Algebra: past paper questions

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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 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 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 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 1Oct/Nov 2025, paper 32Structured3 marks

  Sketch the graph of $y=\mathrm{\mid }x+3a\mathrm{\mid }$, where $a$ is a positive constant.

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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 1Oct/Nov 2023, paper 32Structured5 marks

  Sketch the graph of $y=\mathrm{\mid }4x-2\mathrm{\mid }$.

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• Question 5Oct/Nov 2025, paper 32Structured7 marks

  It is given that $\mathrm{f}(x)=(x-a{)}^2\mathrm{g}(x)$, where $\mathrm{f}(x)$ and $\mathrm{g}(x)$ are polynomials. Show that $(x-a)$ is a factor of ${\mathrm{f}}^{\mathrm{\prime }}(x)$.

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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 3Oct/Nov 2023, paper 32Structured5 marks

  The polynomial $2x^3+a{x}^2-11x+b$ is denoted by $\mathrm{p}(x)$. It is given that $\mathrm{p}(x)$ is divisible by $(2x-1)$ and that when $\mathrm{p}(x)$ is divided by $(x+1)$ the remainder is $12$. Find the values of $a$ and $b$.

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• Question 10May/June 2026, paper 32Structured11 marks

  Express $\frac{x+3}{(x-1)(2x-6)}$ in partial fractions.

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• Question 9Oct/Nov 2025, paper 32Structured9 marks

  Let $\mathrm{f}(x)=\frac{x^2+4ax+6a^2}{(x+2a)(x+3a)}$, where $a$ is a positive constant. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 10Oct/Nov 2024, paper 32Structured13 marks

  A balloon in the shape of a sphere has volume $V$ and radius $r$. Air is pumped into the balloon at a constant rate of $40\pi$ starting when time $t=0$ and $r=0$. At the same time, air begins to flow out of the balloon at a rate of $0.8\pi r$.

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• Question 1Oct/Nov 2023, paper 33Structured4 marks

  Find the set of values of $x$ satisfying the inequality $\mathrm{\mid }{2}^{x+1}-2\mathrm{\mid }<0.5$, giving your answer to 3 significant figures.

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• Question 2May/June 2026, paper 33Structured4 marks

  Find the set of values of $x$ satisfying the inequality $\mathrm{\mid }{3}^{2x-1}-4\mathrm{\mid }>2$.

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• Question 1Oct/Nov 2025, paper 33Structured4 marks

  Solve the inequality $\mathrm{\mid }3x+2\mathrm{\mid }<3\mathrm{\mid }2x-1\mathrm{\mid }$.

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• Question 3Oct/Nov 2023, paper 33Structured5 marks

  The polynomial $2x^3+a{x}^2+bx+6$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. When $\mathrm{p}(x)$ is divided by $(x+2)$ the remainder is $-38$ and when $\mathrm{p}(x)$ is divided by $(2x-1)$ the remainder is $\frac{19}{2}$.

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• Question 9May/June 2026, paper 33Structured10 marks

  Let $\mathrm{f}(x)=\frac{36-4x-3x^2}{(3x+1)(x^2+4)}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 2Oct/Nov 2025, paper 33Structured3 marks

  Find the quotient and the remainder when $3x^4-2x^2$ is divided by $x+1$.

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• Question 8Oct/Nov 2024, paper 33Structured8 marks

  Let $\mathrm{f}(x)=\frac{7a^2}{(a-2x)(3a+x)}$, where $a$ is a positive constant. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 9Oct/Nov 2023, paper 33Structured11 marks

  Let $\mathrm{f}(x)=\frac{17x^2-7x+16}{(2+3x^2)(2-x)}$. Express $\mathrm{f}(x)$ in partial fractions.

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• Question 10Oct/Nov 2025, paper 33Structured8 marksHas a diagram

  Express $\frac{2}{1-9y^2}$ in partial fractions.

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• Question 9Oct/Nov 2024, paper 33Structured8 marks

  Find the quotient and remainder when $x^4+16$ is divided by $x^2+4$.

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• Question 2May/June 2023, paper 31Structured4 marks

  Sketch the graph of $y=\mathrm{\mid }2x+3\mathrm{\mid }$.

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• Question 1May/June 2025, paper 31Structured3 marks

  Sketch the graph of $y=\mathrm{\mid }2x-3\mathrm{\mid }$.

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• Question 1May/June 2024, paper 31Structured4 marks

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

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• Question 3May/June 2023, paper 31Structured4 marks

  Find the coefficient of $x^3$ in the binomial expansion of $(3+x)\sqrt{1+4x}$.

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• Question 5May/June 2025, paper 31Structured5 marks

  The polynomial $3x^3+pa{x}^2+7a^{2}x+q{a}^3$ is denoted by $\mathrm{f}(x)$, where $p$, $q$ and $a$ are constants and $a\mathrm{\ne }0$. When $\mathrm{f}(x)$ is divided by $(x+2a)$ the remainder is $-22a^3$. When $\mathrm{f}(x)$ is divided by $(3x-a)$ the remainder is $-a^3$.

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• Question 11May/June 2024, paper 31Structured11 marks

  In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time $t$ after the first plant is infected there are $x$ infected plants.

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• Question 8May/June 2023, paper 31Structured10 marks

  Let $\mathrm{f}(x)=\frac{3-3x^2}{(2x+1)(x+2{)}^2}$. Express $\mathrm{f}(x)$ in partial fractions.

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