Implicit and parametric differentiationEdexcel International A Level Maths: Subtopic test

1. 1
   A curve $C_1$ has equation $x^3+y^3=9$, and the point $P(1,2)$ lies on $C_1$.

   (a)

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

   [1 mark]

   • A$\frac{x^2}{y^2}$
   • B$-\frac{y^2}{x^2}$
   • C$-\frac{x^2}{y^2}$
   • D$-x^2$

   (b)

   Find the gradient of the normal to $C_1$ at $P$.

   [1 mark]

   • A$4$
   • B$-\frac14$
   • C$\frac14$
   • D$-4$

   (c)

   Find an equation of the tangent to $C_1$ at $P$, in the form $x+by=c$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C_2$ has parametric equations $x=t^2+1,\; y=t^3-3t$, where $t$ is a real parameter.

   (a)

   Find $\frac{dy}{dx}$ in terms of $t$.

   [1 mark]

   • A$\frac{2t}{3t^2-3}$
   • B$\frac{3t^2-3}{2t}$
   • C$3t^2-3$
   • D$\frac{3t^2-3}{2t+1}$

   (b)

   Find the gradient of the tangent to $C_2$ at the point where $t=2$.

   [1 mark]

   • A$\frac49$
   • B$\frac92$
   • C$\frac{9}{5}$
   • D$\frac94$

   (c)

   Find the coordinates of the points on $C_2$ where the tangent is parallel to the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C_3$ has parametric equations $x=3\sin t,\; y=2\cos 2t$, for $0\le t<2\pi$.

   (a)

   Find $\frac{dy}{dx}$ in terms of $t$, giving your answer in its simplest form.

   [3 marks]

   (b)

   Find an equation of the normal to $C_3$ at the point where $t=\frac{\pi}{6}$, giving your answer in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C_4$ has equation $x^2+2xy+2y^2=5$, and the point $P(1,1)$ lies on $C_4$.

   (a)

   Show that $\frac{dy}{dx}=-\frac{x+y}{x+2y}$, and hence find an equation of the normal to $C_4$ at $P$ in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [6 marks]

   (b)

   Find the exact coordinates of the points on $C_4$ where the tangent to the curve is (i) parallel to the $x$-axis, (ii) parallel to the $y$-axis.

   [6 marks]

   Total for question 4: 12 marks

End of questions