PYQ 2018 - VedPrep

IIT JAM MATHEMATICS - 2018
Previous Year Question Paper with Solution.

1. Which one of the following is true?

(a)

(b)

(d) If every proper subgroup of a group is cyclic, then the group is cyclic

Ans. (b)

Sol. Every proper subgroup of Z_n is cyclic.

2. Let a_n = where b₁ = 1, b₂ = 1 and b_{n + 2} = b_n + b_{n + 1},

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Let

3. If is linearly independent set of vectors in a vector space over R then which one of the following sets is also linearly independent?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

Thus, above equations does not have unique solution. So, they are not linearly independent.

These equations hold for α = 1, β = 1, γ = 1 and δ = –1. So, they are not linearly independent.

Hence, they are linearly independent.

4. Let a be a positive real number. If f is a continuous and even function defined on the interval [–a, a] then is equal to

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Let

5. The tangent plane to the surface z = at (1, 1, 2) is given by

(a) x – 3y + z = 0

(b) x + 3y – 2z = 0

(d) 3x – 7y + 2z = 0

Ans. (b)

Sol. We have

Equation to tangent to surface is

6. In R³, the cosine of the acute angle between the surfaces x² + y² + z² – 9 = 0 z – x² – y² + 3 = 0 at the point (2, 1, 2) is

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Let A = x² + y² + z² – 9 = 0

B = –x² – y² + z + 3 = 0

On taking gradient of the given surfaces, we get

and similarly

7. Let be a scalar field be a vector field and let be a constant vector. If represent the position vector then which one of the following is false?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

8. In R², the family of trajectories orthogonal to the family of asteroids x^2/3 + y^2/3 = a^2/3 is given by

(a) x^4/3 + y^4/3 = c^4/3

(b) x^4/3 – y^4/3 = c^4/3

(d) x^2/3 – y^2/3 = c^2/3

Ans. (b)

Sol. x^2/3 + y^2/3 = a^2/3

On differentiating w.r.t. x, we get

For orthogonal Trajectory

9. Consider the vector space V over R of polynomial functions of degree less than or equal to 3 defined on R. Let be defined by (T f)(x) = f(x) – xf(x). Then the rank of T is

(a) 1

(b) 2

(d) 4

Ans. (c)

Sol. Given

10. Let S_n = 1 + Then which one of the following is true for the sequence

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have

Thus, converges to e, which does not belong to Q. Also, we know that, a sequence of real numbers converges iff it is a Cauchy sequence.

Hence, is a Cauchy sequence but does not converge in Q.

11. Let a_n = .

Then which one of the following is true?

(a)

(b)

(c)

(d) lim inf (a_n) = 1 and lim sup (a_n) = 3

Ans. (a)

Sol. We have

As we know that the supremum of a set is its least upper bound and the infimum is the greatest lower bound. The supremum or infimum of a set A is unique if it exists. Moreover, if both exists, then

12. Let Which of the following values of a, b, c do not result in the convergence of the series ?

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

We know that is convergent if p > 1 and divergent if 0 < p < 1.

Hence, option (c) cannot be result in the convergent of the series.

13.

(a) e^–1 – 1

(b) e^–1

(d) 1 + e^–1

Ans. (d)

Sol. We have

14. Let a_n = where Then which one of the following is true?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. We have

The sequence <c_n> is divergent, since it has two convergent subsequence with different limits. Also, <c_n> is divergent if it has a subsequence that tends to ∞ or a subsequence that tends to

15. Suppose that are differentiable functions such that f is strictly increasing and g is strictly decreasing. Define p(x) = f(g(x)) and q(x) = g(f(x)), Then, for t > 0, the sign of is

(a) positive

(b) negative

(d) dependent f and g

Ans. (a)

Sol. We have, f is increasing function,

16. For let f(x) = . Then which one of the following is false?

(a)

(b)

(d) is continuous at x = 0 but not differentiable at x = 0.

Ans. (d)

Sol.

By graph of , it have infinitely many maxima and minima on the (0, 1), (c) is true.

17. Let f(x, y) =

Then which one of the following is true for f at the point (0, 0)?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. We have f(x, y) =

Now, let us check the differentiability of f(x, y) at (0, 0) when .

18. Let and let be a thrice differentiable function. If z = e^u f(v), where u = ax + by and v = ax – by then which one of the following is true?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We have z = e^u f(v), where

u = ax + by and v = ax – by

19. Consider the region D in the yz-plane bounded by the line y = 1/2 and the curve y² + z² = 1, where y > 0. If the region D is revolved about the z-axis in R³ then the volume of the resulting solid is

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Clearly, the region D is the shaded region shown in the following figure.

