# Complex Numbers

## A complex number is a combination of a real and an imaginary number and it is written as z = a + ib, where a, b are real numbers and i is equal to square root of –1.A complex number a + ib becomes a purely imaginary number, when a = 0 and purely real number, when b = 0.In the complex number z = a + ib, the real part is equal to ‘a’ and imaginary part is equal to ‘b’.Two complex numbers z1 = a + ib and z2 = c + id are equal if and only if a = c and b = d.If z is any complex number, then the positive and negative powers of z are z1 = z, z2 = z.z, z3 = z.z.z, and so on.z–1 = 1/z, z–2 = 1/z2, z–3 = 1/z3, and so on. (where, z is any non-zero complex number)Let z = a + ib be a complex number. Then the conjugate of z is equal to a – ib.The modulus of a complex number z = a + ib is denoted by |z| and is defined as square root of sum of the square of a and b.Keywords: Imaginary number, Real number, Modulus, Conjugate.

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• Q1

The equation represents a circle whose centre is

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(0, 3).

##### Explanation:

• Q2

If x + iy = 3/( cos+isin +2) , then 4x – x2 – y2 reduces to

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3.

##### Explanation:

• Q3

The system of equations has

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no solution.

##### Explanation:
The given system of equation represents the system of circles
(x + 1)2 + (y – 1)2 = 2 and x2 + y2  = 9

Distance between the centre is 2 and difference between the radii = 3-2

Therefore, 1st circle lies with 2nd circle. So, the system of equation has no solution.

• Q4

The complex number (1/3 + 3i )3 in the form a + i b is

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-242/27 - i 26.

##### Explanation:

• Q5

The polar form of (–1 – i) is

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