If ω is a solution of x³ – 1 = 0 with  Im(ω) > 0. If a = 2 and b, c (ε ℝ) satisfy the equations  then the value of  is equal to

The complex number z = x + iy which satisfies the equation  lies on

a) The real axis

b) The straight line y = 5

c) Circle passing through origin

d) None of these

If  then

a) Re(z) = 0

b) Im(z) = 0

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

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

The inequality |z – 4| < |z – 2| represents the region given by

a) Re(z) ≥ 0

b) Re(z) < 0

c) Re(z) > 0

d) None of these

The complex numbers sinx + icos2x and cosx – isin2x are conjugate to each other for

a) a = nπ

b) x = 0

c) x =  

d) None of these

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)   

For positive integers n₁ and n₂ the value of the expression  where  is real if and only if

a)

b)

c)

d)

The value of the definite integral  is

a) – 1

b) 2

c)

d)

Let   then the real roots of the equation

 are

a) ± 1

b)

c)

d) 0 and 1

For 0 < a < x the minimum value of the function log_ax + log_xa is 2.

a) True

b) False

Let  for n ≥ 2 and
 

Then equals

a)

b)

c)

d)

Evaluate   where n is a positive integer and t is a parameter independent of x.

a)

b)

c)

d)

If  are the sums of  terms respectively of an arithmetic progression show that .

In an arithmetic progression of which a is the first term, if the sum of the first p terms = 0, show that the sum of the next q terms is equal to

Find an arithmetic progression whose first term is unity such that the second, tenth and thirty fourth sums form a geometric progression.

If α, β are roots of , find the values of
 

If 2x =  and 2y =  find the value of
 

If α, β are imaginary cube roots of unity show that
 

Show that if  is a perfect square, the quantities a, b, c are in harmonic progression.

If the equations
ax + by  1 and
have only one solution prove that
 and  and

If a, b, c, x, y, z are real quantities and
,
show that  and

Show that
 

The arithmetic mean between m and n and the geometric mean between a and b are each equal to   ; find m and n in terms of a and b.