JEE Advanced · Mathematics · 25. AOD

Paragraph:

Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be a thrice differentiable function. Suppose that $F(1)=0, F(3)=-4$ and $F^{\prime}(x)<0$ for all $x \in(1 / 2,3)$. Let $f(x)=x F(x)$ for all $x \in \mathbb{R}$.

Question:

The correct statement(s) is(are)

A $f^{'} (1) < 0$
B $f (2) < 0$
C $f^{'} (x) \neq 0 f o r a n y x \in (1, 3)$
D $f^{'} (x) = 0 f o r s o m e x \in (1, 3)$

Verified Solution

Answer & Solution

Correct Answer

(A) $f^{'} (1) < 0$

Step-by-step Solution

Detailed explanation

F (1) = 0, F (3) = - 4

F^{'} (x) < 0

F (x)

is decreasing, also

F (x) < 0

Now,

f (x) = x F (x)

x \in (\frac{1}{2}, 3) x F (x) < 0

∴ f (2) < 0

Also,

f^{'} (x) = x F^{'} (x) + F (x)

f^{'} (1) = F^{'} (1) < 0

Also,

f (x)

is strictly decreasing
Now,

F (1) = F (x) = 0

∴ f^{'} (x) \neq 0 a s F^{'} (x) < 0 a n d f (x)

is decreasing.

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

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

The tangent to a suitable conic (Column 1) at

(\sqrt{3}, \frac{1}{2})

is found to be

\sqrt{3} x + 2 y = 4

, then which of the following options is the only Correct combination?

JEE Advanced 2017 Hard

Let

E_{1} =

{

x \in ℝ : x \neq 1

and

\frac{x}{x - 1} > 0

} and

E_{2} =

{

x \in E_{1} : \sin^{- 1} (\log_{e} (\frac{x}{x - 1}))

is a real number} (Here the inverse trigonometric function

\sin^{- 1} (x)

assumes values in

[- \frac{π}{2} . \frac{π}{2}]

.)
Let

f : E_{1} \to ℝ

be the function defined by

f (x) = \log_{e} (\frac{x}{x - 1})

And

g : E_{2} \to R

be the function defined by

g (x) = \sin^{- 1} (\log_{e} (\frac{x}{x - 1}))

LIST-I	LIST-II
A. The range of $f$ is	P. $(- \infty, \frac{1}{1 - e}] \cup [\frac{e}{e - 1}, \infty)$
B. The range of $g$ contains	Q. (0, 1)
C. The domain of $f$ contains	R. $[- \frac{1}{2}, \frac{1}{2}]$
D. The domain of $g$ is	S. $(- \infty, 0) \cup (0, \infty)$
	T. $(- \infty, \frac{e}{e - 1}]$
	U. $(- \infty, 0) \cup (\frac{1}{2}, \frac{e}{e - 1}]$

The correct option is:

JEE Advanced 2018 Hard

Paragraph:

Football teams $T_{1}$ and $T_{2}$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_{1}$ winning, drawing and losing a game against $T_{2}$ are $\frac{1}{2}, \frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets $3$ points for a win, $1$ point for a draw and $0$ point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_{1}$ and $T_{2}$, respectively, after two games.

Question:

$P(X=Y)$ is

JEE Advanced 2016 Easy

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Paragraph:

Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function. Suppose that \(F(1)=0, F(3)=-4\) and \(F^{\prime}(x)<0\) for all \(x \in(1 / 2,3)\). Let \(f(x)=x F(x)\) for all \(x \in \mathbb{R}\).

Question:

The correct statement(s) is(are)

Answer & Solution

Step-by-step Solution

See the Complete Solution

Paragraph: Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function. Suppose that \(F(1)=0, F(3)=-4\) and \(F^{\prime}(x)<0\) for all \(x \in(1 / 2,3)\). Let \(f(x)=x F(x)\) for all \(x \in \mathbb{R}\). Question: The correct statement(s) is(are)

Step-by-step Solution

Paragraph:

Let \(F: \mathbb{R} \rightarrow \mathbb{R}\) be a thrice differentiable function. Suppose that \(F(1)=0, F(3)=-4\) and \(F^{\prime}(x)<0\) for all \(x \in(1 / 2,3)\). Let \(f(x)=x F(x)\) for all \(x \in \mathbb{R}\).

Question:

The correct statement(s) is(are)