Question

If `alpha,beta` be the roots of the equation `u^2-2u+2=0` and if `cottheta=x+1,` then `((x+alpha)^n-(x+beta)^n)/(alpha-beta)` is equal to (a) `((sin n theta),(sin^n theta))` (b) `((cosn theta),(cos^n theta))` (c) `((sinn theta),cos^n theta))` (d) `((cosn theta),(sintheta^n theta))`
A. `(sinn ntheta)/(sin^(n) theta)`
B. `(cos ntheta)/( cos^(n)theta)`
C. `(sin ntheta)/(cos^(n) theta)`
D. `(cos ntheta)/(sin^(n) theta)`

Answer 1

Correct Answer - A
`u^(2)-2u+2=0rArru=1pmi`
`rArr ((x+alpha)^(n)-(x+beta)^(n))/(alpha-beta)`
`=([(cottheta-1)+(1+i)]^(n)-[(cottheta-1)+(1-i)]^(n))/(2i)`
`(thereforecottheta-1=x)`
`=((costheta+isintheta)^(n)-(costheta-isintheta)^(n))/(sin^(n)theta 2i)`
`=(2isinntheta)/(sin^(n)theta2i)`
`(sin n theta)/(sin^(n)theta)`