTI 2014]

possible value of third side is (A) 60 74.

(B) 57

(C) 53

(D) 49

Observe the angles 1, 2, 3, 4, 5 and 6 in the square as shown. The measure of angle (1 + 2 + 3 + 4 + 5 + 6) is

[AMTI 2014] 6

1 4

3

(A) 180°

(B) 270°

2

(C) 360°

(D) 225°

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75.

In triangle ABC, B = 2, C. AD is the angle bisector such that DC = AB. Then the measure of A is [AMTI 2014] (A) 60° (B) 72° (C) 84° (D) 108°

76.

ABCD is a square. From the diagonal BD, a length BX is cut off equal to BA. From X, a straight line XY is drawn perpendicular to BD to meet AD at Y. Then AB + AY = [AMTI 2015]

BD (A)

2 BD

(B)

2

(C)

3 BD

(D) BD

77.

AB and AC are tangents at B and C to a circle. D is the mid point of the minor arc BC with respect to the [AMTI 2015] triangle ABC,D is the : (A) orthocenter (B) circumcentre (C) incentre (D) centroid

78.

Two circle of radii in the ratio 1 : 2 touch each other externally. The centre of the small circle is C and that of the big is D. The poin of contact is A. PAQ is a straight line where P is on the smaller circle and Q on the on the bigger circle (PAQ does not pass through C). The angle between the tangent at Q to the bigger [AMTI 2015] circle and the diameter (produced if necessary) of the smaller circle is : (A) 60°

79.

(B) 75°

(C) 80°

(D) none of these

P,Q,R are there points on a circle of centre O. If RS = radius of the circle and PSQ = 12°, then POQ = [AMTI 2015] (A) 36° (B) 42° (C) 48° (D) 54°

EXERCISE – I SU BJ EC TI VE

1.

A square is described on the hypotenuse of a right angled triangle away from the right angled vertex. The ratio of the perpendiculars drawn from the intersection point of the diagonals of this square to the legs of [AMTI 2009] the right angled triangle is _________.

2.

PMT is a tangent to the circle APC at the point P; CN AT is a diameter, to which PN is drawn perpendicular [AMTI 2009] and AM is perpendicular to PT. The ratio of AM to AN is ________.

3.

If the length of a diagonal of a cube is 12 cm, then the area of each of its faces is _______. [AMTI 2009] A point is taken on the hypotenuse of a right triangle equidistant (x) from the legs. The point divides the [AMTI 2009] hypotenuse into parts 30 cm and 40 cm. Then x is equal to ______ cm.

4. 5.

Two circles touch externally at C. AB is a direct common tangent touching the circles at A,B. If AC = 14 [AMTI 2009] cm, BC = 48 cm then the length of the direct common tangent is _______.

6.

In a quadrilateral three consecutive sides are of lengths 2,3 and 4. A circle of radius 1.2 is inscribed in the [AMTI 2009] quadrilateral. The area of the quadrilateral is _______.

7.

The whole surface area of rectangular block is 1332cm2. The length, breadth and height are in the ratio 6 : 5 : 4. The sum of the length, breadth and height is ______centimeters. [AMTI 2010]

8.

Two parallel sides of a trapezoid are 3 and 9, the non parallel sides are 4 and 6. A line parallel to the bases (parallel sides) divides the trapezoid in to two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is _____ . [AMTI 2010]

9.

Triangle ABC had AB = 17, AC = 25 and the altitude to BC has length 15. The sum of the possible value [AMTI 2010] of BC is _____ .

10.

A circle is circumscribed about a triangle with sides 30, 34, 16. It divides the circle into 4 regions with the non triangular regions being A,B,C; C being the largest. Then the value of (C – A – B) is ____ . [AMTI 2010]

11.

In a convex polygon of 16 sides the maximum number of angles which can all be equal to 100 is ___. [AMTI 2010]

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12.

If an arc of circle 1 subtending 600 at the centres, has double the length as the arc subtending 750 at the

area of circle1 centre in circle 2, then area of circle 2 is _____ .

