Question

Let ' $f$ ' be a quadratic polynomial such that
$f(-1-x)=f(-1+x)\forall x\in R\text{ and }$
$(f(1)-5{)}^2+(f(-1)-1{)}^2=f^{\prime }(-1)$

list I		list II
P	The value of $\left[{\sin }^{-1}(f(x))\right]$ whenever defined, is equal to	1	0
Q	The value of $[1+\mathrm{sgn}(f(x))]$ is equal to	2	1
R	The value of $\left[{\tan }^{-1}\left(\frac{1}{f(x)}\right)\right]$ is equal to	3	2
S	The value of $\left[2{\cot }^{-1}\left[\frac{1}{2^{f(x)}}\right]\right]$ is equal to	4	3

[Note: [y] denotes greatest integer less than or equal toy. ]

• A. P-2, Q-3, R-1, S-4
• B. P-2, Q-2, R-3, S-1
• C. P-3, Q-2, R-2, S-3
• D. P-1, Q-2, R-3, S-4

Answer 1

Answer: (A)

$f(x)=a(x+1{)}^2+b$
$f(1)=5,f(-1)=1$
$f(x)=(x+1{)}^2+1=x^2+2x+2$
(P) $f(x)\ge 1\Rightarrow \left[{\sin }^{-1}(f(x))\right]]=\left[\frac{\pi }{2}\right]=1$
(Q) $[1+\mathrm{sgn}(f(x))]=2$
(R) $\left[{\tan }^{-1}\left(\frac{1}{f(x)}\right)\right]=0$
(S) $\left[2{\cot }^{-1}\left[\frac{1}{2^{f(x)}}\right]\right]=3$.