1
JEE Main 2024 (Online) 9th April Morning Shift
+4
-1

If the domain of the function $$f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$$ is $$\mathbf{R}-(\alpha, \beta)$$, then $$12 \alpha \beta$$ is equal to :

A
40
B
36
C
24
D
32
2
JEE Main 2024 (Online) 8th April Evening Shift
+4
-1

Let $$f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0< x \leq \mathrm{a}\end{array}\right.$$ where $$\mathrm{a}> 0$$ and $$\mathrm{g}(x)=(f(|x|)-|f(x)|) / 2$$. Then the function $$g:[-a, a] \rightarrow[-a, a]$$ is

A
neither one-one nor onto.
B
both one-one and onto.
C
one-one.
D
onto
3
JEE Main 2024 (Online) 6th April Evening Shift
+4
-1

If the function $$f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0$$ attains the maximum value at $$x=\frac{1}{\mathrm{e}}$$ then :

A
$$\mathrm{e}^\pi<\pi^{\mathrm{e}}$$
B
$$\mathrm{e}^{2 \pi}<(2 \pi)^{\mathrm{e}}$$
C
$$(2 e)^\pi>\pi^{(2 e)}$$
D
$$\mathrm{e}^\pi>\pi^{\mathrm{e}}$$
4
JEE Main 2024 (Online) 6th April Evening Shift
+4
-1

Let $$f(x)=\frac{1}{7-\sin 5 x}$$ be a function defined on $$\mathbf{R}$$. Then the range of the function $$f(x)$$ is equal to :

A
$$\left[\frac{1}{8}, \frac{1}{5}\right]$$
B
$$\left[\frac{1}{7}, \frac{1}{6}\right]$$
C
$$\left[\frac{1}{7}, \frac{1}{5}\right]$$
D
$$\left[\frac{1}{8}, \frac{1}{6}\right]$$
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