Inverse trigonometric functions: differentiation and integrationAQA A-Level Further Maths: Subtopic test

1. 1
   Let $\mathrm{f}(x)=\arctan(3x)$.

   (a)

   Find $\mathrm{f}'(x)$.

   [1 mark]

   • A$\dfrac{1}{1+9x^2}$
   • B$\dfrac{3}{1+3x^2}$
   • C$\dfrac{3}{1+9x^2}$
   • D$\dfrac{1}{3(1+x^2)}$

   (b)

   Find the value of $\mathrm{f}'\left(\dfrac13\right)$.

   [1 mark]

   • A$\dfrac32$
   • B$\dfrac12$
   • C$\dfrac94$
   • D$3$

   (c)

   Find the equation of the tangent to the curve $y=\mathrm{f}(x)$ at the point where $x=\dfrac13$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\arccos\left(\dfrac{x}{2}\right)$ for $-2\le x\le2$.

   (a)

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

   [1 mark]

   • A$\dfrac{1}{\sqrt{4-x^2}}$
   • B$-\dfrac{1}{\sqrt{1-\frac{x^2}{4}}}$
   • C$-\dfrac{1}{2\sqrt{4-x^2}}$
   • D$-\dfrac{1}{\sqrt{4-x^2}}$

   (b)

   Find the gradient of $C$ at the point where $x=1$.

   [1 mark]

   • A$\dfrac{1}{\sqrt{3}}$
   • B$-\dfrac{1}{\sqrt{3}}$
   • C$-\dfrac{2}{\sqrt{3}}$
   • D$-\dfrac{1}{\sqrt{5}}$

   (c)

   Find the values of $x$ for which the gradient of $C$ is undefined, and describe the tangent to $C$ at these points.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $I=\int_0^1\dfrac{1}{\sqrt{4-x^2}}\,dx$ and $J=\int_0^1\sqrt{4-x^2}\,dx$.

   (a)

   Find the exact value of $I$.

   [3 marks]

   (b)

   Use the substitution $x=2\sin\theta$ to find the exact value of $J$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(x)=\dfrac{1}{x^2-4x+13}$.

   (a)

   (i) Express $x^2-4x+13$ in the form $(x-a)^2+b$.
   (ii) Hence find the exact value of $\int_2^5\mathrm{f}(x)\,dx$.

   [6 marks]

   (b)

   Hence find the exact value of $\int_2^5\dfrac{x+1}{x^2-4x+13}\,dx$.

   [6 marks]

   Total for question 4: 12 marks

End of questions