2017

B OOKLET N O . T EST C ODE : MMA

Forenoon

Questions : 30 Time : 2 hours

Write your Name, Registration Number, Test Centre, Test Code and the Number
of this booklet in the appropriate places on the answersheet.

For each question, there are four suggested answers of which only one
is correct. For each question indicate your choice of the correct answer by
c
darkening the appropriate oval ( ) completely on the answer sheet.

4 marks are allotted for each correct answer,

0 mark for each incorrect answer and
1 mark for each unattempted question.

A LL ROUGH WORK MUST BE DONE ON THIS BOOKLET ONLY.

Y OU ARE NOT ALLOWED TO USE CALCULATORS IN ANY FORM .

STOP! WAIT FOR THE SIGNAL TO START.

MMAo -1
1. The area lying in the first quadrant and bounded by the circle

x2 + y 2 = 4

and lines
x = 0 and x = 1

is given by
√ √ √ √
π 3 π 3 π 3 π 3
(A) + (B) + (C) − (D) +
3 2 6 4 3 2 6 2

2. If (x1 , y1 ) and (x2 , y2 ) are the opposite endpoints of a diameter of a

circle, then the equation of the circle is given by

(A) (x − x1 )(y − y1 ) + (x − x2 )(y − y2 ) = 0

(B) (x − x1 )(y − y2 ) + (x − x2 )(y − y1 ) = 0
(C) (x − x1 )(x − x2 ) + (y − y1 )(y − y2 ) = 0
(D) (x − x1 )(x − x2 ) = (y − y1 )(y − y2 ) = 0

3. If α, β and γ are the roots of the equation x3 + 3x2 − 8x + 1 = 0, then

an equation whose roots are α + 1, β + 1 and γ + 1 is given by

(A) y 3 − 11y + 11 = 0 (B) y 3 − 11y − 11 = 0

(C) y 3 + 13y + 13 = 0 (D) y 3 + 6y 2 + y − 3 = 0

4. Let S ⊆ R. Consider the statement:

“There exists a continuous function f : S → S such that

f (x) 6= x for all x ∈ S.”

This statement is false if S equals

(A) [2, 3] (B) (2, 3] (C) [−3, −2] ∪ [2, 3] (D) (−∞, ∞)

5. If A is a 2 × 2 matrix such that trace A = det A = 3, then what is the

trace of A−1 ?

(A) 1 (B) 1/3 (C) 1/6 (D) 1/2

3
 6. In a class of 80 students, 40 are girls and 40 are boys. Also, exactly 50
students wear glasses. Then the set of all possible numbers of boys
without glasses is

(A) {0, . . . , 30} (B) {10, . . . , 30} (C) {0, . . . , 40} (D) none of these

7. Let n ≥ 3 be an integer. Then the statement

n+1
(n!)1/n ≤
2
is

(A) true for every n ≥ 3

(B) true if and only if n ≥ 5
(C) not true for n ≥ 10
(D) true for even integers n ≥ 6, not true for odd n ≥ 5

8. Let X1 , X2 and X3 be chosen independently from the set {0, 1, 2, 3, 4},

each value being equally likely. What is the probability that the arith-
metic mean of X1 , X2 and X3 is the same as their geometric mean?

1 1 3! 3
(A) (B) (C) (D)
52 53 53 53

9. A function y(x) that satisfies

dy
+ 4xy = x
dx
with the boundary condition y(0) = 0 is

1 2
(A) y(x) = (1 − ex ) (B) y(x) = (1 − e−2x )
4
1 2 1
(C) y(x) = (1 − e2x ) (D) y(x) = (1 − cos x)
4 4

10. The inequality |x2 − 5x + 4| > (x2 − 5x + 4) holds if and only if

(A) 1 < x < 4 (B) x ≤ 1 and x ≥ 4

(C) 1 ≤ x ≤ 4 (D) x takes any value except 1 and 4

4
11. The digit in the unit’s place of the number 20172017 is

(A) 1 (B) 3 (C) 7 (D) 9

12. Which of the following statements is true?

(A) There are three consecutive integers with sum 2015

(B) There are four consecutive integers with sum 2015

(C) There are five consecutive integers with sum 2015

(D) There are three consecutive integers with product 2015

13. An even function f (x) has left derivative 5 at x = 0. Then

(A) the right derivative of f (x) at x = 0 need not exist

(B) the right derivative of f (x) at x = 0 exists and is equal to 5

(C) the right derivative of f (x) at x = 0 exists and is equal to −5

(D) none of the above is necessarily true

14. Let (vn ) be a sequence defined by v1 = 1 and

s  n
2
1
vn+1 = vn +
5

for n ≥ 1. Then lim vn is

n→∞

p p
(A) 5/3 (B) 5/4 (C) 1 (D) nonexistent

15. The diagonal elements of a square matrix M are odd integers while
the off-diagonals are even integers. Then

(A) M must be singular

(B) M must be nonsingular

(C) there is not enough information to decide the singularity of M

(D) M must have a positive eigenvalue

5
16. Let (xn ) be a sequence of real numbers such that the subsequences
(x2n ) and (x3n ) converge to limits K and L respectively. Then

