Differentiating inverse functions and implicit relationshipsEdexcel International A Level Maths: Subtopic test

1. 1
   A curve $C$ has equation $x=\sin 3y$ for $-\frac{\pi}{6}<y<\frac{\pi}{6}$.

   (a)

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

   [1 mark]

   • A$\cos 3y$
   • B$-3\sin 3y$
   • C$3\cos 3y$
   • D$\frac13\cos 3y$

   (b)

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

   [1 mark]

   • A$\frac{1}{3\cos 3y}$
   • B$3\cos 3y$
   • C$\frac{1}{\cos 3y}$
   • D$-\frac{1}{3\sin 3y}$

   (c)

   Find the exact value of $\frac{dy}{dx}$ at the point where $y=\frac{\pi}{18}$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

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

   [1 mark]

   • A$e^{2y}+1$
   • B$2e^{2y}+1$
   • C$2e^{2y}$
   • D$2ye^{2y-1}+1$

   (b)

   Find the gradient of $C$ at the point where $y=0$.

   [1 mark]

   • A$3$
   • B$\frac12$
   • C$1$
   • D$\frac13$

   (c)

   Find an equation of the tangent to $C$ at the point where $y=0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A curve $C$ has equation $x=\tan 2y$ for $-\frac{\pi}{4}<y<\frac{\pi}{4}$.

   (a)

   Show that $\frac{dy}{dx}=\frac{1}{2(1+x^2)}$.

   [3 marks]

   (b)

   Find an equation of the normal to $C$ at the point where $x=\sqrt3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has equation $x=\ln(2y+1)$ for $y>-\frac12$. A calculator may be used.

   (a)

   (i) Find $\frac{dy}{dx}$ in terms of $y$.
   (ii) By making $y$ the subject of the equation, find $\frac{dy}{dx}$ in terms of $x$, and show that it agrees with your answer to (i).

   [6 marks]

   (b)

   The tangent to $C$ at the point where $x=\ln5$ meets the coordinate axes at $A$ and $B$. Find the area of triangle $OAB$, where $O$ is the origin.

   [6 marks]

   Total for question 4: 12 marks

End of questions