[AMTI 2010]

13.

ABCDEF is a non-regular hexagon where all the six sides touch a circle and all the six sides are of equal length. If A = 140° then D = _____ . [AMTI 2011]

14.

The diagonals of a convex quadrilateral are perpendicular. If AB = 4, AD = 5, CD = 6, then length of BC is ____ . [AMTI 2011]

15.

Three circles, each of radius one, have centres at A,B and C. Circles A and B touch each other and circle C touches AB at its midpoint. The area inside circle C and outside circles A and B is ____ . [AMTI 2011]

16.

The numer of rectangles that can be obtained by joining four of the 11 vertices of a 11-sided regular polygon is ____. [AMTI 2011]

17.

Let D,E,F be the midpoint of the sides BC, CA and AB respectively of triangle ABC. AB = 16, BC = 21 and CA = 19. The circum-circles of the triangles BDF and CDE cut at P other than D. Then BPC =___ . [AMTI 2011]

18.

Triangle ABC is equilateral of side length 8cm. Each arc shown in the diagram is an arc of a circle with the opposite vertex of the triangles as its centre. The total area enclosed within the entire figure shown (in [AMTI 2012] cm2) _____ .

19.

ABCD is a square. A line AX meets the diagonal BD at X and AX = 2012 cm. the length of CX (in cm) is : [AMTI 2012]

20.

O is the centre of a circle of radius 15 cm. M is a point at a distance of 5 cm from O. AMB is any chord of the circle through M, then the value of AM × MB is _____ . [AMTI 2012]

21.

An isosceles trapezoid is circumscribed about a circle of radius 2cm and the area of the trapezoid is 20 cm2. The equal sides of the trapezoid have length ____. [AMTI 2012]

22.

A triangle has sides with lengths 13 cm, 14cm and 15 cm. A circle whose centre lies on the longest side touches the other two sides. The radius of the circle is (in cm) ____ . [AMTI 2012]

23.

ABC and ADE are two secants of a circle of radius 3cm. A is at a distance of 5cm from the centre of the circle. The secants include an angle of 30°. The area of the ACE is 10cm2. Then the area of the ADB (in cm2) is ____. [AMTI 2012]

24.

In the adjoining figure two equal circles of radii 2 units each touch. AB is the common diameter. The tangent at B meets the tangent from A to circle at C as shown. If BC = K

2 then the value of K is

[AMTI 2013]

___________ C

A

B

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25.

In the adjoining figure. ABC is equilateral AD, BE and CE are respectively perpendicular to AB, BC and

Area of DEF AC. Then Area of ABC ___________

[AMTI 2013] A

D

F

C

B

E

26.

and BP. Then

27.

1 AP = . Q is the point of intersection of AC AD 2013

ABCD is parallelogram P is a point on AD such that AQ __________ AC

[AMTI 2013]

ABCD is a square. E and F are respectively point on BC and CD such that EAF = 45°. AE and AF cut

Area of AEF the diagonal BD at P, Q respectively. Then Area of APQ = _________ 28.

[AMTI 2013]

In the adjoining figure BAC is a 30° – 60° – 90° triangle. D is the midpoint of AC. The perpendicular at D to AC meets the line parallel to AB through C at E. The line through E perpendicular to DE meets BA produced at F. If DF = 5 x the x = __________

[AMTI 2013]

C

E

D 60° F

29.

A

B 20

PR and PQ are tangent to a circle and QS is a diameter. Then

QPR = ____________ RQS

[AMTI 2013] 30.

ABCD is a square BEFG is another square drawn with the common vertex B such that E, F fall inside the square ABCD. Then DF =

n . AE where n is

[AMTI 2014]

G C

B F E A

D

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31.

In the adjoining figure AB = AC. The exterior angle CAX = 140° D is the point on AB such that CB = CD. DE is drawn parallel to BC to meet AC at E. The measure of the DCE is ________. [AMTI 2014] x

A

E

D

B

32.

