JEE Advanced · Mathematics · 25. AOD

Let $f(x)=(1-x)^{2} \sin ^{2} x+x^{2}$ for all $x \in I R$ and let $g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t$ for all $x \in(1, \infty)$.

Question: Which of the following is true?

A $g$ is increasing on $(1, \infty)$
B $g$ is decreasing on $(1, \infty)$
C $g$ is increasing on $(1,2)$ and decreasing on $(2, \infty)$
D $g$ is decreasing on $(1,2)$ and increasing on $(2, \infty)$

Verified Solution

Answer & Solution

Correct Answer

(B) $g$ is decreasing on $(1, \infty)$

Step-by-step Solution

Detailed explanation

$g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t$, $\therefore \quad g^{\prime}(x)=\left[\frac{2(x-1)}{x+1}-\ln x\right] f(x)$

Here $f(x)>0, \forall x \in(1, \infty)$

Let $h(x)=\frac{2(x-1)}{x+1}-\ln x$

$\therefore \quad h^{\prime}(x)=\frac{4}{(x+1)^{2}}-\frac{1}{x}=\frac{-(x-1)^{2}}{(x+1)^{2} x} < 0, x \in(1, \infty)$

$\Rightarrow h(x)$ is decreasing function.

Hence, for $x>1, h(x) < h(1) \Rightarrow h(x) < 0 \forall x>1$

$\Rightarrow g^{\prime}(x) < 0 \forall x \in(1, \infty)$

Therefore, $g(x)$ is decreasing on $(1, \infty)$.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From JEE Advanced

In a high school, a committee has to be formed from a group of $6$ boys
$M_{1}, M_{2}, M_{3}, M_{4}, M_{5}, M_{6}$ and $5$ girls $G_{1}, G_{2}, G_{3}, G_{4}, G_{5}$ .
(i) Let $α_{1}$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, having exactly $3$ boys and $2$ girls.
(ii) Let $α_{2}$ be the total number of ways in which the committee can be formed such that the committee has at least $2$ members, and having an equal number of boys and girls.
(iii) Let $α_{3}$ be the total number of ways in which the committee can be formed such that the committee has $5$ members, at least $2$ of them being girls.
(iv) Let $α_{4}$ be the total number of ways in which the committee can be formed such that the committee has $4$ members, having at least $2$ girls and such that both $M_{1}$ and $G_{1}$ are NOT in the committee together.

LIST-I LIST-II

A. The value of $α_{1}$ is P. $136$

B. The value of $α_{2}$ is Q. $189$

C. The value of $α_{3}$ is R. $192$

D. The value of $α_{4}$ is S. $200$

T. $381$

U. $461$

The correct option is:

JEE Advanced 2018 Medium

List I describes thermodynamic processes in four different systems. List II gives the magnitudes (either exactly or as a close approximation) of possible changes in the internal energy of the system due to the process.

	List-I		List-II
(I)	$10^{- 3} kg$ of water $100 ° C$ is converted to steam at the same temperature, at a pressure of $10^{5} Pa$ . The volume of the system changes from $10^{- 6} m^{3}$ to $10^{- 3} m^{3}$ in the process. Latent heat of water $= 2250 kJ {kg}^{- 1}$	(P)	$2 kJ$
(II)	$0.2$ moles of a rigid diatomic ideal gas with volume $V$ at temperature $500 K$ undergoes an isobaric expansion to volume $3 V$ . Assume $8.0 J {mol}^{- 1} K^{- 1}$	(Q)	$7 kJ$
(III)	One mole of a monoatomic ideal gas is compressed adiabatically from volume $V = \frac{1}{3} m^{3}$ and pressure $2 kPa$ to volume $\frac{V}{8}$ .	(R)	$4 kJ$
(IV)	Three moles of a diatomic ideal gas whose molecules can vibrate, is given $9 kJ$ of heat and undergoes isobaric expansion.	(S)	$5 kJ$
		(T)	$3 kJ$

Which one of the following options is correct?

JEE Advanced 2022 Medium

Explore more questions on app

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Which of the following is true?

Answer & Solution

Step-by-step Solution

See the Complete Solution

LIST-I	LIST-II
A. The value of $α_{1}$ is	P. $136$
B. The value of $α_{2}$ is	Q. $189$
C. The value of $α_{3}$ is	R. $192$
D. The value of $α_{4}$ is	S. $200$
	T. $381$
	U. $461$

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\). Question: Which of the following is true?

Step-by-step Solution

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Which of the following is true?