1
JEE Main 2024 (Online) 31st January Morning Shift
+4
-1

Let $$g(x)$$ be a linear function and $$f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$$, is continuous at $$x=0$$. If $$f^{\prime}(1)=f(-1)$$, then the value $$g(3)$$ is

A
$$\log _e\left(\frac{4}{9}\right)-1$$
B
$$\frac{1}{3} \log _e\left(\frac{4}{9 e^{1 / 3}}\right)$$
C
$$\log _e\left(\frac{4}{9 e^{1 / 3}}\right)$$
D
$$\frac{1}{3} \log _e\left(\frac{4}{9}\right)+1$$
2
JEE Main 2024 (Online) 27th January Evening Shift
+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 $$x=1$$
3
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
4
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
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