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Math - 2017 Past Year Paper - Question 1

Consider the function f(x, y) = 5 – 4 sin x + y^{2} for 0 < x < 2p and y ∈ R. The set of critical points of f(x, y) consists of

Math - 2017 Past Year Paper - Question 2

Let φ : R → R be a differentiable function such that φ' is strictly increasing with φ(1) = 0. Let a and b denote the minimum and maximum values of φ(x) on the interval [2, 3], respectively.

Then which one of the following is TRUE ?

Math - 2017 Past Year Paper - Question 3

The number of generators of the additive group Z_{36} is equal to

Math - 2017 Past Year Paper - Question 5

Let f : R → R be a twice differentiable function. If g(u, v) = f(u^{2} – v^{2}), then

Math - 2017 Past Year Paper - Question 7

Let f_{1}(x), f_{2}(x), g_{1}(x), g_{2}(x) be differentiable functions on R. be the determinent of the matrix . Then F'(x) is equal to

Math - 2017 Past Year Paper - Question 10

satisfies the assumptions of Rolle’s theorem in the interval [–1, 1], then the ordered pair (p, q) is

Math - 2017 Past Year Paper - Question 11

The flux of the vector field

along the outward normal, across the ellipse x^{2} + 16y^{2} = 4 is equal to

Math - 2017 Past Year Paper - Question 12

Let M be the set of all invertible 5 × 5 matrices with entries 0 and 1. For each

and n_{0}(M) denote the number of 1’s and 0’s in M, respectively. Then

Math - 2017 Past Year Paper - Question 15

The line integral of the vector field

along the boundary of the triangle with vertices (1,0,0), (0,1,0) and (0,0,1), oriented anticlockwise, when viewed from the point (2,2,2) is

Math - 2017 Past Year Paper - Question 16

The area of the surface z = xy/3 intercepted by the cylinder x^{2} + y^{2} ≤ 16 lies in the interval

Math - 2017 Past Year Paper - Question 18

The flux of along the outward normal, across the surface of the solid is equal to

Math - 2017 Past Year Paper - Question 20

Let f : R → [0, ∞) be a continuous function. Then which one of the following is NOT TRUE ?

Math - 2017 Past Year Paper - Question 22

Let P_{3 }denote the real vector space of all polynomials with real coefficients of degree at most 3. Consider the map T : P_{3 }→ P_{3} given by

Math - 2017 Past Year Paper - Question 24

Let S be an infinite subset of R such that S\{a} is compact for some α ∈ S. Then which one of the following is TRUE ?

Math - 2017 Past Year Paper - Question 30

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

satisfying y(0) = 1. Then y(–1) is equal to

*Multiple options can be correct

*Multiple options can be correct

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 33

Let be a function. Then which of the following statements is/are TRUE ?

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 34

If X and Y are n × n matrices with real entries, then which of the following is/are TRUE ?

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 35

Let G be a group of order 20 in which the conjugacy classes have sizes 1, 4, 5, 5, 5. Then which of the followings is/are TRUE ?

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 36

Let {x_{n}} be a real sequence such that Then which of the following statements is/are TRUE ?

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 37

Let S be the set of all rational numbers in (0, 1). Then which of the following statements is/are TRUE ?

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 38

Let M be an n × n matrix with real entries such that M^{3 }= I. Suppose that Mv ≠ v for any nonzero vector v. Then which of the following statements is/are TRUE ?

*Multiple options can be correct

Math - 2017 Past Year Paper - Question 39

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

satisfying the condition y(0) = 2. Then which of the following is/are TRUE ?

*Multiple options can be correct

*Answer can only contain numeric values

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 43

Consider the permutations

in S_{8} . The number of η ∈ S_{8} such that

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 44

Let P be the point on the surface closet to the point (4,2,0). Then the square of the distance between the origin and P is ______.

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 46

Let M be the matrix whose columns are v_{1}, v_{2}, 2v_{1} – v_{2}, v_{1} + 2v_{2} in that order. Then the number of linearly independent solutions of the homogeneous system of linear equations Mx = 0 is __________.

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 48

Let P be a 7 × 7 matrix of rank 4 with real entries. Let a ∈ R^{7} be a column vector. Then the rank of P + aa^{T} is at least ________.

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 49

For x > 0, let |x| denote the greatest integer less than or equal to x. Then

Detailed Solution for Math - 2017 Past Year Paper - Question 49

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 50

The number of subgroups of Z_{7} x Z_{7 }of order 7 is _______.

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 51

Let y(x), x > 0 be the solution of the differential equation

satisfying the conditions y(1) = 1 and y’(1) = 0. Then the value of e^{2}y(e) is ______

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 52

Let T be the smallest positive real number such that the tangent to the helix

at t = T is orthogonal to the tangent at t = 0. Then the line integral of along the section of the helix from t = 0 to t = T is _____

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 54

The maximum order of a permutation s in the symmetric group S_{10 }is ____

*Answer can only contain numeric values

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 56

For a real number x, define [x] to be the smallest integer greater than or equal to x. Then

*Answer can only contain numeric values

Math - 2017 Past Year Paper - Question 57

For x > 1, let

The number of tangents to the curve y = f(x) parallel to the line x + y = 0 is ____

Detailed Solution for Math - 2017 Past Year Paper - Question 57

*Answer can only contain numeric values

*Answer can only contain numeric values

*Answer can only contain numeric values

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