Identify the given quadratic polynomial and its zeroes.

The quadratic polynomial is $x^2 - 3x + k$.
The zeroes are $\alpha = 2$ and $\beta = -5$.

For a quadratic polynomial $ax^2 + bx + c$, the product of the zeroes is given by $\frac{c}{a}$.
In $x^2 - 3x + k$, we have $a=1$, $b=-3$, and $c=k$.
The product of the given zeroes is $2 \times (-5) = -10$.
Equating this to $\frac{c}{a}$:

The correct alternative is (A).
The final answer is $\boxed{\text{(A) -10}}$.

Q.1.ii.

The value of $\sin 30^\circ$ is a fundamental trigonometric ratio.

The correct alternative is (B).
The final answer is $\boxed{\text{(B) } \frac{1}{2}}$.

Q.1.iii.

The two points are $(x_1, y_1) = (-2, 3)$ and $(x_2, y_2) = (4, -1)$.

The distance $D$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

The correct alternative is (B).
The final answer is $\boxed{\text{(B) } \sqrt{52}}$.

Q.1.iv.

Natural numbers are positive integers starting from 1.
The first five natural numbers are $1, 2, 3, 4, 5$.

Sum $= 1 + 2 + 3 + 4 + 5 = 15$.

The mean is the sum of the numbers divided by the count of the numbers.
Count of numbers $= 5$.

The correct alternative is (C).
The final answer is $\boxed{\text{(C) 3}}$.