Question

If $f\left(x\right)=\left(p^{2} - 1\right)\left[\left(tan\right)^{- 1} x\right]+$ $4\left(q^{2} + 2 q - 3\right)\left\{\frac{1}{2 + x^{2}}\right\}$ $+\left(p + q\right)sgn\left(x^{2} - x + 2\right)$ is continuous in $R$ and $f\left(x_{1}\right)=f\left(x_{2}\right)\forall x_{1},x_{2}\in R$ , then largest value of $\left|p + q\right|$ is [Note: $sgn\left(y\right),\left[y\right]$ and $\left\{y\right\}$ denote signum function, greatest integer function and fractional part function respectively.]

• A. $0$
• B. $2$
• C. $4$
• D. $5$

Answer 1

Answer: (C)

Since $f\left(x_{1}\right)=f\left(x_{2}\right)\forall x_{1},x_{2}\in R$
$\therefore f\left(x\right)$ must be constant Hence
$p^{2}-1=0\Rightarrow p=\pm1$
$q^{2}+2q-3=0\Rightarrow q=-3,1$
$\therefore $ Largest possible value of $\left|p + q\right|=4$