Implicit and parametric differentiationEdexcel A-Level Maths: Subtopic test

1. 1
   A curve has equation $x^{2}+y^{2}-4x+6y=12$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

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

   (b)

   The point $(5,1)$ lies on the curve. Find the gradient of the curve at $(5,1)$.

   [1 mark]

   • A$\frac34$
   • B$-\frac34$
   • C$-\frac43$
   • D$-3$

   (c)

   The point $(5,1)$ lies on the curve. Find an equation of the normal to the curve at $(5,1)$, in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has equation $x^{2}+xy+y^{2}=7$.

   (a)

   Find $\frac{dy}{dx}$.

   [1 mark]

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

   (b)

   The point $(1,2)$ lies on $C$. Find the gradient of $C$ at $(1,2)$.

   [1 mark]

   • A$-\frac45$
   • B$\frac45$
   • C$-\frac54$
   • D$-1$

   (c)

   Find the coordinates of the points on $C$ where $\frac{dy}{dx}=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve is given by the parametric equations $x=t^{2}+1$, $y=t^{3}-3t$, where $t$ is a real parameter.

   (a)

   Find $\frac{dy}{dx}$ in terms of $t$, and hence find the gradient of the curve at the point where $t=2$.

   [3 marks]

   (b)

   Find an equation of the normal to the curve at the point where $t=2$, 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 $C$ has parametric equations $x=2\sin t$, $y=3\cos2t$ for $0\le t\le\pi$. The point $P$ on $C$ has parameter $t=\frac\pi6$.

   (a)

   Show that the tangent to $C$ at $P$ has equation $y=-3x+\frac92$.

   [6 marks]

   (b)

   The tangent and the normal to $C$ at $P$ meet the $x$-axis at $T$ and $N$ respectively. Find the area of triangle $PTN$.

   [6 marks]

   Total for question 4: 12 marks

End of questions