Definite Integration Question 15
Question 15
- Consider the statements
$P$ : There exists some $x \in R$ such that,
$$ f(x)+2 x=2\left(1+x^{2}\right) $$
$Q$ : There exists some $x \in R$ such that, $2 f(x)+1=2 x(1+x)$.
Then, (a) both $P$ and $Q$ are true (b) $P$ is true and $Q$ is false (c) $P$ is false and $Q$ is true (d) both $P$ and $Q$ are false
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Answer:
Correct Answer: 15. (c)
Solution:
- Here, $f(x)+2 x=(1-x)^{2} \cdot \sin ^{2} x+x^{2}+2 x$
where, $P: f(x)+2 x=2(1+x)^{2}$
$$ \begin{array}{ll} \therefore & 2\left(1+x^{2}\right)=(1-x)^{2} \sin ^{2} x+x^{2}+2 x \ \Rightarrow & (1-x)^{2} \sin ^{2} x=x^{2}-2 x+2 \ \Rightarrow & (1-x)^{2} \sin ^{2} x=(1-x)^{2}+1 \ \Rightarrow & (1-x)^{2} \cos ^{2} x=-1 \end{array} $$
which is never possible.
$\therefore P$ is false.
Again, let $Q: h(x)=2 f(x)+1-2 x(1+x)$
where, $\quad h(0)=2 f(0)+1-0=1$
$$ h(1)=2 f(1)+1-4=-3 \text {, as } h(0) h(1)<0 $$
$\Rightarrow h(x)$ must have a solution.
$\therefore Q$ is true.
