Questions — AQA Further Paper 2 (110 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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AQA Further Paper 2 Specimen Q7
2 marks Moderate -0.5
  1. In practice the minimum distance predicted by the model is incorrect. Is the minimum distance predicted by the model likely to be too big or too small? Explain your answer with reference to the model. [2 marks]
AQA Further Paper 2 Specimen Q8
5 marks Standard +0.8
Given that \(I_n = \int_0^{\frac{\pi}{2}} \sin^n x \, dx\) \quad \(n \geq 0\) show that \(n I_n = (n-1)I_{n-2}\) \quad \(n \geq 2\) [5 marks]
AQA Further Paper 2 Specimen Q9
6 marks Challenging +1.2
A student claims: "Given any two non-zero square matrices, A and B, then \((\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\)"
  1. Explain why the student's claim is incorrect giving a counter example. [2 marks]
  2. Refine the student's claim to make it fully correct. [1 mark]
  3. Prove that your answer to part (b) is correct. [3 marks]
AQA Further Paper 2 Specimen Q10
8 marks Challenging +1.8
Evaluate the improper integral \(\int_0^{\infty} \frac{4x - 30}{(x^2 + 5)(3x + 2)} \, dx\), showing the limiting process used. Give your answer as a single term. [8 marks]
AQA Further Paper 2 Specimen Q11
8 marks Challenging +1.8
The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics{figure_11} The polar equation of \(C\) is \(r = 4 + 2\cos \theta\), \quad \(-\pi \leq \theta \leq \pi\)
  1. Show that the area of the region bounded by the curve \(C\) is \(18\pi\) [4 marks]
  2. Points \(A\) and \(B\) lie on the curve \(C\) such that \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\) and \(AOB\) is an equilateral triangle. Find the polar equation of the line segment \(AB\) [4 marks]
AQA Further Paper 2 Specimen Q12
11 marks Standard +0.8
\(\mathbf{M} = \begin{pmatrix} -1 & 2 & -1 \\ 2 & 2 & -2 \\ -1 & -2 & -1 \end{pmatrix}\)
  1. Given that 4 is an eigenvalue of M, find a corresponding eigenvector. [3 marks]
  2. Given that \(\mathbf{MU} = \mathbf{UD}\), where D is a diagonal matrix, find possible matrices for D and U. [8 marks]
AQA Further Paper 2 Specimen Q13
7 marks Challenging +1.8
S is a singular matrix such that \(\det \mathbf{S} = \begin{vmatrix} a & a & x \\ x-b & a-b & x+1 \\ x^2 & a^2 & ax \end{vmatrix}\) Express the possible values of \(x\) in terms of \(a\) and \(b\). [7 marks]
AQA Further Paper 2 Specimen Q14
9 marks Challenging +1.2
Given that the vectors a and b are perpendicular, prove that \(|(\mathbf{a} + 5\mathbf{b}) \times (\mathbf{a} - 4\mathbf{b})| = k|\mathbf{a}||\mathbf{b}|\), where \(k\) is an integer to be found. Explicitly state any properties of the vector product that you use within your proof. [9 marks]
AQA Further Paper 2 Specimen Q15
10 marks Challenging +1.3
  1. Show that \((1-\frac{1}{4}e^{2i\theta})(1-\frac{1}{4}e^{-2i\theta}) = \frac{1}{16}(17-8\cos 2\theta)\) [3 marks]
  2. Given that the series \(e^{2i\theta} + \frac{1}{4}e^{4i\theta} + \frac{1}{16}e^{6i\theta} + \frac{1}{64}e^{8i\theta} + \ldots\) has a sum to infinity, express this sum to infinity in terms of \(e^{2i\theta}\) [2 marks]
  3. Hence show that \(\sum_{n=1}^{\infty} \frac{1}{4^{n-1}} \cos 2n\theta = \frac{16\cos 2\theta - 4}{17 - 8\cos 2\theta}\) [4 marks]
  4. Deduce a similar expression for \(\sum_{n=1}^{\infty} \frac{1}{4^{n-1}} \sin 2n\theta\) [1 mark]
AQA Further Paper 2 Specimen Q16
9 marks Challenging +1.8
A designer is using a computer aided design system to design part of a building. He models part of a roof as a triangular prism \(ABCDEF\) with parallel triangular ends \(ABC\) and \(DEF\), and a rectangular base \(ACFD\). He uses the metre as the unit of length. \includegraphics{figure_16} The coordinates of \(B\), \(C\) and \(D\) are \((3, 1, 11)\), \((9, 3, 4)\) and \((-4, 12, 4)\) respectively. He uses the equation \(x - 3y = 0\) for the plane \(ABC\). He uses \(\mathbf{r} - \begin{pmatrix} -4 \\ 12 \\ 4 \end{pmatrix} \times \begin{pmatrix} 4 \\ -12 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) for the equation of the line \(AD\). Find the volume of the space enclosed inside this section of the roof. [9 marks]