The following table shows the mapping x o y. What is the rule for the mapping? In the right-angled triangle POR shown below, R equals

Let's test option B: $x \to 5-x$.

• For $x=0$, $y = 5-0 = 5$. (Matches)
• For $x=2$, $y = 5-2 = 3$. (Matches)
• For $x=4$, $y = 5-4 = 1$. (Matches)
• For $x=6$, $y = 5-6 = -1$. (Matches)
• For $x=8$, $y = 5-8 = -3$. (Matches)

The correct option is B.

2. In the right-angled triangle POR shown below, $\sin \hat{R}$ equals
The triangle POR is right-angled at O.

$\sin(\text{angle}) = \frac{\text{Opposite side}}{\text{Hypotenuse}}$

• The side opposite to angle $\hat{R}$ is OP.
• The hypotenuse is PR. Therefore, $\sin \hat{R} = \frac{OP}{PR}$.

Let's evaluate $\cos \hat{P}$:
$\cos(\text{angle}) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$

• The side adjacent to angle $\hat{P}$ is OP.
• The hypotenuse is PR. Therefore, $\cos \hat{P} = \frac{OP}{PR}$.

Since $\sin \hat{R} = \frac{OP}{PR}$ and $\cos \hat{P} = \frac{OP}{PR}$, it follows that $\sin \hat{R} = \cos \hat{P}$.
The correct option is B.

3. Express 75882 to three significant figures.

The first three significant figures in 75882 are 7, 5, and 8.

The fourth digit is 8. Since 8 is 5 or greater, we round up the third significant figure.

The third significant figure is 8, which rounds up to 9. The digits after the third significant figure become zeros to maintain the place value.
So, 75882 rounded to three significant figures is 75900.
The correct option is C.

4. Find the vector which translates the point (2, 4) to the point (5, 3).

Let the initial point be $A = (2, 4)$.
Let the final point be $B = (5, 3)$.
Let the translation vector be $\vec{v} = \begin{pmatrix} x \\ y \end{pmatrix}$.

The translation means $A + \vec{v} = B$.

For the x-component:

For the y-component:

The translation vector is $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$.
The options for this question are not visible in the image, but the vector is $\boxed{\begin{pmatrix} 3 \\ -1 \end{pmatrix}}$.