1. Let f₁(x), f₂(x), g₁(x), g₂(x) be differentiable functions on

Let F(x) = be the determinant of matrix . Then is equal to

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

So, option (b) is correct answer.

2. Let be a twice differentiable function. If g(u, v) = f(u² – v²) then

= 0

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given be twice differentiable function and g(u, v) = f(u² – v²).

3. Let f(x) = Write L = Then which of the following is true?

(a) L exist but R does not exist

(b) L does not exist but R exists

(c) Both L and R exist

(d) Neither L nor R exists

Ans. (a)

Sol.

Option (a) is correct answer.

4. The number of generators of the additive group is equal to

(a) 6

(b) 12

(c) 18

(d) 36

Ans. (b)

Sol. The number of generators of

= (2² – 2) (3² – 3) = 2(6) = 12.

Hence, option (b) is correct answer.

 

5.

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

(by changing summation in integration by putting k/n = x and 1/n = dx for lower limit).

For k = 1 then k/n = 1/n and for k = n then k/n = 1/n on )

Hence, option (c) is correct answer.

6. If f(x) = satisfies the assumptions of Rolle's theorem in the interval [–1, 1] then the ordered pair (p, q) is

(a) (2, –1)

(b) (–2, –1)

(c) (–2, 1)

(d) (2, 1)

Ans. (d)

Sol. Given f(x) =

f(x) satisfies the assumptions of Rolle's theorem in the interval [–1, 1].

f(x) is continuous in [–1, 1] and f(x) is differentiable in (–1, 1).

Also

Hence, option (d) is correct answer.

7.

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given

We have to find the value of

Now, we start with given

On taking unit on both sides, we get

So, option (a) is correct answer.

8.

(a)

(b) 1 – cos 1

(c) 1 + cos 1

(d)

Ans. (d)

Sol.

Changing order of integration

Option (d) is correct answer.

9. Consider the function f(x, y) = 5 – 4 sin x + y² for The set of critical points of f(x, y) consists of

(a) a point of local maximum and a point of local minimum.

(b) a point of local maximum and a saddle point.

(c) a point of local maximum, a point of local minimum and a saddle point.

(d) a point of local minimum and a saddle point.

Ans. (d)

Sol. Given

For critical points = 0.

Now at A > 0 and AC – B² = 8 sin x > 0 is a point of local minima.

Now at A < 0 and AC – B² < 0 is a saddle point.

Option (d) is correct answer.

10. Let be a differentiable function that is strictly increasing = 0. Let denote the minimum and maximum values of on the interval [2, 3] respectively. Then which one of the following is true?

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given is a differentiable function and is strictly increasing with = 0.

Consider

which is strictly increasing function and

Hence, options (b), (c) and (d) are incorrect.

So, option (a) is correct answer.

11. Let S be an infinite subset of such that is compact for some Then which one of the following is true?

(a) S is a connected set.

(b) S contains no limit points.

(c) S is a union of open intervals.

(d) Every sequence in S has a subsequence converging to an element in S.

Ans. (d)

Sol. Take

But S is not connected, so option (a) is incorrect .

S contains its limit points, so option (b) is incorrect

Lastly, S is not a union of open intervals then option (c) is incorrect.

Hence, option (d) must be the correct answer.

12. Let be a differentiable function such that f(2) = 2 and

for all

(a) 5

(b)

(c) 12

(d) 24

Ans. (d)

Sol. Given

Let h > 0 and in (iv) put x + h in place of x and x in place of y, we get

Option (d) is correct answer.

13. For a > 0, b > 0 let be a planar vector field.

Let be the circle oriented anti-clockwise. Then

(a)

(b)

(c)

(d) 0

Ans. (d)

Sol. Given

M and N are not continuous at (0, 0).

Hence, Green's theorem will not applicable here for C.

Let us enclose the origin by a circle S of radius r. Consider the region R enclosed by curve C' made of C₁, C₂, S, C_r.

Now, M and N are continuous and having continuous partial derivatives in R enclosed by C.

In the figure, S is oriented in the direction on the curve S

Option (d) is correct answer.

14.

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

By using tan^–1x – tan^–1y = tan^–1 where x = n – 1, y = n + 1.

Option (c) is correct answer.

15. The line integral of the vector field along the boundary of the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1), oriented anti-clockwise, when viewed from the point (2, 2, 2) is

(a)

(b) –2

(c)

(d) 2

Ans. (c)

Sol.

Let A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1).

Now, first line integral from A to B is

Equation of the line AB is

t varies from 0 to 1.

Now,

Option (c) is correct answer.

