Questions — Edexcel AEA (167 questions)

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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
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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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Edexcel AEA 2002 June Q1
8 marks Challenging +1.8
Solve the following equation, for \(0 \leq x \leq \pi\), giving your answers in terms of \(\pi\). $$\sin 5x - \cos 5x = \cos x - \sin x.$$ [8]
Edexcel AEA 2002 June Q2
9 marks Challenging +1.8
In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of \(x^2\) is equal to the coefficient of \(x^4\) and the coefficient of \(x^3\) is positive. Find the value of \(p\). [9]
Edexcel AEA 2002 June Q3
11 marks Challenging +1.8
The curve \(C\) has parametric equations $$x = 15t - t^3, \quad y = 3 - 2t^2.$$ Find the values of \(t\) at the points where the normal to \(C\) at \((14, 1)\) cuts \(C\) again. [11]
Edexcel AEA 2002 June Q4
14 marks Hard +2.3
Find the coordinates of the stationary points of the curve with equation $$x^3 + y^3 - 3xy = 48$$ and determine their nature. [14]
Edexcel AEA 2002 June Q5
15 marks Hard +2.3
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation $$y = \sin (\cos x).$$ The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).
  1. Find the coordinates of the points \(A\), \(B\) and \(C\). [3]
  2. Prove that \(B\) is a stationary point. [2]
Given that the region \(OCB\) is convex,
  1. show that, for \(0 \leq x \leq \frac{\pi}{2}\), $$\sin (\cos x) \leq \cos x$$ and $$(1 - \frac{2}{\pi} x) \sin 1 \leq \sin (\cos x)$$ and state in each case the value or values of \(x\) for which equality is achieved. [6]
  2. Hence show that $$\frac{\pi}{4} \sin 1 < \int_0^{\frac{\pi}{2}} \sin(\cos x) \, dx < 1.$$ [4]
Edexcel AEA 2002 June Q6
17 marks Hard +2.3
\includegraphics{figure_2} Figure 2 shows a sketch of part of two curves \(C_1\) and \(C_2\) for \(y \geq 0\). The equation of \(C_1\) is \(y = m_1 - x^{n_1}\) and the equation of \(C_2\) is \(y = m_2 - x^{n_2}\), where \(m_1\), \(m_2\), \(n_1\) and \(n_2\) are positive integers with \(m_2 > m_1\). Both \(C_1\) and \(C_2\) are symmetric about the line \(x = 0\) and they both pass through the points \((3, 0)\) and \((-3, 0)\). Given that \(n_1 + n_2 = 12\), find
  1. the possible values of \(n_1\) and \(n_2\), [4]
  2. the exact value of the smallest possible area between \(C_1\) and \(C_2\), simplifying your answer, [8]
  3. the largest value of \(x\) for which the gradients of the two curves can be the same. Leave your answer in surd form. [5]
Edexcel AEA 2002 June Q7
18 marks Hard +2.3
A student was attempting to prove that \(x = \frac{1}{2}\) is the only real root of $$x^3 + \frac{1}{4}x - \frac{1}{2} = 0.$$ The attempted solution was as follows. $$x^3 + \frac{1}{4}x = \frac{1}{2}$$ $$\therefore \quad x(x^2 + \frac{1}{4}) = \frac{1}{2}$$ $$\therefore \quad x = \frac{1}{2}$$ or $$x^2 + \frac{1}{4} = \frac{1}{2}$$ i.e. $$x^2 = -\frac{1}{4} \quad \text{no solution}$$ $$\therefore \quad \text{only real root is } x = \frac{1}{2}$$
  1. Explain clearly the error in the above attempt. [2]
  2. Give a correct proof that \(x = \frac{1}{2}\) is the only real root of \(x^3 + \frac{1}{4}x - \frac{1}{2} = 0\). [3]
The equation $$x^3 + \beta x - \alpha = 0 \quad \text{(I)}$$ where \(\alpha\), \(\beta\) are real, \(\alpha \neq 0\), has a real root at \(x = \alpha\).
  1. Find and simplify an expression for \(\beta\) in terms of \(\alpha\) and prove that \(\alpha\) is the only real root provided \(|\alpha| < 2\). [6]
An examiner chooses a positive number \(\alpha\) so that \(\alpha\) is the only real root of equation (I) but the incorrect method used by the student produces 3 distinct real "roots".
