Further Pure Mathematics 1 (for Paper 1): past paper questions

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• Question 1Oct/Nov 2024, paper 11Structured10 marks

  The matrix $\mathbf{M}$ represents the sequence of two transformations in the $x$-$y$ plane given by a stretch parallel to the $x$-axis, scale factor $k$ ($k\mathrm{\ne }0$), followed by a shear, $x$-axis fixed, with $(0,1)$ mapped to $(k,1)$.

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• Question 2May/June 2026, paper 11Structured6 marks

  Prove by mathematical induction that $3^{6n}+{92}^n+89$ is divisible by 91 for all positive integers $n$.

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• Question 3May/June 2026, paper 11Structured9 marks

  The matrix $\mathbf{M}$ represents the sequence of two transformations in the $x$-$y$ plane given by a stretch parallel to the $x$-axis, scale factor $k$ ($k\mathrm{\ne }0,1$), followed by a shear, $x$-axis fixed, with $(0,1)$ mapped to $(2,1)$.

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• Question 3Oct/Nov 2024, paper 11Structured10 marks

  The quartic equation $x^4+2x^3-1=0$ has roots $\alpha ,\beta ,\gamma ,\delta$. Find a quartic equation whose roots are ${\alpha }^4,{\beta }^4,{\gamma }^4,{\delta }^4$ and state the value of ${\alpha }^4+{\beta }^4+{\gamma }^4+{\delta }^4$.

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• Question 4May/June 2026, paper 11Structured10 marks

  The cubic equation $5x^3-2x+1=0$ has roots $\alpha ,\beta ,\gamma$. Find a cubic equation whose roots are $\frac{1}{{\alpha }^2},\frac{1}{{\beta }^2},\frac{1}{{\gamma }^2}$.

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• Question 4Oct/Nov 2025, paper 11Structured8 marks

  The quartic equation $x^4+x^3+x^2+x+1=0$ has roots $\alpha ,\beta ,\gamma ,\delta$. Show that a quartic equation with roots $2\alpha +1,2\beta +1,2\gamma +1,2\delta +1$ is $$y^4-2y^3+4y^2+2y+11=0.$$

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• Question 5May/June 2026, paper 11Structured11 marks

  The lines $l_1$ and $l_2$ have equations $\mathbf{r}=-2\mathbf{i}+4\mathbf{j}+4\mathbf{k}+\lambda (3\mathbf{i}-4\mathbf{j}+2\mathbf{k})$ and $\mathbf{r}=-3\mathbf{i}-2\mathbf{j}+4\mathbf{k}+\mu (2\mathbf{i}+4\mathbf{j}+3\mathbf{k})$ respectively.

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• Question 5Oct/Nov 2025, paper 11Structured11 marks

  The plane ${\mathrm{\Pi }}_1$ has equation $\mathbf{r}=-3\mathbf{i}-\mathbf{j}-\mathbf{k}+\lambda (\mathbf{j}+2\mathbf{k})+\mu (\mathbf{i}+3\mathbf{j}+\mathbf{k})$. Find an equation for ${\mathrm{\Pi }}_1$ in the form $ax+by+cz=d$.

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• Question 5Oct/Nov 2023, paper 11Structured15 marks

  Let $k$ be a constant.

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• Question 6May/June 2026, paper 11Structured14 marks

  The curve $C$ has equation $y=\frac{x^2-x+2}{4x+4}$. Find the equations of the asymptotes of $C$.

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• Question 6Oct/Nov 2024, paper 11Structured13 marks

  The curve $C$ has equation $y=\frac{4x^2+x+1}{2x^2-7x+3}$. Find the equations of the asymptotes of $C$.

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• Question 7Oct/Nov 2025, paper 11Structured14 marks

  The curve $C$ has equation $y=\frac{x+2}{x^2+3x+1}$. Find the equations of the asymptotes of $C$.

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• Question 7Oct/Nov 2024, paper 11Structured15 marks

  The lines $l_1$ and $l_2$ have equations $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\lambda (2\mathbf{i}+\mathbf{j}+\mathbf{k})$ and $\mathbf{r}=\mathbf{i}-2\mathbf{j}+9\mathbf{k}+\mu (\mathbf{i}-4\mathbf{j}+2\mathbf{k})$ respectively.

