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Functions question

2018 · Shift 2 · Q33

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PYQ Library
JEE Advanced
Mathematics
Functions
2018 · Shift 2 · Q33

Functions question

2018 · Shift 2 · Q33

JEE AdvancedMathematicsFunctionsMCQ+3 / −1

Let

E_1 = \left\{ x \in \mathbb{R} : x \ne 1 \text{ and } \frac{x}{x-1} \gt 0 \right\}

and

E_2 = \left\{ x \in E_1 : \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right) \text{ is a real number} \right\}

(Here, the inverse trigonometric function

\sin^{-1} x

assumes values in

\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

.) Let

f : E_1 \to \mathbb{R}

be the function defined by

f(x) = \log_e\left(\frac{x}{x-1}\right)

and

g : E_2 \to \mathbb{R}

be the function defined by

g(x) = \sin^{-1}\left(\log_e\left(\frac{x}{x-1}\right)\right)

. | LIST-I | LIST-II | |--------|--------| | P. The range of

f

is | 1.

\left(-\infty, \frac{1}{1-e}\right] \cup \left[\frac{e}{e-1}, \infty\right)

| | Q. The range of

g

contains | 2.

(0, 1)

| | R. The domain of

f

contains | 3.

\left[-\frac{1}{2}, \frac{1}{2}\right]

| | S. The domain of

g

is | 4.

(-\infty, 0) \cup (0, \infty)

| | | 5.

\left(-\infty, \frac{e}{e-1}\right)

| | | 6.

(-\infty, 0) \cup \left(\frac{1}{2}, \frac{e}{e-1}\right]

| The correct option is :

A
P $\to$ 4; Q $\to$ 2; R $\to$ 1 ; S $\to$ 1
B
P $\to$ 3; Q $\to$ 3; R $\to$ 6 ; S $\to$ 5
C
P $\to$ 4; Q $\to$ 2; R $\to$ 1 ; S $\to$ 6
D
P $\to$ 4; Q $\to$ 3; R $\to$ 6 ; S $\to$ 5

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