Question

If `alpha and beta` the roots of the quadratic equation, `x^(2)+x sintheta-2sintheta=0,theta in(0,(pi)/(2)),` then `(alpha^(12)+beta^(12))/((alpha^(-12)+beta^(-12))(alpha-beta)^(24))` is equal to
A. `(2^(12))/((sin theta+8)^(2))`
B. `(2^(6))/((sin theta+8)^(12))`
C. `(2^(12))/((sin theta-4)^(12))`
D. `(2^(12))/((sin theta-8)^(6))`

Answer 1

Correct Answer - A
Given quadratic equation is
`x^(2)+x sin theta -2 sintheta=0,theta in(0,(pi)/(2))`
and its roots are `alpha and beta.`
So, sum of roots `=alpha+beta=sin theta`
and product of roots `=alpha beta=-sin theta`
` implies " "alpha beta =2(alpha+beta)" "...(1)`
Now, the given expression is `(alpha^(12)+beta^(12))/((alpha^(-12)+beta^(-12))(alpha-beta)^(24))`
`=(alpha^(12)+beta^(12))/(((1)/(alpha^(12))+(1)/(beta^(12)))(alpha-beta)^(24))=(alpha^(12)+beta^(12))/(((beta^(12)+alpha^(12))/(alpha^(12)beta^(12)))(alpha-beta)^(24))`
`=[(alphabeta)/((alpha-beta)^(2))]^(12)=((alphabeta)/((alpha+beta)^(2)-4alphabeta))`
`=[(2(alpha+beta))/((alpha+beta)^(2)-8(alpha+beta))]^(12)" "["from Eq."(i)]`
`=((2)/((alpha+beta)-8))=((2)/(-sintheta-8))^(12)[becausealpha+beta=-sintheta]=(2^(12))/((sintheta+8)^(12))`