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Pre-U Pre-U 9794/2 2010 June Q8
14 marks Standard +0.8
The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).
  1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
  2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
    1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
    2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]
Pre-U Pre-U 9794/2 2010 June Q9
15 marks Challenging +1.2
  1. Show that $$\int x^n \ln x \, dx = \frac{x^{n+1}}{(n+1)^2}\left((n+1)\ln a - 1\right) + \frac{1}{(n+1)^2},$$ where \(n \neq -1\) and \(a > 1\). [6]
    1. Determine the \(x\)-coordinate of the point of intersection of the curves \(y = x^3 \ln x\) and \(y = x \ln 2^x\), where \(x > 0\). [2]
    2. Find the exact value of the area of the region enclosed between these two curves, the line \(x = 1\) and their point of intersection. Express your answer in the form \(b + c \ln 2\), where \(b\) and \(c\) are rational. [4]
  3. The curve \(y = (x^3 \ln x)^{0.5}\), for \(1 < x < e\), is rotated through \(2\pi\) radians about the \(x\)-axis. Determine the value of the resulting volume of revolution, giving your answer correct to 4 significant figures. [3]
Pre-U Pre-U 9794/2 2010 June Q10
9 marks Standard +0.8
A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).
  1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
  2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]
Pre-U Pre-U 9794/2 2010 June Q11
10 marks Standard +0.3
Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are given by $$\mathbf{F}_1 = 13\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}, \quad \mathbf{F}_2 = -2\mathbf{i} + 6\mathbf{j} + \mathbf{k},$$ in which the units of the components are newtons. A third force, \(\mathbf{F}_3\), of magnitude 6 N acts parallel to the vector \(2\mathbf{i} - 2\mathbf{j} + \mathbf{k}\).
  1. Find the two possible resultants of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), and show that they have the same magnitude. [5]
A particle, \(P\), of mass 2 kg is initially at rest at the origin. The only forces acting on \(P\) are \(\mathbf{F}_1\) and \(\mathbf{F}_2\).
  1. Find the magnitude of the acceleration of \(P\). [3]
  2. Find the time taken for \(P\) to travel 60 m. [2]
Pre-U Pre-U 9794/2 2010 June Q12
13 marks Standard +0.3
A particle moves along a straight line under the action of a variable force. The acceleration is given by $$a = \begin{cases} 30 - 6t, & \text{for } 0 \leqslant t \leqslant 10 \\ 6t - 90, & \text{for } 10 \leqslant t \leqslant 20 \end{cases}$$ where time \(t\) is measured in seconds and \(a\) in m s\(^{-2}\). The particle is at rest at the origin at \(t = 0\).
    1. Find the velocity \(v\) of the particle in terms of \(t\). Verify that \(v = 0\) when \(t = 10\) and \(t = 20\). [7]
    2. Sketch the velocity-time graph for the motion. [2]
  2. Calculate the total distance travelled by the particle. [4]
Pre-U Pre-U 9794/2 2010 June Q13
8 marks Standard +0.3
A light inextensible string passes over a fixed smooth light pulley. Particles \(A\) and \(B\), of masses 2 kg and 3 kg respectively, are attached to the ends so that the portions of the string below the axis of the pulley are vertical (see diagram). The centre of the horizontal axis of the pulley is 4 m above ground level. \includegraphics{figure_13} The particles are released from rest 1 m above ground level with the string taut.
  1. Determine the acceleration of both particles prior to the impact of \(B\) with the ground. [3]
  2. Determine the greatest height attained by \(A\) above ground level. [3]
  3. If \(B\) rebounds after impact to a first maximum height of 0.05 m, determine the coefficient of restitution between \(B\) and the ground. [2]
Pre-U Pre-U 9794/1 2011 June Q1
3 marks Easy -1.8
Find the equation of the line passing through the points \((-2, 5)\) and \((4, -7)\). Give your answer in the form \(y = mx + c\). [3]
Pre-U Pre-U 9794/1 2011 June Q2
4 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm in which angle \(AOB\) is \(\theta\) radians. The sector has a perimeter of 18 cm.
