1. 1
   A curve has equation $x^{2} + xy + y^{2} = 7$. The point $P(1, 2)$ lies on the curve.

   (a)

   Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.

   [1 mark]

   • A$-\dfrac{2x + y}{x + 2y}$
   • B$\dfrac{2x + y}{x + 2y}$
   • C$-\dfrac{2x + y}{2y}$
   • D$-\dfrac{x}{y}$

   (b)

   Find the gradient of the curve at $P$.

   [1 mark]

   • A$-\dfrac{5}{4}$
   • B$\dfrac{4}{5}$
   • C$-\dfrac{1}{2}$
   • D$-\dfrac{4}{5}$

   (c)

   Find the equation of the normal to the curve at $P$. Give your answer in the form $ax + by + c = 0$, where $a, b, c \in \mathbb{Z}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A spherical balloon is being inflated so that its volume increases at a constant rate of $50\text{ cm}^{3}\text{ s}^{-1}$. The volume of a sphere of radius $r$ is $V = \dfrac{4}{3}\pi r^{3}$ and its surface area is $S = 4\pi r^{2}$.

   (a)

   Find the rate at which the radius is increasing at the instant when the radius is $5$ cm.

   [1 mark]

   • A$\dfrac{3}{10\pi}\text{ cm s}^{-1}$
   • B$2\pi\text{ cm s}^{-1}$
   • C$\dfrac{1}{2\pi}\text{ cm s}^{-1}$
   • D$\dfrac{1}{8\pi}\text{ cm s}^{-1}$

   (b)

   Find the rate at which the surface area is increasing at the instant when the radius is $5$ cm.

   [1 mark]

   • A$40\pi\text{ cm}^{2}\text{ s}^{-1}$
   • B$20\text{ cm}^{2}\text{ s}^{-1}$
   • C$10\text{ cm}^{2}\text{ s}^{-1}$
   • D$100\text{ cm}^{2}\text{ s}^{-1}$

   (c)

   Find the exact radius of the balloon at the instant when the radius is increasing at $0.02\text{ cm s}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A water tank is in the shape of an inverted right circular cone, with its vertex at the bottom. The cone has height $6$ m and the radius of its circular top is $3$ m. Water flows into the empty tank at a constant rate of $0.8\text{ m}^{3}$ per minute. At time $t$ minutes the depth of the water is $h$ m, the radius of the water surface is $r$ m and the volume of water is $V\text{ m}^{3}$.

   (a)

   Show that $V = \dfrac{\pi h^{3}}{12}$.

   [3 marks]

   (b)

   Find the rate at which the depth of the water is increasing when the depth is $2$ m. Hence find the rate at which the radius of the water surface is increasing at this instant.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A lighthouse keeper is in a boat at point $A$, which is $3$ km from the nearest point $P$ on a straight shoreline. She wants to reach a village $Q$ on the shoreline, $4$ km from $P$, as quickly as possible. She rows in a straight line to a point $X$ on the shoreline between $P$ and $Q$, where $PX = x$ km and $0 \le x \le 4$, and then walks along the shore from $X$ to $Q$. She walks at $5\text{ km h}^{-1}$.

   (a)

   She rows at $2\text{ km h}^{-1}$. Find the value of $x$ that minimises her total journey time, justify that it gives the minimum, and find the minimum time exactly. A calculator may be used to compare values.

   [6 marks]

   (b)

   She now uses a motorised boat that travels at $4.5\text{ km h}^{-1}$; her walking speed is unchanged. Show that her journey time is a decreasing function of $x$ for $0 \le x \le 4$. Hence find the route that minimises her journey time, state the minimum time, and explain why solving $T'(x) = 0$ does not give the answer in this case.

   [6 marks]

   Total for question 4: 12 marks

End of questions