Prove that
 

Prove that

Prove that with regard to the quadratic
 
if the equation has one real root then

Prove that with regard to the quadratic
 
if the equation has two equal roots, then  

Find the roots of  and show that the points which represent them are collinear.

Show that the roots of  are the values of  where r = 0, 1, 2 .  .  .  ., n – 1, but omitting  if n is even.

The conjugate of a complex number is . Then the complex number is

a)

b)

c)

d)

The quadratic equations  and  have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

a) 2

b) 1

c) 4

d) 3

If  then the maximum value of |z + 1| is

a) 4

b) 11

c) 6

d) 0

If the difference between the roots of the equation  is less than  then the set of possible values of a is

a)  

b)  

c)  

d)  

The value of  is

a) 1

b) – 1

c) – i

d) i

The values of m for which both the roots of the equation  are greater than – 2 and less than 4 lie in the interval

a) m > 3

b) – 1 < m < 3

c) 1 < m < 4

d) – 2 < m < 0

If  where z is a complex number then the value of   is

a) 54

b) 6

c) 12

d) 18

If the cube roots of unity are  then the roots of the equation  are

a)  

b)  

c)

d)  

If z and  are two non-zero complex numbers such that  then  is equal to

a)

b) 0

c) –π

d)

The value of a for which the sum of the squares of the roots of the equation  assumes the least value is

a) 2

b) 3

c) 0

d) 1

If   and  then z lies on

a) A parabola

b) A straight line

c) A circle

d) An ellipse

If both the roots of the quadratic equation  are less than 5 then k lies in the interval

a) [4, 5]

b)

c)

d) (5, 6]

Let z, w be complex numbers such that  and  then arg (z) equals

a) π/4

b) π/2

c) 3π/4

d) 5π/4

If  and  then  is equal to

a) 1

b) –1

c) 2

d) –2

If  then z lies on

a) The real axis

b) The imaginary axis

c) A circle

d) An ellipse

If  is a root of quadratic equation  then its roots are

a) 0, 1

b) 1, 1

c) 0,    

d)  

If one root of the equation  is 4 while the equation  has equal roots then the value of q is

a) 49/4

b) 12

c) 3

d) 4