A-Level & Admissions

Interactive practice · A-Level Maths

Interactive Exam Questions
AQA A-Level Maths

Ten exam-style questions, worked one line at a time. Open a question, and each line is revealed only after you commit to the previous one, just like building a solution under exam conditions. You're then marked the way AQA marks it: method (M), accuracy (A) and independent (B) marks adding up to the question's real total, with feedback on every slip.

Subject A-Level Maths Further Maths
Board Edexcel OCR AQA
1 · Open a questionEach is an AQA-style question with its real mark total. Try it on paper first.
2 · Build the solutionPick each line; the next appears once you commit. Wrong options mirror real, common mistakes.
3 · Get your marksPress Mark to see your score out of the total, line-by-line M/A/B awards and feedback.

Marks follow AQA conventions: M = method mark (correct approach), A = accuracy mark (correct result; depends on the method), B = an independent mark for correct work or a stated result, R = a reasoning mark for a correct conclusion. Each question shows its total, exactly as on the paper.

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The 10 AQA questions on this page

Every question below is fully interactive above: open it, build the solution line by line, and be marked against an authentic AQA-style mark scheme with per-line feedback. The stems are listed here in full.

  1. Q1. Differentiate a product (with the chain rule) (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Differentiation)
    Given \(y=x^{2}(3x-1)^{4}\), find \(\dfrac{dy}{dx}\) and factorise your answer fully.
    Show worked solution & mark scheme

    \(\frac{d}{dx}(3x-1)^{4}=12(3x-1)^{3}\) (B1). Product rule: \(\frac{dy}{dx}=2x(3x-1)^{4}+12x^{2}(3x-1)^{3}\) (M1 A1). Factor \(2x(3x-1)^{3}\): \((3x-1)+6x=9x-1\), giving \(2x(3x-1)^{3}(9x-1)\) (A1). Total: 4 marks.

  2. Q2. Evaluate a definite integral (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Integration)
    Evaluate \(\displaystyle\int_{1}^{3}\left(8x^{3}-6x\right)\,dx\).
    Show worked solution & mark scheme

    \(\int_{1}^{3}(8x^{3}-6x)\,dx=[\,2x^{4}-3x^{2}\,]_{1}^{3}\) (M1 A1) \(=(162-27)-(2-3)=135-(-1)=\) 136 (dM1 A1). Total: 4 marks.

  3. Q3. Integrate by substitution (Paper 1 (Pure) · typically 5 marks, 5 marks · Pure · Integration)
    Use the substitution \(u=x^{2}+3\) to evaluate \(\displaystyle\int_{0}^{1} 4x\sqrt{x^{2}+3}\,dx\).
    Show worked solution & mark scheme

    \(u=x^{2}+3,\ du=2x\,dx\) (B1), and \(4x\,dx=2\,du\), so the integral is \(\int_{3}^{4}2\sqrt{u}\,du\) (M1) \(=\Big[\tfrac{4}{3}u^{3/2}\Big]_{3}^{4}\) (A1) \(=\tfrac{4}{3}\big(8-3\sqrt3\big)\) (dM1 A1) \(\approx\) 3.74. Total: 5 marks.

  4. Q4. Binomial expansion, first three terms (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Binomial)
    Find the first three terms, in ascending powers of \(x\), of \((3+2x)^{5}\).
    Show worked solution & mark scheme

    \((3+2x)^{5}=3^{5}+\binom{5}{1}3^{4}(2x)+\binom{5}{2}3^{3}(2x)^{2}+\dots\) (M1) \(=243+5(81)(2)x+10(27)(4)x^{2}=\) \(243+810x+1080x^{2}\) (A1, A1). Total: 3 marks.

  5. Q5. Solve an exponential equation (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Logarithms & exponentials)
    Solve \(e^{2x}-7e^{x}+12=0\), giving your answers as exact values.
    Show worked solution & mark scheme

    Let \(y=e^{x}\): \(y^{2}-7y+12=0\) (M1) \(\Rightarrow(y-3)(y-4)=0\) (dM1) \(\Rightarrow y=3,4\) (A1). So \(e^{x}=3\) or \(4\): \(x=\ln 3\) or \(x=\ln 4\) (A1). Total: 4 marks.

  6. Q6. Solve a trigonometric equation (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Trigonometry)
    Solve \(2\cos^{2}\theta-\cos\theta-1=0\) for \(0^{\circ}\le\theta<360^{\circ}\).
    Show worked solution & mark scheme

    \((2\cos\theta+1)(\cos\theta-1)=0\) (M1) \(\Rightarrow\cos\theta=-\tfrac12\) or \(1\) (A1). So \(\theta=120^{\circ},240^{\circ}\) (A1) and \(\theta=0^{\circ}\) (A1). \(\theta=0^{\circ},120^{\circ},240^{\circ}\). Total: 4 marks.

  7. Q7. nth term of an arithmetic sequence (Paper 1 (Pure) · 'write down' · 1 mark, 1 marks · Pure · Sequences & series)
    An arithmetic sequence has first term \(a=7\) and common difference \(d=4\). Write down an expression for the \(n\)th term \(u_{n}\).
    Show worked solution & mark scheme

    \(u_{n}=a+(n-1)d=7+4(n-1)\) (B1), which simplifies to \(4n+3\). Total: 1 mark.

  8. Q8. Resultant force as a vector (Newton's second law) (Paper 2 (Pure & Mechanics) · typically 4 marks, 4 marks · Mechanics · Vectors & forces)
    Two forces \(\mathbf{F}_{1}=(7\mathbf{i}+2\mathbf{j})\,\text{N}\) and \(\mathbf{F}_{2}=(-3\mathbf{i}+6\mathbf{j})\,\text{N}\) act on a particle of mass \(2\,\text{kg}\). Find the magnitude of the particle's acceleration.
    Show worked solution & mark scheme

    Resultant \(\mathbf{R}=(7-3)\mathbf{i}+(2+6)\mathbf{j}=4\mathbf{i}+8\mathbf{j}\) (M1). \(\mathbf{a}=\frac{\mathbf{R}}{m}=2\mathbf{i}+4\mathbf{j}\) (M1). \(|\mathbf{a}|=\sqrt{2^{2}+4^{2}}\) (M1) \(=\sqrt{20}=2\sqrt5\approx\) \(4.47\,\text{m s}^{-2}\) (A1). Total: 4 marks.

  9. Q9. Binomial hypothesis test (Paper 3 (Pure & Statistics) · typically 5 marks, 5 marks · Statistics · Hypothesis testing)
    A seed company claims \(75\%\) of its seeds germinate. A gardener plants \(20\) seeds and \(12\) germinate. Test, at the \(5\%\) level, whether the germination rate is lower than claimed. Let \(p\) be the germination probability and \(X\) the number that germinate.
    Show worked solution & mark scheme

    \(H_{0}:p=0.75,\ H_{1}:p<0.75\) (B1). Under \(H_{0}\), \(X\sim B(20,0.75)\), \(P(X\le 12)\) (M1) \(\approx0.102\) (A1). Since \(0.102>0.05\) (M1), do not reject \(H_{0}\): insufficient evidence that the rate is below \(75\%\) (R1). Total: 5 marks.

  10. Q10. Proof: sum of consecutive odd numbers (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Proof)
    Prove that the sum of any two consecutive odd numbers is a multiple of \(4\).
    Show worked solution & mark scheme

    Let the numbers be \(2n-1\) and \(2n+1\) (B1). Their sum is \((2n-1)+(2n+1)=4n\) (M1). As \(4n=4\times n\) with \(n\) an integer, the sum is a multiple of \(4\) (R1). Proved. Total: 3 marks.

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