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1. Find real values of x and y for which the complex numbers –3 + ix2y and x2 + y + 4i are conjugate of each other.
2. If a + ib = c + i , prove that a2 + b2 = 1 and b = 2c
c - i a c2 –1
3. Express the following complex numbers in the polar form:
(i) 1 + i (ii ) 2 + 6 /3 i
5 + /3 i
4. Find the square root of –7 – 24i.
5. If ( x + iy)1/3 = a + ib, prove that x/a + y/b = 4(a2 – b2)
6. Put the following in the form r(cos q + i sin q), where r is a positive real number and –180 < q <180: ( 1 + 7i )
( 2 – i ) 2
7. Convert the complex number, i – 1 , into polar form.
cos 60 + i sin 60
8. If z = 2 –3i, prove that z2 –4z + 13 = 0.
9. If z1 = 2 + i , z2 = 2 – 3i , z3 = 4 + 5i , evaluate: Im ( z 2 . z3 )
10. If ( a + i)2 = p + iq , show that p2 + q2 = ( a2 + 1 )2
( 2a – i) (4a2 + 1 )
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