Questions — AQA FP2 (145 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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
Sort by: Default | Easiest first | Hardest first
AQA FP2 2013 January Q4
9 marks Standard +0.3
The roots of the equation $$z^3 - 5z^2 + kz - 4 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\).
    1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha\beta\gamma\). [2 marks]
    2. Hence find the value of \(\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2\). [2 marks]
  2. The value of \(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\) is \(-4\).
    1. Explain why \(\alpha\), \(\beta\) and \(\gamma\) cannot all be real. [1 mark]
    2. By considering \((\alpha\beta + \beta\gamma + \gamma\alpha)^2\), find the possible values of \(k\). [4 marks]
AQA FP2 2013 January Q5
11 marks Standard +0.8
  1. Using the definition \(\tanh y = \frac{\text{e}^y - \text{e}^{-y}}{\text{e}^y + \text{e}^{-y}}\), show that, for \(|x| < 1\), $$\tanh^{-1} x = \frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$$ [3 marks]
  2. Hence, or otherwise, show that \(\frac{\text{d}}{\text{d}x}(\tanh^{-1} x) = \frac{1}{1-x^2}\). [3 marks]
  3. Use integration by parts to show that $$\int_{0}^{\frac{1}{4}} \tanh^{-1} x \, \text{d}x = \ln \left(\frac{3^m}{2^n}\right)$$ where \(m\) and \(n\) are positive integers. [5 marks]
AQA FP2 2013 January Q6
8 marks Standard +0.8
A curve is defined parametrically by $$x = t^3 + 5, \quad y = 6t^2 - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).
  1. Show that \(s = \int_{0}^{3} 3t\sqrt{t^2 + A} \, \text{d}t\), stating the value of the constant \(A\). [4 marks]
  2. Hence show that \(s = 61\). [4 marks]
AQA FP2 2013 January Q7
9 marks Standard +0.8
The polynomial \(\text{p}(n)\) is given by \(\text{p}(n) = (n-1)^3 + n^3 + (n+1)^3\).
    1. Show that \(\text{p}(k+1) - \text{p}(k)\), where \(k\) is a positive integer, is a multiple of 9. [3 marks]
    2. Prove by induction that \(\text{p}(n)\) is a multiple of 9 for all integers \(n \geqslant 1\). [4 marks]
  2. Using the result from part (a)(ii), show that \(n(n^2 + 2)\) is a multiple of 3 for any positive integer \(n\). [2 marks]
AQA FP2 2013 January Q8
14 marks Challenging +1.2
  1. Express \(-4 + 4\sqrt{3}\text{i}\) in the form \(r\text{e}^{\text{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). [3 marks]
    1. Solve the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\), giving your answers in the form \(r\text{e}^{\text{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leqslant \pi\). [4 marks]
    2. The roots of the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\) are represented by the points \(P\), \(Q\) and \(R\) on an Argand diagram. Find the area of the triangle \(PQR\), giving your answer in the form \(k\sqrt{3}\), where \(k\) is an integer. [3 marks]
  3. By considering the roots of the equation \(z^3 = -4 + 4\sqrt{3}\text{i}\), show that $$\cos\frac{2\pi}{9} + \cos\frac{4\pi}{9} + \cos\frac{8\pi}{9} = 0$$ [4 marks]
AQA FP2 2011 June Q1
8 marks Moderate -0.3
  1. Draw on the same Argand diagram:
    1. the locus of points for which $$|z - 2 - 5i| = 5$$ [3 marks]
    2. the locus of points for which $$\arg(z + 2i) = \frac{\pi}{4}$$ [3 marks]
  2. Indicate on your diagram the set of points satisfying both $$|z - 2 - 5i| \leqslant 5$$ and $$\arg(z + 2i) = \frac{\pi}{4}$$ [2 marks]
AQA FP2 2011 June Q2
10 marks Standard +0.3
  1. Use the definitions of \(\cosh \theta\) and \(\sinh \theta\) in terms of \(e^\theta\) to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh(x - y)$$ [4 marks]
  2. It is given that \(x\) satisfies the equation $$\cosh(x - \ln 2) = \sinh x$$
    1. Show that \(\tanh x = \frac{5}{4}\). [4 marks]
    2. Express \(x\) in the form \(\frac{1}{2} \ln a\). [2 marks]
AQA FP2 2011 June Q3
6 marks Standard +0.8
  1. Show that $$(r + 1)! - (r - 1)! = (r^2 + r - 1)(r - 1)!$$ [2 marks]
  2. Hence show that $$\sum_{r=1}^{n} (r^2 + r - 1)(r - 1)! = (n + 2)n! - 2$$ [4 marks]
AQA FP2 2011 June Q4
14 marks Standard +0.8
The cubic equation $$z^3 - 2z^2 + k = 0 \quad (k \neq 0)$$ has roots \(\alpha\), \(\beta\) and \(\gamma\).
