Prove that
 
for all

For positive numbers a, b , c prove the inequality
 

Let a, b, c be the side lengths of a triangle with the property that for any positive integer n, the number aⁿ, bⁿ, cⁿ can also be side lengths of a triangle. The triangle is

a) Necessarily isosceles

b) Necessarily equilateral

c) Could be isosceles or equilateral

d) Need not be isosceles or equilateral.

Prove that for any positive integer n
 

Let a₁, a₂, .  .  .  .a_n be positive real numbers such that
 a₁ + a₂+ .  .  .  .+ a_n < 1. Prove that
 

Verify that
 

Find a polynomial with integer coefficient that has the zero

a) x⁴ – 6x⁴ + 6x³ – 12x² + 36x – 1

b) x⁴ + 6x⁴ + 6x³ + 12x² + 36x + 1

c) x⁴ + 6x⁴ + 6x³ + 12x² + 36x – 1

d) x⁶ – 6x⁴  – 6x³ + 12x² – 36x + 1

If then equals

a)

b)

c)

d)

If  are complex numbers such that  then  is

a) Equal to 1

b) Less than 1

c) Greater than 3

d) Equal to 3

Let z₁ and z₂ be n^th roots of unity which subtend a right angle at the origin then n must be of the form

a) 4k + 1

b) 4k + 2

c) 4k + 3

d) 4k

The complex numbers  satisfying  are the vertices of the triangle which is

a) of zero area

b) right angle isosceles

c) equilateral

d) obtuse angled isosceles

Let  then the value of  is

a) 3ω

b) 3ω(ω – 1)

c) 3ω²

d) 3ω(1 – ω)

If |z| = 1 and  then Re (w) is

a) 0

b)

c)

d)

If  is purely real where ω = α + iβ, β ≠ 0 and z ≠ 1 then the set of real values of z is

a)  

b)  

c)  

d)  

A man walks a distance of three units from the origin towards north-east (N direction. From there he walks a distance of 4 units towards north–west (N direction to reach a point P. Then the position of P in the argand plane is

a)

b)

c)

d)

If |z| = 1 and z ≠ ±1 then the value of  lie on

a) a line not passing through the origin

b)

c) the X – axis

d) the Y axis

(One or more than one correct answer)
Let  and  be complex numbers such that  and . If has positive real part and  has negative imaginary part, then  may be

a) Zero

b) Real and positive

c) Real and negative

d) None of these

Show that, if
a, b, c, d ε ℝ

If x = a + b, y = aα + bβ, z = aβ + bα where α, β are cube roots of unity show that .

Find the values of x and y for which the following equation is satisfied

a) x = y = −1

b) x = y = 3

c) x = 1, y = 3

d) x = 3, y = −1

If  are the n roots of unity then show that .

It is given that n is an odd integer greater than 3 and not a multiple of 3. Prove that  is a factor of  

Show that the area of the triangle on the argand diagram formed by the complex numbers z, iz, z + iz is   .