16. The flux of along the outward normal, across the surface of the solid is equal to

(a)

(b)

(c)

(d)

Ans. (d)

Sol. To find flux of across the surface of the solid

Now, flux of is unit normal vector outward to surface S.

By Gauss divergence theorem,

Hence, option (d) is correct answer.

17. The area of the surface z = intercepted by the cylinder lies in the interval

(a)

(b)

(c)

(d)

Ans. (a)

Sol. Given surface z = .

Now, surface area of the given surface intercepted by the cylinder is

Option (a) is correct answer.

18. The flux of the vector field along the outward normal, across the ellipse x² + 16y² = 4 is equal to

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given

We know that the flux across

Take the parametric from of C i.e.,

Option (b) is correct answer.

19. Let be a continuous function. Then which one of the following is not true?

(a)

(b)

(c)

(d)

Ans. (c)

Sol. As given is continuous function.

So, f is also continuous in [–1, 1].

We know for any

Then

As, function is continuous, so either

By above result

i.e., either

then by intermediate value theorem, which states that for any continuous then, if d is a number between f(a), f(b) then there exist such that f(c) = d.

So, there exists such that

f(x) =

So, option (a) is true.

As function is continuous on then function is continuous on [–1, 1]

then either

So, by intermediate value theorem.

We can say that there exist such that f(x) =

Hence, option (b) is correct.

Now, for option (c)

Let

So, it is not true. So, (c) is not true.

Which is the required answer.

Option (d) is correct by first mean value theorem of integral, which states that if f is continuous function on closed and bounded [a, b], then for which

20. Let y(x) be the solution of the differential equation (xy + y + e^–x)dx + (x + e^–x)dy = 0 satisfying y(0) = 1. Then y(–1) is equal to

(a)

(b)

(c)

(d) 0

Ans. (b)

Sol. Given (xy + y + e^–x)dx + (x + e^–x)dy = 0

Mdx + Ndy = 0

i.e., e^x is integrating factor.

Now, multiplying e^x by (1), we get (xy e^x + ye^x + 1)dx + (xe^x + 1)dy = 0

Now, if

the solution is given by

Now given condition is y(0) = 1

Putting in (2), we get 1(0 – 1)e⁰ + 1e⁰ + 0 + 1 = C

So, we get particular solution

y(x – 1)e^x + ye^x + x + y = 1

= (xy – y + y)e^x + x + y = 1

xye^x + x + y = 1

Now to find y(–1), (–1)ye^–1 + y – 1 = 1

–ye^–1 + y = 2

So, option (b) is correct.

21. Let f(x, y) = Then

(a)

(b)

(c)

(d)

Ans. (b)

Sol. Given

Here, put then we get

Now, from here either option (b) or (d) is correct.

Again put

So,

So, option (b) is correct.

22. A particular integral of the differential equation

(a)

(b)

(c)

(d)

Ans. (c)

Sol. Given differential equation is

Option (c) is correct answer.

23. Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each let n₁(M) and n₀(M) denote the number of 1's and 0's in M respectively. Then

(a) 1

(b) 3

(c) 5

(d) 15

Ans. (a)

Sol. Consider the matrix

M =

Clearly, M ∈ M

n₁(M) = 12

n₀(M) = 13

|n₁(M) – n₀(M)| = 1

Option (b), (c) and (d) are incorrect.

Option (a) is correct answer.

24.

Then which one of the following is not true?

(a) Both {a_n} and {b_n} converge, but the limits are not equal.

(b) Both {a_n} and {b_n} converge and the limits are equal.

(c) {b_n} is a decreasing sequence.

(d) {a_n} is an increasing sequence.

Ans. (a)

Sol. Given

We will show:

(i) {a_n} is an increasing sequence.

(ii) {b_n} is a decreasing sequence.

(iii) {a_n} and {b_n} both converge to same limit.

then

So, by principle of Mathematical induction

{a_n} is an increasing sequence

From (1), (2) and (3) if is clear that {b_n} is a decreasing sequence which is bounded below and {a_n} is an increasing sequence which is bounded above.

So, both {a_n} and {b_n} are convergent sequences.

So, {a_n} and {b_n} converge to the same limit.

Hence, option (a) is correct answer.

25.

(a)

(b)

(c)

(d)

Ans. (c)

Sol.

rationalize denominators of each term, we get

Option (c) is correct answer.

26. Let and let L be the curve Then

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

So, x = e^t sin t,  y = e^t cos t,

then dx = e^t(sin t + cos t)dt dy = e^t(cos t – sin t)dt

Now,

On solving we get

Option (d) is correct.

