IIT-JEE 2012 · Paper 2 · Mathematics Q51

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MathematicsSingle correct+3 / -1Question 51 of 60

Passage

Paragraph for Questions 51 and 52

Let $f(x)=(1-x)^2\sin^2x+x^2$ for all $x\in\mathbb R$ , and let $g(x)=\displaystyle\int_1^x\left(\frac{2(t-1)}{t+1}-\ln t\right)f(t)\,dt$ for all $x\in(1,\infty)$ .

Which of the following is true?

g

is increasing on

(1,\infty)

g

is decreasing on

(1,\infty)

g

is increasing on

(1,2)

and decreasing on

(2,\infty)

g

is decreasing on

(1,2)

and increasing on

(2,\infty)

Answer and worked solution

Official answer: B

Derivative. $g'(x)=h(x)f(x)$ with $h(t)=\dfrac{2(t-1)}{t+1}-\ln t$ .

Sign of $f$ . $f(x)=(1-x)^2\sin^2x+x^2>0$ for $x>1$ .

Sign of $h$ . $h(1)=0$ and
$h'(t)=\frac4{(t+1)^2}-\frac1t=\frac{4t-(t+1)^2}{t(t+1)^2}=-\frac{(t-1)^2}{t(t+1)^2}<0\quad(t>1),$
so $h<0$ on $(1,\infty)$ .

So $g'<0$ : $g$ is decreasing on $(1,\infty)$ .

(B)

Topics:Monotonicity Fundamental theorem of calculus

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IIT-JEE 2012 · Paper 2, Mathematics question 51