  1. Find the range of possible values for \(\alpha\). [7]
Edexcel AEA 2004 June Q1
9 marks Challenging +1.8
Solve the equation \(\cos x + \sqrt{(1 - \frac{1}{2} \sin 2x)} = 0\), in the interval \(0° \leq x < 360°\). [9]
Edexcel AEA 2004 June Q2
10 marks Challenging +1.3
  1. For the binomial expansion of \(\frac{1}{(1-x)^2}\), \(|x| < 1\), in ascending powers of \(x\),
    1. find the first four terms,
    2. write down the coefficient of \(x^n\). [2]
  2. Hence, show that, for \(|x| < 1\), \(\sum_{n=1}^{\infty} nx^n = \frac{x}{(1-x)^2}\). [2]
  3. Prove that, for \(|x| < 1\), \(\sum_{n=1}^{\infty} (an+1)x^n = \frac{(a+1)x-x^2}{(1-x)^2}\), where \(a\) is a constant. [4]
  4. Hence evaluate \(\sum_{n=1}^{\infty} \frac{5n+1}{2^{3n}}\). [2]
Edexcel AEA 2004 June Q3
11 marks Challenging +1.8
$$f(x) = x^3 - (k+4)x + 2k,$$ where \(k\) is a constant.
  1. Show that, for all values of \(k\), the curve with equation \(y = f(x)\) passes through the point \((2, 0)\). [1]
  2. Find the values of \(k\) for which the equation \(f(x) = 0\) has exactly two distinct roots. [5]
Given that \(k > 0\), that the \(x\)-axis is a tangent to the curve with equation \(y = f(x)\), and that the line \(y = p\) intersects the curve in three distinct points,
  1. find the set of values that \(p\) can take. [5]
Edexcel AEA 2004 June Q4
12 marks Challenging +1.8
\includegraphics{figure_1} The circle, with centre \(C\) and radius \(r\), touches the \(y\)-axis at \((0, 4)\) and also touches the line with equation \(4y - 3x = 0\), as shown in Fig. 1.
    1. Find the value of \(r\).
    2. Show that \(\arctan \left(\frac{4}{3}\right) + 2 \arctan \left(\frac{1}{2}\right) = \frac{1}{2} \pi\). [8]
The line with equation \(4x + 3y = q\), \(q > 12\), is a tangent to the circle.
  1. Find the value of \(q\). [4]
Edexcel AEA 2004 June Q5
15 marks Challenging +1.8
  1. Given that \(y = \ln [t + \sqrt{(1 + t^2)}]\), show that \(\frac{dy}{dt} = \frac{1}{\sqrt{(1+t^2)}}\). [3]
The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{(1+t^2)}}, \quad y = \ln [t + \sqrt{(1 + t^2)}], \quad t \in \mathbb{R}.$$ A student was asked to prove that, for \(t > 0\), the gradient of the tangent to \(C\) is negative. The attempted proof was as follows: $$y = \ln \left(t + \frac{1}{x}\right)$$ $$= \ln \left(\frac{tx + 1}{x}\right)$$ $$= \ln (tx + 1) - \ln x$$ $$\therefore \frac{dy}{dx} = \frac{t}{tx + 1} - \frac{1}{x}$$ $$= \frac{\frac{t}{x}}{t + \frac{1}{x}} - \frac{1}{x}$$ $$= \frac{t\sqrt{(1+t^2)}}{t + \sqrt{(1+t^2)}} - \sqrt{(1 + t^2)}$$ $$= -\frac{(1+t^2)}{t + \sqrt{(1+t^2)}}$$ As \((1 + t^2) > 0\), and \(t + \sqrt{(1 + t^2)} > 0\) for \(t > 0\), \(\frac{dy}{dx} < 0\) for \(t > 0\).
    1. Identify the error in this attempt.
    2. Give a correct version of the proof. [6]
  2. Prove that \(\ln [-t + \sqrt{(1 + t^2)}] = -\ln [t + \sqrt{(1 + t^2)}]\). [3]
  3. Deduce that \(C\) is symmetric about the \(x\)-axis and sketch the graph of \(C\). [3]
Edexcel AEA 2004 June Q6
17 marks Challenging +1.8
$$f(x) = x - [x], \quad x \geq 0$$ where \([x]\) is the largest integer \(\leq x\). For example, \(f(3.7) = 3.7 - 3 = 0.7\); \(f(3) = 3 - 3 = 0\).
