Math - 2015 Past Year Paper - IIT JAM

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?

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

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

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

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, ∝)?

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

An integrating factor of the differential equation 

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

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

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

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

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 

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 f_zg_w – f_wg_z = 1.
Which one of the following is correct

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

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

  Which one of the following statements is FALSE?

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

Which one of the following statements is true for the series 

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

  is equal to 

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

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

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

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 ?

Then the value of 

Let f: R → R be a function defined by 

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

Which of the following statements is (are) true?

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

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

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?

The initial value problem    has 

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

Let f: R² → R be defined by

At (0,0)

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?

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

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

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 ____________

The number of distinct normal subgroups of S₃ is _____

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 _________

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

If the power series  

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

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

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

let f: R → R be defined by

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

Let f: (0, 1) → R be a continuously differentiable function such that f' has finitely many zeros in (0, 1) and f^' changes 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 ______________

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 ----

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 ______________

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 ____

correct upto three decimal places, is _________

The coefficient of in the Taylor series expansion of the function 

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

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

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 ___________

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 ______