For the following questions answer them individually
If $$f(x) = \begin{cases} 2 + 2x, & -1 \leq x < 0 \\ 1 - \frac{x}{3}, & 0 \leq x \leq 3 \end{cases}$$; $$g(x) = \begin{cases} -x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1 \end{cases}$$, then range of $$(f \circ g(x))$$ is
Consider the function $$f : [\frac{1}{2}, 1] \to R$$ defined by $$f(x) = 4\sqrt{2}x^3 - 3\sqrt{2}x - 1$$. Consider the statements
(I) The curve $$y = f(x)$$ intersects the $$x$$-axis exactly at one point
(II) The curve $$y = f(x)$$ intersects the $$x$$-axis at $$x = \cos\frac{\pi}{12}$$
Then
Suppose $$f(x) = \frac{(2^x + 2^{-x})\tan x \sqrt{\tan^{-1}(x^2 - x + 1)}}{(7x^2 + 3x + 1)^3}$$. Then the value of $$f'(0)$$ is equal to
If the value of the integral $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{(\sin x)^{2023}}}\right) dx = \frac{\pi}{4}(\pi + a) - 2$$, then the value of $$a$$ is
For $$x \in (-\frac{\pi}{2}, \frac{\pi}{2})$$, if $$y(x) = \int \frac{\csc x + \sin x}{\csc x \sec x + \tan x \sin^2 x} dx$$ and $$\lim_{x \to (\frac{\pi}{2})^-} y(x) = 0$$ then $$y(\frac{\pi}{4})$$ is equal to
A function $$y = f(x)$$ satisfies $$f(x)\sin 2x + \sin x - (1 + \cos^2 x)f'(x) = 0$$ with condition $$f(0) = 0$$. Then $$f(\frac{\pi}{2})$$ is equal to
Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero vectors such that $$\vec{b}$$ and $$\vec{c}$$ are non-collinear. If $$\vec{a} + 5\vec{b}$$ is collinear with $$\vec{c}$$, $$\vec{b} + 6\vec{c}$$ is collinear with $$\vec{a}$$ and $$\vec{a} + \alpha\vec{b} + \beta\vec{c} = \vec{0}$$, then $$\alpha + \beta$$ is equal to
Let $$O$$ be the origin and the position vector of $$A$$ and $$B$$ be $$2\hat{i} + 2\hat{j} + \hat{k}$$ and $$2\hat{i} + 4\hat{j} + 4\hat{k}$$ respectively. If the internal bisector of $$\angle AOB$$ meets the line $$AB$$ at $$C$$, then the length of $$OC$$ is
Let $$PQR$$ be a triangle with $$R(-1, 4, 2)$$. Suppose $$M(2, 1, 2)$$ is the mid point of $$PQ$$. The distance of the centroid of $$\Delta PQR$$ from the point of intersection of the line $$\frac{x-2}{0} = \frac{y}{2} = \frac{z+3}{-1}$$ and $$\frac{x-1}{1} = \frac{y+3}{-3} = \frac{z+1}{1}$$ is
A fair die is thrown until $$2$$ appears. Then the probability, that $$2$$ appears in even number of throws, is
