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Interactive practice · Further Maths

Interactive Exam Questions
AQA Further Maths

Ten exam-style AQA Further Maths questions from the compulsory Pure content, 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 Further Maths 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 a correct result or stated fact, dM = a method mark that depends on the previous one. 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. Modulus-argument form of a complex number (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Complex numbers)
    Express \(z=2-2i\sqrt{3}\) in modulus-argument form \(r(\cos\theta+i\sin\theta)\), where \(-\pi<\theta\le\pi\).
    Show worked solution & mark scheme

    \(r=\sqrt{2^{2}+(2\sqrt{3})^{2}}=\sqrt{16}=4\) (B1). \((2,-2\sqrt3)\) is in the fourth quadrant, so \(\theta=-\arctan\sqrt{3}=-\tfrac{\pi}{3}\) (M1). Hence \(z=4\left(\cos\!\left(-\tfrac{\pi}{3}\right)+i\sin\!\left(-\tfrac{\pi}{3}\right)\right)\) (A1). Total: 3 marks.

  2. Q2. Apply De Moivre's theorem (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Complex numbers)
    Use De Moivre's theorem to find \(\left[\,2\left(\cos\dfrac{\pi}{12}+i\sin\dfrac{\pi}{12}\right)\right]^{4}\), giving your answer in the form \(a+bi\).
    Show worked solution & mark scheme

    By De Moivre, \(\left[2\left(\cos\tfrac{\pi}{12}+i\sin\tfrac{\pi}{12}\right)\right]^{4}=2^{4}\left(\cos\tfrac{\pi}{3}+i\sin\tfrac{\pi}{3}\right)\) (M1) \(=16\left(\tfrac12+i\tfrac{\sqrt3}{2}\right)\) (A1) \(=\) \(8+8\sqrt{3}\,i\) (A1). Total: 3 marks.

  3. Q3. Multiply two 2×2 matrices (Paper 1 (Pure) · typically 2 marks, 2 marks · Pure · Matrices)
    Given \(A=\begin{pmatrix}2 & -1\\ 3 & 0\end{pmatrix}\) and \(B=\begin{pmatrix}1 & 4\\ -2 & 5\end{pmatrix}\), find the product \(AB\).
    Show worked solution & mark scheme

    \(AB=\begin{pmatrix}(2)(1)+(-1)(-2) & (2)(4)+(-1)(5)\\ (3)(1)+(0)(-2) & (3)(4)+(0)(5)\end{pmatrix}\) (M1) \(=\) \(\begin{pmatrix}4 & 3\\ 3 & 12\end{pmatrix}\) (A1). Total: 2 marks.

  4. Q4. Determinant and inverse of a 2×2 matrix (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Matrices)
    The transformation matrix is \(M=\begin{pmatrix}4 & 3\\ 2 & 5\end{pmatrix}\). Find \(\det M\) and hence \(M^{-1}\).
    Show worked solution & mark scheme

    \(\det M=(4)(5)-(3)(2)=14\) (B1). Then \(M^{-1}=\tfrac{1}{14}\begin{pmatrix}5 & -3\\ -2 & 4\end{pmatrix}\) (M1) \(=\) \(\begin{pmatrix}\frac{5}{14} & -\frac{3}{14}\\ -\frac{1}{7} & \frac{2}{7}\end{pmatrix}\) (A1). Total: 3 marks.

  5. Q5. Product of the roots of a cubic (Paper 1 (Pure) · typically 1 mark, 1 marks · Pure · Roots of polynomials)
    The cubic \(3x^{3}-4x^{2}+7x-2=0\) has roots \(\alpha,\beta,\gamma\). Write down the value of the product \(\alpha\beta\gamma\).
    Show worked solution & mark scheme

    For \(ax^{3}+bx^{2}+cx+d=0\) the product of the roots is \(\alpha\beta\gamma=-\tfrac{d}{a}\). Here \(a=3,\ d=-2\), so \(\alpha\beta\gamma=-\tfrac{-2}{3}=\) \(\tfrac{2}{3}\) (B1). Total: 1 mark.

