PYQ 2016 - VedPrep

IIT JAM MATHEMATICS - 2016
Previous Year Question Paper with Solution.

1. The sequence {s_n} of real numbers given by

(a) a divergent sequence

(b) an oscillatory sequence

(d) a Cauchy sequence

Ans. (d)

Sol.

Option (d) is correct.

2. Let P be the vector space (over R) of all polynomials of degree < 3 with real coefficients. Consider the linear transformation defined by T(a₀ + a₁ x + a₁x² + a₃x³) = a₃ + a₂x + a₁x² + a₀x³. Then the matrix representation M of T with respect to the ordered basis {1, x, x², x³} satisfies

(a) M² + I₄ = 0

(b) M² – I₄ = 0

(d) M + I₄ = 0

Ans. (b)

Sol.

Hence, M² – I₄ = 0.

Option (b) is correct.

3. Let be a continuous function. Then the integral is equivalent to

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Option (a) is correct.

4. Let be an element of the permutation group S₅. Then the maximum possible order of is

(a) 5

(b) 6

(d) 15

Ans. (a)

Sol. Partition of 5 5 = 3 + 2 LCM – 6 – max

= 3 + 1 + 1 LCM – 3

= 4 + 1 LCM – 4

= 2 + 2 + 1 LCM – 2

= 2 + 1 + 1 + 1 LCM – 2

= 1 + 1 + 1 + 1 + 1 LCM – 1

= 5 LCM – 5

Hence, maximum possible LCM(3, 2) = 6.

So maximum order of element in S₆ is 6.

Option (a) is correct.

5. Let f be a strictly monotonic continuous real valued function defined on [a, b] such that f(a) < a and f(b) > b. Then which one of the following is true?

(a) There exists exactly one such that f (c) = c.

(b) There exists exactly two points such that f (c_i) = c_i, i = 1, 2.

(d) There exist infinitely many points such that f (c) = c.

Ans. (b)

Sol. Define g(x) = f (x) – x then g is continuous function on R

g(a) = f (a) – a < 0

g(b) = f(b) – b > 0

by LMVT (a, b) such that g(c) = 0

If there exist another such that g(d) = 0 then by Rolle's Theorem but g also monotonic function, so there is no another root of g.

Option (b) is correct.

6. The value of is

(a) 0

(b) 4

(d) 2

Ans. (b)

Sol. The value of .

Option (b) is correct.

7. Let . If f(r) = ln r and g(r) = satisfy then h(r) is

(a) r

(b) 2

(d) 2/r

Ans. (c)

Sol. f(r) = ln r

Option (c) is correct.

8. The non-zero value of n for which the differential equation (3xy² + n²x²y)dx + (nx³ + 3x²y)dy = 0, x 0 becomes exact is

(a) –3

(b) –2

(d) 3

Ans. (d)

Sol. The non zero value of n for which the differential equation (3xy² + n²x²y)dx + (nx³ + 3x²y)dy = 0, x 0 becomes exact is

Option (d) is correct.

9. One of the points which lies on the solution curve of the differential equation (y – x)dx + (x + y)dy = 0 with the given condition y (0) = 1 is

(a) (1, –2)

(b) (2, –1)

(d) (–1, 2)

Ans. (c)

Sol. One of the points which lies on the solution curve of the differential equation (y – x)dx + (x + y)dy = 0 with the given condition y(0) = 1 is (y – x)dx + (x + y)dy = 0 using exact

(2, 1) satisfies the equation.

Option (c) is correct.

10. Let S be a closed subset of R, the compact subset of R such that is

(a) closed but not compact

(b) not closed

(d) neither closed nor compact

Ans. (c)

Sol. Let S be a closed subset of R, T a compact subset of R such that is compact.

Option (c) is correct.

11. Let S be the series and T be the series of real numbers. Then which one of the following is true?

(a) Both the series S and T are convergent

(b) S is convergent and T is divergent

(d) Both the series S and T are divergent

Ans. (b)

Sol. S = = convergent.

So, S is convergent, also by using limit comparison test T is divergent.

Option (b) is correct.

12. Let {a_n} be a sequence of positive real numbers satisfying . Then all the terms of the sequence lie in

(a)

(b) [0, 1]

(d) [1, 3]

Ans. (d)

Sol.

Option (d) is correct.

13. The largest eigenvalue of the matrix is

(a) 16

(b) 21

(d) 64

Ans. (b)

Sol. A =

21 is an eigenvalue of A.