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• Question 7Oct/Nov 2023, paper 11Structured15 marks

  The curve $C$ has equation $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=\frac{x^2+2}{x^2-x-2}$. Find the equations of the asymptotes of $C$.

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• Question 2May/June 2026, paper 12Structured6 marks

  Prove by mathematical induction that $3^{6n}+{92}^n+89$ is divisible by 91 for all positive integers $n$.

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• Question 2Oct/Nov 2025, paper 12Structured8 marks

  The cubic equation $x^3+b{x}^2+cx+d=0$, where $b$, $c$ and $d$ are constants, has roots $\alpha$, $\beta$ and $\gamma$.

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• Question 2Oct/Nov 2024, paper 12Structured7 marks

  The line $l_1$ has equation $\mathbf{r}=\mathbf{i}+3\mathbf{j}-\mathbf{k}+\lambda (\mathbf{i}-\mathbf{j}-4\mathbf{k})$. The plane $\mathrm{\Pi }$ contains $l_1$ and is parallel to the vector $2\mathbf{i}+5\mathbf{j}-4\mathbf{k}$.

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• Question 2Oct/Nov 2023, paper 12Structured6 marks

  Prove by mathematical induction that, for all positive integers n, (dⁿ)/(dxⁿ)(x² e^x) = (x² + 2nx + n(n-1))e^x.

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• Question 4Oct/Nov 2025, paper 12Structured10 marks

  Let $k$ and $m$ be non-zero constants.

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• Question 4Oct/Nov 2023, paper 12Structured10 marks

  The cubic equation $27x^3+18x^2+6x-1=0$ has roots $\alpha$, $\beta$, $\gamma$. Show that a cubic equation with roots $3\alpha +1$, $3\beta +1$, $3\gamma +1$ is $$y^3-y^2+y-2=0.$$

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• Question 5Oct/Nov 2023, paper 12Structured13 marks

  The plane ${\mathrm{\Pi }}_1$ has equation $\mathbf{r}=\mathbf{i}-\mathbf{j}-2\mathbf{k}+\lambda (\mathbf{i}-2\mathbf{j}-3\mathbf{k})+\mu (3\mathbf{i}-\mathbf{k})$. Find an equation for ${\mathrm{\Pi }}_1$ in the form $ax+by+cz=d$.

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• Question 6May/June 2026, paper 12Structured14 marks

  The curve $C$ has equation $y=\frac{x^2-x+2}{4x+4}$. Find the equations of the asymptotes of $C$.

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• Question 6Oct/Nov 2025, paper 12Structured14 marks

  The plane $\mathrm{\Pi }$ has equation $x+3y+2z=1$. Find the perpendicular distance from the origin $O$ to the plane $\mathrm{\Pi }$.

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• Question 6Oct/Nov 2024, paper 12Structured15 marks

  The curve $C$ has equation $y=\frac{x^2+3}{x^2+1}$. Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote.

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• Question 7Oct/Nov 2025, paper 12Structured14 marks

  The curve $C$ has equation $y=\frac{x^2+x+1}{x+1}$. Find the equations of the asymptotes of $C$.

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• Question 7Oct/Nov 2023, paper 12Structured16 marks

  The curve $C$ has equation $y=\mathrm{f}(x)$, where $\mathrm{f}(x)=\frac{x^2}{x+1}$. Find the equations of the asymptotes of $C$.

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• Question 2Oct/Nov 2024, paper 13Structured6 marks

  Prove by mathematical induction that, for all positive integers $n$, $$\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}({\tan }^{-1}x)=P_n(x)(1+x^2{)}^{-n},$$ where $P_n(x)$ is a polynomial of degree $n-1$.

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• Question 5Oct/Nov 2024, paper 13Structured13 marks

  Show that the curve with Cartesian equation $$(x^2+y^2{)}^2=6xy$$ has polar equation $r^2=3\sin 2\theta$.

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• Question 6May/June 2026, paper 13Structured14 marks

  The curve $C$ has equation $y=\frac{x+1}{x^2+x+1}$. Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote.

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• Question 6Oct/Nov 2024, paper 13Structured13 marks

  The curve $C$ has equation $y=\frac{4x^2+x+1}{2x^2-7x+3}$. Find the equations of the asymptotes of $C$.

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