Math - 2015 Past Year Paper


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60 Questions MCQ Test IIT JAM Past Year Papers and Model Test Paper (All Branches) | Math - 2015 Past Year Paper

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Math - 2015 Past Year Paper - Question 1

Suppose N is a normal subgroup of a group G. Which one of the following is true?

Math - 2015 Past Year Paper - Question 2

Let y(x) = u(x) sin sin x + v(x) cos x be a solution of the differential equation y” + y = sec x.
Then u(x) is

Math - 2015 Past Year Paper - Question 3

Let a, b, c, d be distinct non- zero real numbers with a + b = c + d. Then an eigenvalue of the
matrix    is

Math - 2015 Past Year Paper - Question 4

Let S be a nonempty subset of R. If S is a finite union of disjoint bounded intervals, then which one of the following is true?

Math - 2015 Past Year Paper - Question 5

Let {xn} be a convergent sequence of real numbers. 

for n ≥ 1, then which one of the following is the limit of this sequence?

Math - 2015 Past Year Paper - Question 6

The volume of the portion of the solid cylinder x2 + y2 ≤ 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane is

Math - 2015 Past Year Paper - Question 7

Let f: R→R be a differentiable function with f(0) = 0. If for all x ∈ R, 1 < f'(x) < 2, then which one of the following statements is true on (0, ∝)?

Math - 2015 Past Year Paper - Question 8

If an integral curve of the differential equation  passes through (0, 0) and (α, 1),
then α is equal to

Math - 2015 Past Year Paper - Question 9

An integrating factor of the differential equation 

Math - 2015 Past Year Paper - Question 10

Let A be a nonempty subset of R Let I(A) denote the set of interior points of A. Then I(A) can be

Math - 2015 Past Year Paper - Question 11

Let S3 be the group of permutations of three distinct symbols. The direct sum    has an element of order

Math - 2015 Past Year Paper - Question 12

The orthogonal trajectories of the family of curves y = C1x3 are

Math - 2015 Past Year Paper - Question 13

Let G be a nonabelian group. Let α ∈ G have order 4 and let β ∈ G have order 3. Then the order of the element αβ in G

Math - 2015 Past Year Paper - Question 14

Let S be the bounded surface of the cylinder x2 + y2 = 1 cut by the planes z = 0 and z = 1 + x. Then the value of the surface integral    is equal to 

Math - 2015 Past Year Paper - Question 15

Suppose that the dependent variables z and w are functions of the independent variables x and y, defined by the equations f(x, y, z, w) = 0 and g(x, y, z, w) = 0, where fzgw – fwgz = 1.
Which one of the following is correct

Math - 2015 Past Year Paper - Question 16

Math - 2015 Past Year Paper - Question 17

Math - 2015 Past Year Paper - Question 18

Let P2(R) be the vector space of polynomials in x of degree at most 2 with real coefficients. Let M2(R) be the vector space of 2 × 2 real matrices. If a linear transformation    is defined as    then

Math - 2015 Past Year Paper - Question 19

Let B1 = {(1, 2), (2, –1)} and B2 = {(1, 0), (0, 1} be ordered bases of R2 . If T : R2 →  R2  is a linear transformation such that 

 

then T(5, 5) is equal to

Math - 2015 Past Year Paper - Question 20

  Which one of the following statements is FALSE?

Math - 2015 Past Year Paper - Question 21

Let f : R → R be a strictly increasing continuous function. If {an} is a sequence in [0, 1], then the sequence {f(an)} is

Math - 2015 Past Year Paper - Question 22

Which one of the following statements is true for the series 

Math - 2015 Past Year Paper - Question 23

Math - 2015 Past Year Paper - Question 24

If y(t) is a solution of the differential equation y” + 4y = 2et, then

  is equal to 

Math - 2015 Past Year Paper - Question 25

For what real values of x and y, does the integral  attain its maximum?

Math - 2015 Past Year Paper - Question 26

The area of the planar region bounded by the curves x = 6y2 – 2 and x = 2y2 is

Math - 2015 Past Year Paper - Question 27

For n ≥ 2, let fn: R → R be given by fn(x) = xn sin x. Then at x = 0, fn has a

Math - 2015 Past Year Paper - Question 28

Math - 2015 Past Year Paper - Question 29

Let G and H be nonempty subsets of R , where G is connected and G U H is not connected.
Which one of the following statements is true for all such G and H ?

