Short Questions Answer Type
i) Find the conjugate of the complex number (2+ 3i)/(2 - 3i). -5/13 -12i/13
ii) Find the modulus. of (1 + i)³/(1 - i³). 2
iii) Find the amplitude of -3-3i. -3π/4
iv) Resolve into factors: a²+ ab + b². (a - bω)(a - bω²)
v) Find the square root of -i . ±(1/√2) (1- i)
vi) Evaluate: (1- ω²)(1- ω⁴)(1- ω⁸)(1- ω¹⁶). 9
vii) If z= x + iy and |z - 2| = |2z - 1|, prove that x²+ y²=1.
viii) Find the smallest positive integer n for which {(1+ i)/(1- i)}ⁿ=. 1. 4
ix) Find the square root of q+ √(q²- 1), 0< q < 1. ±(1/√2) {√(1+ q)+ i √(1- q)}
x) If x = a+ b, y= aω + bω², z= aω²+ bω , find xyz. a³+ b³
xi) Solve: |z| - z = 1+ 2i. (z = x+ iy). z= 3/2 - 2i
xii) The sum and the product of the two complex numbers are respectively 6 and 25. Find the numbers. 3+4i, 3 -4i
xiii) Determine the three cube roots of i. - i, (i+√3)/2, (i - √3)/2
Choose the correct option
i) The absolute value of (-i/2) is
A) i/2 B) 1/2 C) 1/√2 C) none
ii) The value of √-4 . √-9 is
A) 6 B) -6 C) -6 or 6 D) none
iii) The value of ω⁴+ ω⁸+ ω⁻¹. ω⁻² is
A) ω B) ω² C) -1 D) 0
iv) The argument of ia (a < 0), is
A) 0 B) π/2 C) 3π/2 D) π
v) The value of (1- ω+ω²)⁵ + (1+ ω- ω²)⁵.
A) -1 B) 1 C) -32 D) 32
vi) The amplitude of (a + ib)² is
A) tan⁻¹(b/a) B) 2tan⁻¹(b/a) C) 2 tan⁻¹(a/b) D) tan⁻¹(a/b)
vii) If z = x + iy, the value of (amp z + amp of Conj of z) is
A) 0 B) π/2 C) π D) none
viii) The quantity, whose cube root is (1/2) (√3 + i), is
A) -1 B) 1 C) - i D) i.
ix) The value of (1+ ω)(1+ω²)(1+ ω⁴)(1+ ω⁸)......2n factors, is
A) 1 B) 2ⁿ C) -1 D) ωⁿ
x) For any complex number z, the minimum value of |z|+|z -1| is
A) 0 B) 1/2 C) 1 D) 3/2
xi) The real part of (2- i)²/(2+ i) is
A) -2/5 B) -6/5 C) -11/5 D) none
i b ii b iii d iv c v d vi b vii a viii d ix a x c xi d
General Questions
1) Show that, (1- ω+ω²)(1-ω²+ ω⁴)(1- ω⁴ +ω⁸)+ ..... to 2n factors = 2²ⁿ.
2) If X + iY be one of the cube roots of x + iy, prove that, 4(X²- Y²)= x/X + y/Y.
3) If x= a + b, y= aα + bβ, z= aβ + bα, where α, β are complex cube roots of unity, show that, x³+ y³+ z³= 3(a³+ b³).
4) If x= ω²- ω -2, evaluate x⁴+ 3x³+ 2x² --11x -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 x+ iy =√{(a+ ib)/(c + id)}, prove that (x²+ y²)²= (a²+ b²)/(c²+ d²).
7) If a = cos α + i sin α, b= cosβ+ i sinβ, c= cosγ + i sinγ and a+ b +c=0, show that a²+ b²+ c²=0.
8) If z = x + iy and (z -i)/(z -1)= ib, prove that (x - 1/2)²+ (y - 1/2)²= 1/2.
9) If (1+ x + x²)ⁿ = a₀ + a₁x + a₂x² + a₃x³+ ......+ a₂ₙx²ⁿ , then show that a₀ + a₃ + a₆ + .......= 3ⁿ⁻¹.
10) Prove that (a+ bω+ cω²)³+ (a+ bω²+ cω)³ = (2a- b - c)(2b - c - a)(2c - a - b) and 27abc, if a+ b + c =0.
11) If (1+ i)(1+ 2i)(1+ 3i)......(1+ ni)= a+ ib, then show that 2.5.10......(1+ n²)= a²+ b².
12) If If cos k + i sin k and 1+ √(1- a²) = na, prove that (a/2n) (1+ nx)(1+ n/x) = 1+ a cos k.
13) If z = x + iy and arg{(z -1)/(z +1)}=π/4 , show that the locus of (x,y) is a circle.
14) If ω be a Complex cube root of unity, find the simplified value of (a+ bω + cω²)/(c + aω + bω²) + (a+ bω + cω²)/(b + cω + aω²) . -1
15) Prove that the expression x³ᵖ + x³ᑫ⁺¹ + x³ʳ⁺² , where p, q, r are integers, is divisible by x²+ x+1.
16) Show that the points 2+ 3i, 0 ai 1/(-2+ 3i) are collinear.
17) Express a+ ib in the form pω + qω²). (b/√3 - a)ω + (-b/√3 - a)ω² or (- b/√3 - a)ω + (b)√3 - a)ω²
18) Show that the sum and the product of two complex numbers are real if and only if they are conjugate of each other.
19) Solve: z²+ Conj of z=0. (z= x + iy). 0, -1, 1/2 ± i√3/2
20) If a²+ b²+ c²=1 and b + ic = (1+ a)z, then prove that (1+ iz)/(1- iz) = (a+ ib)/(1+ c).
