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

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• Question 1May/June 2026, paper 21Structured4 marks

  The variables $x$ and $y$ satisfy the equation $3^{x+y}=2^{4x+1}$. Using natural logarithms, the equation can be transformed to the form $y=mx+c$. Determine the exact values of $m$ and $c$.

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• Question 1Oct/Nov 2025, paper 21Structured3 marks

  Find $\int 6{\sin }^{2}x\mathrm{d}x$.

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• Question 1Oct/Nov 2024, paper 21Structured5 marks

  The variables $x$ and $y$ satisfy the equation $a^{2y}={\mathrm{e}}^{3x+k}$, where $a$ and $k$ are constants. The graph of $y$ against $x$ is a straight line. Use logarithms to show that the gradient of the straight line is $\frac{3}{2\ln a}$.

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• Question 2May/June 2026, paper 21Structured6 marks

  Solve the equation $\mathrm{\mid }3x+7\mathrm{\mid }=\mathrm{\mid }4x-1\mathrm{\mid }$.

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• Question 2Oct/Nov 2025, paper 21Structured3 marks

  Solve the equation ${\mathrm{e}}^{2x}({\mathrm{e}}^{2x}-8)=48$.

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• Question 2Oct/Nov 2024, paper 21Structured4 marks

  Solve the inequality $\mathrm{\mid }x-7\mathrm{\mid }>4x+3$.

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• Question 3Oct/Nov 2025, paper 21Structured6 marks

  Solve the equation $\mathrm{\mid }2x-3\mathrm{\mid }=\mathrm{\mid }5x+2\mathrm{\mid }$.

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• Question 3Oct/Nov 2023, paper 21Structured6 marks

  Find ${\int }_4^{10}\frac{4}{2x-5}\mathrm{d}x$, giving your answer in the form $\ln a$, where $a$ is an integer.

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• Question 4May/June 2026, paper 21Structured7 marks

  Express $5\cos \theta +3\sin \theta$ in the form $R\cos (\theta -\alpha )$, where $R>0$ and $0^{\circ }<\alpha <{90}^{\circ }$.

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• Question 4Oct/Nov 2024, paper 21Structured8 marks

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=a{x}^3-a{x}^2-15x+18,$$ where $a$ is a constant. It is given that $(x+2)$ is a factor of $\mathrm{p}(x)$. Find the value of $a$.

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• Question 4Oct/Nov 2023, paper 21Structured7 marks

  Sketch, on the same diagram, the graphs of $y=\mathrm{\mid }3x-5\mathrm{\mid }$ and $y=2x+7$.

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• Question 5May/June 2026, paper 21Structured10 marksHas a diagram

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=2x^4+3x^3+4x^2-8x\mathrm{.}$$ Find the quotient when $\mathrm{p}(x)$ is divided by $(2x+3)$, and show that the remainder is 21.

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• Question 5Oct/Nov 2025, paper 21Structured7 marks

  The diagram shows the curve with equation $y=8{\mathrm{e}}^{-\frac{1}{2}x}-1$. The curve meets the axes at the points $A$ and $B$. The shaded region is bounded by the curve and the line segment $AB$.

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• Question 5Oct/Nov 2024, paper 21Structured8 marks

  It is given that ${\int }_a^{a^3}\frac{10}{2x+1}\mathrm{d}x=7$, where $a$ is a constant greater than 1. Show that $a=\sqrt[3]{0.5{\mathrm{e}}^{1.4}(2a+1)-0.5}$.

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• Question 7Oct/Nov 2025, paper 21Structured11 marks

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=2x^4+k{x}^3+k{x}^2+17x+18,$$ where $k$ is a constant. It is given that $(x+2)$ is a factor of $\mathrm{p}(x)$. Find the value of $k$.

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• Question 7Oct/Nov 2023, paper 21Structured11 marks

  The curve with equation ${\mathrm{e}}^{2x}-18x+y^3+y=11$ has a stationary point at $(p,q)$. Find the exact value of $p$.

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• Question 8Oct/Nov 2025, paper 21Structured6 marks

  The equation of a curve is $y=4{\mathrm{e}}^{1-2x}\sqrt{3x-1}$. Find the exact coordinates of the stationary point of the curve.

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• Question 1Oct/Nov 2024, paper 22Structured3 marks

  Use logarithms to show that the equation $5^{8y}=6^{7x}$ can be expressed in the form $y=kx$.

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• Question 1Oct/Nov 2023, paper 22Structured2 marks

  When the polynomial $$a{x}^3+4a{x}^2-7x-5$$ is divided by $(x+2)$, the remainder is 33. Find the value of the constant $a$.

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• Question 1Oct/Nov 2025, paper 22Structured4 marks

  Solve the equation $\ln (3x+5)-\ln (x-2)=4$.

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• Question 2Oct/Nov 2024, paper 22Structured6 marks

  Let $\mathrm{f}(x)=4{\sin }^{2}3x$. Find the value of ${\mathrm{f}}^{\mathrm{\prime }}\left(\frac{1}{4}\pi \right)$.

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• Question 3May/June 2026, paper 22Structured5 marksHas a diagram

  The variables $x$ and $y$ satisfy the equation $y=k{a}^{x+2}$, where $k$ and $a$ are positive constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(1.5,3.22)$ and $(4.0,4.69)$, as shown in the diagram.

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• Question 3Oct/Nov 2024, paper 22Structured6 marks

  A curve has equation $6{\mathrm{e}}^{-x}{y}^2+{\mathrm{e}}^{2x}-12y+7=0$. Find the gradient of the curve at the point $(\ln 3,2)$.

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• Question 3Oct/Nov 2023, paper 22Structured5 marksHas a diagram

  The diagram shows the curve with equation $y=6{\mathrm{e}}^{-\frac{1}{2}x}$. The points on the curve with $x$-coordinates 0 and 2 are denoted by $A$ and $B$ respectively.

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• Question 4Oct/Nov 2024, paper 22Structured7 marks

  Sketch the graphs of $y=1+{\mathrm{e}}^{2x}$ and $y=\mathrm{\mid }x-4\mathrm{\mid }$ on the same diagram.

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• Question 4Oct/Nov 2023, paper 22Structured9 marks

  Sketch, on the same diagram, the graphs of $y=\mathrm{\mid }3-x\mathrm{\mid }$ and $y=9-2x$.

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

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=a{x}^3+b{x}^2-ax+8,$$ where $a$ and $b$ are constants. It is given that $(x+2)$ is a factor of $\mathrm{p}(x)$, and that the remainder is 24 when $\mathrm{p}(x)$ is divided by $(x-2)$.

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• Question 5Oct/Nov 2023, paper 22Structured8 marks

  Find the quotient when $6x^3-5x^2-24x-4$ is divided by $(2x+1)$, and show that the remainder is 6.

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• Question 6May/June 2026, paper 22Structured8 marksHas a diagram

  The diagram shows the curve with equation $y=\mathrm{f}(x)$, where $$\mathrm{f}(x)=a{x}^4+4x^3-3a{x}^2-21x-18$$ and $a$ is a constant. It is given that $(x+2)$ is a factor of $\mathrm{f}(x)$. Find the value of $a$.

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• Question 7May/June 2026, paper 22Structured7 marksHas a diagram

  The diagram shows part of the curve with equation $y=8\cos 2x-\frac{1}{{\cos }^{2}2x}$. The shaded region is bounded by the curve and the $x$-axis. Find the exact area of the shaded region.

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