If the system of equations

$ 3x + y + 4z = 3 $

$ 2x + \alpha y - z = -3 $

$ x + 2y + z = 4 $

has no solution, then the value of $ \alpha $ is equal to:

Let $z$ be the complex number satisfying $|z-5| \leq 3$ and having maximum positive principal argument.

Then $34 \left| \frac{5z - 12}{5iz + 16} \right|^2$ is equal to:

If the area of the region $\{(x, y) : 1-2x \leq y \leq 4-x^2,\; x \geq 0,\; y \geq 0 \}$ is $\dfrac{\alpha}{\beta}$, $\alpha, \beta \in \mathbb{N}, \gcd(\alpha,\beta)=1$, then the value of $(\alpha+\beta)$ is:

The positive integer n, for which the solutions of the equation

$x(x+2) + (x+2)(x+4) + \cdots + (x+2n-2)(x+2n) = \frac{8n}{3}$ are two consecutive even integers, is :