1. Let p and t be positive real numbers. Let D_t be the closed disc of radius t centered at (0, 0), i.e., . Define

I(p, t) = .

Then is finite

(a) for no value of p

(b) only if p < 1

(c) only if p = 1

(d) only if p > 1

Ans. (d)

Sol.

2. Let be a continuous function such that for all

= 0.

Then

(a) There is an f satisfying (*) that takes both positive and negative values.

(b) There is an f satisfying (*) that is 0 at infinitely many points, but is not identically zero.

(c) f must be identically 0 on the whole of γ.

(d) There is an f satisfying (*) that is identically 0 on (0, 1) but not identically 0 on the whole of γ.

Ans. (c)

Sol.

3. For every be a function. From the given choices, pick the statement that is the negation of

"For every and for every real number there exists an integer N > 0 such that < for every integer p > 0".

(a) For ever and for every real number there does not exist any integer N > 0 such that for every integer p > 0.

(b) For every and for every real number there exists an integer N > 0 such that for some integer p > 0.

(c) There exists and there exists a real number such that for every integer N > 0, there exists an integer p > 0 for which the inequality holds.

(d) There exists and there exists a real number such that for every integer N > 0, and for every integer p > 0 the inequality holds.

Ans. (d)

Sol. (d) For every an integer N > 0 such that is convergent.

4. How many elements of the group have order 10?

(a) 10

(b) 4

(c) 5

(d) 8

Ans. (d)

Sol. (d) As number of elements order 'k' in Z_n =

So, in given case 10 divides 50. Hence number of elements of order 10 are given by

Now, = Numbers which are coprime to 10 put less than or equals to 10.

5. Let be a real number. The number of differentiable functions having continuous derivative on [0, 1] and satisfying

is

(a) infinite

(b) exactly one

(c) exactly two

(d) finite but more than two

Ans. (a)

Sol.

6. Let n > 1 be an integer. Consider the following two statements for an arbitrary n × n matrix A with complex entries.

I. If A^k = I_n for some integer then all the eigenvalues of A are k^th roots of unity.

II. If, for some integer all the eigenvalues of A are k^th roots of unity, then A^k = I_n.

Then

(a) I is TRUE but II is FALSE

(b) Neither I nor II is TRUE

(c) Both I and II are TRUE

(d) I is FALSE but II is TRUE

Ans. (a)

Sol. I. A^K = I

We can define a polynomial p(x) = x^k – 1

So, p(A) = A^k – I = 0 because given A^k = I (Identity)

M_A(x)/p(x) by the property of minimal polynomial.

M_A(x)/x^k – 1

If we choose any eigen value of A, then must be root of minimal polynomial because minimal polynomial contain all the eigen values, may be non-repeatly.

So, statement I is correct.

7. Which one of the following subsets of has a non-empty interior?

(a) The set of all irrational numbers in .

(b) The set : x² + bx + 1 = 0 has distinct roots}.

(c) The set : sin(a) = 1}

(d) The set of all rational numbers in .

Ans. (b)

Sol.

8. Let be a continuous function such that P(x) > 0 for all Let y be a twice differentiable function on satisfying for all Suppose that there exist two real numbers, a, b (a < b) such that y(a) = y(b) = 0. Then

(a) y(x) changes sign on (a, b)

(b)

(c)

(d)

Ans. (b)

Sol.

9. For an integer let P_k denote the vector space of all real polynomials in one variable of degree less than or equal to k. Define a linear transformation by

Tf(x) =

Which one of the following polynomials is not in the range of T?

(a) x + x²

(b) x² + x³ + 2

(c) x + 1

(d) x + x³ + 2

Ans. (c)

Sol.

10. Let be a continuous function satisfying f(x) = f(x + 1) for all Then

(a) There exists a unique such that

(b) There is no such that

(c) f is not necessarily bounded above

(d) There exist infinitely many such that

Ans. (a)

Sol. As, function is given continuous and periodic,

i.e.,    f(x) = f(x + 1)

Hence, function must be bounded are there exists many such that

   

So, only option (a) is correct.

11. Consider the following statements.

I. The group has no proper subgroup of finite index.

II. The group has no proper subgroup of finite index.

Which one of the following statements is true?

(a) Neither I nor II is TRUE

(b) Both I and II are TRUE

(c) II is TRUE but I is FALSE

(d) I is TRUE but II is FALSE

Ans. (b)

Sol.

12. Let be defined by . Consider the function defined by

f(x, y) =

Then

(a) f is a continuous function on D and cannot be extended continuously to any point outside D.

(b) f is a discontinuous function on D.

(c) f is a continuous function on D and can be extended continuously to the whole of

(d) f is a continuous function on D and can be extended continuously to

Ans. (d)

Sol.

