1
JEE Advanced 2018 Paper 2 Offline
+3
-1
Let $${E_1} = \left\{ {x \in R:x \ne 1\,and\,{x \over {x - 1}} > 0} \right\}$$ and

$${E_2} = \left\{ \matrix{ x \in {E_1}:{\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right) \hfill \cr is\,a\,real\,number \hfill \cr} \right\}$$

(Here, the inverse trigonometric function $${\sin ^{ - 1}}$$ x assumes values in $$\left[ { - {\pi \over 2},{\pi \over 2}} \right]$$.).

Let f : E1 $$\to$$ R be the function defined by f(x) = $${{{\log }_e}\left( {{x \over {x - 1}}} \right)}$$ and g : E2 $$\to$$ R be the function defined by g(x) = $${\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right)$$.
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
2
JEE Advanced 2018 Paper 2 Offline
+3
-1
In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.

(i) Let $$\alpha$$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 $$\alpha$$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.

i) Let $$\alpha$$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 $$\alpha$$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 such that both M1 and G1 are NOT in the committee together.
LIST-I LIST-II
P. The value of $\alpha_1$ is 1. 136
Q. The value of $\alpha_2$ is 2. 189
R. The value of $\alpha_3$ is 3. 192
S. The value of $\alpha_4$ is 4. 200
5. 381
6. 461
The correct option is
A
P $$\to$$ 4; Q $$\to$$ 6; R $$\to$$ 2; S $$\to$$ 1
B
P $$\to$$ 1; Q $$\to$$ 4; R $$\to$$ 2; S $$\to$$ 3
C
P $$\to$$ 4; Q $$\to$$ 6; R $$\to$$ 5; S $$\to$$ 2
D
P $$\to$$ 4; Q $$\to$$ 2; R $$\to$$ 3; S $$\to$$ 1
3
JEE Advanced 2018 Paper 2 Offline
+3
-1
Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, where a > b > 0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60$$^\circ$$ at one of its vertices N. Let the area of the $$\Delta$$LMN be $$4\sqrt 3$$.

List - I List - II
P. The length of the conjugate axis of H is 1. 8
Q. The eccentricity of H is 2. $${4 \over {\sqrt 3 }}$$
R. The distance between the foci of H is 3. $${2 \over {\sqrt 3 }}$$
S. The length of the latus rectum of H is 4. 4
A
P $$\to$$ 4 ; Q $$\to$$ 2 ; R $$\to$$ 1 ; S $$\to$$ 3
B
P $$\to$$ 4 ; Q $$\to$$ 3 ; R $$\to$$ 1 ; S $$\to$$ 2
C
P $$\to$$ 4 ; Q $$\to$$ 1 ; R $$\to$$ 3 ; S $$\to$$ 2
D
P $$\to$$ 3 ; Q $$\to$$ 4 ; R $$\to$$ 2 ; S $$\to$$ 1
4
JEE Advanced 2018 Paper 2 Offline
+3
-1
Let $${f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to R$$ and $${f_4}:R \to R$$ be functions defined by

(i) $${f_1}(x) = \sin (\sqrt {1 - {e^{ - {x^2}}}} )$$,

(ii) $${f_2}(x) = \left\{ \matrix{ {{|\sin x|} \over {\tan { - ^1}x}}if\,x \ne 0,\,where \hfill \cr 1\,if\,x = 0 \hfill \cr} \right.$$

the inverse trigonometric function tan$$-$$1x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$,

(iii) $${f_3}(x) = [\sin ({\log _e}(x + 2))]$$, where for $$t \in R,\,[t]$$ denotes the greatest integer less than or equal to t,

(iv) $${f_4}(x) = \left\{ \matrix{ {x^2}\sin \left( {{1 \over x}} \right)\,if\,x \ne 0 \hfill \cr 0\,if\,x = 0 \hfill \cr} \right.$$
LIST-I LIST-II
P. The function $$f_1$$ is 1. NOT continuous at $$x = 0$$
Q. The function $$f_2$$ is 2. continuous at $$x = 0$$ and NOT differentiable at $$x = 0$$
R. The function $$f_3$$ is 3. differentiable at $$x = 0$$ and its derivative is NOT continuous at $$x = 0$$
S. The function $$f_4$$ is 4. differentiable at $$x = 0$$ and its derivative is continuous at $$x = 0$$
A
P $$\to$$ 2 ; Q $$\to$$ 3 ; R $$\to$$ 1 ; S $$\to$$ 4
B
P $$\to$$ 4 ; Q $$\to$$ 1 ; R $$\to$$ 2 ; S $$\to$$ 3
C
P $$\to$$ 4 ; Q $$\to$$ 2 ; R $$\to$$ 1 ; S $$\to$$ 3
D
P $$\to$$ 2 ; Q $$\to$$ 1 ; R $$\to$$ 4 ; S $$\to$$ 3
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