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QUESTION: 1

Let f(x) = |x^{2 }– 25| for all x ∈ R. The total number of points of R at which f attains a local extremum (minimum or maximum) is

Solution:

QUESTION: 2

The coefficient of (x – 1)^{2} in the Taylor series expansion of f(x) = xe^{x} (x ∈ R) about the point x = 1 is

Solution:

QUESTION: 3

Solution:

QUESTION: 4

For a, b, c ∈ R, if the differential equation (ax^{2} + bxy + y^{2})dx + (2x^{2} + cxy + y^{2})dy = 0 is exact, then

Solution:

QUESTION: 5

If f(x, y, z) = x^{2}y + y^{2}z + z^{2}x for all (x, y, z) ∈ R^{3} and then the value of

at (1, 1, 1) is

Solution:

QUESTION: 6

The radius of convergence of the power series

Solution:

QUESTION: 7

Let G be a group of order 17. The total number of non- isomorphic subgroups of G is

Solution:

As 17 is a prime, and the order of a subgroup must divide the order of the group, G has only two subgroups, the unity subgroup and G itself. So the answer is 2.

QUESTION: 8

Which one of the following is a subspace of the vector space R^{3} ?

Solution:

QUESTION: 9

Let T : R^{3} → R^{3} be the linear transformation defined by T(x, y, z) = (x + y, z + z, z + x) for all (x, y, z) ∈ R^{3}. Then

Solution:

QUESTION: 10

Let f: R → R be a continuous function satisfying for all x ∈ R . Then the set

is the interval

Solution:

QUESTION: 11

**Directions: Q.11 – Q.35 carry two marks each.**

**The system of linear equations**

**x – y + 2z = b _{1} **

**x + 2y – z = b _{2} **

**2y – 2z = b _{3}
is inconsistent when (b_{1}, b_{2}, b_{3}) equals**

Solution:

QUESTION: 12

be a matrix with real entries. If the sum and the product of all the eigenvalues

of A are 10 30 respectively, then a^{2} + b^{2} equals

Solution:

QUESTION: 13

Consider the subspace W = {(x_{1}, x_{2}, ..., x_{10}) ∈ R^{10 }: x_{ n} = x_{n–1 }+ x_{n–2} for 3 ≤ n ≤ 10} of the vector space R^{10}. The dimension of W is

Solution:

QUESTION: 14

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

x_{2} y”(x) – 2xy’(x) – 4y(x) = 0 for x ∈ [1, 10].

Consider the Wronskian W(x) = y_{1}(x)y_{2}’(x) – y_{2}(x)y_{1}’(x). If W(1) = 1, then W(3) – W(2) equals

Solution:

QUESTION: 15

The equation of the curve passing through the point ,(π/2, 1) and having slope at each

point (x, y) with x ≠ 0 is

Solution:

QUESTION: 16

The value of α ∈ R for which the curves x^{2} + αy^{2} = 1 and y = x^{2} intersect orthogonally is

Solution:

QUESTION: 17

Solution:

QUESTION: 18

Let {x_{n}} be a sequence of real numbers such that where c is a positive real number. Then then sequence {x_{n }/n} is

Solution:

QUESTION: 19

where Then

Solution:

QUESTION: 20

Solution:

QUESTION: 21

The set all limit points of the set

Solution:

QUESTION: 22

Let S = [0, 1] ∪ [2, 3) and let f : S → R be defined by

If T = {f(x) : x ∈ S}, then the inverse function f^{–1} : T → S

Solution:

QUESTION: 23

Let f(x) = x^{3} + x and g(x) = x^{3} – x for all x ∈ R. If f^{–1} denotes the inverse function of f, then the derivative of the composite function g o f^{–1} at the point 2 is

Solution:

QUESTION: 24

For all (x, y) ∈ R^{2} ,

Then at the point (0, 0),

Solution:

QUESTION: 25

For all (x, y) ∈ R^{2}

Solution:

QUESTION: 26

Let f : R → R be a function with continuous derivative such that f ( √2 ) = 2 and f(x) =

for all x ∈ R. Then f(3) equals

Solution:

QUESTION: 27

The value of is

Solution:

QUESTION: 28

If C is a smooth curve in R^{3} from (–1, 0, 1) to (1, 1, –1), then the value of

is

Solution:

QUESTION: 29

Let C be the boundary of the region R = {(x, y) ∈ R^{2} : –1 ≤ y ≤ 1, 0 ≤ x ≤ 1 – y^{2}} oriented in the counterclockwise direction. Then the value of

Solution:

QUESTION: 30

Let G be a cyclic group of order 24. The total number of group isomorphism’s of G onto itself is

Solution:

QUESTION: 31

Let S_{n} be the group of all permutations on the set {1, 2, ..., n} under the composition of mappings.

For n > 2, if H is the smallest subgroup of S_{n} containing the transposition (1, 2) and the cycle (1, 2, ..., n), then

Solution:

QUESTION: 32

Let S be the oriented surface x^{2} + y^{2} + z^{2} = 1 with the unit normal n pointing outward. For the vector field F(x, y, z) = xi + yj + zk, the value of

Solution:

QUESTION: 33

Let f : (0, ∞ ) → R be a differentiable function such that f'(x^{2}) = 1 - x^{3} for all x > 0 and f(1) = 0.

Then f(4) equals

Solution:

QUESTION: 34

Which one of the following conditions on a group G implies that G is abelian?

Solution:

QUESTION: 35

Let S = {x ∈ R : x^{6} – x^{5} ≤ 100} and T = {x^{2 }– 2x : x ∈ (0, ∞)}. Then set S ∩ T is

Solution:

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