13. Consider the function

f(x) = .

Then

(a) f is not continuous at all

(b) f is not continuous at x = 0

(c) all are strict local minima for f

(d) f is continuous at all

Ans. (a)

Sol. We have f(x) =

Option (d) is not true.

Since, option (b), (c), (d) is not true.

Hence, option (a) is true.

14. Let y be the solution of

Then

(a) y is bounded on (–1, 0]

(b)

(c) y attains its minimum at x = 0

(d)

Ans. (c)

Sol. We have

15. Which one of the following statements is true?

(a) Exactly half of the elements in any even order subgroup of S₅ must be even permutations.

(b) There exists a normal subgroup of S₅ of index 7.

(c) There exists a cyclic subgroup of S₅ of order 6.

(d) Any abelian subgroup of S₅ is trivial.

Ans. (c)

Sol. In S₅; only way to get a subgroup of order 6 is to have a disjoint cycle like; (abc) (de)

So we have an order = lcm (3, 2) = 6

However any subgroup of S₅ with distinct elements is always an abelian. i.e. (abc) (de); (acb) (de);... etc.

16. Which one of the following statements is true?

(a)

(b)

(c)

(d)

Ans. (b)

Sol. (a) has uncountable many subgroup and (Q, +) does not have uncountable many subgroup. i.e. They cannot be isomorphic.

Hence, (Q/Z, +) is isomorphic to (Q/2Z, +) (c) (Z, +) is cyclic and (Q, +) is non cyclic. So they cannot be isomorphic.

(d) (Z, +) countable and (R, +) uncountable. So they cannot be isomorphic.

17. Let n ≥ 2 be an integer. Let be the linear transformation defined by

A(z₁, z₂, .... z_n) = (z_n, z₁, z₂, ..., z_{n – 1})

Which one of the following statements is true for every ?

(a) A is nilpotent

(b) All eigenvalues of A are of modulus 1

(c) A is singular

(d) Every eigenvalue of A is neither 0 or 1.

Ans. (b)

Sol. Linear transformation defined by A(z₁, z₂, ...z_n) = (z_n, z₁, z₂,...z_{n – 1}).

We can say, = (1 n n – 1...32)

18. For be an infinitely differentiable function such that all with a < b.

.

Then

(a) f is not a polynomial.

(b) f must be a linear polynomial.

(c) f must be a polynomial of degree less than or equal to 2.

(d) f must be a polynomial of degree greater than 2.

Ans. (c)

Sol.

Hence, option (a) is incorrect and option (b) is also incorrect, also option (d) is incorrect. Correct option is (c).

19. Let be the real vector space of all n × n matrices with real entries,

Let Consider the subspace of W of spanned by {I_n, A, A², ...}. Then the dimension of W over is necessarily

(a)

(b) at most n

(c) n²

(d) n

Ans. (b)

Sol. W = < I, A, A², ... >

A is n × n matrix

C_A(A) = 0

Aⁿ + a_{n – 1} A^{n – 1} + ... + a₁A + a₀I = 0

M_A(A) = 0

⇒ deg M_A(X) n

dim w = deg M_A(x) n

Hence, the dim w over R is at most n.

20. Consider the family of curves x² – y² = ky with parameter . The equation of the orthogonal trajectory to this family passing through (1, 1) is given by

(a) x² + 2xy = 3

(b) x³ + 3xy² = 4

(c) x³ + 2xy² = 3

(d) y² + 2x²y = 3

Ans. (b)

Sol. Given,    x² – y² = ky

Differential w.r. to 'x', we get

21. Define S = . Then

(a)

(b) S = 1

(c)

(d)

Ans. (c)

Sol.

22. Consider the surface is the continuous unit normal field to the surface S with positive z-component, then equals

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

23. Let be a non-constant continuous function such that f ºf = f. Define E_f = . Then

(a) E_f is an interval

(b) E_f is empty

(c) E_f is neither open nor closed

(d) E_f need not be an interval

Ans. (a)

Sol.

24.

(a)

(b) y = 0 has exactly two roots

(c) y = 0 has exactly one root

(d) y = 0 has more than two roots.

Ans. (b)

Sol. Given

25. Let A be an n × n invertible matrix and C be an n × n nilpotent matrix. If X = is a 2n × 2n matrix (each X_ij being n × n) that commutes with the 2n × 2n matrix B = , then

(a) X₁₂ and X₂₂ are necessarily zero matrices

(b) X₁₁ and X₂₂ are necessarily zero matrices

(c) X₁₂ and X₂₁ are necessarily zero matrices

(d) X₁₁ and X₂₁ are necessarily zero matrices.

Ans. (c)

Sol.

