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AQA FP1 2016 June Q4
7 marks Moderate -0.3
  1. Given that \(\sin \frac{\pi}{3} = \cos \frac{\pi}{k}\), state the value of the integer \(k\). [1 mark]
  2. Hence, or otherwise, find the general solution of the equation $$\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}$$ giving your answer, in its simplest form, in terms of \(\pi\). [4 marks]
  3. Hence, given that \(\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}\), show that there is only one finite value for \(\tan x\) and state its exact value. [2 marks]
AQA FP1 2016 June Q5
9 marks Standard +0.8
  1. Use the formulae for \(\sum_{r=1}^n r^2\) and \(\sum_{r=1}^n r\) to show that \(\sum_{r=1}^n (6r - 3)^2 = 3n(4n^2 - 1)\). [5 marks]
  2. Hence express \(\sum_{r=1}^{2n} r^3 - \sum_{r=1}^n (6r - 3)^2\) as a product of four linear factors in terms of \(n\). [4 marks]
AQA FP1 2016 June Q6
9 marks Standard +0.8
A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).
  1. Show that \(a = 2\). [3 marks]
  2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]
AQA FP1 2016 June Q7
10 marks Standard +0.3
  1. Solve the equation \(x^2 + 4x + 20 = 0\), giving your answers in the form \(c + di\), where \(c\) and \(d\) are integers. [3 marks]
  2. The roots of the quadratic equation $$z^2 + (4 + i + qi)z + 20 = 0$$ are \(w\) and \(w^*\).
    1. In the case where \(q\) is real, explain why \(q\) must be \(-1\). [2 marks]
    2. In the case where \(w = p + 2i\), where \(p\) is real, find the possible values of \(q\). [5 marks]
AQA FP1 2016 June Q8
10 marks Standard +0.3
The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\).
    1. Find the matrix \(\mathbf{A}^2\). [1 mark]
    2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
  2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
  3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]
AQA FP1 2016 June Q9
11 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).
  1. Write down the equations of the asymptotes of \(C\). [2 marks]
  2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
  3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
  4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]
AQA FP2 2013 January Q1
7 marks Moderate -0.3
  1. Show that $$12 \cosh x - 4 \sinh x = 4\text{e}^x + 8\text{e}^{-x}$$ [2 marks]
  2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\). [5 marks]
AQA FP2 2013 January Q2
10 marks Standard +0.3
Two loci, \(L_1\) and \(L_2\), in an Argand diagram are given by $$L_1 : |z + 6 - 5\text{i}| = 4\sqrt{2}$$ $$L_2 : \arg(z + \text{i}) = \frac{3\pi}{4}$$ The point \(P\) represents the complex number \(-2 + \text{i}\).
  1. Verify that the point \(P\) is a point of intersection of \(L_1\) and \(L_2\). [2 marks]
  2. Sketch \(L_1\) and \(L_2\) on one Argand diagram. [6 marks]
  3. The point \(Q\) is also a point of intersection of \(L_1\) and \(L_2\). Find the complex number that is represented by \(Q\). [2 marks]
AQA FP2 2013 January Q3
7 marks Standard +0.3
  1. Show that \(\frac{1}{5r-2} - \frac{1}{5r+3} = \frac{A}{(5r-2)(5r+3)}\), stating the value of the constant \(A\). [2 marks]
  2. Hence use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(5r-2)(5r+3)} = \frac{n}{3(5n+3)}$$ [4 marks]
  3. Find the value of $$\sum_{r=1}^{\infty} \frac{1}{(5r-2)(5r+3)}$$ [1 mark]
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]