Find the real values of x and y, if (x⁴ + 2xi) - (3x² + iy) = (3 - 5i) + (1 + 2iy).
(i) Firstly, separate real and imaginary parts of both sides.
(ii) Second, equate the real and imaginary parts of both sides and get equations in terms of x and y.
(iii) Further, solve these equations to get the values of x and y.

3

(a)

If z₁ and z₂ are complex numbers, then prove that Re(z₁ z₂) = Re(z₁) Re(z₂) - Im(z₁) Im(z₂).

3

(a)

Express the following in the form a + ib.
\(\left( -i \right) \left( 3i \right) { \left( -\frac { 1 }{ 6 } i \right) }^{ 3 }\)

3

(a)

Simplify each of the following and put it in the form a + ib.
\({ \left( \frac { 1 }{ 3 } +3i \right) }^{ 3 }\)

3

(a)

Simplify each of the following and put it in the form a + ib.\(\left( 3+\sqrt { -5 } \right) \left( 3-\sqrt { -5 } \right) \)

3

(a)

What is the value of \(\frac { { i }^{ 4x+1 }-{ i }^{ 4x-1 } }{ 2 } ?\)

3

(a)

If \(a+ib=\frac { x+i }{ x-{ i }^{ \prime } } \), where x is real, then prove that \({ a }^{ 2 }+{ b }^{ 2 }=1\)and \(\frac { b }{ a } =\frac { 2x }{ { x }^{ 2 }-1 } \)

3

(a)

Find the modulus and argument of the complex number \(\frac{1+3i}{1-2i}\) and convert them In polar form.

3

(a)

Solve the quadratic equation 5x² - 6x + 2 = 0.

3

(a)

Solve the quadratic equation 8x² - 9x + 3 = 0.

3

(a)