    1. Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha\beta + \beta\gamma + \gamma\alpha\). [2 marks]
    2. Show that \(\alpha^2 + \beta^2 + \gamma^2 = 4\). [2 marks]
    3. Explain why \(\alpha^3 - 2\alpha^2 + k = 0\). [1 mark]
    4. Show that \(\alpha^3 + \beta^3 + \gamma^3 = 8 - 3k\). [2 marks]
  2. Given that \(\alpha^4 + \beta^4 + \gamma^4 = 0\):
    1. show that \(k = 2\); [4 marks]
    2. find the value of \(\alpha^5 + \beta^5 + \gamma^5\). [3 marks]
AQA FP2 2011 June Q5
13 marks Challenging +1.3
  1. The arc of the curve \(y^2 = x^2 + 8\) between the points where \(x = 0\) and \(x = 6\) is rotated through \(2\pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = 2\sqrt{2}\pi \int_0^6 \sqrt{x^2 + 4} \, dx$$ [5 marks]
  2. By means of the substitution \(x = 2 \sinh \theta\), show that $$S = \pi(24\sqrt{5} + 4\sqrt{2} \sinh^{-1} 3)$$ [8 marks]
AQA FP2 2011 June Q6
8 marks Standard +0.3
  1. Show that $$(k + 1)(4(k + 1)^2 - 1) = 4k^3 + 12k^2 + 11k + 3$$ [2 marks]
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$1^2 + 3^2 + 5^2 + \ldots + (2n - 1)^2 = \frac{1}{3}n(4n^2 - 1)$$ [6 marks]
AQA FP2 2011 June Q7
16 marks Challenging +1.3
    1. Use de Moivre's Theorem to show that $$\cos 5\theta = \cos^5 \theta - 10 \cos^3 \theta \sin^2 \theta + 5 \cos \theta \sin^4 \theta$$ and find a similar expression for \(\sin 5\theta\). [5 marks]
    2. Deduce that $$\tan 5\theta = \frac{\tan \theta(5 - 10 \tan^2 \theta + \tan^4 \theta)}{1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$$ [3 marks]
  2. Explain why \(t = \tan \frac{\pi}{5}\) is a root of the equation $$t^4 - 10t^2 + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form. [3 marks]
  3. Deduce that $$\tan \frac{\pi}{5} \tan \frac{2\pi}{5} = \sqrt{5}$$ [5 marks]
AQA FP2 2016 June Q1
6 marks Standard +0.3
  1. Given that \(f(r) = \frac{1}{4r-1}\), show that $$f(r) - f(r+1) = \frac{A}{(4r-1)(4r+3)}$$ where \(A\) is an integer. [2 marks]
  2. Use the method of differences to find the value of \(\sum_{r=1}^{50} \frac{1}{(4r-1)(4r+3)}\), giving your answer as a fraction in its simplest form. [4 marks]
AQA FP2 2016 June Q2
8 marks Standard +0.3
The cubic equation \(3z^3 + pz^2 + 17z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2\mathrm{i}\).