  1. Sketch the graph of \(y = f(x)\) for \(0 \leq x < 4\). [3]
  2. Find the value of \(p\) for which \(\int_2^p f(x) dx = 0.18\). [3]
Given that $$g(x) = \frac{1}{1+kx}, \quad x \geq 0, \quad k > 0,$$ and that \(x_0 = \frac{1}{2}\) is a root of the equation \(f(x) = g(x)\),
  1. find the value of \(k\). [2]
  2. Add a sketch of the graph of \(y = g(x)\) to your answer to part \((a)\). [1]
The root of \(f(x) = g(x)\) in the interval \(n < x < n + 1\) is \(x_n\), where \(n\) is an integer.
  1. Prove that $$2 x_n^2 - (2n - 1)x_n - (n + 1) = 0.$$ [4]
  2. Find the smallest value of \(n\) for which \(x_n - n < 0.05\). [4]
Edexcel AEA 2004 June Q7
19 marks Hard +2.3
Triangle \(ABC\), with \(BC = a\), \(AC = b\) and \(AB = c\) is inscribed in a circle. Given that \(AB\) is a diameter of the circle and that \(a^2\), \(b^2\) and \(c^2\) are three consecutive terms of an arithmetic progression (arithmetic series),
  1. express \(b\) and \(c\) in terms of \(a\), [4]
  2. verify that \(\cot A\), \(\cot B\) and \(\cot C\) are consecutive terms of an arithmetic progression. [3]
In an acute-angled triangle \(PQR\) the sides \(QR\), \(PR\) and \(PQ\) have lengths \(p\), \(q\) and \(r\) respectively.
  1. Prove that $$\frac{p}{\sin P} = \frac{q}{\sin Q} = \frac{r}{\sin R}.$$ [3]
Given now that triangle \(PQR\) is such that \(p^2\), \(q^2\) and \(r^2\) are three consecutive terms of an arithmetic progression,
  1. use the cosine rule to prove that $$\frac{2\cos Q}{q} = \frac{\cos P}{p} + \frac{\cos R}{r}.$$ [6]
  2. Using the results given in parts \((c)\) and \((d)\), prove that \(\cot P\), \(\cot Q\) and \(\cot R\) are consecutive terms in an arithmetic progression. [3]
Edexcel AEA 2008 June Q1
5 marks Standard +0.8
The first and second terms of an arithmetic series are 200 and 197.5 respectively. The sum to \(n\) terms of the series is \(S_n\). Find the largest positive value of \(S_n\). [5]
Edexcel AEA 2008 June Q2
12 marks Challenging +1.8
The points \((x, y)\) on the curve \(C\) satisfy \((x + 1)(x + 2) \frac{dy}{dx} = xy\). The line with equation \(y = 2x + 5\) is the tangent to \(C\) at a point \(P\).
  1. Find the coordinates of \(P\). [4]
  2. Find the equation of \(C\), giving your answer in the form \(y = f(x)\). [8]
Edexcel AEA 2008 June Q3
12 marks Challenging +1.8
  1. Prove that \(\tan 15° = 2 - \sqrt{3}\) [4]
  2. Solve, for \(0 < \theta < 360°\), $$\sin(\theta + 60°) \sin(\theta - 60°) = (1 - \sqrt{3}) \cos^2 \theta$$ [8]
Edexcel AEA 2008 June Q4
13 marks Hard +2.3
\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln(\sec x), \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$$ The points \(A\) and \(B\) are maximum points on \(C\).
  1. Find the coordinates of \(B\) in terms of e. [5]
The finite region \(R\) lies between \(C\) and the line \(AB\).
  1. Show that the area of \(R\) is $$\frac{2}{e} \arccos \left(\frac{1}{e}\right) + 2\ln \left(e + \sqrt{(e^2 - 1)}\right) - \frac{4}{e} \sqrt{(e^2 - 1)}.$$ [arccos \(x\) is an alternative notation for \(\cos^{-1} x\)] [8]
Edexcel AEA 2008 June Q5
14 marks Challenging +1.8
  1. Anna, who is confused about the rules for logarithms, states that $$(\log_3 p)^2 = \log_3 (p^2)$$ and $$\log_3(p + q) = \log_3 p + \log_3 q.$$ However, there is a value for \(p\) and a value for \(q\) for which both statements are correct. Find the value of \(p\) and the value of \(q\). [7]
  2. Solve $$\frac{\log_3(3x^3 - 23x^2 + 40x)}{\log_3 9} = 0.5 + \log_3(3x - 8).$$ [7]
Edexcel AEA 2008 June Q6
15 marks Challenging +1.8
$$f(x) = \frac{ax + b}{x + 2}; \quad x \in \mathbb{R}, x \neq -2,$$ where \(a\) and \(b\) are constants and \(b > 0\).
