Question

Please solve q 8 ?

Answer 1

`12`

Explanation:

any expression of the form `asintheta-bcostheta`

can be written in the form `Rsin(theta-alpha)`

where `theta" and "alpha " are to be determined"`

we have

`4sintheta-3costheta+7--(1)`

so take the first two terms of `(1)`

`4sintheta-3costheta-=Rsin(theta-alpha)`

expand and compare coefficients

`4sintheta-3costheta-=Rsinthetacosalpha-Rsinalphacostheta`

`=>4=Rcosalpha--(2)`

`=>3=Rsinalpha--(3)`

`(2)^2+(3)^2`

`4^2+3^2=R^2cancel((sin^2alpha+cos^2alpha))^1`

`:. R=5`

for teh purpose aof max/min problems `alpha` is not required

`:.4sintheta-3costheta+7-=5sin(theta-alpha)`--(4)

`max(sin(theta-alpha))=1`

`:. max (1)rarr 5sin(theta-alpha)+7=5+7=12`

Answer 2

The correct answer is `option (D)`

Explanation:

Let

`f(theta)=4sintheta-3costheta+7`

Differentiating wrt `theta`

`f'(theta)=4costheta+3sintheta`

The critical points are when

`f'(theta)=0`

`4costheta+3sintheta=0`

`=>`, `tantheta=-4/3`

`=>`, `theta=126.87^@`

Therefore,

`f(126.87^@)=4sin(126.87)-3cos(126.87)+7=12`

Verification (by calculating the decond derivative) :

`f''(theta)=-4sintheta+3costheta`

`f''(126.87^@)=-4sin(126.87^@)+3cos(126.87^@)=-5`

As `f''(theta)<0`, the angle corresponds to a maximum value