1
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1
Let $[x]$ denote the greatest integer function and

$f(x)=\max \{1+x+[x], 2+x, x+2[x]\}, 0 \leq x \leq 2$. Let $m$ be the number of

points in $[0,2]$, where $f$ is not continuous and $n$ be the number of points in

$(0,2)$, where $f$ is not differentiable. Then $(m+n)^{2}+2$ is equal to :
A
3
B
6
C
2
D
11
2
JEE Main 2023 (Online) 13th April Evening Shift
+4
-1

If $$\lim_\limits{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{cx e^{-c x}}{2}}{1-\cos (2 x)}=17$$, then $$5 a^{2}+b^{2}$$ is equal to

A
64
B
68
C
72
D
76
3
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
4
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 $$x=1$$
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