Question 1

In this question, we have to find the real values of x and y. We are given an equation where complex numbers are on both sides. When two complex numbers are equal, their real parts are equal and their imaginary parts are equal. First, we simplify the equation. Then we compare the real parts and the imaginary parts separately. This gives us two simple equations. We solve those equations and get the values of x and y.

Question 2

Here, the value of z is given and we also have z written as x + yi. We are given one more equation: z – 2z = –27 + 15i. We put the value of z into this equation. Then we simplify. After simplifying, we compare the real parts and the imaginary parts. This gives us the values of x and y. It is a simple linear equation.

Question 3

This question has three different parts. In each part, we have (x + iy) whole square given, and on the other side there is a complex number. First, we expand (x + iy)² using the formula (a + b)² = a² + 2ab + b². But here b is iy, so iy whole square becomes –y². So we get x² – y² + 2ixy. Then we compare the real part of this with the real part of the given complex number. And we compare the imaginary part with the imaginary part. This gives us two equations. We solve them to find x and y.

Question 4

In this question, z1 and z2 are given. z1 = 2 + 3i and z2 = –1 – ai. We have to find the real value of a. The condition given is that the imaginary part of z1 multiplied by z2 is equal to 7. So first we multiply z1 and z2. After multiplication, we look at the imaginary part of the answer. We set that imaginary part equal to 7. Then we solve the equation and find the value of a.

Question 5

Here we have two complex numbers: z1 = x + yi and z2 = a + bi. We have to find the values of x, y, a, and b. Two conditions are given. First: z1 + z2 = 10 + 4i. Second: z1 – z2 = 6 + 2i. From the first condition, we get equations for x + a and y + b. From the second condition, we get equations for x – a and y – b. Then we solve these four equations together and find the values of x, y, a, and b.

Question 6

In this question, we have to prove a property of complex numbers. It is about the modulus of complex numbers. We are given two complex numbers z1 and z2. We have to show that some relation between their moduli is true. We simply apply the formula for modulus and simplify. The answer comes out directly. It is a straightforward proof.

Question 7

Here we have to find the square root of complex numbers. There are four parts. For each complex number, we need to find its square root. The method is the same for all. We assume that the square root of (x + iy) is a + ib. Then we square both sides. Then we compare the real parts and the imaginary parts. This gives us two equations. We solve these equations to find a and b. That gives us the square root.

Question 8

In this question, we have to find the square root of 13 – 20√3i. We use the same method as in question 7. We assume the square root is a + ib, square both sides, and compare real and imaginary parts. After finding the square root in the form a + ib, we have to represent it on an Argand diagram. That means we plot the point (a, b) on a graph. The x-axis is the real part and the y-axis is the imaginary part.

Question 9

Here we are given an equation that has complex numbers. We have to find the real values of x and y. First, we simplify the equation. We multiply the complex numbers where needed. Then we add or subtract as given. After simplifying, we compare the left-hand side with the right-hand side. Real parts are equated to real parts. Imaginary parts are equated to imaginary parts. Then we solve the two simple equations to find x and y.

Question 10

This question is similar to question 9. We have an equation with x and y. We simplify the equation first. Then we compare the real part and the imaginary part with the right-hand side. This gives us two equations. We solve them to find x and y.

Question 11

In this question, we have to find the real values of u and v. The equation has a fraction with complex numbers. First, we simplify the fraction. To simplify, we multiply the numerator and denominator by the conjugate of the denominator. Then we compare the result with 4i. The real part should be 0 and the imaginary part should be 4. From this, we get equations and solve for u and v.

Question 12

Here, z1 = 4 + 5i and z2 = a – 2i are given. We have to find the real value of a. The condition is that the real part of z1 multiplied by z2 is equal to 20. First, we multiply z1 and z2. Then we look at the real part of the answer. We set that real part equal to 20. Then we solve the equation and find the value of a.