(A) (xn ) always converges

(B) if K = L, then (xn ) converges
(C) (xn ) may not converge, but K = L
(D) it is possible to have K 6= L

17. Suppose that X is chosen uniformly from {1, 2, ..., 100} and given
X = x, Y is chosen uniformly from {1, 2, ..., x}. Then P (Y = 30) =
 
1 1 1 1
(A) (B) × + ··· +
100 100 30 100

1 1 1 1
(C) (D) × + ··· +
30 100 1 30

18. Consider the following system of equations:

    
1 2 3 4 x1 0
    
 5 6 7 8   x2   0 
 x  =  0
    .
 a 9 b 10 
  3   
6 8 10 13 x4 0

The locus of all (a, b) ∈ R2 such that this system has at least two dis-
tinct solutions for (x1 , x2 , x3 , x4 ) is

(A) a parabola (B) a straight line (C) entire R2 (D) a point

19. If α, β and γ are the roots of x3 − px + q = 0, then the value of the

determinant  
 α β γ 
 
 
 β γ α 
 
 
 γ α β 

is

(A) p (B) p2 (C) 0 (D) p2 + 6q

6
20. The number of ordered pairs (X, Y ), where X and Y are n × n real
matrices such that XY − Y X = I is

(A) 0 (B) 1 (C) n (D) infinite

21. There are four machines and it is known that exactly two of them are
faulty. They are tested one by one in a random order till both the
faulty machines are identified. The probability that only two tests are
required is

1 1 1 1
(A) (B) (C) (D)
2 3 4 6

22. The five vowels—A, E, I, O, U—along with 15 X’s are to be arranged

in a row such that no X is at an extreme position. Also, between any
two vowels there must be at least 3 X’s. The number of ways in which
this can be done is

(A) 1200 (B) 1800 (C) 2400 (D) 3000

23. What is the smallest degree of a polynomial with real coefficients and
having roots 2ω, 2 + 3ω, 2ω 2 , −1 − 3ω and 2 − ω − ω 2 ?

[Here ω 6= 1 is a cube root of unity.]

(A) 5 (B) 7 (C) 9 (D) 10

24. The number of polynomial functions f of degree ≥ 1 satisfying

f (x2 ) = (f (x))2 = f (f (x))

for all real x, is

(A) 0 (B) 1 (C) 2 (D) infinitely many

7
25. For a, b ∈ R, and b > a, the maximum possible value of the integral

Z b
(7x − x2 − 10)dx
a

is

7 9 11
(A) (B) (C) (D) none of these
2 2 2

26. Let n be the number of ways in which 5 men and 7 women can stand
in a queue such that all the women stand consecutively. Let m be the
number of ways in which the same 12 persons can stand in a queue
m
such that exactly 6 women stand consecutively. Then the value of n
is

5 7
(A) 5 (B) 7 (C) (D)
7 5

27. A box contains 5 fair coins and 5 biased coins. Each biased coin has
probability of a head 54 . A coin is drawn at random from the box and
tossed. Then a second coin is drawn at random from the box (without
replacing the first one). Given that the first coin has shown head, the
conditional probability that the second coin is fair, is

20 20 1 7
(A) (B) (C) (D)
39 37 2 13

28. Let H be a subgroup of a group G and let N be a normal subgroup of

G. Choose the correct statement:

(A) H ∩ N is a normal subgroup of both H and N

(B) H ∩ N is a normal subgroup of H but not necessarily of N

(C) H ∩ N is a normal subgroup of N but not necessarily of H

(D) H ∩ N need not be a normal subgroup of either H or N

8
29. Suppose the rank of the matrix
 
1 1 2 2
 
 1 1 1 3 
 
a b b 1

is 2 for some real numbers a and b. Then b equals

(A) 1 (B) 3 (C) 1/2 (D) 1/3

30. The graph of a cubic polynomial f (x) is shown below. If k is a con-

stant such that f (x) = k has three real solutions, which of the follow-
ing could be a possible value of k?

