Question

Consider the statements
P:There exists some x $\in \, R$ such that,
f(x) + 2 x = 2 (1 + x$^2$).
Q:There exists some x $\in \, R$ such that,
$2f(x)+1=2x(1+x)$
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

Answer 1

Answer: (C)

Here, f(x) + 2x= (1 - x)$^2$ sin$^2$x+ x$^2$ + 2x $ \ \ \ \ \ \ \ \ \ \ \ \ ...(i) $
where, P : f(x) + 2x = 2 (1 + x)$^2 \ \ \ \ \ \ \ \ \\ \ ...(ii)$
$\therefore \ \ \ \ \ \ \ 2(1+x^2)=(1-x)^2 sin^2x+x^2+2x$
$\Rightarrow \ \ \ \ (1-x)^2 sin^2 x=x^2-2x+2$
$\Rightarrow \ \ \ \ (1-x)^2 sin^2 x=(1-x)^2+1 \ \ \rightarrow \ \ (1-x)^2 cos^2x=-1$
which is never possible
$\therefore \ \ \ P \ is \ false \ .$
Again, let Q : h(x) = 2 f(x) + 1 - 2x (1 + x)
where, h(0) = 2 f(0) + 1 - 0 = 1
$ \ \ \ \ \ \ \ \ h(1)=2f(1)+1-4=-3, as \ h(0) h(1) <0$
$\Rightarrow \ \ $h(x) m ust have a solution.
$\therefore $Q is true