Algebra and graphs: past paper questions

Where the questions come from

• Question 2May/June 2026, paper 21Structured2 marks

  Solve $7x+4=25$.

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• Question 7Oct/Nov 2025, paper 21Structured4 marks

  Simplify. $\frac{p}{t}÷\frac{2}{t}$

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• Question 3Oct/Nov 2023, paper 21Structured2 marks

  Complete these statements.

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• Question 3May/June 2026, paper 21Structured2 marks

  Find the value of $4x^2+3y$ when $x=5$ and $y=10$.

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• Question 8Oct/Nov 2025, paper 21Structured5 marks

  The cost of one orange is $t$ cents. The cost of one apple is $w$ cents. The total cost of 3 oranges and 1 apple is 51 cents.

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• Question 12Oct/Nov 2024, paper 21Structured3 marks

  Solve the simultaneous equations.

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

  Factorise completely.

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• Question 4May/June 2026, paper 21Structured2 marks

  The diagram shows a rectangle with width $6\text{ cm}$. The width of the rectangle is $\frac{1}{12}$ of the perimeter of the rectangle. Work out the length of the rectangle.

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• Question 16Oct/Nov 2025, paper 21Structured4 marks

  Write down all the inequalities that define the region $R$.

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• Question 13Oct/Nov 2024, paper 21Structured5 marks

  A sequence has $n\text{th}$ term $3n^2-1$. Find the second term in this sequence.

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• Question 10Oct/Nov 2023, paper 21Structured5 marks

  Simplify. $n^5\times n$

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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 17Oct/Nov 2024, paper 21Structured4 marks

  Solve. $$3x^2-7x-16=0$$ You must show all your working and give your answers correct to 2 decimal places.

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• Question 8May/June 2026, paper 21Structured8 marks

  $P$ is the point $(0,4)$ and $Q$ is the point $(1,8)$. Find the coordinates of the midpoint of $PQ$.

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• Question 18Oct/Nov 2025, paper 21Structured4 marks

  $\mathrm{f}(x)=2x+5$ $\mathrm{f}(x)\mathrm{f}(x)-\mathrm{f}\mathrm{f}(x)=a{x}^2+bx+c$ Find the value of $a$, the value of $b$ and the value of $c$.

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• Question 18Oct/Nov 2024, paper 21Structured3 marks

  $$\mathrm{g}(x)=4^{x+3}$$ Find $x$ when $\mathrm{g}(x)=1$.

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• Question 15Oct/Nov 2023, paper 21Structured5 marksHas a diagram

  By shading the unwanted regions of the grid, draw and label the region R which satisfies these inequalities.

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• Question 11May/June 2026, paper 21Structured6 marks

  $A$, $B$ and $C$ are linear sequences. The table gives some information about these sequences.

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• Question 19Oct/Nov 2025, paper 21Structured3 marks

  Solve. $${\left(\frac{1}{3}\right)}^x=9^{x+4}$$ Solve.

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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 20Oct/Nov 2023, paper 21Structured6 marks

  Write as a single fraction in its simplest form.

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

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

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• Question 22Oct/Nov 2025, paper 21Structured5 marks

  Solve. $$\frac{2}{x-1}=\frac{x}{x+2}$$ Solve.

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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 3Oct/Nov 2023, paper 22Structured2 marks

  $v=u+at$ Find the value of $v$ when $u=30$, $a=-2$ and $t=7$.

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• Question 20May/June 2026, paper 21Structured4 marks

  Find the value of ${27}^{-\frac{2}{3}}\times (6^4{)}^{0.5}$.

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• Question 23Oct/Nov 2025, paper 21Structured4 marks

  Find the coordinates of the turning point on the graph of $y=7-2x-x^2$.

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• Question 1Oct/Nov 2024, paper 22Structured2 marks

  These are the first eight terms of a sequence. $$c-3-9-15-21-27-33k$$ Find the value of $c$ and the value of $k$.

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

  Expand and simplify.

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• Question 24May/June 2026, paper 21Structured9 marks

  Find the coordinates of the turning points of the graph of $y=x^3-6x^2+7$.

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