Question

If `sinalpha, sin^2alpha, 1 , sin^4alpha and sin^6alpha` are in A.P., where `-pi < alpha < pi,` then `alpha` lies in the interval
A. `(-pi//2,pi//2)`
B. `(-pi//3,pi//3)`
C. `(-pi//6,pi//6)`
D. none of these

Answer 1

Correct Answer - D
It is given that `sinalpha,sin^(2)alpha,1,sin^(4)alpha,sin^(5)alpha` are in A.P.
`:." "sinalpha,sin^(2)alpha,1` are also in A.P.
`rArr" "2sin^(2)alpha=sinalpha+1`
`2sin^(2)alpha-sinalpha-1=0`
`rArr" "(2sinalpha+a)(sinalpha-1)=0`
`rArr" "sinalpha=-1//2or,sinalpha=1`
`rArr" "alpha=-pi//6or,alpha=pi//2" "[because-piltalphaltpi]`
For `alpha=-pi//6`, the given sequence becomes
`-(1)/(2),(1)/(4),1(1)/(16),-(1)/(32)`.
Clearly, it is not an A.P.
for `alpha=pi//2`, the given sequence becomes 1,1,1,1,1 which is an A.P.
But, `alpha=pi//2` does nnot lie in any of the given intervals.