Option (b) is correct.

14. The value of the integral is

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

Option (c) is correct answer.

15. If the triple integral over the region bounded by the planes 2x + y + z = 4, x = 0, y = 0, z = 0 is given by then the function is

(a) x + y

(b) x – y

(d) y

Ans. (d)

Sol.

Option (d) is correct.

16. The surfaces area of the portion of the plane y + 2z = 2 within the cylinder x² + y² = 3 is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

R_xy is circle x² + y² = 1.

Value of integral is area of the circle =

Option (a) is correct.

17. Let be define by f(x, y) = . Then the value of at the point (0, 0) is

(a) 0

(b) 1

(d) 4

Ans. (b)

Sol.

The value of at the point (0, 0) is 1.

Option (b) is correct answer.

18. The function f (x, y) = 3x²y + 4y³ – 3x² –12y² +1 has a saddle point at

(a) (0, 0)

(b) (0, 2)

(d) (–2, 1)

Ans. (d)

Sol. f(x, y) = 3x²y + 4y² – 3x²–12y²+ 1

for the condition of saddle point rt – s² < 0

= (6y – 6) × (24y – 24) – [6x]²< 0 = 144 (y – 1)²– 36x²< 0

36[4(y – 1)² – x²] < 0, (2, 1) satisfied the point this inequality.

Option (d) is correct answer.

19. Consider the vector field where If the absolute value of the line integral along the closed curve C : x² + y² = a² (oriented counter clockwise) is then is

(a) –2

(b) –1

(d) 2

Ans. (a)

Sol. Here

Solving line integral by parametric form

Therefore, absolute value will be

Option (a) is correct.

20. Let S be the surface of the cone z = bounded by the planes z = 0 and z = 3. Further, let C be the closed curve forming the boundary of the surfaces S. A vector field The absolute value of the integral where is

(a) 0

(b) 9

(d) 18

Ans. (d)

Sol.

because we have to calculate absolute value so need to consider sign of normal

Option (d) is correct answer.

21. Let y(x) be the solution of the differential equation Then y(2) is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Let y(x) be the solution of the differential equation then y(2) is

Option (b) is correct.

22. The general solution of the differential equation with constant coefficients approaches zero as if

(a) b is negative and c is positive

(b) b is positive and c is negative

(d) both b and c are negative

Ans. (c)

Sol. Given differential equation is and corresponding auxiliary equation can be written as then the roots of the above equation can be written as

1. If b < 0 and c > 0, the n nature of roots depends on the sing of b² – 4c. If b² – 4c > 0 then we can write where A is some positive constant. In this case, we obtain < 0 and Hence, On the other hand, if we have b² – 4x < 0 then the roots of auxiliary equation are and the corresponding solution of given ODE is .

Clearly, being b < 0.

2. If b < 0 and c < 0 then by the similar reasoning as done above, we have Hence in this case, we have

3. If b < 0 and c > 0 then nature of roots depends on the sing of b² – 4c. If b² –4c > 0 then we can write where S is some positive constant. In this case, we obtain < 0 and < 0. Hence On the other hand, if we have b² – 4c < 0 then the roots of auxiliary equation are l = and the corresponding solution of given ODE is

4. If b < 0 and a < 0 then by the similar reasoning reasoning as above, we have Hence, in this case, we have

Option (c) is correct.

23. Let denote the set of points x in R such that every neighbourhood of x contains some points of S as well as some points of complement of S. Further, let denote the closure of S. Then which one of the following is false?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Take

and

Option (c) is correct.

24. The sum of the series is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Given series is

Option (c) is correct.

25. Let f(x) = for all Then which one of the following is true?

(a) Maximum value of f (x) is 3/2

(b) Minimum value of f (x) is 1/3

(d) Minimum of f(x) occurs at x = 1

Ans. (a)

Sol. Let

the integral

condition for maximum and minimum f(x) = 0 (for stationary point)

So, maximum value is 3/1.

Option (a) is correct.

26. The matrix M = is a unitary matrix when is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

Option (a) is correct.

27. Let M = and b a non–zero vector such that Mx = b for some . Then the value of x^Tb is

(a)

(b)

(d) 1

Ans. (c)

Sol. given Mx = b is consistent.

Changing in echelon form

Option (c) is correct answer.

28. The number of group homomorphism from the cyclic group to the cyclic group is

(a) 7

(b) 3

(d) 1

Ans. (d)

Sol. Number of group homomorphism from is gcd(m, n).

