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Math - 2019 Past Year Paper - Question 1

Let a_{1} = b_{1} = 0, and for each n ≥ 2, let a_{n} and b_{n} be real numbers given by

Then which one of the following is TRUE about the sequences {a_{n}} and {b_{n}}?

Math - 2019 Past Year Paper - Question 2

Let Let V be the subspace of defined by

Then the dimension of V is

Math - 2019 Past Year Paper - Question 3

Let be a twice differentiable function. Define

f(x,y,z) = g(x^{2} + y^{2} - 2z^{2}).

Detailed Solution for Math - 2019 Past Year Paper - Question 3

Math - 2019 Past Year Paper - Question 4

be sequences of positive real numbers such that nan < bn < n^{2}a_{n} for

all n __>__ 2. If the radius of convergence of the power series then the power series

Detailed Solution for Math - 2019 Past Year Paper - Question 4

Math - 2019 Past Year Paper - Question 5

Let S be the set of all limit points of the set be the set of all positive

rational numbers. Then

Math - 2019 Past Year Paper - Question 6

If x^{h}y^{k} is an integrating factor of the differential equation y(1 + xy) dx + x(1 — xy) dy = 0, then the ordered pair (h, k) is equal to

Detailed Solution for Math - 2019 Past Year Paper - Question 6

Math - 2019 Past Year Paper - Question 7

If y(x) = λe^{2x} + e^{βx}, β ≠ 2, is a solution of the differential equation

satisfying dy/dx (0) = 5, then y(0) is equal to

Detailed Solution for Math - 2019 Past Year Paper - Question 7

Math - 2019 Past Year Paper - Question 8

The equation of the tangent plane to the surface at the point (2, 0, 1) is

Detailed Solution for Math - 2019 Past Year Paper - Question 8

Detailed Solution for Math - 2019 Past Year Paper - Question 9

Math - 2019 Past Year Paper - Question 10

The area of the surface generated by rotating the curve x = y^{3}, 0 ≤ y ≤ 1, about the y-axis, is

Math - 2019 Past Year Paper - Question 11

Let H and K be subgroups of If the order of H is 24 and the order of K is 36, then the order of the subgroup H ∩ K is

Math - 2019 Past Year Paper - Question 12

Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, then

Detailed Solution for Math - 2019 Past Year Paper - Question 12

Math - 2019 Past Year Paper - Question 17

Let be a differentiable function such that f'(x) > f(x) for all and f(0) = 1. Then f( 1) lies in the interval

Detailed Solution for Math - 2019 Past Year Paper - Question 17

Math - 2019 Past Year Paper - Question 18

For which one of the following values of k, the equation 2x^{3} + 3x^{2} − 12x − k = 0 has three distinct real roots?

Detailed Solution for Math - 2019 Past Year Paper - Question 18

Detailed Solution for Math - 2019 Past Year Paper - Question 19

Math - 2019 Past Year Paper - Question 20

Let S be the family of orthogonal trajectories of the family of curves 2x^{2} + y^{2} = k, for and k > 0. If passes through the point (1, 2), then passes through

Detailed Solution for Math - 2019 Past Year Paper - Question 20

Math - 2019 Past Year Paper - Question 21

Let x, x + e^{x} and 1 + x + e^{x} be solutions of a linear second order ordinary differential equation with constant coefficients. If y(x) is the solution of the same equation satisfying y(0) = 3 and y'(0) = 4, then y(1) is equal to

Detailed Solution for Math - 2019 Past Year Paper - Question 21

Math - 2019 Past Year Paper - Question 22

The function f(x,y) = x^{3} + 2xy + y^{3} has a saddle point at

Detailed Solution for Math - 2019 Past Year Paper - Question 22

Math - 2019 Past Year Paper - Question 23

The area of the part of the surface of the paraboloid x^{2 }+ y^{2} + z = 8 lying inside the cylinder x^{2} + y^{2} = 4 is

Detailed Solution for Math - 2019 Past Year Paper - Question 23

Math - 2019 Past Year Paper - Question 24

be the circle (x − 1)^{2} + y^{2} = 1, oriented counterclockwise. Then the value of the line integral

is

Detailed Solution for Math - 2019 Past Year Paper - Question 24

Math - 2019 Past Year Paper - Question 25

be the curve of intersection of the plane x + y + z = 1 and the cylinder x^{2} + y^{2} = 1. Then the value of

is

Detailed Solution for Math - 2019 Past Year Paper - Question 25

Math - 2019 Past Year Paper - Question 26

The tangent line to the curve of intersection of the surface x^{2} + y^{2} − z = 0 and the plane x + y = 3 at the point (1, 1, 2) passes through

Math - 2019 Past Year Paper - Question 27

The set of eigenvalues of which one of the following matrices is NOT equal to the set of eigenvalues of

Math - 2019 Past Year Paper - Question 29

Let {a_{n}} be a sequence of positive real numbers such that a_{1} = 1, for all n ≥ 1.

