Question

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

Q : There exists some $x\in$ such that
$2f\left(x\right)+1=2x(1+x).$
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

The correct option is C P is false and Q is true

$P:x^2+2x+(1–x{)}^2{\sin }^{2}x=2(1+x^2)$
$\Rightarrow {\sin }^{2}x(1–x{)}^2=x^2–2x+2$
$\Rightarrow {\sin }^{2}x(1–x{)}^2=(1–x{)}^2+1$
which can $\text{never}$ be true

$Q:2x^2+2{\sin }^{2}x(1–x{)}^2=2x^2+2x$
$\Rightarrow {\sin }^{2}x(1–x{)}^2=2x$
$\Rightarrow {\sin }^{2}x=\frac{2x}{(1-x{)}^2}$
Clearly, a solution is possible.