1
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
$$R-\{0\}$$
2
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
$$7 p q \,f(0)-9=0$$
3
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

Let $$\beta=\mathop {\lim }\limits_{x \to 0} \frac{\alpha x-\left(e^{3 x}-1\right)}{\alpha x\left(e^{3 x}-1\right)}$$ for some $$\alpha \in \mathbb{R}$$. Then the value of $$\alpha+\beta$$ is :

A
$$\frac{14}{5}$$
B
$$\frac{3}{2}$$
C
$$\frac{5}{2}$$
D
$$\frac{7}{2}$$
4
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1

Let f : R $$\to$$ R be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to :

A
4
B
10
C
11
D
16
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