Question

Column I		Column II
A	$f(x)=\cos \left(\frac{\pi }{\sqrt{3}}\sin x+\sqrt{\frac{2}{3}}\pi \cos x\right)$	P	Domain of $f(x)$ is $(-\infty ,\infty )$
B	$f(x)={\log }_2(|\sin x|+1)$	Q	Range of $f(x)$ contains only one positive integer
C	$f(x)=\cos \{[x]+[-x]\}$	R	$f(x)$ is many-one function
D	$f(x)=\left[\left\{\left|e^x\right|\right\}\right]$	S	$f(x)$ is constant function

where $[x]$ and $\{x\}$ denotes greatest integer and fractional part function.

• A. (A) P, Q, R; (B) P, Q, R; (C) P, Q, R, P; (D) P, R, S
• B. (A) P, Q, R; (B) P, Q, R; (C) P, Q, R, Q; (D) P, R, S
• C. (A) P, Q, R; (B) P, Q, R; (C) P, Q, R, R; (D) P, R, S
• D. (A) P, Q, R; (B) P, Q, R; (C) P, Q, R, S; (D) P, R, S

Answer 1

Answer: (D)

(A) Let $\frac{\pi }{\sqrt{3}}\sin x+\sqrt{\frac{2}{3}}\pi \cos x=t$
$t_{\max }=\pi ;t_{\min }=-\pi$
$f(x)=\cos tt\in [-\pi ,\pi ]$
$f(x)\in [-1,1]\Rightarrow (P),(Q)$
Trigonometric function is periodic
$\therefore f(x)$ is many-one
(R) $\Rightarrow$
(A) $\Rightarrow$ (P), (Q), (R)
(B) Let $|\sin x|+1=t$
$t\in [1,2]$
$f(x)={\log }_{2}t\Rightarrow f(x)\in [0,1]$
$f(x)\text{ contain only one positive integer }$
$\text{ domain is }R\Rightarrow (P),(Q),(R)$
(C) $[x]+[-x]=0x\in I$
$=-1x\notin I$
$\therefore \{[x]+[-x]\}=0$
$\therefore \text{ domain is }(-\infty ,\infty )$
$\text{ Range contains only one integer and also constant function }$
$f(x)\text{ is many-one obviously }\Rightarrow (P),(Q),(R),(S)$
(D) We know $\left|e^x\right|\in (0,\infty )\forall x\in R$
$\therefore \left\{\left|e^x\right|\right\}\in [0,1)$
$\therefore \left[\left\{\left|e^x\right|\right\}\right]\in \{0\}$
$f(x)\text{ is constant function }$
$\text{ domain is }R$
$\text{ obviously }f(x)\text{ is many-one }\Rightarrow (P),(R),(S)]$