1. Tardigrade
  2. Question
  3. Mathematics
  4. Let f: R arrow[α, ∞), f( x )= x 2+3 ax + b , g ( x )= sin -1 ( x /4)(α ∈ 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 operatornamegof( x ) is defined for x ∈[-1,1] then possible integral values of b can be Q - 1 C Let a =2, α=-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 ° g(x) is/are S 1

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

Inverse Trigonometric Functions

Solution:

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

$\operatorname{gof}(x)=\sin ^{-1}\left(\frac{f(x)}{4}\right) \text { is defined } \forall x \in[-1,1] $
$\Rightarrow -1 \leq \frac{ f ( x )}{4} \leq 1 \forall x \in[-1,1]$
$-4 \leq x ^2-3 x + b \leq 4 \forall x \in[-1,1]$
$f _{\text {min }}= f (+1)= b -2 $
$f _{\max }= f (-1)= b +4$
$\Rightarrow b-2 \geq-4 \text { and } b+4 \leq 4 $
$b \geq-2 \text { and } b \leq 0 \Rightarrow b \in[-2,0] $
$\therefore b =-2,-1,0 \Rightarrow P , Q , R $
image
(C)$f: R \rightarrow[-8, \infty), f(x)=x^2+6 x+b $
$f_{\operatorname{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+3 t+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}$
$\left. h ( t )\right|_{\max }$ occurs when $t =\frac{\pi}{2}$ and $\left. h ( t )\right|_{\min }$ occurs when $t =-\frac{3}{2}$
$\left.\therefore h ( t )\right|_{\min }=-\frac{1}{4}$ and $\left. h ( t )\right|_{\max }=\frac{\pi^2}{4}+\frac{3 \pi}{2}+2$
$\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 $0,1 \Rightarrow R , S$ ]