So group homomorphism from is gcd(4, 7) = 1.

Option (d) is correct.

29. In the permutation group S_n(n > 5), if H is the smallest subgroup containing all the 3-cycles then which one of the following is true?

(a) Order of H is 2

(b) Index of H in S_n is 2

(d) H = S_n

Ans. (b)

Sol. that is index 2.

Option (b) is correct.

30. Let be defined as f(x) = . Then, at x = 0, the function f is

(a) continuous and differentiable when a = 0

(b) continuous and differentiable when a > 0

(d) continuous and differentiable when a < –1

Ans. (b)

Sol. It is continuous if and differentiable if

Option (b) is correct.

31. Let {s_n} be a sequence of positive real numbers satisfying 2s_{n + 1} = are the roots of the equation x² – 2x + = 0 and then which of the following statement (s) is/are true?

(a) {s_n} is monotonically decreasing

(b) {s_n} is monotonically increasing

(c)

(d)

Ans. (a), (c)

Sol. Sequence {s_n}, s_{n + 1}> 0 where 2s_{n + 1} = are the roots of x² – 2x + = 0.

Further, we have because {s_n} is monotonically decreasing.

Hence

Option (a), (c) is correct answer.

32. The value(s) of the integral is/are

(a) 0 when n is even

(b) 0 when n is odd

(d) when n is odd

Ans. (a), (d)

Sol.

Option (a), (d) is correct.

33. Let be defined by

f(x, y) = .

Then at the point (0, 0), which of the following statement(s) is/are true?

(a) f is not continuous

(b) f is continuous

(d) Both first order partial derivatives off exist

Ans. (b), (d)

Sol.

and

Option (b), (d) is correct.

34. Consider the vector field on an open connected set . Then which of the following statement(s) is/are true?

(a) Divergence of is zero on S.

(b) The line integral of is independent of path in S.

(d) The line of is zero around any piecewise smooth closed path in S.

Ans. (b), (c), (d)

Sol.

Option (b), (c), (d) is correct.

35. Consider the differential equation . Then which of the following statement(s) is/are true?

(a) The solution is unbounded when

(b) The solution is unbounded when

(d) The solution is bounded when

Ans. (c), (d)

Sol. Consider the differential equation

Integrating factor

Solution of the differential equation

The solution is unbounded when the solution is when

Option (c), (d) is correct.

36. Which of the following statement(s) is/are true?

(a) There exists a connected set in R which is not compact.

(b) Arbitrary union of closed intervals in R need not be compact.

(d) Every bounded infinite subset V of R has a limit point in V itself.

Ans. (a), (b)

Sol.

Option (a), (b) is correct.

37. Let P(x) = . Then which of the following statement(s) is/are true?

(a) The equation P(x) = 0 has exactly one solution in .

(b) P(x) is strictly increasing for all .

(d) P(x) is strictly decreasing for all .

Ans. (a), (d)

Sol.

If P(x) is strictly decreasing function let

Option (a), (d) is correct.

38. Let G be a finite group and O(G) denotes its order. Then which of the following statement(s) is/are true?

(a) G is abelian if O(G) denotes = pq where p and q are distinct primes.

(b) G is abelian if every non identity element of G is of order 2.

(d) G is abelian if O(G) = p³ where p is prime.

Ans. (b), (c)

Sol. For option (a), counter example is D₅, so option (a) is incorrect.

For option (d), counter example is D₄, so option (d) is also incorrect.

For option (b), we know that if every non identity element of G is of order 2 then G must be abelian because all elements of the group is self inverse. Hence option (b) is correct.

For option (c), if the quotient group G/Z(G) is cyclic then G must be abelian where Z(G) is the centre of G.

Option (b), (c) is correct.

39. Consider the set V = . For which of the following choice(s) the set V becomes a two dimensional subspace of ?

(a)

(b)

(c)

(d)

Ans.

Sol.

40. Let S = . Then which of the following statement(s) is/are true?

(a) S is closed

(b) S is not open

(d) 0 is a limit point of S

Ans. (b), (d)

Sol. Let S = then '0' is limit point of S and S is not connected because S is discrete and neither open nor closed.

Option (b), (d) is correct.

41. Let {s_n} be a sequence of real numbers given by . Then the least upper bound of the sequence {s_n} is ________ .

Ans. 0.5

Sol. {s_n} where s_n = then least upper bound of {s_n} is the of the set A = .

