1
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

$$\text { If } \lim _\limits{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3} \text {, then } 2 \alpha-\beta \text { is equal to : }$$

A
2
B
1
C
5
D
7
2
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
Consider the function.

$$f(x)=\left\{\begin{array}{cc} \frac{\mathrm{a}\left(7 x-12-x^2\right)}{\mathrm{b}\left|x^2-7 x+12\right|} & , x<3 \\\\ 2^{\frac{\sin (x-3)}{x-[x]}} & , x>3 \\\\ \mathrm{~b} & , x=3, \end{array}\right.$$

where $[x]$ denotes the greatest integer less than or equal to $x$. If $\mathrm{S}$ denotes the set of all ordered pairs (a, b) such that $f(x)$ is continuous at $x=3$, then the number of elements in $\mathrm{S}$ is :
A
Infinitely many
B
4
C
2
D
1
3
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
If $\mathrm{a}=\lim\limits_{x \rightarrow 0} \frac{\sqrt{1+\sqrt{1+x^4}}-\sqrt{2}}{x^4}$ and $\mathrm{b}=\lim\limits _{x \rightarrow 0} \frac{\sin ^2 x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $a b^3$ is :
A
36
B
25
C
32
D
30
4
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
Let $[x]$ denote the greatest integer function and

$f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. Let $m$ be the number of

points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in

$(0,2)$, where $f$ is not differentiable. Then $(m+n)^{2}+2$ is equal to :
A
3
B
6
C
2
D
11
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