Math - 2018 Past Year Paper - IIT JAM MCQ

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?