1
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

$$\mathop {\lim }\limits_{t \to 0} {\left( {{1^{{1 \over {{{\sin }^2}t}}}} + {2^{{1 \over {{{\sin }^2}t}}}}\, + \,...\, + \,{n^{{1 \over {{{\sin }^2}t}}}}} \right)^{{{\sin }^2}t}}$$ is equal to

A
$${{n(n + 1)} \over 2}$$
B
n
C
n$$^2$$ + n
D
n$$^2$$
2
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1

Let $$f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$$

Then at $$x=0$$

A
$$f$$ is continuous but $$f'$$ is not continuous
B
$$f$$ and $$f'$$ both are continuous
C
$$f$$ is continuous but not differentiable
D
$$f'$$ is continuous but not differentiable
3
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1

$$\text { Let the function } f(x)=\left\{\begin{array}{cl} \frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & ;\text { if } x \neq 0 \\ 10 & ; \text { if } x=0 \end{array} \text { be continuous at } x=0 .\right.$$

Then $$\alpha$$ is equal to

A
10
B
$$-$$10
C
5
D
$$-$$5
4
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

If $$\lim\limits_{x \rightarrow 0} \frac{\alpha \mathrm{e}^{x}+\beta \mathrm{e}^{-x}+\gamma \sin x}{x \sin ^{2} x}=\frac{2}{3}$$, where $$\alpha, \beta, \gamma \in \mathbf{R}$$, then which of the following is NOT correct?

A
$$\alpha^{2}+\beta^{2}+\gamma^{2}=6$$
B
$$\alpha \beta+\beta \gamma+\gamma \alpha+1=0$$
C
$$\alpha\beta^{2}+\beta \gamma^{2}+\gamma \alpha^{2}+3=0$$
D
$$\alpha^{2}-\beta^{2}+\gamma^{2}=4$$
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