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Math - 2016 Past Year Paper - Question 2

Let be the vector space (over R) of all polynomials of degree ≤ 3 with real coefficients. Consider the linear transformation T: P → P defined by

Then the matrix representation M of T with respect to the ordered basis {1, x, x^{2}, x^{2} }satisfies

Math - 2016 Past Year Paper - Question 3

Let f : [-1,1] → R be a continuous function. Then the integral

is equal to

Math - 2016 Past Year Paper - Question 4

Let σ be an element of the permutation group S_{5 } Then the maximum possible order of σ is

Math - 2016 Past Year Paper - Question 5

Let f be a strictly monotonic continuous real valued function defined on [a,b] such that f(a) < a and f(b) >b Then which one of the following is TRUE?

Math - 2016 Past Year Paper - Question 8

The nonzero value of n for which the differential equation

becomes exact is

Math - 2016 Past Year Paper - Question 9

One of the points which lies on the solution curve of the differential equation

with the given condition y(0) = 1, is

Math - 2016 Past Year Paper - Question 10

Let S be a closed set of R, T a compact set of R such that S ∩ T ≠ Ø. Then S ∩ T is

Math - 2016 Past Year Paper - Question 11

Let S be the series

and T be the series

of real numbers. Then which one of the following is TRUE?

Math - 2016 Past Year Paper - Question 12

Let {a_{n}} be a sequence of positive real numbers satisfying

Then all the terms of the sequence lie in

Math - 2016 Past Year Paper - Question 15

If the triple integral over the region bounded by the planes 2x + y + z = 4, x = 0 , y = 0, z = 0

is given by then the function

Math - 2016 Past Year Paper - Question 16

The surface area of the portion of the plane y + 2z = 2 within the cylinder x^{2} + y^{2} = 3 is

Math - 2016 Past Year Paper - Question 18

The function f(x,y) = 3x^{2}y + 4y^{3} -3x^{2} - 12y^{2} + 1 has a saddle point at

Math - 2016 Past Year Paper - Question 20

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

Then y(2) is

Math - 2016 Past Year Paper - Question 22

The general solution of the differential equation with constant coefficients

approaches zero as x → ∞ if

Math - 2016 Past Year Paper - Question 27

a non-zero vector such that Mx = b for some x ∈ R^{3}

The value of x^{T}b is

Math - 2016 Past Year Paper - Question 28

The number of group homomorphisms from the cyclic group Z_{4} to the cycle group Z_{7} is

Math - 2016 Past Year Paper - Question 29

In the permutation group S_{n }(n ≥ 5) , if H is the smallest subgroup containing all the 3-cycles, then which one of the following is TRUE?

Detailed Solution for Math - 2016 Past Year Paper - Question 30

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 31

be a sequence of positive real numbers satisfying

If α and β are the roots of the equation

then which of the following statement(s) is(are) TRUE ?

*Multiple options can be correct

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 33

Then at the point (0, 0) which of the following statement(s) is(are) TRUE ?

*Multiple options can be correct

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 35

Consider the differential equation

Then which of the following statement(s) is(are) TRUE?

*Multiple options can be correct

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 37

Then which of the following statement(s) is(are)

TRUE?

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 38

Let G be a finite group and o(G) denotes its order. Then which of the following statement(s) is(are) TRUE?

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 39

Consider the set For which of the following choice(s) the set V becomes a two dimensional subspace of R^{3} over R ?

*Multiple options can be correct

Math - 2016 Past Year Paper - Question 40

Then which of the following statement(s) is(are) TRUE?

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 41

Let {S_{n}} be a sequence of real numbers given by

Then the least upper bound of the sequence {S_{n}} is ____

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 42

Let {S_{k}} be a sequence of real numbers, where

then is

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 43

Let be a non-zero vector and Then the dimension of the vector space over R is

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 44

Let f be a real valued function defined by

Then the value of at any point (x,y) where x > 0, y > 0, is _____

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 45

Then the value of integral from (0, 0) to (1, 1) along the path

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 46

is expressed as

where ξ lies between 2 and x , then the value of c is ____

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 47

Let y_{1}(x) , y_{2}(x), y_{2}(x) be linearly independent solutions of the differential equation

If the Wronskian W(y_{1 },y_{2 },y_{3}) is of the form ke^{bx} for some constant k, then the value of b is____________

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 49

Let f : (0,∝ ) → R be a continuous function such that

Then the value of

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 50

Let G be a cyclic group of order 12. Then the number of non-isomorphic subgroups of G is ____________

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 52

Let R be the region enclosed by and . Then the value of

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 53

Let

Then Mx = 0 has infinitely many solutions if trace(M) is

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 54

Let C be the boundary of the region enclosed by y = x^{2} ,y = x + 2, and x = 0. Then the value of the line integral

where C is traversed in the counter clockwise direction, is _____

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 55

Let S be the closed surface forming the boundary of the region V bounded by x^{2} + y^{2} = 3,

z = 0, z = 6, Then the value of

where n^{^} is the unit outward drawn normal to the surface S, is ______

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 56

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

Then y(x) attains its maximum value at x = ____________

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 58

Let H denote the group of all 2x 2 invertible matrices over Z_{5} under usual matrix multiplication. Then the order of the matrix in H is ----

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 59

N(A) the null space of A and R(B) the range space of B. Then the dimension of N(A )∩ R(B)over R is __

*Answer can only contain numeric values

Math - 2016 Past Year Paper - Question 60

The maximum value of f(x,y) = x^{2} + 2y^{2} subject to the constraint y - x^{2} + 1 = 0 is ___________

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