1. Let and V be the vector space of all X such that AX = 0. Then dim(V) is

(a) 0

(b) 1

(c) 3

(d) 3

Ans. (b)

Sol. Here AX = 0 dim(V) = nullity of (A)

Find rank of A,

rank (A) = 2 dim(A) = 3 = Rank(A) + nullity (A)

nullity (A) = 1 dim (V) = 1

2. The value of n for which the divergence of the function vanishes is

(a) 1

(b) –1

(c) 3

(d) –3

Ans. (c)

Sol.

3. Let A and B be subsets of Which of the following is NOT necessarily true?

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

4. Let [x] dentoe the greatest integer function of x. The value of for which the function

is continuous at x = 0 is

(a) 0

(b) sin(–1)

(c) sin 1

(d) 1

Ans. (c)

Sol. –1 < –x² < 0 for –1 < x < 1 (x 0)

[–x²] = –1 for –1 < x < 1 x 0

Therefore,

5. Let the function f(x) be defined by

for x in (0, 1). Then

(a) f is continuous at every point in (0, 1)

(b) f is discontinuous at every point in (0, 1)

(c) f is discontinuous only at one point in (0, 1)

(d) f is continuous only at one point in (0, 1)

Ans. (d)

Sol. At any

i.e., f is continuous only at .

6. The value of the integral is

(a) 0

(b) 7/9

(c) 14/9

(d) 28/9

Ans. (d)

Sol.

7. Let Then is

(a)

(b)

(c)

(d) 0

Ans. (b)

Sol.

8. Let p be a prime number. Let G be the group of all 2 × 2 matrices over with determinant 1 under matrix multiplication. The order of G is

(a) (p – 1) p (p + 1)

(b) p² (p – 1)

(c) p³

(d) p³ (p – 1) + p

Ans. (a)

Sol. (p – 1) (p) (p + 1)

9. Let V be the vector space of all 2 × 2 matrices over . Consider the subspaces

If m = dim(W₁ W₂) and n = dim(W₁ + W₂), then the pair (m, n) is

(a) (2, 3)

(b) (2, 4)

(c) (3, 4)

(d) (1, 3)

Ans. (b)

Sol.

dim(W₁ W₂) = 2, dim(W₁) = 3, dim(W₂) = 3

we know,

dim(W₁ + W₂) = dim(W₁) + dim(W₂) – dim(W₁ W₂)

dim(W₁ + W₂) = 3 + 3 – 2 = 4, m = dim(W₁ W₂) = 2, n = dim(W₁ + W₂) = 4

pair (m, n) = (2, 4)

10. Let P_n be the real vector space of all polynomials of degree at most n. Let D : P_n P_n–1 and T : P_n P_n+1 be the linear transformations defined by

D(a₀ + a₁x + a₂x² +.....+ a_nxⁿ) = a₁ + 2a₂x +.....+ na_nx^n–1,

T(a₀ + a₁x + a₂x² +.....+ a_nxⁿ) = a₀x + a₁x² +.....+ a_nxⁿ⁺¹,

respectively. If A is the matrix representation of the transformation DT – TD : P_n P_n with respect to the standard basis of P_n, then the trace of A is

(a) –n

(b) n

(c) n + 1

(d) –(n+1)

Ans. (c)

Sol. Consider (DT)P_n = D(T(P_n))

= D(a₀x + a₁x² +.....+a_nxⁿ⁺¹) = a₀ + 2a₁x +.....+ (n + 1) a_nxⁿ

Consider (TD)(P_n) = T(D(P_n))

= T(a₁ + 2a₂x +.....+ n a_nx^n–1) = a₁x + 2a₂x² +.....+ n a_nxⁿ)

(DT – TD)P_n = (DT)P_n – (TD)P_n

= (a₀ + 2a₁x +.....+ (n + 1)a_nxⁿ) – (a₁x + 2a₂x² +.....+ n a_nxⁿ)

= a₀ + (2 – 1)a₁x + (3 – 2)a₂x².....+ ((n + 1)– n) a_nxⁿ

= a₀ + a₁x +.....+ a_nxⁿ

11. The equation of the curve satisfying and passing through the origin is _{—————}.

Sol.

ue^–x = xe^–x + e^–x + c

but it passes through origin x = 0, y = 0

1 = 0 + 1 + c c = 0

sec y = 1 + x.

12. Let f be a continuously function such that for all x (0, ). The value of is _{—————}.

Sol.

13. Let for x 0, y 0, z 0. Let w = f(u, v), where f is a real valued function defined on having continuous first order derivatives. The value of at the point (1, 2, 3) is _{—————}.

Sol.

14. The set of points at which the function f(x, y) = x⁴ + y⁴ – x² – y² + 1, (x, y) attains local maximum is_{—————}.

Sol. f(x, y) = x⁴ + y⁴ – x² – y² + 1

So, f attains local maximum at (0, 0) only.

15. Let C be the boundary of the region in the first quadrant bounded by y = 1 – x², x = 0 and y = 0, oriented counter-clockwise. The value of _{—————}.

Ans.

Sol. (using Green's theorem)

16. Let . If is the Taylor's formula for f about x = 0 with maximum possible value of n, then the value of for 0 < x 1 is _{—————}.

Ans.

Sol.

f is differentiable thrice on [–1, 1], but fourth derivative of f does not exists at x = 0

The value of n = 2

17. Let and let C be the curve of intersection of the plane z = x + 4 and the cylinder x² + y² = 4, oriented counter-clockwise. The value of is _{—————}.

Ans.

Sol.

