1
JEE Main 2022 (Online) 29th June Morning Shift
+4
-1

The domain of the function $${\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)$$ is :

A
$$R - \left\{ { - {1 \over 2},{1 \over 2}} \right\}$$
B
$$( - \infty , - 1] \cup [1,\infty ) \cup \{ 0\}$$
C
$$\left( { - \infty ,{{ - 1} \over 2}} \right) \cup \left( {{1 \over 2},\infty } \right) \cup \{ 0\}$$
D
$$\left( { - \infty ,{{ - 1} \over {\sqrt 2 }}} \right] \cup \left[ {{1 \over {\sqrt 2 }},\infty } \right) \cup \{ 0\}$$
2
JEE Main 2022 (Online) 28th June Morning Shift
+4
-1

Let a function f : N $$\to$$ N be defined by

$$f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right.$$

then, f is

A
one-one but not onto
B
onto but not one-one
C
neither one-one nor onto
D
one-one and onto
3
JEE Main 2022 (Online) 26th June Evening Shift
+4
-1

Let f : R $$\to$$ R be defined as f (x) = x $$-$$ 1 and g : R $$-$$ {1, $$-$$1} $$\to$$ R be defined as $$g(x) = {{{x^2}} \over {{x^2} - 1}}$$.

Then the function fog is :

A
one-one but not onto
B
onto but not one-one
C
both one-one and onto
D
neither one-one nor onto
4
JEE Main 2022 (Online) 26th June Morning Shift
+4
-1

Let $$f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\}$$. If $${f^{n + 1}}(x) = f({f^n}(x))$$ for all n $$\in$$ N, then $${f^6}(6) + {f^7}(7)$$ is equal to :

A
$${7 \over 6}$$
B
$$- {3 \over 2}$$
C
$${7 \over {12}}$$
D
$$- {{11} \over {12}}$$
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