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 (-x 4 -x 2 -1) 3.Find the integral solutions of the equations a) (1-i) n =2 n 4.If 1 ω, ω 2 are the three cube roots of the unity then, prove that a) (a+b ω+c ω 2 ) 3 +(a+b ω 2 +c ω) 3 = (2a-b-c)(2b-c-a)(2c-a-b)=27abc if a+b+c = 0 b) 1. (2- ω)(2- ω 2 )+ 2. (3- ω)(3- ω 2 ) +3 (4- ω)(4- ω 2 )+…………+(n-1)(n- ω)(n- ω 2 ) = {n 2 (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/a 2 -b 2 7. if iz 3 +z 2 -z + i = 0 then show that IzI = 1 8. If IzI = 1 then prove that arg(z 2 +conj z) = ½ arg (z) 9.If α, β are the roots of the equation x 2 -2x+4=0 prove that α n +β n = 2 n+1 cos ). 10. Construct an equation whose roots are nth power of roots of equation x 2 -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 Z 1 Z 2 I 2 - IZ 1 – Z 2 I 2 = (1-IZ 1 I 2 )(1 – IZ 2 I 2 ) 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) A 2 +B 2 = e -π B 16. let Z 1 and Z 2 be the roots of the equation z 2 + pz+q = 0 , where the coefficient p and q may be complex number . Let A and B represents Z 1 and Z 2 respectively . If P 2 = 4q cos 2 α/2

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