Question

Assertion :The function $y=f\left(x\right)$, defined parametrically as $y=t^2+t|t|,x=2t-|t|,t\in R,$ is continuous for all real $x.$ Reason: $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}2x^2,& x\ge 0\end{array}\\ \begin{array}{cc}0,& x<0\end{array}\end{array}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

We have, $y=t^2+t\left|t\right|$ and $x=2t-\left|t\right|$

when $t\ge 0$

$x=2t-t=t,y=t^2+t^2=2t^2$

$\therefore x=t$ and $y=2t^2$

$\Rightarrow y=2x^2\forall x\ge 0$

Also, when $t<0$

$x=2t+t=3t$ and $y=t^2-t^2=0$

$\Rightarrow y=0$ for all $x<0$

Hence, $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}2x^2,& x\ge 0\end{array}\\ \begin{array}{cc}0,& x<0\end{array}\end{array}$ which is clearly continuous for all $x$.