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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]
OCR H240/02 2020 November Q1
9 marks Easy -1.3
  1. Differentiate the following with respect to \(x\).
    1. \((2x + 3)^7\) [2]
    2. \(x^3 \ln x\) [3]
  2. Find \(\int \cos 5x \, dx\). [2]
  3. Find the equation of the curve through \((1, 3)\) for which \(\frac{dy}{dx} = 6x - 5\). [2]
OCR H240/02 2020 November Q2
4 marks Moderate -0.8
Simplify fully \(\frac{2x^3 + x^2 - 7x - 6}{x^2 - x - 2}\). [4]
OCR H240/02 2020 November Q3
7 marks Moderate -0.3
In this question you should assume that \(-1 < x < 1\).
  1. For the binomial expansion of \((1 - x)^{-2}\)
    1. find and simplify the first four terms, [2]
    2. write down the term in \(x^n\). [1]
  2. Write down the sum to infinity of the series \(1 + x + x^2 + x^3 + \ldots\). [1]
  3. Hence or otherwise find and simplify an expression for \(2 + 3x + 4x^2 + 5x^3 + \ldots\) in the form \(\frac{a - x}{(b - x)^2}\) where \(a\) and \(b\) are constants to be determined. [3]
OCR H240/02 2020 November Q4
5 marks Standard +0.3
In this question you must show detailed reasoning. Solve the equation \(3\sin^4 \phi + \sin^2 \phi = 4\), for \(0 \leq \phi < 2\pi\), where \(\phi\) is measured in radians. [5]
OCR H240/02 2020 November Q5
5 marks Standard +0.3
  1. Determine the set of values of \(n\) for which \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) are positive integers. [3]
A 'Pythagorean triple' is a set of three positive integers \(a\), \(b\) and \(c\) such that \(a^2 + b^2 = c^2\).
  1. Prove that, for the set of values of \(n\) found in part (a), the numbers \(n\), \(\frac{n^2 - 1}{2}\) and \(\frac{n^2 + 1}{2}\) form a Pythagorean triple. [2]
OCR H240/02 2020 November Q6
3 marks Standard +0.3
Prove that \(\sqrt{2} \cos(2\theta + 45°) = \cos^2 \theta - 2\sin \theta \cos \theta - \sin^2 \theta\), where \(\theta\) is measured in degrees. [3]
OCR H240/02 2020 November Q7
8 marks Moderate -0.8
\(A\) and \(B\) are fixed points in the \(x\)-\(y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of
  1. the quantity \(|\mathbf{a} - \mathbf{b}|\), [1]
  2. the vector \(\frac{1}{2}(\mathbf{a} + \mathbf{b})\). [1]
The circle \(P\) is the set of points with position vector \(\mathbf{p}\) in the \(x\)-\(y\) plane which satisfy $$\left|\mathbf{p} - \frac{1}{2}(\mathbf{a} + \mathbf{b})\right| = \frac{1}{2}|\mathbf{a} - \mathbf{b}|.$$
  1. State, in terms of \(\mathbf{a}\) and \(\mathbf{b}\),
    1. the position vector of the centre of \(P\), [1]
    2. the radius of \(P\). [1]
It is now given that \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}\).
  1. Find a cartesian equation of \(P\). [4]
OCR H240/02 2020 November Q8
9 marks Standard +0.3
The rate of change of a certain population \(P\) at time \(t\) is modelled by the equation \(\frac{dP}{dt} = (100 - P)\). Initially \(P = 2000\).
  1. Determine an expression for \(P\) in terms of \(t\). [7]
  2. Describe how the population changes over time. [2]
OCR H240/02 2020 November Q9
4 marks Easy -1.3
The histogram shows information about the numbers of cars in five different price ranges, sold in one year at a car showroom. \includegraphics{figure_9} It is given that 66 cars in the price range £10000 to £20000 were sold.
  1. Find the number of cars sold in the price range £50000 to £90000. [1]
  2. State the units of the frequency density. [1]
  3. Suggest one change that the management could make to the diagram so that it would provide more information. [1]
  4. Estimate the number of cars sold in the price range £50000 to £60000. [1]
OCR H240/02 2020 November Q10
7 marks Moderate -0.3
Pierre is a chef. He claims that 90% of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5% significance level. [7]
OCR H240/02 2020 November Q11
9 marks Moderate -0.3
As part of a research project, the masses, \(m\) grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.
Mass (g)\(50 \leq m < 150\)\(150 \leq m < 200\)\(200 \leq m < 250\)\(250 \leq m < 350\)
Frequency162318355165
  1. Calculate estimates of the mean and standard deviation of these masses. [2]
The masses, \(x\) grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable \(X \sim N(200, 3600)\).
  1. Use the model to find \(P(150 < X < 210)\). [1]
  2. Use the model to determine \(x_1\) such that \(P(160 < X < x_1) = 0.6\), giving your answer correct to five significant figures. [3]
It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.
  1. Use these results to show that the model may not be appropriate. [1]
  2. Suggest a different value of a parameter of the model in the light of these results. [2]
OCR H240/02 2020 November Q12
6 marks Moderate -0.3
In the past, the time for Jeff's journey to work had mean 45.7 minutes and standard deviation 5.6 minutes. This year he is trying a new route. In order to test whether the new route has reduced his journey time, Jeff finds the mean time for a random sample of 30 journeys using the new route. He carries out a hypothesis test at the 2.5% significance level. Jeff assumes that, for the new route, the journey time has a normal distribution with standard deviation 5.6 minutes.
  1. State appropriate null and alternative hypotheses for the test. [2]
  2. Determine the rejection region for the test. [4]