1.Express in the form of A+iB

a) ((1+2i)/(1+i))ⁿ for n = ±1, ±2, ±3, ±4………..

b) ln(1+itan α)

2.Find the sqrt of

a) x+ sqrt (-x⁴-x²-1)

3.Find the integral solutions of the equations

a) (1-i)ⁿ=2ⁿ

4.If 1 ω, ω² are the three cube roots of the unity then, prove that

a) (a+b ω+c ω²)³+(a+b ω²+c ω)³= (2a-b-c)(2b-c-a)(2c-a-b)=27abc if a+b+c = 0

b) 1. (2- ω)(2- ω²)+ 2. (3- ω)(3- ω²) +3 (4- ω)(4- ω²)+…………+(n-1)(n- ω)(n- ω²) = {n²(n+1)²/4 } – n

5.If sin α +sin β + sin γ = cos α +cos β + cos γ = 0 then show that ,

Σcos 3α = 3 cos (α+β+γ) and Σsin 3α = 3 sin (α+β+γ)

6. prove that tan (i ln {(a-ib)/(a+ib)} = 2ab/a²-b²

7. if iz³+z²-z + i = 0 then show that IzI = 1

8. If IzI = 1 then prove that arg(z²+conj z) = ½ arg (z)

9.If α, β are the roots of the equation x²-2x+4=0 prove that αⁿ+βⁿ = 2ⁿ⁺¹ cos ).

10. Construct an equation whose roots are nth power of roots of equation x²-2x cos θ +1 =0

11.If x=e^iθ and ²) = nc-1` then prove that 1+c cos θ = (1+nx)(1+)

12. prove that I1-conj Z₁Z₂I² - IZ₁ – Z₂I² = (1-IZ₁I²)(1 – IZ₂I²)

13. If ABC is atriangle and triangles BCX , CAY , ABZ are drawn on BC , CA , AB , directly similar to one another . Provce that the centroid of XYZ and ABC coincides.

14. IF z = x+iy and w= , IwI = 1 , then find the locus of z.

15.If i^{i^i^i^……n} = A+iB principle values only being considered then prove that

a) tan πA = B/A

b) A²+B² = e^{-π B}

16.let Z₁ and Z₂ be the roots of the equation z²+ pz+q = 0 , where the coefficient p and q may be complex number . Let A and B represents Z₁ and Z₂ respectively . If

P² = 4q cos²α/2

Abhishek Mishra

13 Years agoGrade 12