Then the sum of the series lies in the interval

Math - 2019 Past Year Paper - Question 30

Let {a_{n}} be a sequence of positive real numbers. The series converges if the series

Detailed Solution for Math - 2019 Past Year Paper - Question 30

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 31

Let G be a noncyclic group of order 4. Consider the statements I and II:

I. There is NO injective (one-one) homomorphism from

II. There is NO surjective (onto) homomorphism from

Then

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 32

Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy^{-1}, g(x) = x^{-1} and h = g _{° }g.

Then

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 33

Let S and T be linear transformations from a finite dimensional vector space V to itself such that S(T(v)) = 0 for all v ∈ V. Then

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 34

Let be differentiable vector fields and let g be a differentiable scalar function. Then

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 35

Consider the intervals S = (0, 2] and T = [1, 3). Let S° and T° be the sets of interior points of S and T, respectively. Then the set of interior points of S\T is equal to

Math - 2019 Past Year Paper - Question 36

Let {a_{n}} be the sequence given by

Then which of the following statements is/are TRUE about the subsequences {a_{6n−1}} and {a_{6n+4}}?

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 37

Let f(x) = cos(|π - x|) + (π - n) sin |x| and g(x) = x^{2} If h(x) = f(g(x)), then

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 38

be given by f(x) = (sin x)^{π} − π sin x + π.

Then which of the following statements is/are TRUE?

*Multiple options can be correct

*Multiple options can be correct

Math - 2019 Past Year Paper - Question 40

Let {a_{n}} be the sequence of real numbers such that a_{1} = 1 and a_{n+1 }= a_{n} + a^{2}_{n} for all n __>__ 1.

Then

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 41

Let x be the 100-cycle (1 2 3 ⋯ 100) and let y be the transposition (49 50) in the permutation group S_{100}. Then the order of xy is ______

Detailed Solution for Math - 2019 Past Year Paper - Question 41

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 42

Let W_{1} and W_{2} be subspaces of the real vector space defined by

W_{1} = {(x_{1,}x_{2}, ...,x_{100}) : x_{i} = 0 if i is divisible by 4},

W_{2} = { (x_{1};x_{2}, ....x_{100}) : x_{i} = 0 if i is divisible by 5}.

Then the dimension of W_{1} ∩ W_{2} is ____

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 43

Consider the following system of three linear equations in four unknowns x_{1}, x_{2}, x_{3} and x_{4}

x_{1} + x_{2} + x_{3} + x_{4} = 4,

x_{1} + 2x_{2} + 3x_{3} + 4x_{4} = 5,

x_{1} + 3x_{2} + 5x_{3} + kx_{4} = 5.

If the system has no solutions, then k = _____

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 44

Let and be the ellipse

oriented counter clockwise. Then the value of (round off to 2 decimal places) is_______________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 45

The coefficient of in the Taylor series expansion of the function about x = π/2, is ____________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 46

Let be given by

Then

max {f(x): x ∈ [0,1]} - min [f(x):x ∈ [0,1]}

is ___________

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 48

Let

Then the directional derivative of f at (0, 0) in the direction of

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 49

The value of the integral (round off to 2 decimal places) is ___________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 50

The volume of the solid bounded by the surfaces x = 1 - y^{2} and x = y^{2} - 1, and the planes z = 0 and z = 2 (round off to 2 decimal places) is

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 51

The volume of the solid of revolution of the loop of the curve y^{2} = x^{4} (x + 2) about the x-axis (round off to 2 decimal places) is ___________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 52

The greatest lower bound of the set (round off to 2 decimal places) is ______________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 53

Let G = : n __<__ 55, gcd(n, 55) = 1} be the group under multiplication modulo 55. Let x ∈ G be such that x^{2} = 26 and x > 30. Then x is equal to _______

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 54

The number of critical points of the function is ___________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 55

The number of elements in the set {x ∈ S_{3}: x^{4} = e), where e is the identity element of the permutation group S_{3}, is

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 56

If is an eigenvector corresponding to a real eigenvalue of the matrix then z − y is equal to__________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 57

Let M and N be any two 4 × 4 matrices with integer entries satisfying

Then the maximum value of det(M) + det(N) is ___________

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 58

Let M be a 3 × 3 matrix with real entries such that M^{2} = M + 2I, where I denotes the 3 × 3 identity matrix. If α, β and γ are eigenvalues of M such that αβγ = -4, then α+β+γ is equal to ______

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 59

Let y(x) = xv(x) be a solution of the differential equation If v(0) = 0 and v(1) = 1, then v(-2) is equal to ______

*Answer can only contain numeric values

Math - 2019 Past Year Paper - Question 60

If y(x) is the solution of the initial value problem then y(ln 2) is (round off to 2 decimal places) equal to ____________

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