1
JEE Advanced 2020 Paper 1 Offline
+3
-1
If the function f : R $$\to$$ R is defined by f(x) = |x| (x $$-$$ sin x), then which of the following statements is TRUE?
A
f is one-one, but NOT onto
B
f is onto, but NOT one-one
C
f is BOTH one-one and onto
D
f is NEITHER one-one NOR onto
2
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) = $${\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right)$$.

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
3
JEE Advanced 2017 Paper 2 Offline
+3
-1
Let S = {1, 2, 3, .........., 9}. For k = 1, 2, .........., 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 =
A
210
B
252
C
126
D
125
4
JEE Advanced 2014 Paper 2 Offline
+3
-1
Let f1 : R $$\to$$ R, f2 : [0, $$\infty$$) $$\to$$ R, f3 : R $$\to$$ R, and f4 : R $$\to$$ [0, $$\infty$$) be defined by

$${f_1}\left( x \right) = \left\{ {\matrix{ {\left| x \right|} & {if\,x < 0,} \cr {{e^x}} & {if\,x \ge 0;} \cr } } \right.$$

f2(x) = x2 ;

$${f_3}\left( x \right) = \left\{ {\matrix{ {\sin x} & {if\,x < 0,} \cr x & {if\,x \ge 0;} \cr } } \right.$$

and

$${f_4}\left( x \right) = \left\{ {\matrix{ {{f_2}\left( {{f_1}\left( x \right)} \right)} & {if\,x < 0,} \cr {{f_2}\left( {{f_1}\left( x \right)} \right) - 1} & {if\,x \ge 0;} \cr } } \right.$$

A
P - 3, Q - 1, R - 4, S - 2
B
P - 1, Q - 3, R - 4, S - 2
C
P - 3, Q - 1, R - 2, S - 4
D
P - 1, Q - 3, R - 2, S - 4
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