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Pre-U Pre-U 9794/2 Specimen Q8
14 marks Standard +0.8
    1. Find the general solution of the differential equation $$x \frac{dy}{dx} = y(1 + x \cot x),$$ expressing \(y\) in terms of \(x\). [5]
    2. Find the particular solution given that \(y = 1\) when \(x = \frac{1}{2}\pi\). [2]
  2. The real variables \(x\) and \(y\) are related by \(x^2 - y^2 = 2ax - b\), where \(a\) and \(b\) are real constants.
    1. Show that \(\frac{dy}{dx} = 0\) can only be solved for \(x\) and \(y\) if \(b \geqslant a^2\). [5]
    2. Show that \(y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2\). [2]
Pre-U Pre-U 9794/2 Specimen Q9
15 marks Challenging +1.3
Two curves are defined by \(y = x^k\) and \(y = x^{\frac{1}{k}}\), for \(x \geqslant 0\), where \(k > 0\).
  1. Prove that, except for one value of \(k\), the curves intersect in exactly two points. [4]
The two curves enclose a finite region \(R\).
  1. Find the area, \(A\), of \(R\), giving your answer in the form \(A = f(k)\) and distinguishing clearly between the cases \(k < 1\) and \(k > 1\). [4]
  2. Determine the set of values of \(k\) for which \(A \leqslant 0.5\). [3]
  3. The function \(f\) is given by \(f : x \mapsto A\) with \(k > 1\). Prove that \(f\) is one-one and determine its inverse. [4]
Pre-U Pre-U 9794/2 Specimen Q10
7 marks Moderate -0.3
  1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]
Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).
  1. Find the speeds of the particles immediately after the collision. [4]
  2. Determine the coefficient of restitution between the particles. [2]
Pre-U Pre-U 9794/2 Specimen Q11
11 marks Standard +0.3
A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.
  1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]
The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.
  1. Find the position vector of the particle \(3\) seconds later. [5]
Pre-U Pre-U 9794/2 Specimen Q12
11 marks Standard +0.3
A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.
  1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]
After the particles have moved a distance of \(12\) m the string is cut.
  1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
  2. Find the value of \(t\) when \(P\) comes to rest. [3]
Pre-U Pre-U 9794/2 Specimen Q13
11 marks Standard +0.3
A gunner fires one shell from each of two guns on a stationary ship towards a vertical cliff \(AB\) of height \(100\) m whose foot \(A\) is at a horizontal distance \(600\) m from the point of projection.
  1. Given that the shell from the first gun hits the cliff, travelling horizontally, at a point \(45\) m above \(A\), determine the initial velocity of the shell. Express your answer in the form \(a\mathbf{i} + b\mathbf{j}\), where \(a\) and \(b\) are integers. [6]
  2. The shell from the second gun hits the cliff at its top point \(B\). Given that the initial speed of the shell is \(300\) m s\(^{-1}\), determine the possible angles of projection. [5]
Pre-U Pre-U 9795 Specimen Q1
4 marks Standard +0.3
The region \(R\) of an Argand diagram is defined by the inequalities $$0 \leqslant \arg(z + 4\mathrm{i}) \leqslant \frac{1}{4}\pi \quad \text{and} \quad |z| \leqslant 4.$$ Draw a clearly labelled diagram to illustrate \(R\). [4]
Pre-U Pre-U 9795 Specimen Q2
6 marks Challenging +1.2
It is given that $$\mathrm{f}(n) = 7^n (6n + 1) - 1.$$ By considering \(\mathrm{f}(n + 1) - \mathrm{f}(n)\), prove by induction that \(\mathrm{f}(n)\) is divisible by 12 for all positive integers \(n\). [6]
Pre-U Pre-U 9795 Specimen Q3
6 marks Standard +0.3
Solve exactly the equation $$5 \cosh x - \sinh x = 7,$$ giving your answers in logarithmic form. [6]
Pre-U Pre-U 9795 Specimen Q4
6 marks Standard +0.3
Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]
Pre-U Pre-U 9795 Specimen Q5
7 marks Standard +0.8
Find the general solution of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 6\frac{\mathrm{d}y}{\mathrm{d}x} + 9y = 72\mathrm{e}^{3x}.$$ [7]
Pre-U Pre-U 9795 Specimen Q6
8 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows a sketch of the curve \(C\) with polar equation \(r = a \cos^2 \theta\), where \(a\) is a positive constant and \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\). [2]
  2. The equation of \(C\) may be expressed in the form \(r = \frac{1}{2}a(1 + \cos 2\theta)\). Using this form, show that the area of the region enclosed by \(C\) is given by $$\frac{1}{16}a^2 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (3 + 4 \cos 2\theta + \cos 4\theta) \, \mathrm{d}\theta,$$ and find this area. [6]
Pre-U Pre-U 9795 Specimen Q7
8 marks Challenging +1.2
The equation $$8x^3 + 12x^2 + 4x - 1 = 0$$ has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(2\alpha + 1, 2\beta + 1, 2\gamma + 1\) is $$y^3 - y - 1 = 0.$$ [3] The sum \((2\alpha + 1)^n + (2\beta + 1)^n + (2\gamma + 1)^n\) is denoted by \(S_n\). Find the values of \(S_3\) and \(S_{-2}\). [5]
