Question

The number of integral values of `k` for which the equation `7cos x +5 sinx=2k+1` has a solution is (1) `4` (2) `8` (3) `10` (4) `12`

Answer 1

`7cosx+5sinx=2k+1`
`7/sqrt(7^2+5^2)cosx+5/sqrt(7^2+5^2)sinx=(2k+1)/sqrt(7^2+5^2)`
`tanphi=7/5`
`sin(x+phi)=(2k+1)/sqrt74`
`-1<=sin(x+phi)<=1`<br> `=-1<=(2k+1)/sqrt74<-1`<br> after pitting the values.
4+4=8.
option 2 is correct.