Question(s) from Search: IIT

Let A be a set containing n elements. A subset P of A is constructed at random. The set A is reconstructed by replacing the elements of P. A subset of Q of A is again chosen at random. Find the probability that P and Q have no elements in common.

The derivative of an even function is always an odd function.

a) False

b) True

If  then

a) Re(z) = 0

b) Im(z) = 0

c) Re(z) = 0, Im(z) > 0

d) Re(z) > 0, Im(z) < 0

a) True

b) False

The derivative of  with respect to  at x =  is

a) 0

b) 1

c) 2

d) 4

If f (x) is differentiable and  , then  equals

a)

b)

c)

d)

 equals

a)

b)

c)

d) 4 f (2)

Let z and ω be two non zero complex numbers such that |z| = |ω| and Arg(z) + Arg(ω) = π then z equals

a)  ω

b)  

c)  

d)   

The function  is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0 is

a) a – b

b) a + b

c) lna – lnb

d) None of these

Find the value of

a)

b)

c)

d)

The set of lines  where  is concurrent at the point . . .

If tan θ =  then sin θ is

a)  but not  

b)  or

c)  but not −

d) None of these

Find the sum of the series
 

The set of all points where the function  is differentiable is

a)

b) [0, ∞)

c)  

d)  (0, ∞)

e)  None of these

Given a function f (x) such that
i) it is integrable over every interval on the real axis and
ii) f (t + x) = f (x) for every x and a real t, then show that the integral  is independent of a.

If the algebraic sum of the perpendicular distance from the point
(2, 0), (0, 2) and (1, 1) to a variable straight line be zero then the line passes through a fixed point whose coordinates are

The general solution of
 is

a)

b)

c)

d)

The function f(x) =  denotes the greatest integer function is discontinuous at

a) All x

b) All integer points

c) No x

d) x which is not an integer

If f (x) and g (x) are continuous functions on (0, a) satisfying f (x) = f (a – x) and g (x) + g (a – x) = 2 then show that

The equation of the circles through (1, 1) and the point of intersection of
 
is

a)

b)

c)

d) None of these

The general value of θ satisfying the equation
 is

a)

b)

c)

d)

A cubic f (x) vanishes at x = −2 and has a relative minimum/maximum at x = −1 and . If , find the cube f (x).

a) x³ + x² + x + 1

b) x³ + x² − x + 1

c) x³ − x² + x + 2

d) x³ + x² − x + 2

If a circle passes through the points (a, b) and cuts the circle  orthogonally, then the equation of the locus of its centre is

a)

b)

c)

d)

In ΔPQR, angle R . If tan  and tan  are roots of the equation

a)

b)

c)

d)

Prove that
where  and n is an even integer.