Question

Consider the statements : P : There exists some x IR such that f(x) + 2x = 2(1+x2) Q : There exists some x IR 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.
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

Correct Answer - C
We have,
`f(x) = (1-x)^(2) sin^(2) x+x^(2)` for all `x in R`
`therefore f(x) + 2x = 2(1+x^(2))`
`rArr (1-x)^(2) sin^(2) x + x^(2) + 2x = 2(1+x^(2))`
`rArr (1-x)^(2) sin^(2) x = x^(2) - 2x + 2`
`rArr (1-x)^(2) sin^(2) x =(x-1)^(2)+1`
`rArr (1-x)^(2) sin^(2) x - (x-1)^(2) = 1`
`rArr (1-x)^(2) (1- sin^(2)s) = - 1`
`rArr (1-x)^(2) cos^(2) c = -1`, which is not true for any `x in R`.
So, statement P is not