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)$$\begin{gathered} {}^2 \\ ...(ii) \end{gathered}$$
$\therefore 2(1+x^2)=(1-x{)}^{2}si{n}^{2}x+x^2+2x$
$\Rightarrow (1-x{)}^{2}si{n}^{2}x=x^2-2x+2$
$\Rightarrow (1-x{)}^{2}si{n}^{2}x=(1-x{)}^2+1\to (1-x{)}^{2}co{s}^{2}x=-1$
which is never possible
$\therefore Pisfalse.$
Again, let Q : h(x) = 2 f(x) + 1 - 2x (1 + x)
where, h(0) = 2 f(0) + 1 - 0 = 1
$h(1)=2f(1)+1-4=-3,ash(0)h(1)<0$
$\Rightarrow$h(x) m ust have a solution.
$\therefore$Q is true