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 nonzero 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
Marks:1Answer:
(0, 3).
Explanation:

Q2
If x + iy = 3/( cos+isin +2) , then 4x – x^{2} – y^{2} reduces to
Marks:1Answer:
3.
Explanation:

Q3
The system of equations has
Marks:1Answer:
no solution.
Explanation:
The given system of equation represents the system of circles
(x + 1)^{2} + (y – 1)^{2} = 2 and x^{2} + y^{2} = 9
Distance between the centre is 2 and difference between the radii = 32Therefore, 1^{st} circle lies with 2^{nd} circle. So, the system of equation has no solution.

Q4
The complex number (1/3 + 3i )^{3} in the form a + i b is
Marks:1Answer:
242/27  i 26.
Explanation:

Q5
The polar form of (–1 – i) is
Marks:1Answer:
(2)[cos(3/4)+isin(3/4)].
Explanation: