Question

$I$ :Range of $2\cos x-3{\cos }^{2}x+5$ is $[0,4]$.

$II$ : Range of $2\cos x-5\sin x$ is [3, 5]
Which of the above statement is true ?

• A. only I
• B. only II
• C. Both I and II
• D. Neither I nor II

Answer 1

The correct option is D Neither I nor II

I. $f\left(x\right)=2\cos x-3{\cos }^{2}x+5$
Put $y=\cos x,yϵ[-1,1]$
$\therefore f\left(y\right)=-3y^2+2y+5$
$f^{\prime }\left(y\right)=0\Rightarrow$ for finding maxima or minima
$-6y+2=0$
$y=\frac{1}{3}$
at $y=\frac{1}{3},\cos x=\frac{1}{3}$
$f\left(y\right)=2\times \frac{1}{3}-3\left(\frac{1}{9}\right)+5$
$=\frac{1}{3}+5=\frac{16}{3}$
at $\cos x=-1,f\left(x\right)=-2-3+5=0$
at $\cos x=+1,f\left(x\right)=2-3+5=4$
$\Rightarrow$ Range of $f\left(x\right)ϵ[0,\frac{16}{3}]$
II. $2\cos x-5\sin x$
range $ϵ[-\sqrt{2^2+5^2},\sqrt{2^2+5^2}]$
$ϵ[-\sqrt{29},\sqrt{29}]$
Neither I nor II True.