Math - 2018 Past Year Paper - IIT JAM

Let a be a positive real number. If f is a continuous and even function defined on the interval [–a, a], then  is equal to :

The tangent plane to the surface  at (1, 1, 2) is given by

In , the cosine of the acute angle between the surfaces x² + y² + z² - 9 = 0 and z - x² - y² + 3 = 0 at the point (2, 1, 2) is :

Let f :  be a scalar field,  be a vector filed and let  be a constant vector. If  represents the position vector  then which one of the following is FALSE?

In , the family of trajectories orthogonal to the family of asteroids x^2/3 + y^2/3 = a^2/3 is given by

Consider the vector space V over of polynomial functions of degree less then or equal to 3 defined on . Let T : V → V be defined by (Tf)(x) = f(x) – xf’(x). Then the rank of T is

Let  Then which one of the following is True for the sequence ? 

Let  
Then which one of the following is TRUE?

Let a, b, c . Which of the following values of a, b, c do NOT result in the convergence of the series

Let  Then the sum of the series  is:

Let  and let  where  Then which one of the following is true?

Suppose that f, g  are differentiable functions such that f is strictly increasing and g is strictly decreasing. Define p(x) = f(g(x)) and q(x) = g(f(x)),  Then, for t > 0, the sign of  is :

For  Then which one of the following is false?

Let  Then which one of the following is true for f at the point (0, 0)?

Let a, b  be a thrice differentiable function. If z = e^u f(v), where u = ax + by and v = ax – by, then which one of the following is true?

Consider the region D in the yz plane bounded by the line y = 1/2 and the curve y² + z² = 1, where y≥0. If the region D is revolved about the z- axis in , then the volume of the resulting solid is :

If  where C is the boundary of the triangular region bounded by the lines x = 0, y = 0 and x + y = 1 oriented in the anti- clockwise direction, is :

Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?

Let y(x) be the solution of the differential equation dy/dx + = , for x ≥ 0, y(0) = 0, where  Then y(x) =

An integrating factor of the differential equation  is:

A particular integral of the differential equation is:

Let G be a group satisfying the property that f :  is a homomorphism implies f(g) = 0,  Then a possible group G is :

Let H be the quotient group  Consider the following statements.
I. Every cyclic subgroup of H is finite.
II. Every finite cyclic group is isomorphic to a subgroup of H.
Which one of the following holds?

Let I denote the 4 × 4 identity matrix. If the roots of the characteristic polynomial of a 4 × 4 matrix M are  , then M⁸ =

Consider the group  under component- wise addition. Then which of the following is a subgroup of  ?

Let f :  be a function and let J be a bounded open interval in Define

Which one of the following is false?

For  Then which one of the following is true?

Let f :  be defined by f ( x) =  On which of the following interval(s) is f one- one?

The solution(s) of the differential equation  satisfying y(0) = 0 is (are)

Suppose f, g, h are permutations of the set  where
f interchanges ∝ and β but fixes γ and δ
g interchanges β and γ but fixes ∝ and δ,
h interchanges γ and δ but fixes ∝ and β.
Which of the following permutations interchange(s) ∝ and δ but fix(es) β and γ?

Let P and Q be two non- empty disjoint subsets of Which of the following is (are) false?

Let  denote the group of non- zero complex numbers under multiplication. Suppose  Which of the following is (are) subgroup(s) of ?

Suppose  Consider the following system of linear equations. x + y + z = α, x + βy + z = γ, x + y + αz = β. If this system has at least one solution, then which of the following statements is (are) true?

Let m,   Then which of the following is (are) Not possible?

One among the following is the correct explanation of pedal equation of an polar curve, r = f (θ), p = r sin(∅) (where p is the length of the perpendicular from the pole to the tangent & ∅ is the angle made by tangent to the curve with vector drawn to curve from pole)is _______. 

Which of the following subsets of is (are) connected?

Let S be a subset of ¡ such that 2018 is an interior point of S. Which of the following is (are) true?

The order of the element (1 2 3)(2 4 5)(4 5 6) in the group S₆ is _______ .

Let  Then the absolute value of the directional derivative of φ in the direction of the line  at the point (1, – 2, 1) is _______ .

Let  Then the value of  is _______ .

Let  be given by

Then  at the point (0, 0) is _______ .

Let  for (x, y)  Then f_x(1, 1) + f_y(1, 1) = _______ .

Let  be continuous on  and differentiable on   then f(6) = _______ .

Let  Then the radius of convergence of the power series  about x = 0 is _______ .

Let A₆ be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in A₆ is _______ .

Let W₁ be the real vector space of all 5 × 2 matrices such that the some of the entries in each row is zero. Let W₂ be the real vector space of all 5 × 2 matrices such that the sum of the entries in each column is zero. Then the dimension of the space is _______ .

The coefficient of x⁴ in the power series expansion of e^{sin x} about x = 0 is _______ (correct up to three decimal places).

Let  where k,  Then  is _______ . (correct up to one decimal places).

Let  be such that f" is continuous on and f(0) = 1, f'(0) = 0 and f"(0) = – 1. Then  is _______ (correct up to three decimal places).

Suppose x, y, z are positive real number such that x + 2y + 3z = 1. If M is the maximum value of xyz², then the value of 1/M is _______ .

If the volume of the solid in  bounded by the surfaces x = – 1, x = 1, y = – 1, y = 1, z = 2, y² + z² = 2 is α– π, then a = _______ .

If  then the value of  is _______ .

The value of the integral  is _______ . (correct up to three decimal places).

Suppose  is a matrix of rank 2. Let T :  be the linear transformation defined by T(P) = QP. Then the rank of T is _______ .

The area of the parametrized surface  is _______ (correct up to two decimal places).

If x(t) is the solution to the differential equation  satisfying x(0) = 1, then the value of x (√2 ) is _______ (correct up to two decimal places).

If y(x) = v(x) sec x is the solution of y" – (2 tan x)y' + 5y = 0, satisfying y(0) = 0 and y '(0) =  √6, then v  is _______ . (correct up to two decimal places).