The set of values of `lambda in R` such that `sin^2theta+costheta=lambdacos^2theta` holds for some `theta,` is (a)`(-oo,1]` (b) (`-oo,-1]` (c)`varphi` (d) `[-1,oo` )

Options

A. `(-oo,1]`
B. `(-oo,-1]`
C. `phi`
D. `[-1,oo)`

Answer

D

Explanation

We have
`tan^2theta+sectheta=lambda`
`rArr sec^2theta+sectheta-(lambda+1)=0`
`rArr sectheta=-(-1pmsqrt(4lambda+5))/2le-1`
For real `sectheta`,
`4lambda+5ge0rArrlambdage(-5)/4 ...(i)`
Also,` secthetage or secthetale-1`
`rArr(-1pmsqrt(4lambda+5))/2ge1or (-1pmsqrt(4lambda+5))/2ge-1`
`rArr4alambda+5ge9or4lambda+5ge1`
`rArrlambdage1orlambdage-1 ...(2)`
`:. " From(1) and (2), we get "lambdain[-1,oo)`.