Pure Mathematics 3 (for Paper 3): past paper questions

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• Question 1Oct/Nov 2025, paper 31Structured4 marks

  Find the exact value of ${\int }_1^2\ln 3x\mathrm{d}x$.

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• Question 1Oct/Nov 2024, paper 31Structured5 marks

  The polynomial $4x^3+a{x}^2+5x+b$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(2x+1)$ is a factor of $\mathrm{p}(x)$.

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• Question 1Oct/Nov 2023, paper 31Structured5 marks

  Find the exact coordinates of the points on the curve $y=\frac{x^2}{1-3x}$ at which the gradient of the tangent is equal to 8.

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• Question 2May/June 2026, paper 31Structured5 marks

  The variables $x$ and $y$ satisfy the equation $y^3=A{\mathrm{e}}^{kx}$, where $A$ and $k$ are constants. The graph of $\ln y$ against $x$ is a straight line passing through the points $(-1.32,5.09)$ and $(0.25,1.16)$. Find the values of $A$ and $k$.

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• Question 2Oct/Nov 2025, paper 31Structured5 marks

  Show that the equation ${\log }_4(2x+1)=2{\log }_4(3x-1)-2$ can be written as a quadratic equation in $x$.

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• Question 2Oct/Nov 2024, paper 31Structured5 marks

  Find the exact value of ${\int }_1^3{x}^2\ln 3x\mathrm{d}x$.

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• Question 3May/June 2026, paper 31Structured6 marks

  The equation of a curve is $y=\frac{2x^2+7x}{3x^2-1}$. Find an equation of the normal to the curve at the point where $x=1$.

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• Question 3Oct/Nov 2025, paper 31Structured8 marksHas a diagram

  Express $3\sqrt{2}\sin (x+{45}^{\circ })+\cos x$ in the form $R\cos (x-\alpha )$, where $R>0$ and $0^{\circ }<\alpha <{90}^{\circ }$.

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• Question 3Oct/Nov 2024, paper 31Structured4 marks

  The equation of a curve is $\ln (x+y)=3x^{2}y$. Find the gradient of the curve at the point $(1,0)$.

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• Question 3Oct/Nov 2023, paper 31Structured4 marks

  The variables $x$ and $y$ are related by the equation $y=a{b}^x$, where $a$ and $b$ are constants. The diagram shows the result of plotting $\ln y$ against $x$ for two pairs of values of $x$ and $y$. The coordinates of these points are $(1,3.7)$ and $(2.2,6.46)$.

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• Question 4May/June 2026, paper 31Structured5 marks

  The square roots of $-71-(20\sqrt{3})\mathrm{i}$ can be expressed in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are real and exact. By first forming a quartic equation, find the square roots of $-71-(20\sqrt{3})\mathrm{i}$.

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• Question 4Oct/Nov 2023, paper 31Structured5 marks

  The complex number $u$ is defined by $u=\frac{3+2\mathrm{i}}{a-5\mathrm{i}}$, where $a$ is real. Express $u$ in the Cartesian form $x+\mathrm{i}y$, where $x$ and $y$ are in terms of $a$.

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• Question 5Oct/Nov 2025, paper 31Structured6 marksHas a diagram

  The shaded region on the Argand diagram shows points representing complex numbers $z$ defined by two inequalities. The shaded region is bounded by a circle and a line parallel to the imaginary axis. The boundaries of the region are included in the shaded region.

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• Question 5Oct/Nov 2024, paper 31Structured7 marks

  By sketching a suitable pair of graphs, show that the equation $2+{\mathrm{e}}^{-0.2x}=\ln (1+x)$ has only one root.

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• Question 5Oct/Nov 2023, paper 31Structured6 marks

  Given that $$\sin (x+\frac{1}{6}\pi )-\sin (x-\frac{1}{6}\pi )=\cos (x+\frac{1}{3}\pi )-\cos (x-\frac{1}{3}\pi ),$$ find the exact value of $\tan x$.

