SATHEE: Definite Integration Question 15

Definite Integration Question 15

Question 15

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, $2 f (x) + 1 = 2 x (1 + x)$ .

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

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Answer:

Correct Answer: 15. (c)

Solution:

Here, $f (x) + 2 x = (1 - x)^{2} \cdot \sin^{2} x + x^{2} + 2 x$

where, $P : f (x) + 2 x = 2 (1 + x)^{2}$

$\begin{array}{ll} ∴ & 2 (1 + x^{2}) = (1 - x)^{2} \sin^{2} x + x^{2} + 2 x \Rightarrow & (1 - x)^{2} \sin^{2} x = x^{2} - 2 x + 2 \Rightarrow & (1 - x)^{2} \sin^{2} x = (1 - x)^{2} + 1 \Rightarrow & (1 - x)^{2} \cos^{2} x = - 1 \end{array}$

which is never possible.

$∴ P$ is false.

Again, let $Q : h (x) = 2 f (x) + 1 - 2 x (1 + x)$

where, $h (0) = 2 f (0) + 1 - 0 = 1$

$h (1) = 2 f (1) + 1 - 4 = - 3, as h (0) h (1) < 0$

$\Rightarrow h (x)$ must have a solution.

$∴ Q$ is true.

Definite Integration Question 15

Question 15

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Definite Integration Question 15

Question 15

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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