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# 1.Express in the form of  A+iBa)    ((1+2i)/(1+i))n for n = ±1, ±2, ±3, ±4………..b)    ln(1+itan α)2.Find the sqrt ofa)   x+ sqrt (-x4-x2-1)3.Find the integral solutions of the equationsa)  (1-i)n=2n4.If 1 ω, ω2 are the three cube roots of the unity then, prove thata)  (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 = 0b)  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 } – n5.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-b27. if iz3+z2-z + i = 0 then show that IzI =  18.  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 =011.If x=eiθ 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 ii^i^i^……n = A+iB  principle values only being considered then prove thata) tan πA  =  B/Ab)  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

8 years ago

Harsh Patodia
IIT Roorkee
907 Points
HI student

((1+2i)/(1+i))n
Rationalize and then make it in euler form i.e rei
and then apply demoivre’s theorem to get the answer in a +ib

Please post 1 question at a time. Put other question separately on different post
6 years ago
Ravi
69 Points
9 Complete the question. Cos(?)

10 Find the roots which will becos θ+i sinθ,cos θ-i sinθ

i.e. e, e-iθ

Their nth powers will beeinθ, e-inθ ...(I)

Form the new eqn by (x-a)(x-b)=0

where a,b represent the nth powers of roots(from eqn I)
6 years ago
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### Course Features

• 731 Video Lectures
• Revision Notes
• Test paper with Video Solution
• Mind Map
• Study Planner
• NCERT Solutions
• Discussion Forum
• Previous Year Exam Questions