1
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1

Suppose $$f: \mathbb{R} \rightarrow(0, \infty)$$ be a differentiable function such that $$5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}$$. If $$f(3)=320$$, then $$\sum_\limits{n=0}^{5} f(n)$$ is equal to :

A
6875
B
6525
C
6575
D
6825
2
JEE Main 2023 (Online) 29th January Morning Shift
+4
-1

Let $$x=2$$ be a root of the equation $$x^2+px+q=0$$ and $$f(x) = \left\{ {\matrix{ {{{1 - \cos ({x^2} - 4px + {q^2} + 8q + 16)} \over {{{(x - 2p)}^4}}},} & {x \ne 2p} \cr {0,} & {x = 2p} \cr } } \right.$$

Then $$\mathop {\lim }\limits_{x \to 2{p^ + }} [f(x)]$$, where $$\left[ . \right]$$ denotes greatest integer function, is

A
2
B
1
C
0
D
$$-1$$
3
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1

If the function $$f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$$

is continuous at $$x = {\pi \over 2}$$, then $$9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$$ is equal to

A
11
B
10
C
8
D
2e$$^4$$ + 8
4
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

The value of $$\mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + 3} - \sqrt {{n^4} + 5n + 4} }}$$ is :

A
$${3 \over {2\sqrt 2 }}$$
B
$${3 \over 2}(\sqrt 2 + 1)$$
C
$$3(\sqrt 2 + 1)$$
D
$${{\sqrt 2 + 1} \over 2}$$
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