  6. Q6. Proof by induction: \(\sum r(r+1)\) (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Proof by induction)
    Prove by induction that \(\displaystyle\sum_{r=1}^{n} r(r+1)=\tfrac{1}{3}n(n+1)(n+2)\) for all integers \(n\ge 1\).
    Show worked solution & mark scheme

    \(n=1\): LHS \(=1\times2=2=\tfrac13(1)(2)(3)=\) RHS (B1). Assume true for \(n=k\); then \(\sum_{r=1}^{k+1} r(r+1)=\tfrac13 k(k+1)(k+2)+(k+1)(k+2)\) (M1) \(=\tfrac13(k+1)(k+2)(k+3)\) (dM1). This is the formula with \(n=k+1\); true for \(n=1\) and \(n=k\Rightarrow n=k+1\), so true for all \(n\ge1\) by induction (B1). Total: 4 marks.

  7. Q7. Sum a series by the method of differences (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Series)
    Show that \(\dfrac{1}{(2r-1)(2r+1)}=\dfrac{1}{2}\left(\dfrac{1}{2r-1}-\dfrac{1}{2r+1}\right)\) and hence find \(\displaystyle\sum_{r=1}^{n}\dfrac{1}{(2r-1)(2r+1)}\).
    Show worked solution & mark scheme

    \(\tfrac12\left(\tfrac{1}{2r-1}-\tfrac{1}{2r+1}\right)=\tfrac12\cdot\tfrac{2}{(2r-1)(2r+1)}=\tfrac{1}{(2r-1)(2r+1)}\) (M1). The sum telescopes (M1): \(\tfrac12\left(1-\tfrac{1}{2n+1}\right)\) (A1) \(=\tfrac12\cdot\tfrac{2n}{2n+1}=\) \(\dfrac{n}{2n+1}\) (A1). Total: 4 marks.

  8. Q8. Maclaurin series of \(\ln(1+3x)\) (Paper 1 (Pure) · typically 4 marks, 4 marks · Pure · Maclaurin series)
    Find the Maclaurin series for \(f(x)=\ln(1+3x)\) up to and including the term in \(x^{3}\).
    Show worked solution & mark scheme

    \(f'(x)=\tfrac{3}{1+3x},\ f''(x)=-9(1+3x)^{-2},\ f'''(x)=54(1+3x)^{-3}\) (M1). At \(x=0\): \(f=0,\ f'=3,\ f''=-9,\ f'''=54\) (A1). Then \(f(x)=3x+\tfrac{-9}{2}x^{2}+\tfrac{54}{6}x^{3}\) (M1) \(=\) \(3x-\tfrac{9}{2}x^{2}+9x^{3}\) (A1). Total: 4 marks.

  9. Q9. Volume of revolution about the x-axis (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Further calculus)
    The region bounded by \(y=\dfrac{2}{x}\), the \(x\)-axis and the lines \(x=1\) and \(x=3\) is rotated through \(2\pi\) about the \(x\)-axis. Find the exact volume of the solid formed.
    Show worked solution & mark scheme

    \(V=\pi\displaystyle\int_{1}^{3}\left(\tfrac2x\right)^{2}dx=\pi\int_{1}^{3}4x^{-2}\,dx\) (M1) \(=\pi\left[-\tfrac4x\right]_{1}^{3}\) (M1) \(=\pi\left(-\tfrac43+4\right)=\) \(\dfrac{8\pi}{3}\) (A1). Total: 3 marks.

  10. Q10. Angle between two lines (scalar product) (Paper 1 (Pure) · typically 3 marks, 3 marks · Pure · Vectors)
    Lines \(L_{1}\) and \(L_{2}\) have direction vectors \(\mathbf{a}=\begin{pmatrix}2\\-1\\2\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}1\\2\\2\end{pmatrix}\). Find the acute angle \(\theta\) between the lines, to the nearest degree.
    Show worked solution & mark scheme

    \(\mathbf{a}\cdot\mathbf{b}=2-2+4=4\), \(|\mathbf{a}|=|\mathbf{b}|=\sqrt9=3\) (M1). So \(\cos\theta=\tfrac{4}{3\times3}=\tfrac49\) (M1), giving \(\theta=\arccos\tfrac49\approx\) \(64^{\circ}\) (A1). Total: 3 marks.

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