20. If where C is the boundary of the triangular region bounded by the lines x = 0, y = 0 and x + y = 1 oriented in the anti-clockwise direction is

(a) 5/2

(b) 3

(d) 5

Ans. (b)

Sol.

21. Let U, V and W be finite dimensional real vector spaces be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S) then which one of the following is true?

(a) nullity of T = nullity of S

(b)

(d) If dimension of V = 4, dimension of U = 3 and T is one-one, then P is identically zero.

Ans. (c)

Sol.

22. Let y(x) be the solution of the differential equation + y = f(x), for y(0) = 0 where f(x) = . Then y(x) =

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We have

23. An integrating factor of the differential equation = 0 is

(a) x²

(b) 3 log_ex

(d) 2 log_ex

Ans. (c)

Sol.

24. A particular integral of the differential equation is

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given differential equation is

25. Let G be a group satisfying the property that is a homomorphism implies f(g) = 0, G. Then a possible group G is

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

26. Let H be the quotient group of Consider the following statements.

I. Every cyclic subgroup of H is finite.

II. Every finite cyclic group is isomorphic to a subgroup of H.

Which one of the following holds?

(a) I is TRUE but II is FALSE

(b) II is TRUE but I is FALSE

(d) Neither I nor II is true

Ans. (c)

Sol. H =

To verify

I. Every cyclic subgroup of H is finite.

II. Every finite cyclic group is isomorphic to a subgroup of H.

II. Let G be any finite cyclic group, say G = <a> such that O(a) = n.

As we know that every cyclic group of order n is isomorphic to Z_n.

From Eqs. (i) and (ii), we get

which is a subgroup of H.

Hence, statement II is also true.

27. Let I denote the 4 × 4 identity matrix. If the roots of the characteristic polynomial of 4 × 4 matrix M are then M⁸ =

(a) I + M²

(b) 2I + M²

(d) 3I + 2M²

Ans. (c)

Sol. We have M =

28. Consider the group under component-wise addition. Then which of the following is a subgroup of

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

29. Let be a function and let J be a bounded open interval in R. Define

Which one of the following is FALSE?

(a)

(b) If f is a bounded function in J and such that the length of the interval J_n tends to 0 as

(d) If f is continuous at a point then for any given there exists an intervals such that

Ans. (b)

Sol.

Since, we have

be a function and J be a bounded open interval in R, define as

then we conclude that

the length of the internal.

If f is discontinuous at a point

If f is continuous at a point then for any given there exists an internal such that

30. For f₂(x) = log_e(1 + 2x) and f₃(x) = 2x. Then which one of the following is true?

(a)

(b) f₁(x) < f₃(x) < f₂(x) for x > 0

(c)

(d) f₂(x) < f₁(x) < f₃(x) for x > 0

Ans. (c)

Sol. f₁(x) = , f₂(x) = log_e(1 + 2x), f₃(x) = 2x

graph of f₁(x), f₂(x) and f₂(x) are

Clearly, from graph f₃(x) > f₂(x) > f₁(x) for x > 0

Hence, option (c) will be correct.

31. Let be defined by f(x) = x + . On which of the following interval(s) f is one-one?

(a)

(b) (0, 1)

(d)

Ans. (b)

Sol. We have

It is clearly from graph f is one-one in (0, 1).

32. The solution(s) of the differential equation = (sin 2x) y^1/3 satisfying y(0) = 0 is/are

(a) y(x) = 0

(b)

(c)

(d)

Ans. (a), (b), (c)

Sol. We have

On integrating both sides, we get

Hence, the equation becomes

Also, y(x) = 0

y(0) = 0.

Hence, option (a), (b) and (c) are correct.

33. Suppose f, g, h are permutations of the set where

f interchanges

g interchanges

h interchanges

Which of the following permutations interchanges(s)

(a)

(b)

(c)

(d)

Ans. (a), (d)

Sol.

34. Let P and Q be two non-empty disjoint subsets of R. Which of the following is/are false?

(a) If P and Q are compact, then is also compact

(b) If P and Q are connected, then is also not connected

(d) If and P are open, then Q is open

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

Sol. (a) If P and Q are two compact set in R.

Hence, this statement is false.

Option (b), (c), (d) are false.

35. Let denote the group of non-zero complex numbers under multiplication. Suppose Which of the following is/are subgroup(s) of

(a)

(b)

(c)

(d)

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

Sol.

So, there exist

Hence, option (a) is not true.

But options (c) and (d) will be true.

36. Suppose Consider the following system of linear equations:

If this system has at least one solution, then which of the following statements is/are true?

(a)

(b)

(c)

(d)

Ans. (a), (b)

Sol. We have

For atleast one solution.

Atleast one row must be zero.