27. The interval of convergence of the power series is

(a)

(b)

(c)

(d)

Ans. (d)

Sol. Given

Put 4x – 12 = y then we get

Now,

So, it is convergent if

Now,

We get as it is alternating series by Leibnitz rule it is convergent, no need to verify at

(according to options)

So, option (d) is correct answer.

28. Let P₃ denote the real vector space of all polynomials with real coefficient of degree at most 3. Consider the map given by Then

(a) T is neither one-one nor onto

(b) T is both one-one and onto

(c) T is one-one but not onto

(d) T is onto but one-one

Ans. (b)

Sol. Given

Take B = {1, x, x², x³} be the basis of P₃

T(1) = 0 + 1 = 1

T(x) = x

T(x²) = 2 + x²

T(x³) = 6x + x³

Now, [T : B] = which is a non-singular matrix

T is non-singular transformation

So, T is one-one and onto also.

Option (b) is correct answer.

29. Which one of the following is true?

(a) Every sequence that has a convergent subsequence is a cauchy sequence.

(b) Every sequence that has a convergent subsequence is a bounded sequence.

(c) The sequence {sin n} has a convergent subsequence.

(d) The sequence has a convergent subsequence.

Ans. (c)

Sol. Option (a) and (b) are clearly rejected as

Let

Clearly, {a_n} has a convergent subsequence {a_2m}; but as it is non-convergent real sequence, hence it is not a Cauchy sequence.

Also, hence {a_n} is also not bounded.

Option (a) and (b) is incorrect.

And option (d) is also rejected as is a strictly increasing sequence.

Also, is a strictly increasing sequence.

is a strictly increasing sequence which is bounded above as.

Hence, do not have a convergent subsequence.

Option (d) is also incorrect.

Option (c) is correct answer.

30. Let M = . Then

(a) does not exist

(b)

(c)

(d)

Ans. (c)

Sol. Given

Option (c) is correct answer.

31. Let S be the set of all rational numbers in (0, 1). Then which of the following statements is/are true?

(a) S is a closed subset of

(b) S is not a closed subset of

(c) S is an open subset of

(d) Every x ∈ (0, 1)/S is a limit point of S

Ans. (b), (d)

Sol.

Every irrational in (0, 1) is a limit point of S, but S contains no irrational number.

Hence, S is not a closed subset of R.

So, option (a) is incorrect and option (b) is correct.

Option C is incorrect as, for any

Option (d) is correct due to the fact that S is dense in (0, 1).

 Option (b), (d) are correct answer.

32. Let be such that every solution of = 0

(a) 3k² + l < 0 and k > 0

(b) k² + l > 0 and k < 0

(c)

(d) k² – l > 0, k > 0 and l > 0

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

Sol. Given differential equation is

Every solution of above equation approaches to zero as x approaches to infinity

i.e.,

Take k = 1, 1 = 1

Now, from option (a) 3k² + l = 3 + 1 = 4 > 0.

Option (a) is incorrect.

From option (b), k² + l = 2 > 0 and k = 1 > 0.

Option (b) is also incorrect.

From option (d), k² – l = 0 > 0 which is not true.

Option (d) is also incorrect.

Now, (D² + 2kD + l)y = 0.

This question is wrong.

33. Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the following is/are true?

(a) G contains a normal subgroup of order 5

(b) G contains a non-normal subgroup of order 5

(c) G contains a subgroup of order 10

(d) G contains a normal subgroup of order 4

Ans. (a), (c)

Sol. O(G) = 20 = 2².5

In order to find number of Sylow 5 subgroup(s), use Sylow's third theorem.

Using Cauchy's theorem

G has subgroup of order 2

Let H is subgroup of order 5 and K is subgroup of order 2,

H is normal subgroup of G

So HK is normal subgroup of G

(a) and (c) are correct

G may not contain subgroup of order 4 as if G contains normal subgroup of order 4, (say) L then O(H) = 5, O(L) = 4 and H L = {e}

HL is abelian HL = G is abelian but G is not abelian

So may not possible.

34. If X and Y are n × n matrices with real entries then which of the following is/are true?

(a) If P^–1XP is a diagonal for some real invertible matrices P then there exists a basis for consisting of eigenvectors of X.

(b) If X is diagonal with distinct diagonal entries and XY = YX then Y is also diagonal.

(c) If X² is diagonal then X is diagonal.

(d) If X is diagonal and XY = YX for all Y then .

Ans. (a), (b), (d)

Sol. For option (a)

Given P^–1X P is diagonal matrix for some real invertible matrix 'P' where P = [p₁, p₂, ..., p_n]_{n × n}

where p₁ is the i^th column of 'P'.