26. Let g be an element of S₇ such that g commutes with the element (2, 6, 4, 3). The number of such g is

(a) 48

(b) 6

(c) 4

(d) 24

Ans. (d)

Sol. S₇, (2 6 4 3)

27. Let be a bijective map such that

The number of such bijective maps is

(a) zero

(b) infinite

(c) exactly one

(d) finite but more than one

Ans. (a)

Sol.

28. Consider the two series

Which one of the following holds?

(a) Both I and II converge

(b) I converges and II diverges

(c) I diverges and II converges

(d) Both I and II diverge.

Ans. (d)

Sol. We have

Therefore, both series I and II are diverges.

29.

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

30. Let G be a finite abelian group of odd order. Consider the following two statements:

I. The map defined by f(g) = g² is a group isomorphism.

II. The product

(a) Both I and II are TRUE

(b) Neither I nor II is TRUE

(c) II is TRUE but I is FALSE

(d) I is TRUE but II is FALSE.

Ans. (a)

Sol. Given order of G = 2K + 1, KEN abelian

31. Let G be a finite group of order 28. Assume that G contains a subgroup of order 7. Which of the following statements is/are true?

(a) G contains atleast two subgroup of order 7

(b) G contains no normal subgroup of order 7

(c) G contains a unique subgroup of order 7

(d) G contains a normal subgroup of order 7.

Ans. (b), (c)

Sol. We have O(G) = 28 = 2² × 7

32. Let be a differentiable function on (a, b). Which of the following statements is/are true?

(a) If for some then there exists a such that f(x) > f(x₀) for all

(b) If for some then f is increasing in a neighbourhood of x₀

(c) in (a, b) implies that f is increasing in (a, b)

(d) f is increasing in (a, b) implies that in (a, b)

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

Sol.

 

Option (d) is wrong.

33. Let V be a finite dimensional vector space and be a linear transformation. Let R(T) denote the range of T and N(T) denote the null space of T. If rank(T) = rank(T²), then which of the following is/are necessarily true?

(a) N(T) = {0}

(b) N(T) = N(T²)

(c)

(d) R(T) = R(T²)

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

Sol.

34. Consider the four functions from

f₁(x) = x⁴ + 3x³ + 7x + 1

f₂(x) = x³ + 3x² + 4x

f₃(x) = arctan(x)

and f₄(x) =

Which of the following subsets of are open?

(a) The range of f₄

(b) The range of f₂

(c) The range of f₁

(d) The range of f_3.

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

Sol. Given,   f₁(x) = x⁴ + 3x³ + 7x + 1

Option (c) is wrong.

f₂(x) = x³ + 3x³ + 4x

Option (a) is true.

35. Which of the following subsets of is/are connected?

(a)

(b)

(c)

(d)

Ans. (a), (b)

Sol. For solving this problem, we should take all (x) values, which satisfy the given property. So, in first option, we should be differentiate the given equation or function w.r.t. x,

It is clear that the function is increasing function and continuous, so if we calculate their root of (x³ + x + 1), then one root is attains at X-axis, (cubic root), therefore

Therefore, option (a) is correct.

For option (b), x³ – 1 > 0

If we draw the graph x³,

But here x³ – 1 then it shifts at 1.

Therefore, option (b) is also correct.

For option (c)

Irrational function are not the connected function.

So, it is not correct.

For option (d)

The graph of this function is given by

[because polynomial is cubic, so the range set is whole R]

So, this is disconnected function.

So, the right answer for this problem is option (a) and (b).

36. Consider the two functions f(x, y) = x + y and g(x, y) = xy – 16 defined on Then

(a) The function g has a global extreme value at (0, 0) subject to the condition f = 0.

(b) The function f has no global extreme value subject to the condition g = 0.

(c) The function f attains global extreme values at (4, 4) and (–4, –4) s.t. the condition g = 0.

(d) The function g has no global extreme value subject to the condition f = 0.

Ans. (a), (d)

Sol.

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

37. Let . Consider the two functions u, v : defined by

u(x, y) = x² – y² and v(x, y) = xy

Consider the gradients of the function u and v respectively. Then

(a)

(b)

(c)

(d)

Ans. (a), (c)

Sol.

Hence, are perpendicular at each point of D.

Therefore, option (a) is correct and option (b), (d) wrong.

Now, (c)

38. Let m > 1 and n > 1 be integers. Let A be an m × n matrix such that for some m × 1 matrix b₁, the equation Ax = b₁ has infinitely many solutions. Let b₂ denote an m × 1 matrix different from b₁. Then Ax = b₂ has

(a) finitely many solutions for some b₂

(b) no solution for some b₂

(c) infinitely many solutions for some b₂

(d) a unique solution for some b₂.