6
3
0
-3
-6
-9
-12

(A) 3 (B) 0 (C) −7 (D) −3

9
 2017

B OOKLET N O . T EST C ODE : MMA

Forenoon

Questions : 30 Time : 2 hours

Write your Name, Registration Number, Test Centre, Test Code and the Number
of this booklet in the appropriate places on the answersheet.

For each question, there are four suggested answers of which only one
is correct. For each question indicate your choice of the correct answer by
c
darkening the appropriate oval ( ) completely on the answer sheet.

4 marks are allotted for each correct answer,

0 mark for each incorrect answer and
1 mark for each unattempted question.

A LL ROUGH WORK MUST BE DONE ON THIS BOOKLET ONLY.

Y OU ARE NOT ALLOWED TO USE CALCULATORS IN ANY FORM .

STOP! WAIT FOR THE SIGNAL TO START.

MMAe -1
1. If A is a 2 × 2 matrix such that trace A = det A = 3, then what is the
trace of A−1 ?

(A) 1 (B) 1/3 (C) 1/6 (D) 1/2

2. If α, β and γ are the roots of the equation x3 + 3x2 − 8x + 1 = 0, then

an equation whose roots are α + 1, β + 1 and γ + 1 is given by

(A) y 3 − 11y + 11 = 0 (B) y 3 − 11y − 11 = 0

(C) y 3 + 13y + 13 = 0 (D) y 3 + 6y 2 + y − 3 = 0

3. If α, β and γ are the roots of x3 − px + q = 0, then the value of the

determinant  
 α β γ 
 
 
 β γ α 
 
 
 γ α β 

is

(A) p (B) p2 (C) 0 (D) p2 + 6q

4. The number of polynomial functions f of degree ≥ 1 satisfying

f (x2 ) = (f (x))2 = f (f (x))

for all real x, is

(A) 0 (B) 1 (C) 2 (D) infinitely many

5. If (x1 , y1 ) and (x2 , y2 ) are the opposite endpoints of a diameter of a

circle, then the equation of the circle is given by

(A) (x − x1 )(y − y1 ) + (x − x2 )(y − y2 ) = 0

(B) (x − x1 )(y − y2 ) + (x − x2 )(y − y1 ) = 0
(C) (x − x1 )(x − x2 ) + (y − y1 )(y − y2 ) = 0
(D) (x − x1 )(x − x2 ) = (y − y1 )(y − y2 ) = 0

3
6. Consider the following system of equations:
    
1 2 3 4 x1 0
    
 5 6 7 8   x2   0 
 a 9 b 10   x  =  0  .
    
  3   
6 8 10 13 x4 0

The locus of all (a, b) ∈ R2 such that this system has at least two dis-
tinct solutions for (x1 , x2 , x3 , x4 ) is

(A) a parabola (B) a straight line (C) entire R2 (D) a point

7. The inequality |x2 − 5x + 4| > (x2 − 5x + 4) holds if and only if

(A) 1 < x < 4 (B) x ≤ 1 and x ≥ 4

(C) 1 ≤ x ≤ 4 (D) x takes any value except 1 and 4

8. Suppose the rank of the matrix

 
1 1 2 2
 
 1 1 1 3 
 
a b b 1

is 2 for some real numbers a and b. Then b equals

(A) 1 (B) 3 (C) 1/2 (D) 1/3

9. A box contains 5 fair coins and 5 biased coins. Each biased coin has
probability of a head 54 . A coin is drawn at random from the box and
tossed. Then a second coin is drawn at random from the box (without
replacing the first one). Given that the first coin has shown head, the
conditional probability that the second coin is fair, is

20 20 1 7
(A) (B) (C) (D)
39 37 2 13

4
10. Let (xn ) be a sequence of real numbers such that the subsequences
(x2n ) and (x3n ) converge to limits K and L respectively. Then

(A) (xn ) always converges

(B) if K = L, then (xn ) converges
(C) (xn ) may not converge, but K = L
(D) it is possible to have K 6= L

11. Let X1 , X2 and X3 be chosen independently from the set {0, 1, 2, 3, 4},
each value being equally likely. What is the probability that the arith-
metic mean of X1 , X2 and X3 is the same as their geometric mean?
1 1 3! 3
(A) (B) (C) (D)
52 53 53 53
12. Let S ⊆ R. Consider the statement:

“There exists a continuous function f : S → S such that

f (x) 6= x for all x ∈ S.”

