For the following questions answer them individually
For $$\alpha, \beta, \gamma \neq 0$$. If $$\sin^{-1}\alpha + \sin^{-1}\beta + \sin^{-1}\gamma = \pi$$ and $$(\alpha + \beta + \gamma)(\alpha - \gamma + \beta) = 3\alpha\beta$$, then $$\gamma$$ equal to
If $$f(x) = \frac{4x+3}{6x-4}, x \neq \frac{2}{3}$$ and $$(f \circ f)(x) = g(x)$$, where $$g: \mathbb{R} - \left\{\frac{2}{3}\right\} \to \mathbb{R} - \left\{\frac{2}{3}\right\}$$, then $$(g \circ g \circ g)(4)$$ is equal to
Let $$g(x)$$ be a linear function and $$f(x) = \begin{cases} g(x), & x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{1/x}, & x > 0 \end{cases}$$, is continuous at $$x = 0$$. If $$f'(1) = f(-1)$$, then the value of $$g(3)$$ is
The area of the region $$\left\{(x, y): y^2 \leq 4x, x < 4, \frac{xy(x-1)(x-2)}{(x-3)(x-4)} > 0, x \neq 3\right\}$$ is
The solution curve of the differential equation $$y\frac{dx}{dy} = x(\log_e x - \log_e y + 1), x > 0, y > 0$$ passing through the point $$(e, 1)$$ is
Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = \frac{\tan x + y}{\sin x(\sec x - \sin x \tan x)}, x \in \left(0, \frac{\pi}{2}\right)$$ satisfying the condition $$y\left(\frac{\pi}{4}\right) = 2$$. Then, $$y\left(\frac{\pi}{3}\right)$$ is
Let $$\vec{a} = 3\hat{i} + \hat{j} - 2\hat{k}$$, $$\vec{b} = 4\hat{i} + \hat{j} + 7\hat{k}$$ and $$\vec{c} = \hat{i} - 3\hat{j} + 4\hat{k}$$ be three vectors. If a vector $$\vec{p}$$ satisfies $$\vec{p} \times \vec{b} = \vec{c} \times \vec{b}$$ and $$\vec{p} \cdot \vec{a} = 0$$, then $$\vec{p} \cdot (\hat{i} - \hat{j} - \hat{k})$$ is equal to
The distance of the point $$Q(0, 2, -2)$$ from the line passing through the point $$P(5, -4, 3)$$ and perpendicular to the lines $$\vec{r} = -3\hat{i} + 2\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 5\hat{k}), \lambda \in \mathbb{R}$$ and $$\vec{r} = \hat{i} - 2\hat{j} + \hat{k} + \mu(-\hat{i} + 3\hat{j} + 2\hat{k}), \mu \in \mathbb{R}$$ is
Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability, that first drawn marble is red and second drawn marble is white, is
Three rotten apples are accidently mixed with fifteen good apples. Assuming the random variable $$x$$ to be the number of rotten apples in a draw of two apples, the variance of $$x$$ is
