Implicit and parametric differentiationAQA A-Level Maths: Subtopic test

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

   (a)

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

   [1 mark]

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

   (b)

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

   [1 mark]

   • A$\frac34$
   • B$-\frac43$
   • C$\frac23$
   • D$\frac43$

   (c)

   Find the coordinates of the points on the curve where the tangent is parallel to the $y$-axis.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $xy^2+x^2=6$ and passes through the point $(2,1)$.

   (a)

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

   [1 mark]

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

   (b)

   Find the gradient of the curve at $(2,1)$.

   [1 mark]

   • A$\frac54$
   • B$-1$
   • C$-\frac54$
   • D$-\frac52$

   (c)

   Find the equation of the tangent to the curve at $(2,1)$, in the form $ax+by=c$ with integer coefficients.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve is defined by the parametric equations $x=t^2+2t$ and $y=t^3-3t$, where $t$ is a real parameter.

   (a)

   Show that $\frac{dy}{dx}=\frac{3(t-1)}{2}$ for $t\ne-1$.

   [3 marks]

   (b)

   Find the equation of the tangent to the curve at the point where $t=2$, in the form $ay=bx+c$ with integer coefficients.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has parametric equations $x=4\cos t$ and $y=3\sin t$, for $0\le t<2\pi$.

   (a)

   (i) Find $\frac{dy}{dx}$ in terms of $t$. (ii) Find the equation of the tangent to $C$ at the point where $t=\frac{\pi}{3}$, in the form $ax+by=c$, giving exact values.

   [6 marks]

   (b)

   (i) Show that a Cartesian equation of $C$ is $\frac{x^2}{16}+\frac{y^2}{9}=1$. (ii) Use implicit differentiation to find $\frac{dy}{dx}$ in terms of $x$ and $y$. (iii) Find the coordinates of the points on $C$ where the gradient is $-1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions