A-Level & Admissions

Interactive practice · A-Level Maths

Interactive Exam Questions
OCR A-Level Maths

Work through exam-style solutions one line at a time. At each step, choose the line you think comes next, some of the wrong options are exactly what you'd write after a slip earlier on. Press Mark to see which steps you got right and why the others are wrong. This is practice, not a study guide: do it actively.

Subject A-Level Maths Further Maths
Board Edexcel OCR AQA
1 · Read the questionEach card is an OCR-style question. Try it on paper first.
2 · Choose each linePick the next correct step. Wrong options mirror real, common mistakes.
3 · Mark & learnSee which steps were right, with feedback on every slip, then reveal the full solution.

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The 10 OCR 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 OCR-style mark scheme with per-line feedback. The stems are listed here in full.

  1. Q1. Differentiate using the chain rule (Paper 1 (Pure) · typically 2 marks, 2 marks · Pure · Differentiation)
    Given \(y=(2x^{3}-5)^{4}\), find \(\dfrac{dy}{dx}\).
    Show worked solution & mark scheme

    Chain rule: \(\frac{dy}{dx}=4(2x^{3}-5)^{3}\times 6x^{2}\) (M1) \(=\) \(24x^{2}(2x^{3}-5)^{3}\) (A1). Total: 2 marks.

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

    \(\int_{1}^{3}(3x^{2}-4x+1)\,dx=[\,x^{3}-2x^{2}+x\,]_{1}^{3}\) (M1 A1) \(=(27-18+3)-(1-2+1)\) (dM1) \(=12-0=\) 12 (A1). Total: 4 marks.

  3. Q3. Integration by substitution (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Integration)
    Find \(\displaystyle\int 3x^{2}\,(x^{3}+2)^{4}\,dx\) using the substitution \(u=x^{3}+2\).
    Show worked solution & mark scheme

    With \(u=x^{3}+2,\ du=3x^{2}\,dx\) (M1), the integral becomes \(\int u^{4}\,du\) (M1) \(=\dfrac{u^{5}}{5}+c\) (A1) \(=\) \(\dfrac{(x^{3}+2)^{5}}{5}+c\) (A1). Total: 4 marks.

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

    \((3+2x)^{4}=3^{4}+\binom{4}{1}3^{3}(2x)+\binom{4}{2}3^{2}(2x)^{2}+\dots\) (M1) \(=\) \(81+216x+216x^{2}\) (A1, A1). Total: 3 marks.

  5. Q5. Solve a logarithmic equation (Paper 2 (Pure) · typically 4 marks, 4 marks · Pure · Logarithms)
    Solve \(\log_{2}x+\log_{2}(x-2)=3\).
    Show worked solution & mark scheme

    \(\log_{2}[x(x-2)]=3\) (M1) \(\Rightarrow x(x-2)=2^{3}=8\) (M1) \(\Rightarrow x^{2}-2x-8=0\Rightarrow(x-4)(x+2)=0\) (A1). Reject \(x=-2\): \(x=4\) (A1). Total: 4 marks.

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

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

  7. Q7. Magnitude and unit vector (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Vectors)
    The vector \(\mathbf{a}=5\mathbf{i}-12\mathbf{j}\). Find (a) \(|\mathbf{a}|\) and (b) the unit vector in the direction of \(\mathbf{a}\).
    Show worked solution & mark scheme

    (a) \(|\mathbf{a}|=\sqrt{5^{2}+(-12)^{2}}\) (M1) \(=\sqrt{169}=\) 13 (A1).

    (b) Unit vector \(=\dfrac{\mathbf{a}}{|\mathbf{a}|}=\) \(\tfrac{1}{13}(5\mathbf{i}-12\mathbf{j})\) (A1). Total: 3 marks.

  8. Q8. Proof: sum of two consecutive odd numbers (Paper 1 (Pure) · typically 2 marks, 2 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 \(2n+3\) (B1). Then \((2n+1)+(2n+3)=4n+4=4(n+1)\), which is a multiple of 4 (B1). Proven. Total: 2 marks.

  9. Q9. Binomial distribution probability (Paper 2 (Statistics) · typically 2 marks, 2 marks · Statistics · Binomial distribution)
    The random variable \(X\sim B(12,\,0.25)\). Find \(P(X=3)\), giving your answer to 4 decimal places.
    Show worked solution & mark scheme

    \(P(X=3)=\binom{12}{3}(0.25)^{3}(0.75)^{9}=220\times0.015625\times0.0750847\approx\) 0.2581 (4 d.p.) (M1 A1). Total: 2 marks.

  10. Q10. Newton's second law (Paper 3 (Mechanics) · typically 3 marks, 3 marks · Mechanics · Forces)
    A box of mass \(4\,\text{kg}\) rests on a smooth horizontal floor. A horizontal force of \(10\,\text{N}\) acts on it. Find (a) the acceleration and (b) the speed after \(3\,\text{s}\) from rest.
    Show worked solution & mark scheme

    (a) Smooth floor, so \(F=ma\Rightarrow 10=4a\) (M1) \(\Rightarrow a=\) \(2.5\ \text{m s}^{-2}\) (A1).

    (b) From rest, \(v=u+at=0+2.5\times3=\) \(7.5\ \text{m s}^{-1}\) (A1). Total: 3 marks.

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