Ans. (b), (c)

Sol.

39. Consider the equation x²⁰²¹ + x²⁰²⁰ + ... + x – 1 = 0. Then

(a) exactly one real root is positive

(b) no real root is positive

(c) all real roots are positive

(d) exactly one real root is negative

Ans. (a), (b)

Sol. Given that,

   f(x) = x²⁰²¹ + x²⁰²⁰ + x²⁰¹⁹ + .... x – 1 = 0

We have to calculate the positive root and number of positive roots of the function.

In this function the sign change on time only.

We know that if a function change the sign more that 1, then we have positive roots in between them for example, if the function change its sign three (3) times than the number of positive roots in between 1 to 3.

Here, the function change its sign exactly one time.

Therefore, it has exactly one positive root.

For calculating the negative roots, we check for

f(– x) = ( – x)²⁰²¹ + (– x)²⁰²⁰ + ... + (– x) – 1

= – x²⁰²¹ + x²⁰²⁰ + ... – x – 1

Here, the sign change 2020 times.

So, therefore the function have exactly one real root is positive and all real roots are positive.

40. Let be a function with the property that for every the value of the expression is finite. Define g(y) = Then

(a)

(b) g is odd if f is even

(c) g is even if f is even

(d)

Ans. (c)

Sol.

Hence, option (c) is correct and option (b) is wrong.

41. The number of group homomorphisms from the group to the group S₃ is _________.

Ans. 4

Sol.

Now Z₄ has 3 subgroups and as it is a cyclic abelian so all subgroups are normal.

Hence they are

42. Consider the subset Let

P(x, y) = .

For If C denotes the unit circle traversed in the counter-clockwise direction, the value of

is _________.

Ans. (–2)

Sol. Given

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

43. Let y : be a differentiable function satisfying

Then y(2) equals _________.

Ans. 3

Sol. Given

44. Consider the set A = has a real root}. The number of connected components of A is _________.

Ans. (2)

Sol. Given, set A = {a R : x² = a(a + 1) (a + 2)} has a real root.

Here,

45. The value of is _________.

Ans. 7

Sol.

46. Let B = and define u(x, y, z) = sin((1 – x² – y² – z²)²) for (x, y, z) B. Then the value of is _________.

Ans. 0

Sol.

47. The number of cycles of length 4 in S₅ is _________.

Ans. 90

Sol. The number of cycles of length 4 in S₆ is

48. The value of is _________.

Ans. 1

Sol.

49. Let V be the real vector space of all continuous functions such that the restriction of f to the interval [0, 1] is a polynomial of degree less than or equal to 2, the restriction of f to the interval [1, 2] is a polynomial of degree less than or equal to 3 and f(0) = 0. Then the dimension of V is equal to _________.

Ans. 6

Sol. We have

50. Let be a vector field in and C be a continuously differentiable path with the starting point (0, 1) and the end point . Then equals _________.

Ans. 1

Sol.

51. Consider those continuous functions that have the property that given any

The number of such function is _________.

Ans. 0

Sol.

52. The number of elements of order two in the group S₄ is equal to _________.

Ans. 9

Sol. S₄ all one-one onto mappings of type; (1, 2, 3, 4) {1, 2, 3, 4}.

So order of S₄ = 4!

Partitions possible are;

Number of elements of order k in the group

Now for 2 + 1 + 1;

Number of elements of order two in S₄.

Ans for 2 + 2,

Number of elements of order 2 in S₄

53. The least possible value of k, accurate up to two decimal places, for which the following problem

has a solution is _________.

Ans. 4

Sol.

54. The largest positive number a such that

for every strictly increasing surjective continuous function is _________.

Ans. 1

Sol.

55. Define the sequence

s_n =

Define The number of limit points of the sequence is _________.

Ans. 8

Sol. Given A =

56. The determinant of the matrix

is _________.

Ans. 2021

Sol.

57. The value of

is _________.

Ans. 15

Sol.

58. Let S be the surface defined by

Let be the continuous unit normal field to the surface S with positive z-component. Then the value of

is _________.

Ans. 0

Sol. Given that

Similarly, we calculate the S₃, S₄ .... S_m, therefore the values put in Eq. (i),

Here, there is result, which we known that

So, we can say that

is increasing sequence which is not bounded above, this implies that sequence does not have limit point.

Therefore, the number of limit points of the sequence {σ_m} is (0).

59. Let A = . Then the largest eigenvalue of A is _________.

Ans. 2

Sol.

60. Let A = . Consider the linear map T_A from the real vector space to itself defined by T_A(X) = AX – XA, for all The dimension of the range of T_A is _________.

Ans. 0

Sol.