IIT-JEE 2012 · Paper 2 · Mathematics Q52

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MathematicsSingle correct+3 / -1Question 52 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 $x\in(1,\infty)$ .

Consider the statements:

P: There exists some $x\in\mathbb R$ such that $f(x)+2x=2(1+x^2)$

Q: There exists some $x\in\mathbb R$ such that $2f(x)+1=2x(1+x)$

Then

both P and Q are true

P is true and Q is false

P is false and Q is true

both P and Q are false

Answer and worked solution

Official answer: C

Statement P. $f(x)+2x=2+2x^2$ becomes
$(1-x)^2\sin^2x=x^2-2x+2=(x-1)^2+1\implies-(1-x)^2\cos^2x=1,$
impossible (the left side is $\le0$ ). P is false.

Statement Q. $2f(x)+1=2x+2x^2$ becomes $k(x)=2(1-x)^2\sin^2x-2x+1=0$ . Now $k(0)=1>0$ and $k(1)=-1<0$ , so by the intermediate value theorem a root exists. Q is true.

(C)

Topics:Rolle's and mean value theorems Trigonometric equations

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IIT-JEE 2012 · Paper 2, Mathematics question 52