Algebraic manipulation: past paper questions

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• Question 16May/June 2026, paper 21Structured2 marks

  Expand and simplify $(3x+a)(x+4a)$.

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• Question 17Oct/Nov 2025, paper 21Structured5 marks

  $I=M(k^2+c^2)$ Find the value of $I$ when $M=7,k=3$ and $c=2$.

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• Question 5Oct/Nov 2023, paper 21Structured3 marks

  Factorise completely.

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• Question 6May/June 2026, paper 22Structured5 marks

  $E=5c^2-d^2$ Find the value of $E$ when $c=\sqrt{18}$ and $d=2$.

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• Question 25Oct/Nov 2025, paper 21Structured4 marks

  Simplify. $$\frac{10ax+6bx-25a-15b}{4x^2-25}$$ Simplify.

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• Question 22Oct/Nov 2024, paper 21Structured6 marksHas a diagram

  The graph of $y=(x+2)(x-1{)}^2$ is shown on the grid. Show that $y=(x+2)(x-1{)}^2$ can be written as $y=x^3-3x+2$.

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• Question 10Oct/Nov 2023, paper 22Structured2 marks

  Expand and simplify.

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• Question 10Oct/Nov 2025, paper 22Structured5 marks

  $b=dm+2mk$ $d=3.14$, $m=7.92$ and $k=10.16$. By rounding each value correct to 1 significant figure, work out an estimate for $b$.

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• Question 23Oct/Nov 2024, paper 21Structured2 marks

  $$(x-5{)}^2+k=x^2-px-21$$ Find the value of $p$ and the value of $k$.

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• Question 14Oct/Nov 2023, paper 22Structured3 marks

  $y=2w^2-x$ Rearrange the formula to make $w$ the subject. $y=2w^2-x$ Rearrange the formula to make $w$ the subject.

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• Question 15Oct/Nov 2025, paper 22Structured3 marks

  Factorise. $x^2-64$

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• Question 11Oct/Nov 2024, paper 22Structured5 marks

  Factorise fully.

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• Question 24Oct/Nov 2023, paper 22Structured4 marks

  Simplify. $$\frac{ax-2a-x+2}{a^2-1}$$ Simplify.

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• Question 21May/June 2026, paper 22Structured3 marks

  Write $x^2+8x-7$ in the form $(x+a{)}^2+b$ where $a$ and $b$ are integers.

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• Question 15Oct/Nov 2025, paper 23Structured4 marks

  Factorise. $x^2-7x+12$

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• Question 15Oct/Nov 2023, paper 23Structured3 marks

  $T=\sqrt{3d-e}$ Rearrange the formula to make $d$ the subject. $T=\sqrt{3d-e}$ Rearrange the formula to make $d$ the subject.

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• Question 17May/June 2026, paper 23Structured6 marks

  $c=4a^2+2b$ Find the value of $c$ when $a=-2$ and $b=3$.

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• Question 6Oct/Nov 2025, paper 41Structured1 mark

  Expand. $$g(3-2g)$$ Expand.

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• Question 20Oct/Nov 2023, paper 23Structured3 marks

  $(x+a)(x+2)(2x+3)$ is equivalent to $2x^3+b{x}^2+cx-18$. Find the value of each of $a$, $b$ and $c$.

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• Question 26May/June 2026, paper 23Structured4 marks

  Write $3x^2-6x+5$ in the form $a(x+b{)}^2+c$.

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• Question 12Oct/Nov 2024, paper 23Structured2 marks

  $$V=\sqrt[3]{\frac{x}{y}}$$ Rearrange the formula to write $x$ in terms of $V$ and $y$.

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• Question 2Oct/Nov 2023, paper 41Structured12 marks

  $s=\frac{1}{2}a{t}^2$ Find the value of $s$ when $a=9.8$ and $t=20$.

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• Question 22May/June 2026, paper 42Structured3 marks

  Write $\frac{8x^2+24x}{2x^2-18}$ as a single fraction in its simplest form.

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• Question 23Oct/Nov 2024, paper 23Structured2 marks

  Write $x^2+8x-7$ in the form $(x+a{)}^2+b$.

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• Question 17Oct/Nov 2025, paper 43Structured3 marks

  Make $t$ the subject of the formula. $$x=\frac{7t}{5-t}$$ Make $t$ the subject of the formula. $$x=\frac{7t}{5-t}$$

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• Question 25Oct/Nov 2024, paper 23Structured4 marks

  Simplify. $$\frac{3x-2-3xy+2y}{1-y^2}$$ Simplify.

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• Question 1Oct/Nov 2024, paper 41Structured19 marks

  Write 70 as a product of its prime factors.

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• Question 18May/June 2023, paper 22Structured4 marks

  Make $x$ the subject of the formula. $$c=\frac{3x}{2x-5}$$ Make $x$ the subject of the formula. $$c=\frac{3x}{2x-5}$$

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• Question 23May/June 2025, paper 21Structured3 marks

  Simplify. $$\frac{2x^2+10x}{x^2-25}$$ Simplify.

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• Question 6Oct/Nov 2024, paper 43Structured9 marks

  Simplify ${\displaystyle \frac{24u}{5y}}\times {\displaystyle \frac{10}{3u}}$.

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