Question

Which of the following option(s) is (are) CORRECT ?

• A. Fundamental period of the function $f\left(x\right)={\sin }^{2}2x+{\cos }^{4}2x+2$ is $\frac{\pi }{4}$
• B. Number of integral values of $a$ for which $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ is defined for all $x\in R$ is $3$
• C. Number of integral values of $a$ for which $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ is defined for all $x\in R$ is $7$
• D. Fundamental period of the function $f\left(x\right)={\sin }^{2}2x+{\cos }^{4}2x+2$ is $\frac{\pi }{2}$

Answer 1

Answer: (A), (B)

The correct options are
A Fundamental period of the function $f\left(x\right)={\sin }^{2}2x+{\cos }^{4}2x+2$ is $\frac{\pi }{4}$
B Number of integral values of $a$ for which $f\left(x\right)=\log \left({\log }_{1/3}\right({\log }_7(\sin x+a)\left)\right)$ is defined for all $x\in R$ is $3$

$f\left(x\right)={\sin }^{2}2x+{\cos }^{4}2x+2$
$={\sin }^{2}2x+(1-{\sin }^{2}2x{)}^2+2$
$={\sin }^{2}2x+(1-{\sin }^{2}2x{)}^2+2$
$=3-{\sin }^{2}2x+{\sin }^{4}4x$
$=3-{\sin }^{2}2x(1-{\sin }^{2}2x)$
$=3-{\sin }^{2}2x{\cos }^{2}2x$
$=3-\frac{{\sin }^{2}4x}{4}$

Period of ${\sin }^{2}x$ is $\pi$
$\therefore$ Period of $f\left(x\right)$ is $\frac{\pi }{4}.$

$f$ is defined if ${\log }_{1/3}\left({\log }_7\right(\sin x+a\left)\right)>0$
$\Rightarrow 0<{\log }_7(\sin x+a)<1$
$\Rightarrow 1<\sin x+a<7$
$\Rightarrow 1-\sin x<a<7-\sin x\forall x\in R$
The inequality is true for all reals.
So, $a>\underset{x\in R}{max}\{1-\sin x\}=2$
and $a<\underset{x\in R}{min}\{7-\sin x\}=6$
$\therefore a\in (2,6)$