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• Question 6May/June 2026, paper 31Structured10 marks

  A curve has the equation $y={\mathrm{e}}^{(x^2+4x)}{\cos }^{2}x$, where $\cos x\mathrm{\ne }0$. Show that the $x$-coordinates of the stationary points of this curve satisfy the equation $\tan x=x+2$.

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• Question 6Oct/Nov 2025, paper 31Structured5 marks

  Solve the quadratic equation $(2+\mathrm{i})w^2+4w+2-\mathrm{i}=0$.

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• Question 7May/June 2026, paper 31Structured6 marksHas a diagram

  The shaded region on the Argand diagram shows points representing the complex numbers, $z$, defined by two inequalities. The shaded region is bounded by the circle and the line parallel to the imaginary axis shown in the diagram and includes its boundaries.

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• Question 7Oct/Nov 2024, paper 31Structured9 marks

  Let $\mathrm{f}(x)=\frac{5x^2+8x+5}{(1+2x)(2+x^2)}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 7Oct/Nov 2023, paper 31Structured7 marks

  The variables $x$ and $\theta$ satisfy the differential equation $$\frac{x}{\tan \theta }\frac{\mathrm{d}x}{\mathrm{d}\theta }=x^2+3.$$ It is given that $x=1$ when $\theta =0$. Solve the differential equation, obtaining an expression for $x^2$ in terms of $\theta$.

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• Question 8May/June 2026, paper 31Structured6 marks

  Find the exact value of ${\int }_0^{1}3x^2{\mathrm{e}}^{3x}\mathrm{d}x$.

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• Question 8Oct/Nov 2025, paper 31Structured8 marks

  The variables $x$ and $y$ satisfy the differential equation $$(x^2+1)\frac{\mathrm{d}y}{\mathrm{d}x}=kx{\mathrm{e}}^{2y},$$ where $k$ is a constant. It is given that $y=0$ when $x=0$ and that $y=-\frac{1}{2}$ when $x=1$.

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• Question 8Oct/Nov 2024, paper 31Structured9 marks

  Given that $z=1+y\mathrm{i}$ and that $y$ is a real number, express $\frac{1}{z}$ in the form $a+b\mathrm{i}$, where $a$ and $b$ are functions of $y$.

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• Question 8Oct/Nov 2023, paper 31Structured8 marks

  By sketching a suitable pair of graphs, show that the equation $$\sqrt{x}={\mathrm{e}}^x-3$$ has only one root.

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• Question 10May/June 2026, paper 31Structured10 marks

  The polynomial $a{x}^3-11x^2+bx+4$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(x-2)$ is a factor of $\mathrm{p}(x)$, and that when ${\mathrm{p}}^{\mathrm{\prime }}(x)$ is divided by $(x-2)$ the remainder is 24.

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• Question 10Oct/Nov 2025, paper 31Structured11 marks

  Let $\mathrm{f}(x)=\frac{x^3+2x-11}{(3+x)(2+x^2)}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 10Oct/Nov 2024, paper 31Structured10 marksHas a diagram

  A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time $t$ minutes, the depth of the water in the tank is $h$ metres.

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• Question 10Oct/Nov 2023, paper 31Structured11 marks

  Let $\mathrm{f}(x)=\frac{24x+13}{(1-2x)(2+x{)}^2}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 1Oct/Nov 2024, paper 32Structured4 marks

  Expand $(9-3x{)}^{\frac{1}{2}}$ in ascending powers of $x$, up to and including the term in $x^2$, simplifying the coefficients.

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• Question 11Oct/Nov 2023, paper 31Structured10 marksHas a diagram

  In the diagram, $OABCDEFG$ is a cuboid in which $OA=3\text{ units}$, $OC=2\text{ units}$ and $OD=2\text{ units}$. Unit vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ are parallel to $OA$, $OD$ and $OC$ respectively. $M$ is the midpoint of $EF$.

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