    1. Write down the value of another non-real root, \(\beta\), of this equation. [1 mark]
    2. Hence find the value of \(\alpha\beta\). [1 mark]
  2. Find the value of the third root, \(\gamma\), of this equation. [3 marks]
  3. Find the values of \(p\) and \(q\). [3 marks]
AQA FP2 2016 June Q3
10 marks Challenging +1.3
The arc of the curve with equation \(y = 4 - \ln(1-x^2)\) from \(x = 0\) to \(x = \frac{3}{4}\) has length \(s\).
  1. Show that \(s = \int_0^{\frac{3}{4}} \frac{\sqrt{1+x^2}}{1-x^2} \, dx\). [4 marks]
  2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer. [6 marks]
AQA FP2 2016 June Q4
6 marks Standard +0.8
  1. Given that \(y = \tan^{-1} \sqrt{3x}\), find \(\frac{dy}{dx}\), giving your answer in terms of \(x\). [2 marks]
  2. Hence, or otherwise, show that \(\int_{\frac{1}{3}}^1 \frac{1}{(1+3x)\sqrt{x}} \, dx = \frac{\sqrt{3}\pi}{n}\), where \(n\) is an integer. [4 marks]
AQA FP2 2016 June Q5
12 marks Standard +0.3
  1. Find the modulus of the complex number \(-4\sqrt{3} + 4\mathrm{i}\), giving your answer as an integer. [2 marks]
  2. The locus of points, \(L\), satisfies the equation \(|z + 4\sqrt{3} - 4\mathrm{i}| = 4\).
    1. Sketch the locus \(L\) on the Argand diagram below. [3 marks]
    2. The complex number \(w\) lies on \(L\) so that \(-\pi < \arg w \leq \pi\). Find the least possible value of \(\arg w\), giving your answer in terms of \(\pi\). [2 marks]
  3. Solve the equation \(z^3 = -4\sqrt{3} + 4\mathrm{i}\), giving your answers in the form \(re^{\mathrm{i}\theta}\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). [5 marks]
AQA FP2 2016 June Q6
14 marks Challenging +1.2
  1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that $$x = \ln(y + \sqrt{y^2 + 1}).$$ [4 marks]
  2. A curve has equation \(y = 6\cosh^2 x + 5\sinh x\).
    1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number. [5 marks]
    2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh^{-1} 2\) has area \(A\). Show that $$A = a\cosh^{-1} 2 + b\sqrt{3} + c$$ where \(a\), \(b\) and \(c\) are integers. [5 marks]
AQA FP2 2016 June Q7
6 marks Standard +0.3
Given that \(p \geq -1\), prove by induction that, for all integers \(n \geq 1\), $$(1 + p)^n \geq 1 + np$$ [6 marks]
AQA FP2 2016 June Q8
13 marks Challenging +1.8
  1. By applying de Moivre's theorem to \((\cos \theta + \mathrm{i} \sin \theta)^4\), where \(\cos \theta \neq 0\), show that $$(1 + \mathrm{i} \tan \theta)^4 + (1 - \mathrm{i} \tan \theta)^4 = \frac{2\cos 4\theta}{\cos^4 \theta}$$ [3 marks]
  2. Hence show that \(z = \mathrm{i} \tan \frac{\pi}{8}\) satisfies the equation \((1 + z)^4 + (1 - z)^4 = 0\), and express the three other roots of this equation in the form \(\mathrm{i} \tan \phi\), where \(0 < \phi < \pi\). [2 marks]
  3. Use the results from part (b) to find the values of:
    1. \(\tan^2 \frac{\pi}{8} \tan^2 \frac{3\pi}{8}\); [4 marks]
    2. \(\tan^2 \frac{\pi}{8} + \tan^2 \frac{3\pi}{8}\). [4 marks]