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Let f(x) = (1 - x)2sin2x + x2 for all x ...

Let f(x) = (1 - x)2sin2x + x2 for all x ∈ R, and let g(x) = ∫(2(t - 1)/(t + 1) - ln t)f(t)dt for t ∈ [1, x] for all x ∈ (1, ∞). Consider the statements: P: There exists some x ∈ R, such that f(x) + 2x = 2(1 + x2) Q: There exists some x ∈ R, such that 2f(x) + 1 = 2x(1 + x) (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

Text Solution

Verified by Experts

The correct Answer is:
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)
`:. 2 (1+x^(2))=(1-x)^(2) sin^(2) x+x^(2)+2x`
`rArr (1-x)^(2)sin^(2)x = x^(2)-2x+2`
`rArr(1-x)^(2)sin^(2)x = (1-x)+1`
`rArr (1 -x)^(2) cos^(2)=-1`
which is never possible .
`:.` 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) = f (1) + 1 4 =- 3 , as h (0) h (1) `lt` 0
`rArr` h (x) must have a solution.
`:.` Q is true.
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