Math - 2017 Past Year Paper
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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 4
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 6
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 8
Then which one of the following is TRUE ?
Math - 2017 Past Year Paper - Question 9
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 13
Math - 2017 Past Year Paper - Question 14
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 17
be the circle oriented anti- clockwise. Then
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 19
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 21
The interval of convergence of the power series
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 23
Let f(x, y) = for (x, y) ≠ (0, 0). Then
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 25
Math - 2017 Past Year Paper - Question 26
Then which one of the followings is NOT TRUE ?
Math - 2017 Past Year Paper - Question 27
Math - 2017 Past Year Paper - Question 28
Which one of the following is TRUE ?
Math - 2017 Past Year Paper - Question 29
A particular integral of the differential equation
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
Math - 2017 Past Year Paper - Question 31
Then which of the following statements is/are TRUE ?
*Multiple options can be correct
Math - 2017 Past Year Paper - Question 32
The volume of the solid is expressible as
*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 ?
A. If f is differentiable at (0, 0), then all directional derivatives of f exist at (0, 0).
B. If all directional derivatives of f exist at (0, 0), then f is differentiable at (0, 0).
C. If all directional derivatives of f exist at (0, 0), then f is continuous at (0, 0).
D. If the partial derivatives exist and are continuous in a disc centered at (0,
0) then f is differentiable at (0, 0).
*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
Math - 2017 Past Year Paper - Question 40
*Answer can only contain numeric values
Math - 2017 Past Year Paper - Question 41
Evaluation of 8^{3} × 8^{2} × 8^{-5} is.......
*Answer can only contain numeric values
Math - 2017 Past Year Paper - Question 42
Let G be a subgroup of Then the order of G is -----
*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 :- 55
Solution :- lim x -->0 1/x => 1/0
= ∞ i.e Apply L-hospital rule
lim x-->0 x[1/x + 2/x + 3/x + 4/x +.......................10/x]
lim x→0 x[10*11]/2 * 1/x
= [10*11]/2 => 55
*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 ____
*Answer can only contain numeric values
*Answer can only contain numeric values
Math - 2017 Past Year Paper - Question 59
The radius of convergence of the power series
*Answer can only contain numeric values
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