Question

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

Let $\left[k\right]$ denote the greatest integer less than or equal to $k$ and $\text{sgn}$ denote the signum function.
$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)=\text{sgn}(x^2-ax+1)\text{has exactly one point of discontinuity,}& \text{(P)}& 1\\ & \text{then value(s) of}a\text{can be}& & \\ \text{(B)}& \text{If}f\left(x\right)=[2+3|n|\sin x]\text{has exactly}11\text{points of discontinuity in}& \text{(Q)}& 2\\ & x\in (0,\pi ),\text{then}n\text{cannot be}& & \\ \text{(C)}& \text{If}f\left(x\right)=|||x|-2|-p|\text{has exactly three points of non-differentiability,}& \text{(R)}& -1\\ & \text{then value(s) of}p\text{can be}& & \\ \text{(D)}& \text{If}\underset{x\to -\infty }{lim}\frac{\sqrt{4x^2+3}-x}{3x-2}=L,\text{then}L\text{equals}& \text{(S)}& -2\\ & & \text{(T)}& 3\end{array}$

Which of the following is a CORRECT combination?

• A. $\left(C\right)\to \left(R\right),\left(S\right);\left(D\right)\to \left(S\right)$
• B. $\left(C\right)\to \left(Q\right),\left(R\right),\left(S\right);\left(D\right)\to \left(R\right)$
• C. $\left(C\right)\to \left(R\right),\left(S\right);\left(D\right)\to \left(R\right)$
• D. $\left(C\right)\to \left(R\right),\left(T\right);\left(D\right)\to \left(S\right)$

Answer 1

The correct option is C $\left(C\right)\to \left(R\right),\left(S\right);\left(D\right)\to \left(R\right)$

$\left(C\right)$
$f\left(x\right)=|||x|-2|-p|$
Let $y=||x|-2|$

For three points of non-differentiability,
$-p\ge 0$
$\Rightarrow p\le 0$

$\left(D\right)$
$L=\underset{x\to -\infty }{lim}\frac{\sqrt{4x^2+3}-x}{3x-2}$
Replace $x$ by $-x$
$L=-\underset{x\to \infty }{lim}\frac{\sqrt{4x^2+3}+x}{3x+2}=-\underset{x\to \infty }{lim}\frac{\sqrt{4+\frac{3}{x^2}}+1}{3+\frac{2}{x}}L=-1$