1
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let f : N $$\to$$ R be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers x and y. If f(1) = 2, then the value of $$\alpha$$ for which

$$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)}$$

holds, is :

A
2
B
3
C
4
D
6
2
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let f : R $$\to$$ R be defined as $$f(x) = {x^3} + x - 5$$. If g(x) is a function such that $$f(g(x)) = x,\forall 'x' \in R$$, then g'(63) is equal to ________________.

A
$${1 \over {49}}$$
B
$${3 \over {49}}$$
C
$${43 \over {49}}$$
D
$${91 \over {49}}$$
3
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let f(x) be a polynomial function such that $$f(x) + f'(x) + f''(x) = {x^5} + 64$$. Then, the value of $$\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}$$ is equal to:

A
$$-$$15
B
$$-$$60
C
60
D
15
4
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1 Let $$f:R \to R$$ and $$g:R \to R$$ be two functions defined by $$f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$$ and $$g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$$. Then, for which of the following range of $$\alpha$$, the inequality $$f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$$ holds ?

A
(2, 3)
B
($$-$$2, $$-$$1)
C
(1, 2)
D
($$-$$1, 1)
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