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)

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.

The sum of four numbers in arithmetic progression is 48 and the product of their extremes is to the product of their means as 27 to 35. Find the numbers.

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.