Pre-U Pre-U 9795 Specimen Q8
9 marks Standard +0.3
The curve \(C\) has equation $$y = \frac{x^2 - 2x - 3}{x + 2}.$$
  1. Find the equations of the asymptotes of \(C\). [4]
  2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]
Pre-U Pre-U 9795 Specimen Q9
9 marks Challenging +1.3
Given that \(w_n = 3^{-n} \cos 2n\theta\) for \(n = 1, 2, 3, \ldots\), use de Moivre's theorem to show that $$1 + w_1 + w_2 + w_3 + \ldots + w_{N-1} = \frac{9 - 3\cos 2\theta + 3^{-N+1} \cos 2(N-1)\theta - 3^{-N+2} \cos 2N\theta}{10 - 6\cos 2\theta}.$$ [7] Hence show that the infinite series $$1 + w_1 + w_2 + w_3 + \ldots$$ is convergent for all values of \(\theta\), and find the sum to infinity. [2]
Pre-U Pre-U 9795 Specimen Q10
10 marks Standard +0.3
  1. Find the inverse of the matrix \(\begin{pmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & 2 \end{pmatrix}\), and hence solve the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 10,
    3x + 8y + 2z &= 8. \end{align} [5]
  2. Find the value of \(k\) for which the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 15,
    3x + 8y + 3z &= k, \end{align} is consistent. Find the solution in this case and interpret it geometrically. [5]
Pre-U Pre-U 9795 Specimen Q11
10 marks Challenging +1.8
A group \(G\) has distinct elements \(e, a, b, c, \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation. Prove that if $$a \circ a = b, \quad b \circ b = a$$ then the set of elements \(\{e, a, b\}\) forms a subgroup of \(G\). [5] Prove that if $$a \circ a = b, \quad b \circ b = c, \quad c \circ c = a$$ then the set of elements \(\{e, a, b, c\}\) does not form a subgroup of \(G\). [5]
Pre-U Pre-U 9795 Specimen Q12
12 marks Standard +0.3
With respect to an origin \(O\), the points \(A, B, C, D\) have position vectors $$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$ respectively. Find
  1. a vector perpendicular to the plane \(OAB\), [2]
  2. the acute angle between the planes \(OAB\) and \(OCD\), correct to the nearest \(0.1°\), [3]
  3. the shortest distance between the line which passes through \(A\) and \(B\) and the line which passes through \(C\) and \(D\), [4]
  4. the perpendicular distance from the point \(A\) to the line which passes through \(C\) and \(D\). [3]
Pre-U Pre-U 9795 Specimen Q13
12 marks Standard +0.8
Given that \(y = \cos\{\ln(1 + x)\}\), prove that
  1. \((1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} = -\sin\{\ln(1 + x)\}\), [1]
  2. \((1 + x)^2 \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + (1 + x)\frac{\mathrm{d}y}{\mathrm{d}x} + y = 0\). [2]
Obtain an equation relating \(\frac{\mathrm{d}^3 y}{\mathrm{d}x^3}\), \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) and \(\frac{\mathrm{d}y}{\mathrm{d}x}\). [2] Hence find Maclaurin's series for \(y\), up to and including the term in \(x^3\). [4] Verify that the same result is obtained if the standard series expansions for \(\ln(1 + x)\) and \(\cos x\) are used. [3]
Pre-U Pre-U 9795 Specimen Q14
13 marks Challenging +1.8
Let \(J_n = \int_1^{\mathrm{e}} (\ln x)^n \, \mathrm{d}x\), where \(n\) is a positive integer. By considering \(\frac{\mathrm{d}}{\mathrm{d}x}(x(\ln x)^n)\), or otherwise, show that $$J_n = \mathrm{e} - nJ_{n-1}.$$ [4] Let \(J_n = \frac{J_n}{n!}\). Show that $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots + \frac{1}{10!} = \frac{1}{\mathrm{e}}(1 + J_{10}).$$ [6] It can be shown that $$\sum_{r=2}^{n} \frac{(-1)^r}{r!} = \frac{1}{\mathrm{e}}(1 + (-1)^n J_n)$$ for all positive integers \(n\). Deduce the sum to infinity of the series $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots,$$ justifying your conclusion carefully. [3]
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]
Edexcel AEA 2014 June Q5
15 marks Challenging +1.8
The square-based pyramid \(P\) has vertices \(A, B, C, D\) and \(E\). The position vectors of \(A, B, C\) and \(D\) are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) respectively where $$\mathbf{a} = \begin{pmatrix} -2 \\ 3 \\ -1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 5 \\ 8 \\ -6 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} 6 \\ 1 \\ 1 \end{pmatrix}$$
  1. Find the vectors \(\overrightarrow{AB}\), \(\overrightarrow{AC}\), \(\overrightarrow{AD}\), \(\overrightarrow{BC}\), \(\overrightarrow{BD}\) and \(\overrightarrow{CD}\). [3]
  2. Find
    1. the length of a side of the square base of \(P\),
    2. the cosine of the angle between one of the slanting edges of \(P\) and its base,
    3. the height of \(P\),
    4. the position vector of \(E\).
    [9] A second pyramid, identical to \(P\), is attached by its square base to the base of \(P\) to form an octahedron.
  3. Find the position vector of the other vertex of this octahedron. [3]