1
JEE Main 2024 (Online) 27th January Evening Shift
MCQ (Single Correct Answer)
+4
-1

Consider the function $$f:(0,2) \rightarrow \mathbf{R}$$ defined by $$f(x)=\frac{x}{2}+\frac{2}{x}$$ and the function $$g(x)$$ defined by

$$g(x)=\left\{\begin{array}{ll} \min \lfloor f(t)\}, & 0<\mathrm{t} \leq x \text { and } 0 < x \leq 1 \\ \frac{3}{2}+x, & 1 < x < 2 \end{array} .\right. \text { Then, }$$

A
$$g$$ is continuous but not differentiable at $$x=1$$
B
$$g$$ is continuous and differentiable for all $$x \in(0,2)$$
C
$$g$$ is not continuous for all $$x \in(0,2)$$
D
$$g$$ is neither continuous nor differentiable at $$x=1$$
2
JEE Main 2024 (Online) 27th January Evening Shift
MCQ (Single Correct Answer)
+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
3
JEE Main 2024 (Online) 27th January Morning Shift
MCQ (Single Correct Answer)
+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
4
JEE Main 2024 (Online) 27th January Morning Shift
MCQ (Single Correct Answer)
+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
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