1
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
$$\alpha^{2}-\beta^{2}+\gamma^{2}=4$$
2
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1

The number of points, where the function $$f: \mathbf{R} \rightarrow \mathbf{R}$$,

$$f(x)=|x-1| \cos |x-2| \sin |x-1|+(x-3)\left|x^{2}-5 x+4\right|$$, is NOT differentiable, is :

A
1
B
2
C
3
D
4
3
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

The function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ defined by

$$f(x)=\lim\limits_{n \rightarrow \infty} \frac{\cos (2 \pi x)-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}$$ is continuous for all x in :

A
$$R-\{-1\}$$
B
$$\mathbb{R}-\{-1,1\}$$
C
$$R-\{1\}$$
D
$$R-\{0\}$$
4
JEE Main 2022 (Online) 27th July Evening Shift
+4
-1

If for $$\mathrm{p} \neq \mathrm{q} \neq 0$$, the function $$f(x)=\frac{\sqrt[7]{\mathrm{p}(729+x)}-3}{\sqrt[3]{729+\mathrm{q} x}-9}$$ is continuous at $$x=0$$, then :

A
$$7 p q \,f(0)-1=0$$
B
$$63 q \,f(0)-\mathrm{p}^{2}=0$$
C
$$21 q \,f(0)-\mathrm{p}^{2}=0$$
D
$$7 p q \,f(0)-9=0$$
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