1) Find the conjugate of the complex number (2+3i)/(2-3i). 5/13 - 12i/13
2) find the modulus of (1-i)³/(1-i³). 2
3) Find the amplitude of -3- 3i. -3π/4
4) Resolve into factors: a²+ ab+ b². (a- bw)(a- bw²)
5) Find the square root of -i. ±1/√2(1- i)
6) Evaluate : (1- w²)(1- w⁴)(1- w⁸)(1- w¹⁶). 9
7) If z= x+ iy and |z- 2| = |2z - 1|, Prove that, x² + y² = 1.
8) Find the smallest positive integers n for which {(1+i)/(1 - i)}ⁿ = 1. 4
9) Find the square root of q+ √(q² -1), 0< q <1. ±1/√2 {√(1+q)+ i √(2-q)}
10) If x= a+ b, y= aw + bw², z= aw² + be, find xyz. a³ + b³
11) Solve: |z| - z = 1+ 2i, (z= x+ i y). 3/2 - 2i
12) The sum and the product of two complex numbers are respectively 6 and 25. Find the numbers. 3±4i
13) Determine the three cube roots of i. i, (i+√3)/2
14) If z= x + i y and (z -1)/(z+1) is purely imaginary, prove that, |z| = 1.
15) The absolute value of (i/2) is.
A) i/2. B) 1/2. C) 1/√2 D) none
16) The value of √4 . √9 is
A) 6 B) -6. C) -6 or 6 D) none
17) The value of w⁴+ w⁸+ 1/w .1/w² is
A) w B) w². C) -1 D) 0.
18) The argument of ia(a< 0), is.
A) 0 B) π/2 C) 3π/2. D) π
19) The value of (1- w +w²)⁵ + (1+ w -w²)⁵ is..
A) -1 B) 1 C) -32 D) 32.
20) The amplitude of (a+ i b)² is
A) tan⁻¹(b/a) B) 2tan⁻¹(b/a).
C) 2tan⁻¹(a/b) D) tan⁻¹(a/b)
21) If z= x + i y, the value of (amp z + amp z') is.
A) 0 B) π/2 C) π D) none.
22) the quantity, whose cube root is 1/2 (√3 + i), is.
A) -1 B) 1 C) -i D) i.
23) The value of (1+ w)(1+ w²)(1+ w⁴)(1- w⁸)....2n factors, is
A) 1. B) 2ⁿ C) -1 D) wⁿ
24) For any complex number z, the minimum value of |z | + |z - 1| is...
A) 0 B) 1/2 C) 1. D) 3/2
25) The real part of (2- i)²/(2+ i) is
A) -2/5 B) -6/5 C) -11/5 D) none
B) GENERAL QUESTIONS:
1) Show that (1- w+ w²) (1- w²+w⁴) (1- w⁴+ w⁸)....2n factors = 2²ⁿ.
2) If X+ i Y be one of the cube root of x + i y, prove that, 4(X² - Y²)= x/X + y/Y.
3) If x= a + i b, y= a¢+ b$, z= a$ + b¢, where ¢, $ are complex cube roots of unity, show that x³ + y³ + z³ = 3(a³ + b³).
4) If x= w² - w - 2, evaluate x⁴+ 3x³ + 2x² -11 x -4. 3
5) If x= 2 - i √3, Find the value of k from the equation 2x⁴ - 5x³ - 3x² + 41x + k= 0. -35
6) If cos a + i sin a, b= cos b+ I sin b, c= cos d + i sin d and a+ b+c= 0, prove that a²+ b²+ c²= 0.
7) If z= x + i y and (z- i)/(z + i) = i b, show that (x - 1/2)² + (y - 1/2)² = 1/2.
8) Prove that (a + bw + cw²) + (a + bw² + cw)³ = (2a- b - c)(2b - c - a)(2c - a- b) and 27abc, if a+ b+ c= 0.
9) If z= x + i y and arg{(z -1)/(z+ 1)} = π/4, Show that the locus of (x,y) is a circle.
10) If w be a Complex cube root of unity, find the simplified value of (a +bw + cw²)/(c + aw+ bw²) + (a +bw + cw²)/(b + cw+ aw²). -1
11) Show that the points 2+ 3i , 0 and 1/(-2 + 3i) are collinear.
12) Express a+ i b in the form pw + qw². (b/√3 - a)w + (-b/√3 - a)w² or (-b/√3 - a)w + (b/√3 - a)w²
13) solve: z² + z'= 0 (z= x+ i y). 0, -1, 1/2 ± √3 i/2.
14) If a²+ b²+ c²= 1 and b+ ic= (1+ a)z, then prove that (1+ i z)/(1- i z) = (a + i b)/(1+ c).
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