For the following questions answer them individually
If the system of equations $$\\2x + 3y - z = 5\\ x + \alpha y + 3z = -4\\ 3x - y + \beta z = 7\\$$ has infinitely many solutions, then $$13\alpha\beta$$ is equal to:
Let $$f: \mathbb{R} \to \mathbb{R}$$ and $$g: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} \log_e x, & x > 0 \\ e^{-x}, & x \leq 0 \end{cases}$$ and $$g(x) = \begin{cases} x, & x \geq 0 \\ e^x, & x < 0 \end{cases}$$. Then, $$g \circ f: \mathbb{R} \to \mathbb{R}$$ is:
Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} \frac{a - b\cos 2x}{x^2}, & x < 0 \\ x^2 + cx + 2, & 0 \leq x \leq 1 \\ 2x + 1, & x > 1 \end{cases}\\$$ If $$f$$ is continuous everywhere in $$\mathbb{R}$$ and $$m$$ is the number of points where $$f$$ is NOT differentiable then $$m + a + b + c$$ equals:
If $$5f(x) + 4f\left(\frac{1}{x}\right) = x^2 - 2$$, $$\forall x \neq 0$$ and $$y = 9x^2 f(x)$$, then $$y$$ is strictly increasing in:
The value of the integral $$\int_0^{\pi/4} \frac{x\,dx}{\sin^4 2x + \cos^4 2x}$$ equals:
The area enclosed by the curves $$xy + 4y = 16$$ and $$x + y = 6$$ is equal to:
Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = 2x(x+y)^3 - x(x+y) - 1$$, $$y(0) = 1$$. Then, $$\left(\frac{1}{\sqrt{2}} + y\left(\frac{1}{\sqrt{2}}\right)\right)^2$$ equals:
Let $$\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}$$, $$\vec{b} = \hat{i} + 2\hat{j} - 4\hat{k}$$ and $$\vec{c} = ((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}$$. Then $$\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})$$ is equal to:
If the shortest distance between the lines $$\frac{x-\lambda}{-2} = \frac{y-2}{1} = \frac{z-1}{1}$$ and $$\frac{x-\sqrt{3}}{1} = \frac{y-1}{-2} = \frac{z-2}{1}$$ is $$1$$, then the sum of all possible values of $$\lambda$$ is:
A bag contains $$8$$ balls, whose colours are either white or black. $$4$$ balls are drawn at random without replacement and it was found that $$2$$ balls are white and other $$2$$ balls are black. The probability that the bag contains equal number of white and black balls is:
