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 t t \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 ]=0 x \in I } $
$ =-1 x \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 )]$