Trigonometry: past paper questions

Where the questions come from

• 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 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 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 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 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 3Oct/Nov 2025, paper 23Structured6 marks

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

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• Question 7Oct/Nov 2024, paper 22Structured9 marks

  Express $4\sin \theta \sin (\theta +{60}^{\circ })$ in the form $$a+R\sin (2\theta -\alpha ),$$ where $a$ and $R$ are positive integers and $0^{\circ }<\alpha <{90}^{\circ }$.

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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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• Question 1Oct/Nov 2023, paper 23Structured3 marks

  It is given that $\theta$ is an acute angle in degrees such that $\sin \theta =\frac{2}{3}$. Find the exact value of $\sin (\theta +{60}^{\circ })$.

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• Question 5May/June 2025, paper 21Structured9 marks

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=a{x}^3+b{x}^2-ax-24,$$ where $a$ and $b$ are constants. It is given that $(2x-3)$ is a factor of $\mathrm{p}(x)$ and that the remainder is $-15$ when $\mathrm{p}(x)$ is divided by $(x+1)$.

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• Question 6May/June 2023, paper 21Structured7 marks

  Show that ${\int }_{\frac{1}{4}\pi }^{\frac{1}{3}\pi }(4{\cos }^{2}2x+\frac{1}{{\cos }^{2}x})\mathrm{d}x=\frac{3}{4}\sqrt{3}+\frac{1}{6}\pi -1$.

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• Question 5May/June 2025, paper 22Structured9 marks

  The polynomial $\mathrm{p}(x)$ is defined by $\mathrm{p}(x)=a{x}^4+b{x}^3+13x^2-35x+15$, where $a$ and $b$ are constants. It is given that $(2x-1)$ and $(x-3)$ are factors of $\mathrm{p}(x)$. Find the values of $a$ and $b$.

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• Question 7May/June 2023, paper 21Structured11 marks

  Express $7\cos \theta +24\sin \theta$ in the form $R\cos (\theta -\alpha )$, where $R>0$ and $0^{\circ }<\alpha <{90}^{\circ }$.

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• Question 7May/June 2025, paper 22Structured11 marks

  Express $4\cos \theta \sin (\theta +{30}^{\circ })$ in the form $R\cos (2\theta -\alpha )+k$, where $R>0$, $0^{\circ }<\alpha <{90}^{\circ }$ and $k$ is a constant.

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