So, columns of P are nothing but eigen vector of X

Since 'P' is invertible matrix

  Columns of P are linearly independent

  {p₁, p₂, ..., p_n} are linearly independent

  {p₁, p₂, ..., p_n} are basis of as they are linearly independent and 'n' in numbers

So, there exist a basis of consisting of eigen vectors of X

So, option (a) is correct.

For option (b),

Given X is diagonal matrix with distinct diagonal entries and XY = YX

Also, we can generalize it for n >3.

So, option (b) is correct

For option (c)

Take

is diagonal matrix but X is not diagonal matrix

So, option (c) is incorrect.

For option (d)

Given X is diagonal matrix,

So, option (d) is correct.

35. Let be a function. Then which of the following statements is/are true?

(a) If f is differentiable at (0, 0) then all directional derivatives of f exist at (0, 0).

(b) If all directional derivatives of f exist at (0, 0) then f is differentiable at (0, 0).

(c) If all directional derivatives of f exist at (0, 0) then f is continuous at (0, 0).

(d) If the partial derivatives exist and are continuous in a disc centered at (0, 0) then f is differentiable at (0, 0).

Ans. (a), (d)

Sol. For option (a)

If function is differentiable at (x₀, y₀) then all partial derivatives of f exist all (x₀, y₀)

So, option (a) is true.

For option (b) and (c).

Take

This function is not continuous at (0, 0), so not differentiable at (0, 0), but all directional derivatives exist at (0, 0).

To check directional derivatives:

So, all directional derivatives exist.

For option (d):

It is necessary condition for function to be differentiable.

Necessary Condition: If and if all partial derivatives of f at point (x₀, y₀) and are continuous in a disc centered at (x₀, y₀), then function is differentiable at (x₀, y₀).

36. Let y(x) be the solution of the differential equation

satisfying the condition y(0) = 2. Then which of the following is/are true?

(a) The function y(x) is not bounded above

(b) The function y(x) is bounded

(c)

(d)

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

Sol. Given = (y – 1)(y – 3) and y(0) = 2

By variable separable

On integrating both sides, we get

which is required particular solution.

So, we get bounded solution.

So, option (b), (c) and (d) are correct.

37. The volume of the solid is expressible as

(a)

(b)

(c)

(d)

Ans. (a), (b), (d)

Sol. Given region is

Now volume of any solid is

Option (a) is correct.

Option (d) is also correct.

38. Let {x_n} be a real sequence such that Then which of the following statements is/are true?

(a) If x₁ = 1/2 then {x_n} converges to 1.

(b) If x₁ = 1/2 then {x_n} converges to 2.

(c) If x₁ = 3/2 then {x_n} converges to 1.

(d) If x₁ = 3/2 then {x_n} converges to –3.

Ans. (a), (c)

Sol. Given [x_n] be a real sequence and

Now let x_n converges to l as n tends to

i.e., then on taking on both sides, we get

If x₁ > 0, then (in given) x_n> 0. So, x_n cannot converge to negative n. So, x_n cannot converge to –3.

So, option (d) cannot be true.

Now, we prove it increasing by induction

Result is true for n = 1

Let us assume that result is true for n = k

So, we proved that {x_n} is increasing sequence or {(x_{n + 1} – x_n)} is always positive.

So, {x_n} is convergent either to 1 or to 2.

So, if x_n converges to 2, then for some becomes negative, which is not true.

So, {x_n} is convergent to 1.

Case-II: If

We shall prove, it is a decreasing sequence using principle of mathematical induction

For n = 1

Result is true for n = 1.

Assume that x_k – x_{k – 1} < 0

So, we get that {x_n} is decreasing

So, only possibility for the sequence is that x_n converges to 1.

So, option (a) and (c) are correct.

39. Let M a n × n matrix with real entries such that M³ = I. Suppose that for any non-zero vector Then which of the following statements is/are true?

(a) M has real eigenvalues.

(b) M + M^–1 has real eigenvalues.

(c) n is divisible by 2.

(d) n is divisible by 3.

Ans. (b), (c)

Sol. Given that M³ = 1 and M is n × n matrix

Also, 1 is not eigen value of M.

Now, minimal polynomial of M divides M³ – 1.

Let be eigen value of M.

Then, minimal polynomial can't have factor

The only possibility of

Eigen value of are

Option (a) is incorrect.

As, complex roots always appear in conjugate pairs for an equation with real coefficients.

Here, the order of matrix must be divisible by 2.