  1. Find \(f^{-1}(x)\). [2]
  2. Hence, or otherwise, find the value of \(a\) so that \(f(x) = x\). [2]
The curve \(C\) has equation \(y = f(x)\) and \(f(x)\) satisfies \(f(x) = x\).
  1. On separate axes sketch
    1. \(y = f(x)\), [3]
    2. \(y = f(x - 2) + 2\). [3]
On each sketch you should indicate the equations of any asymptotes and the coordinates, in terms of \(b\), of any intersections with the axes. The normal to \(C\) at the point \(P\) has equation \(y = 4x - 39\). The normal to \(C\) at the point \(Q\) has equation \(y = 4x + k\), where \(k\) is a constant.
  1. By considering the images of the normals to \(C\) on the curve with equation \(y = f(x - 2) + 2\), or otherwise, find the value of \(k\). [5]
Edexcel AEA 2008 June Q7
22 marks Challenging +1.8
Relative to a fixed origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are $$\overrightarrow{OA} = -3\mathbf{i} + \mathbf{j} - 9\mathbf{k}, \quad \overrightarrow{OB} = \mathbf{i} - \mathbf{k}, \quad \overrightarrow{OC} = 5\mathbf{i} + 2\mathbf{j} - 5\mathbf{k} \text{ respectively}.$$
  1. Find the cosine of angle \(ABC\). [4]
The line \(L\) is the angle bisector of angle \(ABC\).
  1. Show that an equation of \(L\) is \(\mathbf{r} = \mathbf{i} - \mathbf{k} + t(\mathbf{i} + 2\mathbf{j} - 7\mathbf{k})\). [4]
  2. Show that \(|\overrightarrow{AB}| = |\overrightarrow{AC}|\). [2]
The circle \(S\) lies inside triangle \(ABC\) and each side of the triangle is a tangent to \(S\).
  1. Find the position vector of the centre of \(S\). [7]
  2. Find the radius of \(S\). [5]
Edexcel AEA 2014 June Q1
5 marks Standard +0.8
The function f is given by $$f(x) = \ln(2x - 5), \quad x > 2.5$$
  1. Find \(f^{-1}(x)\). [2] The function g has domain \(x > 2\) and $$g(x) = \ln\left(\frac{x + 10}{x - 2}\right), \quad x > 2$$
  2. Find \(g(x)\) and simplify your answer. [3]
Edexcel AEA 2014 June Q2
6 marks Challenging +1.2
Given that $$3\sin^2 x + 2\sin x = 6\cos x + 9\sin x \cos x$$ and that \(-90° < x < 90°\), find the possible values of \(\tan x\). [6]
Edexcel AEA 2014 June Q3
11 marks Standard +0.8
  1. On separate diagrams sketch the curves with the following equations. On each sketch you should mark the coordinates of the points where the curve crosses the coordinate axes.
    1. \(y = x^2 - 2x - 3\)
    2. \(y = x^2 - 2|x| - 3\)
    3. \(y = x^2 - x - |x| - 3\)
    [7]
  2. Solve the equation $$x^2 - x - |x| - 3 = x + |x|$$ [4]
Edexcel AEA 2014 June Q4
13 marks Hard +2.3
Given that $$(1 + x)^n = 1 + \sum_{r=1}^{\infty} \frac{n(n-1)...(n-r+1)}{1 \times 2 \times ... \times r} x^r \quad (|x| < 1, x \in \mathbb{R}, n \in \mathbb{R})$$
  1. show that $$(1 - x)^{-\frac{1}{2}} = \sum_{r=0}^{\infty} \binom{2r}{r}\left(\frac{x}{4}\right)^r$$ [5]
  2. show that \((9 - 4x^2)^{-\frac{1}{2}}\) can be written in the form \(\sum_{r=0}^{\infty} \binom{2r}{r} \frac{x^{2r}}{3^{r}}\) and give \(q\) in terms of \(r\). [3]
  3. Find \(\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r}{9} \times \left(\frac{x}{3}\right)^{2r-1}\) [3]
  4. Hence find the exact value of $$\sum_{r=1}^{\infty} \binom{2r}{r} \times \frac{2r\sqrt{5}}{9} \times \frac{1}{5^r}$$ giving your answer as a rational number. [2]