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

Let $$f$$ and $$g$$ be two functions defined by

$$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ |x-1|, & x \geq 0\end{array}\right.$$ and $$\mathrm{g}(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ 1, & x \geq 0\end{array}\right.$$

Then $$(g \circ f)(x)$$ is :

A
continuous everywhere but not differentiable at $$x=1$$
B
differentiable everywhere
C
not continuous at $$x=-1$$
D
continuous everywhere but not differentiable exactly at one point
2
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 $$x=1$$
3
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
$$2 \alpha$$
4
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
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