## Rajasthan Board RBSE Class 11 Maths Chapter 5 Complex Numbers Ex 5.1

Question 1.

Write the following in simplest form:

(i) i^{52}

(ii) √-2 √-3

(iii) (1 + i)^{5} (1 – i)^{5}

Solution:

Question 2.

Find the additive and multiplicative inverse of following numbers:

(i) 1 + 2i

(ii) 1/(3 + 4i)

(iii) (3 + i)^{2}

Solution:

(i) Let z = 1 + 2i

Then additive inverse = -z = -(1 + 2i) = -1 – 2i

And multiplicative inverse = \(\frac { 1 }{ z }\)

Question 3.

Find the conjugate number of complex number \(\frac { \left( 2+i \right) ^{ 3 } }{ 3+i }\)

Solution:

Question 4.

Find the modulus of the following:

(i) 4 + i

(ii) -2 – 3i

(iii) 1/(3 – 2i).

Solution:

Question 5.

If a^{2} + b^{2} = 1 then find the value of \(\frac { 1+b+ia }{ 1+b-ia }\)

Solution:

Question 6.

If a = cos θ + i sin θ, then find the value of \(\frac { (1+a) }{ (1-a) }\)

Solution:

Question 7.

Find the value of x and y which satisfy the equation

Solution:

Comparing the real and imaginary part on both sides

4x + 9y – 3 = 0 and 4x + 9y = 3 …..(i)

2x – 7y – 3 = 10 and 2x – 7y = 13 ……(ii)

Solving equation (i) and (ii), we get

x = 3 and y = -1

Question 8.

If z_{1} and z_{2} are any two complex numbers, then prove that

|z_{1} + z_{2}|^{2} + |z_{1} – z_{2}|^{2} = 2|z_{1}|^{2} + 2|z_{2}|^{2}.

Solution:

Question 9.

Solution:

Question 10.

If (x + iy)^{1/3} = a + ib, then prove that \(\frac { x }{ a }\) + \(\frac { y }{ b }\) = 4(a^{2} – b^{2}).

Solution:

Question 11.

Solution: