1. 1
   The curve $C$ has equation $y=x^{2}-4x+1$. The point $P(3,\,-2)$ lies on $C$.

   (a)

   Find the gradient of the tangent to $C$ at $P$.

   [1 mark]

   • A$2$
   • B$-2$
   • C$6$
   • D$-\frac{1}{2}$

   (b)

   Find the gradient of the normal to $C$ at $P$.

   [1 mark]

   • A$\frac{1}{2}$
   • B$-2$
   • C$-\frac{1}{2}$
   • D$2$

   (c)

   Find the equation of the tangent to $C$ at $P$. Give your answer in the form $y=mx+c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $y=\frac{2}{x}$, $x\neq0$, passes through the point $R(2,\,1)$.

   (a)

   Find the equation of the tangent to the curve at $R$.

   [1 mark]

   • A$y=\frac{1}{2}x$
   • B$y=2x-3$
   • C$y=-x+3$
   • D$y=-\frac{1}{2}x+2$

   (b)

   The normal to the curve at $R$ meets the $x$-axis at the point $S$. Find the $x$-coordinate of $S$.

   [1 mark]

   • A$4$
   • B$1.5$
   • C$3$
   • D$-1.5$

   (c)

   Find the coordinates of the other point on the curve at which the tangent is parallel to the tangent at $R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=x^{3}-2x^{2}+1$. The point $P(2,\,1)$ lies on $C$.

   (a)

   Find the equation of the tangent to $C$ at $P$.

   [3 marks]

   (b)

   The tangent at $P$ meets $C$ again at the point $Q$. Show that the $x$-coordinate of $Q$ satisfies $x^{3}-2x^{2}-4x+8=0$, and hence find the coordinates of $Q$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The side profile of a skateboard ramp is modelled by $h(x)=-0.02x^{3}+0.3x^{2}$, for $0\le x\le10$, where $h(x)$ is the height, in metres, of the ramp surface above horizontal ground at a horizontal distance $x$ metres from the start of the ramp. The ground is the line $y=0$. The point $A$ on the ramp surface has $x=8$.

   (a)

   Find the equation of the tangent to the ramp surface at $A$. Give your answer in the form $y=mx+c$.

   [6 marks]

   (b)

   A straight support strut is fixed along the normal to the ramp surface at $A$ and runs down to the ground.
   (i) Find the equation of the line along which the strut lies.
   (ii) Find the length of the strut.
   You may use a calculator in this part.

   [6 marks]

   Total for question 4: 12 marks

End of questions