Further Pure Mathematics 2 (for Paper 2): past paper questions

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• Question 2Oct/Nov 2023, paper 21Structured5 marks

  Find the roots of the equation $(z+5\mathrm{i}{)}^3=4+4\sqrt{3}\mathrm{i}$, giving your answers in the form $r\cos \theta +\mathrm{i}(r\sin \theta -5)$, where $r>0$ and $0<\theta <2\pi$.

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• Question 2May/June 2026, paper 21Structured7 marks

  Let $I_n={\int }_0^1(1+t{)}^n\sin \pi t\mathrm{d}t$, where $n$ is a non-negative integer. Show that, for $n⩾2$, ${\pi }^2{I}_n=\pi (2^n+1)-n(n-1)I_{n-2}$.

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• Question 3Oct/Nov 2025, paper 21Structured9 marks

  The integral $I_n$ is defined by $I_n={\int }_0^1(1+x^2{)}^n\mathrm{d}x$. By considering $\frac{\mathrm{d}}{\mathrm{d}x}(x(1+x^2{)}^n)$, or otherwise, show that $$(2n+1)I_n=2^n+2n{I}_{n-1}\mathrm{.}$$ [5]

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• Question 4Oct/Nov 2025, paper 21Structured10 marks

  Find the particular solution of the differential equation $$5\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+2\frac{\mathrm{d}y}{\mathrm{d}x}+y=x^2+5x+3,$$ given that, when $x=0$, $y=\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

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• Question 4Oct/Nov 2023, paper 21Structured10 marks

  Find the particular solution of the differential equation $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+2\frac{\mathrm{d}y}{\mathrm{d}x}+3y=27x^2,$$ given that, when $x=0$, $y=2$ and $\frac{\mathrm{d}y}{\mathrm{d}x}=-8$.

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• Question 5Oct/Nov 2024, paper 21Structured10 marks

  Find the particular solution of the differential equation $$6\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+x=t^2+t+1,$$ given that, when $t=0$, $x=12$ and $\frac{\mathrm{d}x}{\mathrm{d}t}=-6$.

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• Question 7Oct/Nov 2024, paper 21Structured10 marks

  Show that an appropriate integrating factor for $$\sqrt{x^2+16}\frac{\mathrm{d}y}{\mathrm{d}x}+y=x\sqrt{x^2+16}$$ is $\frac{1}{4}x+\frac{1}{4}\sqrt{x^2+16}$.

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• Question 7May/June 2026, paper 21Structured10 marks

  State the sum of the series $z+z^2+z^3+\dots +z^n$, for $z\mathrm{\ne }1$.

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• Question 8Oct/Nov 2025, paper 21Structured12 marks

  State the sum of the series $z+z^2+\dots +z^n$, for $z\mathrm{\ne }1$.

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• Question 8Oct/Nov 2024, paper 21Structured14 marks

  By considering the binomial expansion of (z + 1/z )⁷, where z = cosθ + i sinθ, use de Moivre's theorem to show that cos⁷θ = a cos 7θ + b cos 5θ + c cos 3θ + d…

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• Question 8Oct/Nov 2023, paper 21Structured15 marksHas a diagram

  State the sum of the series $1+z+z^2+\dots +z^{n-1}$, for $z\mathrm{\ne }1$.

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• Question 1Oct/Nov 2024, paper 22Structured4 marks

  Find the value of ${\int }_6^7\frac{1}{\sqrt{(x-5{)}^2-1}}\mathrm{d}x$, giving your answer in the form $\ln (a+\sqrt{b})$, where $a$ and $b$ are integers to be determined.

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• Question 1Oct/Nov 2023, paper 22Structured5 marks

  Find the Maclaurin’s series for $\ln (x+2)+\ln (x^2+5)$ up to and including the term in $x^2$.