18. Let f and g the function from \{0, 1} to defined by for x \ {0, 1}. The smallest group of functions from \{0, 1} to containing f and g under composition of functions is isomorphic to _{—————}.

Sol.

Now, any group which contains two elements with order 2 and 3. Then the smallest group which contains these two elements will be isomorphic to S₃.

So, f g g f not commutative.

Since non-abelian group of order 6 is S₃.

19. The orthogonal trajectory of the family of curve which passes through (1, 1) is _{—————}.

Sol.

Step-I: Find then differential equation (1) w.r.t. x

Step-II: For orthogonal trajectory change

but it passes through (1, 1) 1 = 1.c c = 1

Orthogonal trajectory is y = x²

20. The function to which the power series converges is _{—————}.

Sol.

(differentiating w.r.t.y)

(putting – x² = y)

21. Let and for Show that the sequence {s_n} is convergent, and find its limit.

Sol. Let

Going on this way, we can say s_n < a, n

Since s_n < a 1 form (1),

So, the limit is lim s_n =

22. Evaluate by changing the order of integration.

Sol. Let

Boundaries of the region R:

By changing the order of integration of I,

23. Find the general solution of the differential equation

Sol.

(D₁(D₁ – 1)(D₁ – 2) + D₁(D₁ – 1) –6D₁ + 6 = 0

D₁(D₁ – 1)(D₁ – 2)+ D₁(D₁ – 1) –6(D₁ – 1) = 0

(D₁ – 1)(D₁(D₁ – 2) + D₁ – 6) = 0

D₁ = 1, –2, 3

CF = c₁e^x + c₂e^–2x + c₃e^3x

24. Let S₁ be the hemisphere x² + y² + z² = 1, z > 0 and S₂ be the closed disc x² + y² 1 in the xy-plane. Using Gauss' divergence theorem, evaluate where

Also evaluate

Sol. Let V be the volume enclosed by S₁ and S₂. By Gauss divergence theorem,

(converting to polar)

25. Let

Show that the first order partial derivatives of f with respect to x and y exist at (0, 0). Also show that f is not continuous at (0, 0).

Sol.

Hence, f_x(0, 0) and f_y (0, 0) exist.

So, if the limit exists, would be 0.

Let a neighbourhood of (0, 0) be x² + y² <

(changing into polar coodinate)

26. Let A be an n × n diagonal matrix with characteristic polynomial (x – a)^p (x – b)^q, where a and b are distinct real numbers. Let V be the real vector space of all n × n matrices B such that AB = BA. Determine the dimension of V.

Sol. Characteristic polynomial of A = (x – a)^p (x – b)^q A has p diagonal elements equal to a and q diagonal elements equal to b and p + q = n. For our convenience, we can take first p-diagonal elements as a and remaining diagonal elements as b. So, A = diag (a,a,a,......,a,b,b,,b,......,b). Let B = [b_ij]_{n × n}•

AB = BA

(i) ab_ij = bb_ij for i = 1, 2,......, p; j = p + 1, p + 2,......, n.

(ii) bb_ij = ab_ij for i = p + 1, p + 2,......, n; j = 1, 2, 3,......, p.

(i) b_ij = 0 for i = 1, 2,......, p; j = p + 1, p + 2,......, n. (as a ≠ b)

(ii) b_ij = 0 for i = p + 1, p + 2,......, n; j = 1, 2, 3,......p. (as a ≠ b)

b_ij = 0 for p·q + pq = 2pq elements of B. So, number of independent elements in B are n² – 2pq. So, dimension of B is equal to (n² – 2pq).

27. Let A be an n × n real symmetric matrix with n distinct eigenvalues. Prove that there exists an orthogonal matrix P such that AP = PD, where D is a real diagonal matrix.

Sol. Let eigenvalues of A be r = 1,2,3,......,n, for r k. Let eigenvector of A be x_r corresponding to r = 1,2,3,......,n. Ax_i = and Ax_j = .

(A is symmetric, so A = A^T)

Hence, AP = PD, where P is an orthogonal matrix.

28. Let K be a compact subset of with nonempty interior. Prove that K is of the form [a, b] or of the form where {I_n} is a countable disjoint family of open intervals with end points in K.

Sol. Since  be a metric space with usual metric defined on it.

And K be a compact subset of with nonempty interior. K be also a metric space with usual metric.

And a metric space is compact if and only if it is sequentially compact.

Means each convergent sequence has it's limit point in metric space. And also we know that real line only closed sets has all it's limit point. So every convergent will converge in itself and so K must be a closed and bounded set. [since also in a finite dimensional metric space every set is closed and bounded iff it is compact]

And closed set on either in form of a closed interval or union of closed interval.

So K must be of the form [a, b] or of the form [a, b] / UI_n, where {I_n} is a countable disjoint family of open intervals with end points in K.

29. Let f : [a, b] be a continuous function such that f is differentiable in (a, c) and (c, b), a < c < b. If exists, then prove that f is differentiable at c and

Sol. Given (i) f : [a, b] be continuous function

(ii) f is differentiable in (a, c) and (c, b), a, < c < b

To prove (a) f is differentiable at c

Now from (*)

is differentiable at c and this proves part (b).

30. Let G be a finite group, and let be an automorphism of G such that (x) = x if and only if x = e, where e is the identity element in G. Prove that every g G can be represented as g = x^{–1(x) for some x G. Moreover, if ((x)) = x for every x G, then show that G is abelian.}

Sol.

[say g = x^–1 y for more g G].

[By closure property of Group]:

Moreover, given that,

Hence G be an abelian group.