1. grad MP= gradPN

**Answer**

$k=-3$

**Explanation**

Given 2 points on any line, we can determine the equation of the line in the form $$y=mx+c$$ where

$m$ is the gradient

$c$ is the $y$-intercept

To calculate the gradient, $m$ between $P$ and $N$, we use the formula:

\[ \dfrac{y_2-y_1}{x_2-x_1}\]

\[\dfrac{4-2}{3-0}=\dfrac{2}{3}\]

Since $c$ is the $y$-intercept i.e. the pont on the line when $x=0$ and in this case the point has the coordinates $P(0;2)$ thus $c=2$

$\therefore y=\dfrac{2}{3}x+2$

We are told that the 3 points $MP$ and $N$ are collinear, so it implies that the lines $MP$ and $PN$ have the same gradient, so:

\[\dfrac{2}{3}=\dfrac{2-0}{0-k}\]

$\therefore k=-3$

2.

**Answer**

$= 119.74^{\circ}$

**Explanation**

Given the gradients of the two straight lines we can calculate the values of $\alpha$ and $\beta$.

To determine the value of $\alpha$ we use the fact that have a right-angle formed between the two axes, thus

\[\tan{\alpha}=\dfrac{O}{A} = \dfrac{2}{3} \]

which is just the gradient that we calculated in the first question, so

\[\alpha = \tan^{-1}\left({\frac{2}{3}}\right) = 33.69^{\circ}\]

Using the same thinking, we can infer that

\[\tan{\beta}=\dfrac{O}{A} = \dfrac{-1}{2} \]

so

\[\beta = \tan^{-1}\left({\frac{-1}{2}}\right) = -26.57^{\circ}+180^{\circ} = 153.43^{\circ} \]

hence

$$\theta = \beta - \alpha $$

$$= 153.43^{\circ} - 33.69^{\circ} $$

$$= 119.74^{\circ}$$

3.

**Answer**

11 units

**Explanation**

We know that the coordinates for the point $M$ are $(-3,0)$, if we determine the coordinates of $R$ then we will be able to get the length of $MR$.

Since $R$ is on the line $y= \frac{-1}{2}x+4$, and knowing that the value of $y$ at that point is $0$, i.e.

\[ 0 = \frac{-1}{2}x+4 \]

Solving for $x$ gives us $8$, thus $R(8;0)$

Now we can get the length of the the line $MR$

\[x_R - x_M = 8 - (-3) = 11 \]

4.

**Answer**

22 square units

**Explanation**

Remember that the area of a triangle is given by the formula

\[ A = \dfrac{1}{2} \times B \times H \]

where $B$ is the base and $H$ is the height of the triangle. (Remember that the base and the height of any triangle are always at $90^{\circ}$ to each other)

\[ A_{\triangle{MNR}} = \dfrac{1}{2} \times 11 \times 4 = 22 \]