C

ABCD is a rectangle DEFC is a parallelogram. ABEF is a straight line. Area of the quadrilateral CGEF [AMTI 2014]

is ___________ 5 cm

B

A

E

F

G 18 cm D

40 cm

C

33.

ABC is an isosceles triangle of base 30 cm. The altitude to the base is 20 cm. The length of the other altitude (in cm) is __________ . [AMTI 2015]

34.

In a right triangle a semi-circle is incribed so that its diameter lies on the hypotenuse and its centre divides the hypotenuse in to two segments of length 15cm and 20cm. The length of the arc of the semicircle between the points at which the legs touch the semicircle is K . Then the numerical value of [AMTI 2015] K is __________ .

35.

ABC is a triangle. D is the midpoint of AB. E is the midpoint of DB. F is the midpoint of BC. If the area of ABC is 64cm2, then the area of AEF(in cm2) is __________ . [AMTI 2015]

36.

In ADE, ADE = 140°. B and C are points on AD and AE respectively. A,B,C,D,E are all distinct. If AB [AMTI 2015] = BC = CD = DE then EAD is equal to __________ .

37.

Viswa is playing with three rectangular boxes which are identical. The dimensions of each are 50 cm × 5 cm × 2cm. He glues them in several ways. After gluing the maximum surface area he could get [AMTI 2015] is __________ cm2.

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INEQUALITIES EXERCISE – I OBJECTIVE

1.

The positive numbers x and y satisfy xy = 1. The minimum value of

(A)

1 2

(B)

5 8

1 x

(C) 1

4

1 +

4y 4

is

[AMTI 2005]

(D) no minimum

2.

a,b,c,d are four distinct positive real numbers such that a > b, c < d, b > c and d < a. Then [AMTI 2007] (A) b < d < a always (B) d < b < a always (C) d > c and d < b (D) a is the greatest always

3.

If a,b,c,d are non zero positive real and a + b + c + d = s 1 then the minimum value of s  s  s  s    1   1   1   1 is a  b  c  d 

(A) 4

4.

[AMTI 2007]

(B) S

(D) S4

(C) 81

The number of pairs of prime numbers (p, q) satisfying the condition

1 1 5 51 < + < will be p q 100 6 [AMTI 2008]

(A) 49 5.

(B) 24

n 1 n 4 n7 n  10 n  13 + + + + n n3 n6 n9 n  12 1

B=

n 1

1 +

n2

(A) A = B

7.

1 +

n5

1 +

n8

1 +

n  11

(B) A = 2B

then

[AMTI 2011]

(C) A < B

The number of integers n which satisfy (n2 – 2) (n2 – 20) < 0 is (A) 3 (B) 4 (C) 5

The value of the expression

(A) 8.

(D) 48

n is a natural number greater than 1, and A=

6.

(C) 50

x for all x > 0

(D) A > B [AMTI 2011] (D) 6

( x  2 )2  8 x 2 ( x ) is equal to x

(B) – x for 0 < x < 2

When b  0, then 12a2b3 – a6 – b9 (A) always is less than or equal to 64 (C) always negative

(C)

[AMTI 2011]

x for 0 < x < 2

(D) – x for all x > 0 [AMTI 2011]

(B) always greater than 64 (D) always lies in the interval [60,64]

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EXERCISE –II SU BJ EC TI VE

1.

n(n  2) < 2009 is _______

The number of positive integers satisfying the inequality 2000 <

[AMTI 2009] 2.

Given a, b, c  0 and a + b + c = 1 then the maximum value of

3.

If a,b are positive and a + b = 1 the minimum value of a4 + b4 is _________

[AMTI 2010]

4.

If | x | + x + y = 10, x + | y | – y = 12 then x + y = ________

[AMTI 2010]

5.

When x is real, the greatest possible value of 10x – 100x is

[AMTI 2011]

6.

x1, y1, x2, y2 are real numbers. If x12 + x 22  2 and y12 + y 22  4, the maximum value of the expression x1y1 + x2y2 is [AMTI 2011]

7.

x,y,z are real numbers such that (x + y)2 = 16, (y + z)2 = 36, (z + x)2 = 81, x + y + z > 3. The number of [AMTI 2012] possible values of (x + y + z) is

8.

