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