Question

Assertion :Let $F\left(x\right)={\int }_1^{x^2}\cos \sqrt{t}dt$.
STATEMENT-1 : $F^{{}^{\prime }}\left(x\right)=\cos x$ Reason: STATEMENT-2 : If $f\left(x\right)={\int }_a^x\varphi \left(t\right)dt$ then $f^{{}^{\prime }}\left(x\right)=\varphi \left(x\right)$

• A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for Statement-1
• C. Statement-1 is True, Statement-2 is False
• D. Statement-1 is False, Statement-2 is True

Answer 1

The correct option is D Statement-1 is False, Statement-2 is True

For $F\left(x\right)={\int }_{g\left(x\right)}^{h\left(x\right)}f\left(t\right)dt\Rightarrow \frac{d\left(F\left(x\right)\right)}{dx}=h^{\prime }\left(x\right)f\left(h\left(x\right)\right)-g^{\prime }\left(x\right)f\left(g\left(x\right)\right)$

$F\left(x\right)={\int }_1^{x^2}\cos \sqrt{t}dt\Rightarrow F^{\prime }\left(x\right)=2x\cos \sqrt{x^2}=2x\cos x$

And $f\left(x\right)={\int }_a^x\varphi \left(t\right)dt{f}^{\prime }\left(x\right)=\varphi \left(x\right)$

Hence, Statement-1 is False, Statement-2 is True.