The number of integers ‘n’ which satisfy the inequality (n2–2) (n2 – 20) < 0 is __________ .

3

ab +

3

bc +

3

c  a is ______ [AMTI 2009]

[AMTI 2015]

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[34]

MISCELLANEOUS EXERCISE – I OBJECTIVE

Let A = {a, b, c} and B = {a, b, d, e,f } How many sets C consisting of characters from the english alphabet can be constructed so that C  B and such that A  C has one element and C A. [AMTI 2004] (A) 16 (B) 14 (C) 8 (D) 6

2.

Five children each owned a different number of rupees. The ratio of any one’s fortune to the fortune of every child poorer than himself was an integer. The combined fortune of the children was 847 rupees. The least number of rupees that a child had was [AMTI 2004] (A) 12 Rs. (B) 10 Rs. (C) 7 Rs. (D) 5 Rs.

3.

During holidays, five people A,B,C,D and E went swimming regularly. Each time they went, exactly one of them was missing. A went the least number of times (5 times) and E most often (8 times). What can [AMTI 2004] we say about the number of times B, C and D went ? (A) each went six times (B) each went seven times (C) 2 went 6 times and one went 7 times (D) 2 went 7 times and 1 went 6 times

4.

If two successive discounts of % and b% are given on the sale of a certain article, the single equivalent discount is [AMTI 2005]

U

1.

(A) a + b +

ab 100

(B) a + b –

ab 100

(C) a – b +

ab 100

(D) a – b –

ab 100

5.

In how many different ways can 3 children share eight identical sweets so that each child gets at least [AMTI 2005] one ? (A) 21 (B) 24 (C) 36 (D) 45

6.

Seven consecutive numbers are chosen. From these seven numbers, two numbers are chosen. What is [AMTI 2005] the probability that the sum of the two numbers chosen is divisible by 7 ? (A)

1 7

(B)

2 7

(C)

3 7

(D)

4 7

7.

Eighteen balls are placed in 4 boxes so that no box is empty. Then [AMTI 2006] (A) At least one box has at least 5 balls (B) At least one box has exactly 5 balls (C) Exactly one box has at least 5 balls (D) Exactly one box has exactly 5 balls

8.

Harsha looks at a calendar for the year 20xy. He notices that April. 20xy has exactly 4 Mondays and 4 [AMTI 2006] Fridays. Then 28 April 20xy would fall on a : (A) Sunday only (B) Friday, Monday or Tuesday (C) Monday only (D) Monday, Sunday or Tuesday only

9.

ABC can walk at the rates 3,4 and 5 km an hour respectively. They start from a place at 1,2,3 hrs respectively. When B catches A, B sends him back with a message to C. At what time C gets the message ? [AMTI 2007] (A) 5 hrs. 15 mts (B) 3 hrs 30 mts (C) 6 hrs 15 mts (D) 5 hrs 45 mts

10.

An island is inhabited by both liars and knights. Every knight always tells the truth and each liar always lies. One day 12 islanders gathered together and issued a few statements. Two people said “Exactly two of us are liars”. Another four people said “Exacly four of us are liars”. The remaining six said “Exactly six of us are liars”. How many liars are there. [AMTI 2008] (A) 10 (B) 8 (C) 6 (D) 4

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11.

In one move the king can go to any adjacent square, along a row, column or diagonal. How many routes with the minimum number of moves are there for the king to travel from the top left square to the bottom [AMTI 2008] right square ? K

K

(A) 35 12.

(B) 2

(C) 15

(D) 4

Four students Anu, Banu, Chandu, Dheeraj make the following statements. Anu : Banu, Chandu and Dheeraj are Math Majors. Banu : Anu, Chandu and DHeeraj are commerce majors. Chandu : Both Anu and Banu are lying DHeeraj : Anu, Banu and Chandu are telling the truth Then the number of students telling the truth is. (A) 0 (B) 1 (C) 2 (D) 3

[AMTI 2009]

13.