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• Question 2Oct/Nov 2024, paper 22Structured7 marks

  The curve $C$ has equation $$4y^2+4\ln (xy)=1.$$ Show that, at the point $(2,\frac{1}{2})$ on $C$, $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{6}$.

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• Question 2May/June 2026, paper 22Structured7 marks

  Let $I_n={\int }_0^1(1+t{)}^n\sin \pi t\mathrm{d}t$, where $n$ is a non-negative integer. Show that, for $n⩾2$, ${\pi }^2{I}_n=\pi (2^n+1)-n(n-1)I_{n-2}$.

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• Question 3Oct/Nov 2025, paper 22Structured9 marks

  The variables $x$ and $y$ are such that $y=2$ when $x=-1$ and $$x^{2}y+(x+y{)}^3=3.$$ Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{4}$ when $x=-1$.

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• Question 4Oct/Nov 2025, paper 22Structured8 marks

  By considering the binomial expansion of (z + 1/z )⁶, where z = cosθ + i sinθ, use de Moivre's theorem to show that cos⁶θ = a cos 6θ + b cos 4θ + c cos 2θ + d…

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• Question 5Oct/Nov 2025, paper 22Structured10 marks

  Find the general solution of the differential equation $$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+4\frac{\mathrm{d}y}{\mathrm{d}x}+3y=5\cos x\mathrm{.}$$

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• Question 5Oct/Nov 2024, paper 22Structured10 marksHas a diagram

  Find the particular solution of the differential equation $$3\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}+2\frac{\mathrm{d}y}{\mathrm{d}x}+y=x^2,$$ given that, when $x=0$, $y=\frac{\mathrm{d}y}{\mathrm{d}x}=0$.

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• Question 7Oct/Nov 2025, paper 22Structured11 marks

  Show that $\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}x\sqrt{4-x^2}+2{\sin }^{-1}\left(\frac{1}{2}x\right))=\sqrt{4-x^2}$.

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• Question 3Oct/Nov 2025, paper 23Structured9 marks

  The integral $I_n$ is defined by $I_n={\int }_0^1(1+x^2{)}^n\mathrm{d}x$. By considering $\frac{\mathrm{d}}{\mathrm{d}x}(x(1+x^2{)}^n)$, or otherwise, show that $$(2n+1)I_n=2^n+2n{I}_{n-1}\mathrm{.}$$

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• Question 3May/June 2026, paper 23Structured6 marks

  Let $I_n={\int }_0^a(\ln (x+1){)}^n\mathrm{d}x$, where $n$ is a non-negative integer and $a>0$. By considering the derivative of $(x+1)(\ln (x+1){)}^n$, or otherwise, show that for $n⩾1$ $$I_n=(a+1)(\ln (a+1){)}^n-n{I}_{n-1}\mathrm{.}$$

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• Question 5Oct/Nov 2024, paper 23Structured10 marks

  Find the particular solution of the differential equation $$6\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+x=t^2+t+1,$$ given that, when $t=0$, $x=12$ and $\frac{\mathrm{d}x}{\mathrm{d}t}=-6$.

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• Question 4May/June 2026, paper 23Structured10 marks

  Find the particular solution of the differential equation $$49\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+14\frac{\mathrm{d}x}{\mathrm{d}t}+x=196{\mathrm{e}}^{\frac{1}{7}t},$$ given that, when $t=0$, $x=7$ and $\frac{\mathrm{d}x}{\mathrm{d}t}=0$.

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• Question 6May/June 2026, paper 23Structured10 marks

  Show that an appropriate integrating factor for $$(1-x^2)\frac{\mathrm{d}y}{\mathrm{d}x}+y=(1-x^2)\sqrt{1-x}$$ is $\sqrt{\frac{1+x}{1-x}}$.

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• Question 7May/June 2026, paper 23Structured12 marksHas a diagram

  Use the substitution $u=1-x^2$ to find $\int \frac{x}{\sqrt{1-x^2}}\mathrm{d}x$.

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