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. (a) Both I and II are trueB. (b) I is true and II is falseC. (c) I is false and II is trueD. (d) Both I and II are false

Consider the statements : P : There exists some x IR such that f(x) +

1.	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. (a) Both I and II are trueB. (b) I is true and II is falseC. (c) I is false and II is trueD. (d) Both I and II are false
Answer» Correct Answer - (c) Here, `f(x)+2x=(1-x)^(2) cdot sin ^(2) x+x^(2) +2x …(i)` where, `I: f(x)+2x=2(1+x)^(2) …(ii)` `therefore 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)^(2)+1` `rArr (1-x)^(2) cos^(2) x=-1` which is never possible. `therefore I` is false. Again, let `h(x)=2f(x)+1-2x(1+x)` where, `h(0)=2f(0)+1-0=1` `h(1)=2(1) +1-4=-3 as [h(0)h(1)lt0]` `rArr h(x)` must have a solution. `therefore II` is true.

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