This statement is false if S equals

(A) [2, 3] (B) (2, 3] (C) [−3, −2] ∪ [2, 3] (D) (−∞, ∞)

13. Let n ≥ 3 be an integer. Then the statement

n+1
(n!)1/n ≤
2
is

(A) true for every n ≥ 3

(B) true if and only if n ≥ 5
(C) not true for n ≥ 10
(D) true for even integers n ≥ 6, not true for odd n ≥ 5

14. In a class of 80 students, 40 are girls and 40 are boys. Also, exactly 50
students wear glasses. Then the set of all possible numbers of boys
without glasses is

(A) {0, . . . , 30} (B) {10, . . . , 30} (C) {0, . . . , 40} (D) none of these

5
15. The diagonal elements of a square matrix M are odd integers while
the off-diagonals are even integers. Then

(A) M must be singular

(B) M must be nonsingular
(C) there is not enough information to decide the singularity of M
(D) M must have a positive eigenvalue

16. The five vowels—A, E, I, O, U—along with 15 X’s are to be arranged

in a row such that no X is at an extreme position. Also, between any
two vowels there must be at least 3 X’s. The number of ways in which
this can be done is

(A) 1200 (B) 1800 (C) 2400 (D) 3000

17. An even function f (x) has left derivative 5 at x = 0. Then

(A) the right derivative of f (x) at x = 0 need not exist

(B) the right derivative of f (x) at x = 0 exists and is equal to 5
(C) the right derivative of f (x) at x = 0 exists and is equal to −5
(D) none of the above is necessarily true

18. There are four machines and it is known that exactly two of them are
faulty. They are tested one by one in a random order till both the
faulty machines are identified. The probability that only two tests are
required is

1 1 1 1
(A) (B) (C) (D)
2 3 4 6
19. A function y(x) that satisfies
dy
+ 4xy = x
dx
with the boundary condition y(0) = 0 is

1 2
(A) y(x) = (1 − ex ) (B) y(x) = (1 − e−2x )
4
1 2 1
(C) y(x) = (1 − e2x ) (D) y(x) = (1 − cos x)
4 4

6
20. Let H be a subgroup of a group G and let N be a normal subgroup of
G. Choose the correct statement:

(A) H ∩ N is a normal subgroup of both H and N

(B) H ∩ N is a normal subgroup of H but not necessarily of N
(C) H ∩ N is a normal subgroup of N but not necessarily of H
(D) H ∩ N need not be a normal subgroup of either H or N

21. Let n be the number of ways in which 5 men and 7 women can stand
in a queue such that all the women stand consecutively. Let m be the
number of ways in which the same 12 persons can stand in a queue
m
such that exactly 6 women stand consecutively. Then the value of n
is

5 7
(A) 5 (B) 7 (C) (D)
7 5

22. What is the smallest degree of a polynomial with real coefficients and
having roots 2ω, 2 + 3ω, 2ω 2 , −1 − 3ω and 2 − ω − ω 2 ?
[Here ω 6= 1 is a cube root of unity.]

(A) 5 (B) 7 (C) 9 (D) 10

23. Which of the following statements is true?

(A) There are three consecutive integers with sum 2015

(B) There are four consecutive integers with sum 2015
(C) There are five consecutive integers with sum 2015
(D) There are three consecutive integers with product 2015

24. Suppose that X is chosen uniformly from {1, 2, ..., 100} and given
X = x, Y is chosen uniformly from {1, 2, ..., x}. Then P (Y = 30) =
 
1 1 1 1
(A) (B) × + ··· +
100 100 30 100

1 1 1 1
(C) (D) × + ··· +
30 100 1 30

7
25. The graph of a cubic polynomial f (x) is shown below. If k is a con-
stant such that f (x) = k has three real solutions, which of the follow-
ing could be a possible value of k?

6
3
0
-3
-6
-9
-12

(A) 3 (B) 0 (C) −7 (D) −3

26. The area lying in the first quadrant and bounded by the circle

x2 + y 2 = 4

and lines
x = 0 and x = 1

is given by
√ √ √ √
π 3 π 3 π 3 π 3
(A) + (B) + (C) − (D) +
3 2 6 4 3 2 6 2

27. For a, b ∈ R, and b > a, the maximum possible value of the integral
Z b
(7x − x2 − 10)dx
a

is

7 9 11
(A) (B) (C) (D) none of these
2 2 2

28. The digit in the unit’s place of the number 20172017 is

(A) 1 (B) 3 (C) 7 (D) 9

8
29. Let (vn ) be a sequence defined by v1 = 1 and
s  n
2
1
vn+1 = vn +
5

for n ≥ 1. Then lim vn is

n→∞

p p
(A) 5/3 (B) 5/4 (C) 1 (D) nonexistent

30. The number of ordered pairs (X, Y ), where X and Y are n × n real
matrices such that XY − Y X = I is

(A) 0 (B) 1 (C) n (D) infinite