Q1
Answer
The points $A$ and $B$ have coordinates $(3k-4,-2)$ and $(1,k+1)$ respectively, where $k$ is a constant.
  1. Given that the gradient of $AB$ is $-\dfrac{3}{2}$, show that $k=3$.
  2. Find an equation of the line through $A$ and $B$.
  3. Find an equation of the perpendicular bisector of $A$ an $B$. Leave your answer in the form $ax+by+c=0$, where $a$, $b$, and $c$ are integers.
  1. gradient: $\dfrac{k+1 - (-2)}{1 - (3k-4)} = -\dfrac{3}{2}$
  2. $y = -\dfrac{3}{2}x + \dfrac{11}{2}$
  3. $y = \dfrac{2}{3}x - 1$
Q2
Answer
  1. Sketch the graph of $y=\log_9(x+a)$, where $a > 0$ and $x > -a$. Label any asymptotes and points of intersection with the axes. Leave your answers in terms of $a$ where necessary.
  2. Describe, with a reason, the relationship between the graph of $y=\log_9(x+a)$ and $y=\log_9(x+a)^2$.
  1. Sketches
  2. Stretch parallel to $y$ axis, scale factor 2
Q3
Answer
Shown below is the graph of $y=\mathrm{f}(x)$. Sketch the graphs of $y=\mathrm{f}(2x)$ and $y=\mathrm{f}(-x)$, labelling the coordinates of the images of the points $A$, $B$, $C$ and $D$ in each case.
$\mathrm{f}(2x)$: $A(-0.5,0)$, $B(0,-2)$, $C(0.5, -4)$, $D(1,0)$ and $\mathrm{f}(-x)$: $A(1,0)$, $B(0,-2)$, $C(-1,4)$, $D(-2,0)$
Q4
Answer
A buoy is a device which floats on the surface of the sea and moves up and down as waves pass. For a certain buoy, its height, $y$ metres above its position in still water, is modelled by $y = \sin 180t^{\circ}$, where $t$ is the time in seconds.
  1. Sketch a graph showing the height of the buoy above its still water level for $0 \leq t \leq 10$ showing the coordinates of points of intersection with the $t$-axis.
  2. Write down the number of times the buoy is $0.4$ m above its still water position during the first 10 seconds.
  3. Give one reason why this model might not be realistic.
  1. Sketch
  2. 10
  3. Not all waves will be the same