Math - 2017 Past Year Paper - IIT JAM

Let f : R → R be a twice differentiable function. If g(u, v) = f(u² – v²), then

Let f₁(x), f₂(x), g₁(x), g₂(x) be differentiable functions on R.   be the determinent of the matrix   . Then F'(x) is equal to 

Then which one of the following is TRUE ?

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

The flux of the vector field

along the outward normal, across the ellipse x² + 16y² = 4 is equal to

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

and n₀(M) denote the number of 1’s and 0’s in M, respectively. Then

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

The area of the surface z = xy/3 intercepted by the cylinder x² + y²  ≤ 16 lies in the interval

be the circle oriented anti- clockwise. Then

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

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

The interval of convergence of the power series 

Let P₃ denote the real vector space of all polynomials with real coefficients of degree at most 3. Consider the map T : P₃ → P₃ given by 

Let f(x, y) =   for (x, y) ≠ (0, 0). Then

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 ?

Then which one of the followings is NOT TRUE ?

Which one of the following is TRUE ?

A particular integral of the differential equation 

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

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

Then which of the following statements is/are TRUE ?

The volume of the solid     is expressible as

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

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

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 ?

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

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

Let M be an n × n matrix with real entries such that M³ = I. Suppose that Mv ≠ v for any nonzero vector v. Then which of the following statements is/are TRUE ?

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

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

 Evaluation of 8³ × 8² × 8^-5 is.......

Let G be a subgroup of    Then the order of G is -----

Consider the permutations  

   in S₈ . The number of η ∈  S₈  such that  

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 ______.

  Let M be the matrix whose columns are v₁, v₂, 2v₁ – v₂, v₁ + 2v₂ in that order. Then the number of linearly independent solutions of the homogeneous system of linear equations Mx = 0 is __________.

Let P be a 7 × 7 matrix of rank 4 with real entries. Let a ∈ R⁷ be a column vector. Then the rank of P + aa^T is at least ________.

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

The number of subgroups of Z₇ x Z₇ of order 7 is _______.

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 e²y(e) is ______

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 _____

The maximum order of a permutation s in the symmetric group S₁₀ is ____

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

For x > 1, let

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

The radius of convergence of the power series