1
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

$$\text { Let } f(x)=a x^{2}+b x+c \text { be such that } f(1)=3, f(-2)=\lambda \text { and }$$ $$f(3)=4$$. If $$f(0)+f(1)+f(-2)+f(3)=14$$, then $$\lambda$$ is equal to :

A
$$-$$4
B
$$\frac{13}{2}$$
C
$$\frac{23}{2}$$
D
4
2
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1

The function $$f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$$, is :

A
increasing in $$\left(-\frac{1}{2}, 1\right)$$
B
decreasing in $$\left(\frac{1}{2}, 2\right)$$
C
increasing in $$\left(-1,-\frac{1}{2}\right)$$
D
decreasing in $$\left(-\frac{1}{2}, \frac{1}{2}\right)$$
3
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

Considering only the principal values of the inverse trigonometric functions, the domain of the function $$f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{2}+3}\right)$$ is :

A
$$\left(-\infty, \frac{1}{4}\right]$$
B
$$\left[-\frac{1}{4}, \infty\right)$$
C
$$(-1 / 3, \infty)$$
D
$$\left(-\infty, \frac{1}{3}\right]$$
4
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}$$ and $$g: \mathbf{R} \rightarrow \mathbf{R}$$ be such that $$g(f(x))=x$$ for all $$x \in \mathbf{R}$$. If $$\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots, \mathrm{a}_{\mathrm{n}}$$ be in arithmetic progression with mean zero, then the value of $$f\left(g\left(\frac{1}{\mathrm{n}} \sum\limits_{i=1}^{\mathrm{n}} f\left(\mathrm{a}_{i}\right)\right)\right)$$ is equal to :

A
0
B
3
C
9
D
27
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