Find the value of y in the figure below, given that O is the centre of the circle.

In a circle, the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle (circumference).
From the diagram, the angle at the center is $55^\circ$ (labeled at O), and the angle at the circumference is $y^\circ$ (labeled at P). However, based on the options provided, it is highly probable that the $55^\circ$ is the angle at the circumference ($\angle QPR$) and $y$ is the angle at the center ($\angle QOR$). We will proceed with this common interpretation for such problems when options are integers.

If the angle at the circumference $\angle QPR = 55^\circ$, then the angle at the center $\angle QOR$ is twice this value.

This corresponds to option B.

The final answer is $\boxed{\text{B. } 110^\circ}$.

4. In the figure below, AD is a tangent to the circle ABC, $|AB| = |AC|$ and $\angle BAC = 2x^\circ$. Find $\angle DAC$.

Given that $|AB| = |AC|$, triangle ABC is an isosceles triangle. Therefore, the base angles are equal: $\angle ABC = \angle ACB$.

The sum of angles in a triangle is $180^\circ$.

Substitute $\angle BAC = 2x^\circ$ and $\angle ACB = \angle ABC$:

The Alternate Segment Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
Here, AD is the tangent and AC is the chord. The angle in the alternate segment is $\angle ABC$.
Therefore, $\angle DAC = \angle ABC$.

This corresponds to option B.

The final answer is $\boxed{\text{B. } (90 - x)^\circ}$.

5. A fair die is rolled once. What is the probability of obtaining a 3 or 4?

A fair die has 6 faces, so the possible outcomes are $\{1, 2, 3, 4, 5, 6\}$.
Total number of outcomes = 6.

We want to obtain a 3 or a 4. The favorable outcomes are $\{3, 4\}$.
Number of favorable outcomes = 2.

The probability of an event is given by the formula:

This corresponds to option D.

The final answer is $\boxed{\text{D. } \frac{1}{3}}$.

6. Find the midpoint of a line joining the points A(2, 2) and B(6, 4).

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $(x_m, y_m)$ is given by:

Given $A(2, 2)$ and $B(6, 4)$, we have $x_1=2, y_1=2, x_2=6, y_2=4$.

The midpoint is $(4, 3)$.
This corresponds to option A.

The final answer is $\boxed{\text{A. } (4, 3)}$.

7. Find the distance between the points A(3, -3) and B(4, -3).

For two points $(x_1, y_1)$ and $(x_2, y_2)$, the distance $d$ is given by:

Given $A(3, -3)$ and $B(4, -3)$, we have $x_1=3, y_1=-3, x_2=4, y_2=-3$.

This corresponds to option D.

The final answer is $\boxed{\text{D. } 1}$.

8. If the equation $4x^2 - 4\sqrt{P}x + 3 = 0$ has equal roots, find the value of $P$.

A quadratic equation in the form $ax^2 + bx + c = 0$ has equal roots if its discriminant ($\Delta$) is equal to zero.

The equation is $4x^2 - 4\sqrt{P}x + 3 = 0$.
Comparing this to $ax^2 + bx + c = 0$, we have:
$a = 4$
$b = -4\sqrt{P}$
$c = 3$

This corresponds to option C.

The final answer is $\boxed{\text{C. } 3}$.