Option (c) is correct.

e.g., If M is 2 × 2 matrix satisfying M² + M + 1 = 0

Then n = 2 which is not divisible by 3

Option (d) is incorrect

As, M satisfy M² + M + I = 0

Taking M^–1 on both sides

M + M^–1 = –I –I has distinct eigen value –1 all eigen values are real.

Option (b) is correct

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

40. For define the map Let

.

For Then which of the following statements is/are true?

(a)

(b)

(c) For every

(d)

Ans. (a), (c)

Sol.

LHS = RHS

Hence, G is associative.

Hence, identity exist.

Note G is not group. As, there does not exist inverses of elements of the type

Options (a) and (c) are correct.

41. Let P be a 7 × 7 matrix of rank 4 with real entries. Let be a column vector. Then the rank of P + aa^T is atleast ________.

Ans. 3

Sol. rank(P) = 4 and rank(aat^T) = 1

42. Let . Let M be the matrix whose columns are in that order. Then the number of linearly independent solutions of the homogeneous system of linear equation Mx = 0 is _________.

Ans. 2

Sol.

Clearly, column 3^rd and 4^th are linear combination of

For homogeneous system Mx = 0

Number of linearly independent solutions = rank M = 2.

43. If the orthogonal trajectories of the family of ellipses x² + 2y² = c₁, c₁ > 0 are given by y = _________.

Ans. 2

Sol. Given family of ellipses is x² + 2y² = c₁, c₁ > 0 which is orthogonal to

Product of slopes of both the families is equal to –1.

44. The number of subgroups of of order 7 is _________.

Ans. 8

Sol. The number of subgroups of of order 7 is 8.

45. = _________.

Ans. 0.5

Sol. Key idea: Maclaurin's Series of f(x) = sin(x) – x cos(x) is used here

46. Let G be a subgroup of generated by . Then the order of G is _________.

Ans. 6

Sol. Let

Therefore, G must be a group of order 6.

47.

Ans. 1260

Sol. Consider

We know that relations between Beta-Gamma function.

48. Consider the permutations in S₈. The number is equal to _________.

Ans. 0

Sol.

Two permutations in S₈ are conjugate if they have some cyclic decomposition.

49. For x > 0, let [x] denote the greatest integer less than or equal to x. Then = _________.

Ans. 55

Sol.

Clearly, for any x > 0;

Hence, by Squeeze principle

50. Let P be the point on the surface z = closest to the point (4, 2, 0). Then the square of the distance between the origin and P is _________.

Ans. 10

Sol. To find: Point 'P' on the surface z = which is nearest to the Point (4, 2, 0).

Consider f(x, y, z) = (x – 4)² + (y – 2)² + z² i.e., f is the square of the distance from point (x, y, z) to the origin

Now, we have to minimize f.

51. Let T be the smallest positive real number such that the tangent to the helix

at t = T is orthogonal to the tangent at t = 0. Then the line integral of along the section of the helix from t = 0 t = T is _________.

Ans. 2.094

Sol. Given equation of helix is r(t) =

Tangents of r(t) at t = T and t = 0 are orthogonal.

52. For a real number x, define [x] to be the smallest integer greater than or equal to x.

Then

Ans. 3

Sol. where [x] is the smallest integer greater than or equal to x.

53. Let y(x), x > 0 be the solution of the differential equation

satisfying the conditions y(1) = 1 and Then the value of e²y(e) is _________.

Ans. 3

Sol. Given differential equation is

Above equation is of the form Cauchy homogeneous equation with variable coefficients.

54. If y(x) = then y(1) = _________

Ans. 1.359

Sol.

Differentiate using Leibnitz Rule

55. Let f(x) =

Then (f(x₀))²(1 + (p² – 1) sin²x₀) = _________.

Ans. 1

Sol. Given

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

Ans. 1

Sol. Given power series is

If we put x = 1 then we get which is divergent.

So, radius of convergent is 9 < 1 < 1.

So, by D'-Alembert's ratio test is convergent for 0 < l < 1.

So, radius of convergent of given power is 1.

57. The maximum order of a permutation in the symmetric group S₁₀ is _________.

Ans. 30

Sol. Maximum order of a permutation in S₁₀ is 30.

58. For x > 1, let f(x) =

The number of tangents to the curve y = f(x) parallel to the line x + y = 0 is _________.

Ans. 2

Sol. We have

Slope of the line x + y = 0 is –1.

Now, we have to find number of tangents to the curve

On solving, we get two tangents to the curve y = f(x) parallel to the line x + y = 0.

59.

Ans. 6

Sol. Let A =

It is a companion matrix.

60.

Then

Ans. 1

Sol.