Further Pure 2: Further calculusEdexcel A-Level Further Maths: Topic test

1. 1
   For integers $n\geq0$, let $L_n=\int_1^e(\ln x)^n\,dx$.

   (a)

   What is the value of $L_0$?

   [1 mark]

   • A$1$
   • B$e$
   • C$e-1$
   • D$0$

   (b)

   What is the value of $L_1$?

   [1 mark]

   • A$1$
   • B$e-1$
   • C$e$
   • D$0$

   (c)

   Show that $L_n=e-nL_{n-1}$ for $n\geq1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $C$ has equation $y=\frac{x^2}{4}-\frac12\ln x$ for $1\leq x\leq2$.

   (a)

   What is $\frac{dy}{dx}$?

   [1 mark]

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

   (b)

   Which expression is equal to $1+\left(\frac{dy}{dx}\right)^2$?

   [1 mark]

   • A$\left(\frac{x}{2}-\frac{1}{2x}\right)^2$
   • B$\left(\frac{x}{2}+\frac{1}{2x}\right)^2$
   • C$\frac{x^2}{4}+\frac{1}{4x^2}$
   • D$1+\frac{x^2}{4}+\frac{1}{4x^2}$

   (c)

   Find the exact length of $C$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A spiral has polar equation $r=e^{2\theta}$ for $0\leq\theta\leq1$.

   (a)

   Show that $r^2+\left(\frac{dr}{d\theta}\right)^2=5e^{4\theta}$.

   [3 marks]

   (b)

   Hence find the exact length of the spiral.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For integers $n\geq0$, let $C_n=\int_0^1\cosh^nx\,dx$. The curve $y=\cosh x$ for $0\leq x\leq1$ is rotated through $2\pi$ radians about the $x$-axis.

   (a)

   Show that, for $n\geq2$, $nC_n=\sinh1\cosh^{n-1}1+(n-1)C_{n-2}$.

   [6 marks]

   (b)

   Use the result in part (a) to find the exact area of the curved surface formed.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   For integers $n\geq0$, let $Q_n=\int_0^1x^n\sqrt{1-x}\,dx$. It is given that $Q_n=\frac{2n}{2n+3}Q_{n-1}$ for $n\geq1$.

   (a)

   What is the value of $Q_0$?

   [1 mark]

   • A$\frac13$
   • B$\frac23$
   • C$\frac32$
   • D$1$

   (b)

   What is the value of $Q_1$?

   [1 mark]

   • A$\frac25$
   • B$\frac23$
   • C$\frac49$
   • D$\frac{4}{15}$

   (c)

   Find the exact value of $Q_3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve has parametric equations $x=t-\sin t$, $y=1-\cos t$ for $0\leq t\leq2\pi$.

   (a)

   Which expression is equal to $\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2$?

   [1 mark]

   • A$2-2\cos t$
   • B$2+2\cos t$
   • C$1$
   • D$2-\cos t$

   (b)

   Which expression is equal to $\sqrt{2-2\cos t}$ for $0\leq t\leq2\pi$?

   [1 mark]

   • A$2\cos\frac t2$
   • B$4\sin^2\frac t2$
   • C$2\sin\frac t2$
   • D$\sin t$

   (c)

   Find the exact length of the curve.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   For integers $n\geq0$, let $E_n=\int_0^{\frac{\pi}{4}}\sec^nx\,dx$.

   (a)

   Use integration by parts to show that, for $n\geq2$, $E_n=\left[\sec^{n-2}x\tan x\right]_0^{\frac{\pi}{4}}-(n-2)\int_0^{\frac{\pi}{4}}\sec^{n-2}x\tan^2x\,dx$.

   [3 marks]

   (b)

   Hence show that $(n-1)E_n=2^{\frac{n-2}{2}}+(n-2)E_{n-2}$ for $n\geq2$, and find the exact value of $E_4$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The astroid arc $C$ has parametric equations $x=\cos^3t$, $y=\sin^3t$ for $0\leq t\leq\frac{\pi}{2}$.

   (a)

   Find the exact length of $C$.

   [6 marks]

   (b)

   The arc $C$ is rotated through $2\pi$ radians about the $x$-axis. Find the exact area of the surface generated.

   [6 marks]

   Total for question 8: 12 marks

End of questions