Ananya wrote all the integers of 1 to 6 digits. She uses only the two digits 0 and 5. In all, how many 0,s did she use ? (leading zeros not allowed in the numbers) [AMTI 2009] (A) 256 (B) 128 (C) 129 (D) 200

14.

In a kilometer race Ram beats Shyam by 25 meters or 5 seconds. The time taken by Ram to complete [AMTI 2010] the race is (A) 1 minute (B) 5 minutes and 30 seconds (C) 3 minutes and 15 seconds

15.

(D) 4 minutes and 10 seconds

If the average of n observations (where n is odd) arranged in ascending order is a, the average of the first

n 1 n 1 n 1 observation is b and that of the last observations is c, then the th observation is 2 2 2 [AMTI 2009] (A) (n + 1) (b + c) – na

(B)

n(a  b  c ) 3

(C)

(n  1)(b  c )  2na 2

(D)

abc 3n

16.

A, B and C run for a race which is a straight road of x meters. A beats B by 30 meters B beats C by 20 meters, A beats C by 48 meters. Then x (in meters) is [AMTI 2014] (A) 150 (B) 200 (C) 300 (D) 500

17.

By rearranging the digits of the integer 1288, we get a total of 12 different integers including 1288. The sum of all these twelve integers is [AMTI 2014] (A) 577727 (B) 63327 (C) 72227 (D) 466627

18.

There are 2014 people sitting around a big round table dinner. Each person shakes hands with everybody except the persons sitting on both sides of him. The total number of handshakes that takes place is [AMTI 2014] (A) 1007  2014 (B) 2014  2012 (C) 1007  2011 (D) 1007  2012

19.

A merchant has 100 kg sugar, part of which he sells at 7% profit and the rest at 17% profit. He gais 10% [AMTI 2015] on the whole. The amount of sugar (in kg) he sold at 7% profit is : (A) 60 (B) 50 (C) 80 (D) 70

20.

A train leaves a station 1 hour before the scheduled time. The driver decreases the speed by 4 km/h. At the next station 120 km away, the train reached on scheduled time. The original speed of the train is (in km/h : [AMTI 2015] (A) 24 (B) 36 (C) 18 (D) 22

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EXERCISE –II SUBJECTIVE

1.

In a chess tournament players get 1 point for a win 0 for a loss and

1 point for a draw. In a tournament 2

where every player plays against every other player exactly once, the top four scores were 5 and 2

1 . The lowest score in the tournament was ______. 2

1 1 , 4 ,4 2 2

[AMTI 2011]

2.

Seven digit numbers are formed by the digits 1,2,3,4,5,6 and 7. in each number no digit is repeated.Prove that among all these numbers there is no number which is a multiple of another number. [AMTI 2012]

3.

The contents of two vessels containing water and milk in the ratio 1 : 2 and 2 : 5 are mixed in the ratio 1 : 4 The resulting mixture will have water and milk in the ratio _______. [AMTI 2014]

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ANSWER KEY NUMBER SYSTEM EXERC ISE-1 Que s. Ans. Que s. Ans. Que s. Ans. Que s. Ans. Que s. Ans. Que s. Ans.

1 A 16 A 31 D 46 A 61 B 76 C

2 B 17 A 32 B 47 C 62 C 77 A

3 B 18 D 33 C 48 C 63 A 78 D

4 D 19 B 34 A 49 A 64 A 79 C

5 A 20 C 35 D 50 B 65 B 80 A

6 B 21 A 36 B 51 A 66 D

7 C 22 C 37 D 52 D 67 C

8 A 23 C 38 B 53 B 68 B

9 D 24 A 39 C 54 C 69 A

10 A 25 D 40 C 55 C, D 70 C

11 D 26 A 41 B 56 B 71 D

12 B 27 C 42 A 57 C 72 B

13 A 28 D 43 C 58 B 73 D

14 C 29 B 44 B 59 D 74 B

15 A 30 D 45 A 60 B 75 D

13 D 28 B 43 C 58 C

14 B 29 D 44 C 59 C

15 C 30 C 45 D 60 C

EXERC ISE-2

1.

