1. 1
   The function $f$ is defined for $x\in\mathbb{R}$ by $f(x)=x^{2}-2x-8$.

   (a)

   Write down the coordinates of the local maximum point on the graph of $y=|f(x)|$.

   [1 mark]

   • A$(1,-9)$
   • B$(-1,9)$
   • C$(1,9)$
   • D$(0,8)$

   (b)

   Find the number of solutions of the equation $|f(x)|=5$.

   [1 mark]

   • A4
   • B2
   • C3
   • D6

   (c)

   Write down the $x$-intercepts and the $y$-intercept of the graph of $y=f(|x|)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $h$ is continuous and differentiable for all $x\in\mathbb{R}$. Its only stationary points are a local maximum at $(-1,4)$ and a local minimum at $(2,-3)$, and its only zeros are $x=-4$, $x=1$ and $x=3$.

   (a)

   Write down the coordinates of the local minimum point on the graph of $y=h(2x-1)$.

   [1 mark]

   • A$(3,-3)$
   • B$(0.5,-3)$
   • C$(1.5,-6)$
   • D$(1.5,-3)$

   (b)

   Which of the following is the complete set of vertical asymptotes of the graph of $y=\dfrac{1}{h(x)}$?

   [1 mark]

   • A$x=-1$, $x=2$
   • B$x=-4$, $x=1$, $x=3$
   • C$x=-4$, $x=3$
   • D$x=0$

   (c)

   Find the coordinates of the stationary points on the graph of $y=\dfrac{1}{h(x)}$, and state the nature of each.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The functions $f$ and $g$ are defined for $x\in\mathbb{R}$ by $f(x)=|2x-5|$ and $g(x)=|x+1|$.

   (a)

   Solve the equation $f(x)=g(x)$.

   [3 marks]

   (b)

   Hence, or otherwise, solve the inequality $f(x)>g(x)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The displacement, in centimetres, of a float from its rest level $t$ seconds after observation begins is modelled by $d(t)=2\sin\left(\frac{\pi t}{3}\right)+1$, for $0\le t\le6$. Displacement above the rest level is positive. The distance of the float from its rest level is $|d(t)|$.

   (a)

   (i) Find the values of $t$ for which $d(t)=0$.\n(ii) Write down the range of $d$, and hence find the range of $|d(t)|$ and the range of $[d(t)]^{2}$ for $0\le t\le6$.

   [6 marks]

   (b)

   A sensor records whenever the float is at least $2$ cm from its rest level. Solve $|d(t)|\ge2$ for $0\le t\le6$, and hence find the fraction of the $6$ seconds for which the sensor records.

   [6 marks]

   Total for question 4: 12 marks

End of questions