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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 trueB. P is true and Q is falseC. P is false and Q is trueD. both P and Q are false |
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Answer» Correct Answer - 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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