Notice that

Hence,

Correct answer is 0.5.

42. Let {s_k} be a sequence of real numbers where is ________.

Ans. 1

Sol.

Correct answer is 1.

43. Let x = be a non-zero vector and A = . Then the dimension of the vector space is ________ .

Ans. 2

Sol.

According to question we have to find nullity of A =

i.e., nullity of xx^T so change in echelon form.

Correct answer is 2.

44. Let f be a real valued function defined by f (x, y) = 2 ln(x²y²e^y/x), x > 0, y > 0 then the value of at any point (x,y) where x > 0, y > 0 is_________.

Ans. 8

Sol. f(x, y) = 2log( x² y² e^y/x) = u

v is a homogeneous function of degree 4

Correct answer is 8.

45. Let be a vector field all (x, y) with x > 0 and Then the value of the line integral from (0, 0) to (1, 1) along the path C : x = t², y = t³, is _________.

Ans. 1.5

Sol.

Correct answer is 1.5.

46. If defined by f(x) = is expressed as f(x) = where lies between 2 and x then the value of c is _________.

Ans. –1

Sol.

Correct answer is –1.

47. Let y₁(x), y₂(x) and y₃(x) be linearly independent solutions of the differential equation

If the Wronskian W(y₁, y₂, y₃) is of the form ke^bx for some constant k, then the value of b is _________.

Ans. k = 2, b = 6.

Sol. Let y₁( x), y₂( x), y₃(x) be linearly independent solutions of the differential equation

Solving the auxiliary equation m – 6m² + 11m – 6 = 0; m = 1, 2, 3.

y = C₁e^x + C₂e^2x + C₃e^3x and y₁ = e^x; y₂ = e^2x; y₃ = e^3x.

Clearly compare that W (y₁, y₂, y₃) = 2e^6x.

Correct answer is k = 2, b = 6.

48. The radius of convergence of the power series is _________.

Ans. 0.5

Sol.

Radius of convergence = 1/2 = 0.5.

Correct answer is 0.5.

49. Let be a continuous function such that 4x sin 2x + 2 cos 2x. Then the value of is _________.

Ans. 0.25

Sol. + 4x sin 2x + 2 cos 6x.

Differentiating both sides

Correct answer is 0.25.

50. Let G be a cyclic group of order 12. Then the number of non-isomorphic subgroups of G is________.

Ans. 6

Sol. G = Z₁₂ all subgroup are to Z₁, Z₂, Z₃, Z₄, Z₆, Z₁₂.

So in total non isomorphic subgroup are ... 6.

Correct answer is 6.

51. The value of is _________.

Ans. 1

Sol.

Correct answer is 1.

52. Let R be the region enclosed by x² + 4y²> 1and x² + y²< 1. Then the value of is _________.

Ans. 0.75

Sol.

Region is symmetric is all four quadrant is also |xy| is symmetric.

Correct answer is 0.75.

53. Let M = Then Mx = 0 has infinitely many solution if tr(M) is _________.

Ans. 3

Sol.

54. Let C be the boundary of the region enclosed by y = x², y = x + 2 and x = 0. Then the value of the line integral where C is traversed in the counter clockwise direction is _________.

Ans. 0.8

Sol. C₁ : y = x²; dy = 2xdx

Correct answer is 0.8.

55. Let S be the closed surface forming the boundary of the region V bounded by x² + y² = 3, z = 0, z = 6. A vector field is defined over V with . Then the value of where is the unit outward drawn normal to the surface S is _________.

Ans. 72

Sol.

Correct answer is 72.

56. Let y(x) be the solution of the differential equation Then y(x) attains its maximum value at x = _________.

Ans. 0.28

Sol.

(D²y + 5Dy + 6y) = 0; (D² + 5D + 6)y = 0.

Auxiliary equation (m² + 5m + 6) = 0; (m² + 3m + 2m + 6) = 0; m = –2, m = –3.

If roots are real and distinct C.F. = C₁e^–2x + C₂e^–3x; y = C₁e^–2x + C₂e^–3x

If y (0) = 1, 1 = C₁ + C₂

Correct answer is 0.28.

57. The value of the double integral _________.

Ans. 2

Sol.

Correct answer is 2.

58. Let H denote the group of all 2 × 2 invertible matrices over Z₅ under usual matrix multiplication. Then the order of the matrix in H is _________.