Algebra: 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 2Oct/Nov 2024, paper 21Structured4 marks

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

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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 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 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 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 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 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 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 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 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 2Oct/Nov 2024, paper 23Structured4 marks

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

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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 23Structured8 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 2May/June 2025, paper 21Structured6 marks

  Use logarithms to solve the inequality $4^x<0.05$.

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• Question 3May/June 2024, paper 21Structured8 marks

  Sketch on the same diagram the graphs of $y=\mathrm{\mid }3x-8\mathrm{\mid }$ and $y=5-x$.

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• Question 5Oct/Nov 2023, paper 23Structured9 marks

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=6x^3+a{x}^2+bx-20,$$ where $a$ and $b$ are constants. It is given that $(x+2)$ is a factor of $\mathrm{p}(x)$ and that the remainder is $-11$ when $\mathrm{p}(x)$ is divided by $(x+1)$.

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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 4May/June 2023, paper 21Structured8 marks

  The polynomial $\mathrm{p}(x)$ is defined by $$\mathrm{p}(x)=2x^3+3x^2+kx-30,$$ where $k$ is a constant. It is given that $(x-3)$ is a factor of $\mathrm{p}(x)$. Find the value of $k$.

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• Question 2May/June 2025, paper 22Structured5 marks

  Sketch on the same diagram the graphs of $y=\mathrm{\mid }2x-9\mathrm{\mid }$ and $y=4x-5$.

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• Question 1May/June 2024, paper 22Structured4 marks

  Solve the inequality $\mathrm{\mid }5x+7\mathrm{\mid }>\mathrm{\mid }2x-3\mathrm{\mid }$.

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• Question 4May/June 2023, paper 22Structured7 marksHas a diagram

  The diagram shows the graph of $y=3-{\mathrm{e}}^{-\frac{1}{2}x}$. On the diagram, sketch the graph of $y=\mathrm{\mid }5x-4\mathrm{\mid }$, and show that the equation $3-{\mathrm{e}}^{-\frac{1}{2}x}=\mathrm{\mid }5x-4\mathrm{\mid }$ has exactly two real roots.

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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 5May/June 2024, paper 22Structured8 marks

  The polynomial $\mathrm{p}(x)$ is defined by $\mathrm{p}(x)=9x^3+18x^2+5x+4$. Find the quotient when $\mathrm{p}(x)$ is divided by $(3x+2)$, and show that the remainder is 6.

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