37. Let Then which of the following is/are not possible?

(a) rank (PQ) = n

(b) rank (QP) = m

(d) rank (QP) = , the smallest integer larger than or equal to

Ans. (a), (d)

Sol.

38. If then which among the following is/are true?

(a)

(b) = 0 along any simple closed curve C

(d)

Ans. (a), (b), (c)

Sol. We have

39. Which of the following subsets of R is/are connected)?

(a)

(b)

(c)

(d)

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

Sol.

which cannot be written as union of two disjoint open sets, so this is a connected set.

40. Let S be a subset of γ such that 2018 is an interior point of S. Which of the following is/are true?

(a) S contains an interval

(b) There is a sequence in S which does not converge to 2018

(d) There is a point such that | z – 2018 | = 0.002018

Ans. (a), (b), (c)

Sol.

There exist r > 0 such that

Hence, option (a), (b), (c) is true.

Hence, option (d) is false.

41. The order of the element (1 2 3) (2 4 5) (4 5 6) in the group of S₆ is _________.

Ans. 4

Sol.

Let is first convert it in disjoint cycles

42. Let Then the absolute value of the directional derivative of in the direction of the line at the point (1, –2, 1) is _________.

Ans. 7

Sol.

So, direction derivation of at (1, –2, 1) in the direction of line is

43. Let f(x) = for 0 < x < 2. Then the value of is _________.

Ans. 1

Sol. We have f(x) =

44. Let be given by

f(x, y) =

Then at the point (0, 0) is _________.

Ans. 0

Sol. We have f(x, y) =

45. Let f(x, y) = for x > 0, y > 0. Then f_x(1, 1) + f_y(1, 1) = _________.

Ans. 3

Sol. We have

46. Let be continuous on and differentiable on . If f(x) = then f(6) = _________.

Ans. 9

Sol. We have

On integrating both sides, we get

47. Let a_n = . Then the radius of convergence of the power series about x = 0 is _________.

Ans. 2

Sol.

48. Let A₆ be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in A₆ is _________.

Ans. 0

Sol.

49. Let W₁ be the real vector space of all 5 × 2 matrices such that the sum of the entries in each row is zero. Let W₂ be the real vector space of all 5 × 2 matrices such that the sum of the entries in each column is zero. Then the dimension of the space is _________

Ans. 4

Sol. We have all 5 × 2 matrices.

We know that all 5 × 2 matrices contains null space has a basis formed by the set {(1, 0), (0, 1)}.

50. The coefficient of x⁴ in the power series expansion of e^{sin x} about x = 0 is _________ (correct up to three decimal places).

Ans. –0.125

Sol. We have f(x) = e^{sin x}.

Power series expansion of f(x) at x = 0 is

51. Let a_k = (–1)^{k – 1}, s_n = a₁ + a₂ + ... + a_n and = (s₁ + s₂ + ... + s_n)/n, where k, is _________ (correct up to one decimal place).

Ans. 0.5

Sol. We have, a_k = (–1)^{k – 1}

s_n = a₁ + a₂ + ... + a_n

s₁ = a₁ = 1

s₂ = a₁ + a₂ = 1 – 1 = 0

s₃ = a₁ + a₂ + a₃ = 1 – 1 + 1 = 1

s_n = a₁ + a₂ + a₃ + ... + a_n

52. Let be such that is continuous on and Then is _________ (correct up to three decimal places).

Ans. 0.367

Sol. Consider f(x) = 1 – .

Clearly, f(x) satisfy all the given conditions.

53. Suppose x, y, z are positive real numbers such that x + 2y + 3z = 1. If M is the maximum value of xyz² then the value of is _________.

Ans. 1152

Sol.

54. If the volume of the solid in bounded by the surfaces

x = – 1, x = 1, y = –1, y = 1, z = 2, y² + z² = 2

is = _________.

Ans. 6

Sol. We have, x = –1, x = 1, y = –1, y = 1, z = 2, y² + z² = 2

Volume of the solid bounded by surface is

55.

Ans. 3

Sol. We have

put (sin t – cos t) = x

(cos t + sin t)dt = dx

sin 2t = 1 – x²

56. The value of the integral is _________ (correct up to three decimals places).

Ans. 0.236

Sol.

By changing the order of integration

57. Suppose is matrix of rank 2. Let be the linear transformation defined by T(P) = QP. Then the rank of T is _________.

Ans. 6

Sol. By using Rank-Nullity Theorem we get, rank of T = 6.

58. The area of the parametrized surface

is _________ (correct up to two decimal places).

Ans. 6.50

Sol. We have

Area of parametrized surface

59. If x(t) is the solution to the differential equation = x²t³ + xt for t > 0 satisfying x(0) = 1 then the value of is __________ (correct up to two decimal places).