1
JEE Main 2023 (Online) 11th April Morning Shift
+4
-1

Let $$f(x)=\left[x^{2}-x\right]+|-x+[x]|$$, where $$x \in \mathbb{R}$$ and $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then, $$f$$ is :

A
continuous at $$x=0$$, but not continuous at $$x=1$$
B
continuous at $$x=0$$ and $$x=1$$
C
continuous at $$x=1$$, but not continuous at $$x=0$$
D
not continuous at $$x=0$$ and $$x=1$$
2
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1

If $$\alpha > \beta > 0$$ are the roots of the equation $$a x^{2}+b x+1=0$$, and $$\lim_\limits{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^{2}+b x+a\right)}{2(1-\alpha x)^{2}}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right), \text { then } \mathrm{k} \text { is equal to }$$ :

A
$$2 \beta$$
B
$$\beta$$
C
$$\alpha$$
D
$$2 \alpha$$
3
JEE Main 2023 (Online) 8th April Morning Shift
+4
-1

$$\lim_\limits{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^{2}(3 x)\right.}{\cos ^{3}(4 x)}\right)\left(\frac{\sin ^{3}(4 x)}{\left(\log _{e}(2 x+1)\right)^{5}}\right)\right)$$ is equal to _____________.

A
15
B
18
C
9
D
24
4
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1

Let $$a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}$$ be $$\mathrm{n}$$ positive consecutive terms of an arithmetic progression. If $$\mathrm{d} > 0$$ is its common difference, then

$$\lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left(\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots \ldots \ldots+\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_{n}}}\right)$$ is

A
$$\frac{1}{\sqrt{d}}$$
B
1
C
0
D
$$\sqrt{d}$$
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