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Math - 2018 Past Year Paper - Question 2

Let where b_{1} = 1, b_{2} = 1 and b_{n+2} = b_{n} + b_{n+1}, Then is:

Math - 2018 Past Year Paper - Question 3

If {v_{1}, v_{2}, v_{3}} is a linearly independent set of vectors in a vector space over then which one of the following sets is also linearly independent?

Math - 2018 Past Year Paper - Question 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 :

Math - 2018 Past Year Paper - Question 5

The tangent plane to the surface at (1, 1, 2) is given by

Math - 2018 Past Year Paper - Question 6

In , the cosine of the acute angle between the surfaces x^{2} + y^{2} + z^{2} - 9 = 0 and z - x^{2} - y^{2} + 3 = 0 at the point (2, 1, 2) is :

Math - 2018 Past Year Paper - Question 7

Let f : be a scalar field, be a vector filed and let be a constant vector. If represents the position vector then which one of the following is FALSE?

Math - 2018 Past Year Paper - Question 8

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

Math - 2018 Past Year Paper - Question 9

Consider the vector space V over of polynomial functions of degree less then or equal to 3 defined on . Let T : V → V be defined by (Tf)(x) = f(x) – xf’(x). Then the rank of T is

Math - 2018 Past Year Paper - Question 10

Let Then which one of the following is True for the sequence ?

Math - 2018 Past Year Paper - Question 12

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

Math - 2018 Past Year Paper - Question 14

Let and let where Then which one of the following is true?

Math - 2018 Past Year Paper - Question 15

Suppose that f, g 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 :

Math - 2018 Past Year Paper - Question 17

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

Math - 2018 Past Year Paper - Question 18

Let a, b 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?

Math - 2018 Past Year Paper - Question 19

Consider the region D in the yz plane bounded by the line y = 1/2 and the curve y^{2} + z^{2} = 1, where y≥0. If the region D is revolved about the z- axis in , then the volume of the resulting solid is :

Math - 2018 Past Year Paper - Question 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 :

Math - 2018 Past Year Paper - Question 21

Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U 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?

Math - 2018 Past Year Paper - Question 22

Let y(x) be the solution of the differential equation dy/dx + = , for x ≥ 0, y(0) = 0, where Then y(x) =

Math - 2018 Past Year Paper - Question 23

An integrating factor of the differential equation is:

Math - 2018 Past Year Paper - Question 24

A particular integral of the differential equation is:

Math - 2018 Past Year Paper - Question 25

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

Math - 2018 Past Year Paper - Question 26

Let H be the quotient group 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?

Math - 2018 Past Year Paper - Question 27

Let I denote the 4 × 4 identity matrix. If the roots of the characteristic polynomial of a 4 × 4 matrix M are , then M^{8} =

Math - 2018 Past Year Paper - Question 28

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

Math - 2018 Past Year Paper - Question 29

Let f : be a function and let J be a bounded open interval in Define

Which one of the following is false?

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 31

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

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 32

The solution(s) of the differential equation satisfying y(0) = 0 is (are)

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 33

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

f interchanges ∝ and β but fixes γ and δ

g interchanges β and γ but fixes ∝ and δ,

h interchanges γ and δ but fixes ∝ and β.

Which of the following permutations interchange(s) ∝ and δ but fix(es) β and γ?

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 34

Let P and Q be two non- empty disjoint subsets of Which of the following is (are) false?

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 35

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

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 36

Suppose Consider the following system of linear equations. x + y + z = α, x + βy + z = γ, x + y + αz = β. If this system has at least one solution, then which of the following statements is (are) true?

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 37

Let m, Then which of the following is (are) Not possible?

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 38

One among the following is the correct explanation of pedal equation of an polar curve, r = f (θ), p = r sin(∅) (where p is the length of the perpendicular from the pole to the tangent & ∅ is the angle made by tangent to the curve with vector drawn to curve from pole)is _______.

Detailed Solution for Math - 2018 Past Year Paper - Question 38

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 39

Which of the following subsets of is (are) connected?

*Multiple options can be correct

Math - 2018 Past Year Paper - Question 40

Let S be a subset of ¡ such that 2018 is an interior point of S. Which of the following is (are) true?

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 41

The order of the element (1 2 3)(2 4 5)(4 5 6) in the group S_{6} is _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 42

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

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 44

Let be given by

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

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 45

Let for (x, y) Then f_{x}(1, 1) + f_{y}(1, 1) = _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 46

Let be continuous on and differentiable on then f(6) = _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 47

Let Then the radius of convergence of the power series about x = 0 is _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 48

Let A_{6} be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in A_{6} is _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 49

Let W_{1} be the real vector space of all 5 × 2 matrices such that the some of the entries in each row is zero. Let W_{2} 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 _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 50

The coefficient of x^{4} in the power series expansion of e^{sin x} about x = 0 is _______ (correct up to three decimal places).

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 51

Let where k, Then is _______ . (correct up to one decimal places).

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 52

Let be such that f" is continuous on and f(0) = 1, f'(0) = 0 and f"(0) = – 1. Then is _______ (correct up to three decimal places).

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 53

Suppose x, y, z are positive real number such that x + 2y + 3z = 1. If M is the maximum value of xyz^{2}, then the value of 1/M is _______ .

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 54

If the volume of the solid in bounded by the surfaces x = – 1, x = 1, y = – 1, y = 1, z = 2, y^{2} + z^{2} = 2 is α– π, then a = _______ .

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 56

The value of the integral is _______ . (correct up to three decimal places).

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 57

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

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 58

The area of the parametrized surface is _______ (correct up to two decimal places).

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 59

If x(t) is the solution to the differential equation satisfying x(0) = 1, then the value of x (√2 ) is _______ (correct up to two decimal places).

*Answer can only contain numeric values

Math - 2018 Past Year Paper - Question 60

If y(x) = v(x) sec x is the solution of y" – (2 tan x)y' + 5y = 0, satisfying y(0) = 0 and y '(0) = √6, then v is _______ . (correct up to two decimal places).

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