Math - 2015 Past Year Paper - Question 30

Then the value of 

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 31

Let f: R → R be a function defined by 

In which of the following interval(s), f takes the value 1?

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 32

Which of the following statements is (are) true?

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 33

Which of the following conditions implies (imply) the convergence of a sequence {xn} of real numbers?

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 34

 

If C is the curve of intersection of the surfaces x2 + y2 = 1 and y + z = 2, then which of the following is (are) equal to

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 35

Let V be the set of 2 × 2 matrices with complex entries such that a11 + a22 = 0. Let W be the set of matrices in V with . Then, under usual matrix addition and scalar multiplication, which of the following is (are) true?

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 36

The initial value problem    has 

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 37

Which of the following statements is (are) true on the interval (0, π/2) ?

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 38

Let f: R2 → R be defined by

  

At (0,0)

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 39

Let f, g : [0, 1] → [0, 1] be functions. Let R(f) and R(g) be the ranges of f and g, respectively.
Which of the following statements is (are) true?

*Multiple options can be correct
Math - 2015 Past Year Paper - Question 40

Let f: (–1, 1) → R be the function defined by    Then

*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 41

Let C be the straight line segment from P(0, π) to (4, π/2) in the xy- plane. Then the value of


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 42

Let S be the portion of the surface  bounded by the planes x = 0, x = 2, y = 0, and y = 3. The surface area of S, correct upto three decimal places, is ____________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 43

The number of distinct normal subgroups of S3 is _____


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 44

If the directional derivative of f at (0, 0) exists along the direction cos αi sin α j , where sin a ≠  0, then the value of cot a is _________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 45

The maximum rate of change of f at (π/4 , 0, 1) correct upto three decimal places, is ________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 46

If the power series  

converges for |x| < c and diverges for |x| > c, then the value of c, correct upto three decimal places, is--


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 47

If 52015 ≡  n. ( mod 11) and n ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then n is equal to _________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 48

If the set    is linearly dependent in the vector space of all 2 × 2 matrices with real entries, then x is equal to ___


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 49

let f: R → R be defined by

The number of points at which f is continuous, is ______________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 50

Let f: (0, 1) → R be a continuously differentiable function such that f' has finitely many zeros in (0, 1) and fchanges sign at exactly two of these points. Then for any y ∈ R , the maximum number of solutions to f(x) = y in (0, 1) is ______________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 51

Let R be the planar region bounded by the lines x = 0, y = 0 and the curve x2 + y2 = 4, in the first quadrant. Let C be the boundary of R, oriented counter- clockwise. Then the value of

 is ----


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 52

Suppose G is a cyclic group and σ, τ ∈ G are such that order(σ) = 12 and order (τ) = 21. Then the order of the smallest group containing σ and τ is ______________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 53


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 54

Let M2(R) be the vector space of 2 × 2 real matrices. Let V be a subspace of M2(R) defined by

Then the dimension  is ____


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 55

correct upto three decimal places, is _________


Detailed Solution for Math - 2015 Past Year Paper - Question 55



We now change the order of integration. Then the integral equals




 

*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 56

The coefficient of in the Taylor series expansion of the function 

 

about the point  π/4 correct upto three decimal places, is ______


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 57

  as the power series expansion    then a5  correct upto three decimal places, is equal to _


Detailed Solution for Math - 2015 Past Year Paper - Question 57

We have
a1x + a2x2 + a3x3 + a4x4 + a5x5 +..
On differentiation, it gives, by Leibnitz rule on the left, and termwise differentiation on the right,
ex = 1 +
cosx
Then

The coefficient of t4 in (*) is 5a5.

The coefficient of x4 in (**) is 1/2 + 1/4!, i.e., 1/2 + 1/24 = 13/24.

Comparing these coefficients (note that symbols x and t play the same role), we get
13/24 = 5a5
Which given a5 = 1/5. 13/24 ≈ 0.108.

*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 58

Let ℓ be the length of the portion of the curve x = x(y) between the lines y = 1 and y = 3, where x(y) satisfies

The value of ℓ, correct upto three decimal places, is ___________


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 59


*Answer can only contain numeric values
Math - 2015 Past Year Paper - Question 60

Let P and Q be two real matrices of size 4 × 6 and 5 × 4, respectively. If rank(Q) = 4 and rank(QP) = 2, then rank(P) is equal to ______


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