Q102 / 155 · WBJEE 2018 · Mathematics

Let $f(x)=\left\{\begin{array}{cc}-2 \sin x, & \text { if } x \leq-\frac{\pi}{2} \\ A \sin x+B, & \text { if }-\frac{\pi}{2} < x < \frac{\pi}{2} \text { . Then, } \\ \cos x, & \text { if } x \geq \frac{\pi}{2}\end{array}\right.$

  • A
    $f$ is discontinuous for all $A$ and $B$
  • Correct answer
    B
    $f$ is continuous for all $A=-1$ and $B=1$
  • C
    $f$ is continuous for all $A=1$ and $B=-1$
  • D
    $f$ is continuous for all real values of $A B$

Solution

At $\quad x=-\frac{\pi}{2}$ $$ \begin{array}{l} \text { LHL }=-2 \\ \text { RHL }=-A+B \\ \text { For continuity, } \mathrm{LHL}=\mathrm{RHL}=f(-\pi / 2) \end{array} $$ $\Rightarrow \quad-A+B=2$ At $x=\pi / 2$ $$ \begin{aligned} \mathrm{LHL} &=A+B \\ \mathrm{RHL} &=0 \\ \text { For continuity, } \mathrm{LHL} &=\mathrm{RHL}=f(\pi / 2) \end{aligned} $$ $\Rightarrow \quad A+B=0$ On solving Eqs. (i) and (ii), we get $$ A=-1 \text { and } B=1 $$

Hence, the correct answer is option: 'B'.

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