Question

STATEMENT - 1 : ${\int }_0^1\frac{cosx}{1+x^2}dx>\frac{\pi }{4}cos1$
STATEMENT - 2 : If $f\left(x\right)$ and $g\left(x\right)$ are continuous on $[a,b]$, then ${\int }_a^{b}f\left(x\right)g\left(x\right)dx=f\left(c\right){\int }_a^{b}g\left(x\right)dx$ for some $c\in (a,b)$.

• A. statement 1 is true and statement 2 is correct explanation of statement 1
• B. statement 1 is true and statement 2 is not a correct explanation of statement 1
• C. statement 1 is false and statement 2 is true
• D. statement 1 is true and statement 2 is false

Answer 1

Answer: (A)

statement 1 is true and statement 2 is correct explanation of statement 1

Statement - 2, being the statement of generalized mean value theorem, is true.
Using statement - 2, there exists $c\in (0,1)$ such that
${\int }_0^1\frac{\cos x}{1+x^2}dx=\cos c{\int }_0^1\frac{1}{1+x^2}dx=\frac{\pi }{4}\cos c$
Clearly, $\cos c>\cos 1$ for all $c\in (0,1)$
$\Rightarrow \frac{\pi }{4}\cos c>\frac{\pi }{4}\cos 1\Rightarrow {\int }_0^1\frac{\cos x}{1+x^2}dx>\frac{\pi }{4}\cos 1$
So statement - 1 is true. Also, statement - 2 is a correct explanation for statement -1.