Question

Let $f:R\to [\alpha ,\infty ),f(x)=x^2+3ax+b,g(x)={\sin }^{-1}\frac{x}{4}(\alpha \in R)$

Column I		Column II
A	The possible integral values of ' $a$ ' for which $f(x)$ is many one in interval $[-3,5]$ is/are	P	- 2
B	Let $a=-1$ and $\mathrm{gof}(x)$ is defined for $x\in [-1,1]$ then possible integral values of $b$ can be	Q	- 1
C	Let $a=2,\alpha =-8$ the value(s) of $b$ for which $f(x)$ is surjective is/are	R	0
D	If $a=1,b=2$, then integers in the range of $f\circ g(x)$ is/are	S	1

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

Answer 1

Answer: (D)

(A) $-3<-\frac{3a}{2}<5\Rightarrow -\frac{10}{3}<a<2$
$\therefore a\in \{-3,-2,-1,0,1\}\Rightarrow P,Q,R,S$
(B) $f(x)=x^2-3x+b$

$\mathrm{gof}(x)={\sin }^{-1}\left(\frac{f(x)}{4}\right)\text{ is defined }\forall x\in [-1,1]$
$\Rightarrow -1\le \frac{f(x)}{4}\le 1\forall x\in [-1,1]$
$-4\le x^2-3x+b\le 4\forall x\in [-1,1]$
$f_{\text{min }}=f(+1)=b-2$
$f_{\max }=f(-1)=b+4$
$\Rightarrow b-2\ge -4\text{ and }b+4\le 4$
$b\ge -2\text{ and }b\le 0\Rightarrow b\in [-2,0]$
$\therefore b=-2,-1,0\Rightarrow P,Q,R$

(C)$f:R\to [-8,\infty ),f(x)=x^2+6x+b$
$f_{\mathrm{mim}}=f(-3)=b-9$
$b-9=-8\Rightarrow b=1\Rightarrow S$
(D)$h(x)=f\circ g(x)={\left({\sin }^{-1}\frac{x}{4}\right)}^2+3{\sin }^{-1}\frac{x}{4}+2$
$h(t)=t^2+3t+2,t={\sin }^{-1}\frac{x}{4},t\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
$={\left(t+\frac{3}{2}\right)}^2-\frac{1}{4}$
${h(t)|}_{\max }$ occurs when $t=\frac{\pi }{2}$ and ${h(t)|}_{\min }$ occurs when $t=-\frac{3}{2}$
${\therefore h(t)|}_{\min }=-\frac{1}{4}$ and ${h(t)|}_{\max }=\frac{{\pi }^2}{4}+\frac{3\pi }{2}+2$
$\therefore$ Range of $f\circ g(x)=\left[-\frac{1}{4},\frac{{\pi }^2}{4}+\frac{3\pi }{2}+2\right]$
$\therefore$ Possible integers are $0,1\Rightarrow R,S$ ]