5149

2.

6929

3.

530

4.

{a,e}

5.

3

6.

16

7.

9

8.

36

9.

885309

10.

28053

11.

544

12.

333/67

13.

4024

14.

5

15.

95

16.

7

17. 21.

10 142857

18. 22.

12 625

19. 23.

1236 757

20. 24.

10 1

25.

726

26.

27.

15

28.

841 +

29.

6

30.

7

11

ALGE BR A EXERC ISE-1

Que. Ans. Que. Ans. Que. Ans. Que. Ans.

1 C 16 B 31 C 46 B

2 D 17 A 32 B 47 D

3 A 18 C 33 B 48 D

4 D 19 C 34 C 49 C

5 D 20 D 35 A 50 C

6 A 21 D 36 D 51 B

7 A 22 A 37 A 52 C

8 A 23 C 38 C 53 D

9 24 C 39 C 54 B

10 C 25 C 40 C 55 A

11 D 26 D 41 D 56 C

12 A 27 A 42 A 57 B

EXERC ISE-2

1. 5. 9. 13. 17.

2009 35 1 n=4 2013

2. 6. 10. 14. 18.

3 45 –16 x = 3, y = 1 0

3. 7. 11. 15. 19.

2008 22 3 5 one pair i.e(12,3)

4. 8. 12. 16.

4 3 9 no real value

20.

3

21.

2 3

22.

4

23.

1

24.

1 2015

25.

2430

26.

28

27.

5

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GEOMETRY EXERC ISE-1 Que s. Ans. Que s. Ans. Que s. Ans. Que s. Ans. Que s. Ans. Que s. Ans.

1 B 16 A 31 A 46 C 61 C 76 D

2 A 17 C 32 A 47 D 62 D 77 C

3 C 18 D 33 A 48 A 63 A 78 D

4 D 19 D 34 C 49 D 64 A 79 A

5 B 20 B 35 D 50 C 65 D

6 A 21 B 36 A 51 A 66 B

7 D 22 C 37 C 52 A 67 C

8 B 23 A 38 A 53 D 68 B

9 C 24 A 39 B 54 C 69 C

10 B 25 D 40 B 55 A 70 A

11 C 26 A 41 A 56 A 71 C

12 A 27 A 42 B 57 D 72 D

13 B 28 A 43 D 58 C 73 A

14 B 29 C 44 A 59 C 74 B

15 B 30 C 45 B 60 A 75 B

EXERC ISE-2

1. 5.

1:1 50

2. 6.

1:1 7.2 sq.units

3. 7.

4 cm2 45

4. 8.

24 1 : 4 or 4 : 1

9.

40

10.

240

11.

2

12.

25 4

13.

100°

14.

3 3

15.

2

16.

0

17.

2BAC

18.

32( –

19.

2012

20.

200

21. 25. 29. 33. 37.

5 cm 3:1 2:1 24 2120

22. 26. 30. 34.

56/9 1:2014 2 6

23. 27. 31. 35.

8/5

24. 28. 32. 36.

2 7 400 10

1 4

6.

3)

30° 24

INEQUALITIES EXERC ISE-1

Ques. Ans.

1 C

2 D

3 C

4 A

5 C

6 D

7 B

8 A

18 5

5.

EXERC ISE-2

1.

9

2.

3

7.

3

8.

6

18

1 8

3.

4.

3

MISCELLANEOUS EXERC ISE-1

Que s. Ans. Que s. Ans.

1 11 D

2 C 12 B

3 C 13 C

4 B 14 C

5 A 15 C

6 A 16 C

7 A 17

8 B 18 C

9 A 19 D

10 C 20 A

EXERC ISE-2

1.

1

1 1 , 1, 2 2

3.

31/74

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[39]