Question

Assertion :The area bounded by the curves $y=x^2-3$ and $y=kx+2$ is the least if $k=0$ Reason: The area bounded by the curves $y=x^2-3$ and $y=kx+2$ is $\sqrt{k^2+20}$

• 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 C Assertion is correct but Reason is incorrect

$A={\int }_{\alpha }^{\beta }(kx+2-x^2+3)dx$
$={(\frac{k{x}^2}{2}-\frac{x^3}{3}+5x)}_{\alpha }^{\beta }$
$=(\frac{k(\alpha +\beta )}{2}-({(\alpha +\beta )}^2-\alpha \beta )\frac{1}{3}+5)(\beta -\alpha )$$=\sqrt{k^2+20}[\frac{k^2}{2}-\left(\frac{k^2+5}{3}\right)+5]=\frac{1}{6}{(k^2+20)}^{3/2}$
Hence statement I is true & II is false