  1. Show that \(\theta = \frac{18 - 2r}{r}\). [2]
  2. Find the area of the sector in terms of \(r\), simplifying your answer. [2]
Pre-U Pre-U 9794/1 2011 June Q3
3 marks Moderate -0.5
Solve the equation \(3 + 2x = |7 - 4x|\). [3]
Pre-U Pre-U 9794/1 2011 June Q4
6 marks Moderate -0.8
  1. Show that \(4 \ln x - \ln(3x - 2) - \ln x^2 = \ln\left(\frac{x^2}{3x - 2}\right)\), where \(x > \frac{2}{3}\). [3]
  2. Hence solve the equation \(4 \ln x - \ln(3x - 2) - \ln x^2 = 0\). [3]
Pre-U Pre-U 9794/1 2011 June Q5
5 marks Standard +0.3
A circle has equation \(x^2 + y^2 = 16\). Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines \(x = 1\) and \(x = 2\) is rotated through \(2\pi\) radians about the \(x\)-axis. [5]
Pre-U Pre-U 9794/1 2011 June Q6
7 marks Standard +0.3
  1. Sketch, on a single diagram, the graphs of \(y = e^{3x}\) and \(y = x\) and state the number of roots of the equation \(e^{3x} = x\). [3]
  2. Use the Newton-Raphson method with \(x_0 = 0\) to determine the value of a root of the equation \(e^{3x} = x\) correct to 3 decimal places. [4]
Pre-U Pre-U 9794/1 2011 June Q7
7 marks Standard +0.3
  1. Given that the point \((-1, -2, 4)\) lies on both the lines $$\mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ a \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 4 \\ b \end{pmatrix} + \mu \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix},$$ find \(a\) and \(b\). [3]
  2. Find the acute angle between the lines. [4]
Pre-U Pre-U 9794/1 2011 June Q8
8 marks Standard +0.3
  1. Find and simplify the first three terms in the expansion of \((1 - 4a)^{\frac{1}{2}}\) in ascending powers of \(a\), where \(|a| < \frac{1}{4}\). [4]
  2. Hence show that the roots of the quadratic equation \(x^2 - x + a = 0\) are approximately \(1 - a - a^2\) and \(a + a^2\), where \(a\) is small. [4]
Pre-U Pre-U 9794/1 2011 June Q9
9 marks Standard +0.8
  1. Prove that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\) and deduce that $$\sin \theta + \sin 3\theta = 4 \sin \theta \cos^2 \theta.$$ [5]
  2. Hence find the values of \(\theta\) such that \(0° < \theta < 180°\) that satisfy the equation $$\cot^2 \theta = \sin \theta + \sin 3\theta.$$ [4]
Pre-U Pre-U 9794/1 2011 June Q10
9 marks Moderate -0.3
  1. The complex number \(z\) is such that \(|z| = 2\) and \(\arg z = -\frac{3}{4}\pi\). Find the exact value of the real part of \(z\) and of the imaginary part of \(z\). [2]
  2. The complex numbers \(u\) and \(v\) are such that $$u = 1 + ia \quad \text{and} \quad v = b - i,$$ where \(a\) and \(b\) are real and \(a < b\). Given that \(uv = 7 + 9i\), find the values of \(a\) and \(b\). [7]
Pre-U Pre-U 9794/1 2011 June Q11
9 marks Standard +0.3
An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).
  1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
  2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]
Pre-U Pre-U 9794/1 2011 June Q12
10 marks Moderate -0.8
Find the general solution of the differential equation $$\frac{dy}{dx} = \frac{x}{x(1 + x^2)}$$ giving your answer in the form \(y = f(x)\). [10]
Pre-U Pre-U 9794/1 2011 June Q13
7 marks Moderate -0.3
  1. A random sample of young people in a certain town comprised 312 boys and 253 girls. Denoting a boy's age by \(x\) years and a girl's age by \(y\) years, the following data were obtained: $$\sum x = 4618, \quad \sum x^2 = 68812, \quad \sum y = 3719, \quad \sum y^2 = 55998.$$
    1. Calculate the mean and standard deviation of the ages of the boys in the sample and also of the girls in the sample. [3]
    2. Use these results to comment on the distribution of the ages of the boys and girls in the sample. [1]
  2. How many arrangements of the letters of the word DEFEATED are there in which the Es are separated from each other? [3]
Pre-U Pre-U 9794/1 2011 June Q14
9 marks Standard +0.3
  1. The table below relates the values of two variables \(x\) and \(y\).
    \(x\)1\(A\)\(A + 3\)10
    \(y\)2\(A - 1\)\(A\)5
    \(A\) is a positive integer and \(\sum xy = 92\).
    1. Calculate the value of \(A\). [3]
    2. Explain how you can tell that the product-moment correlation coefficient is 1. [1]
  2. A music society has 300 members. 240 like Puccini, 100 like Wagner and 50 like neither.
    1. Calculate the probability that a member chosen at random likes Puccini but not Wagner. [3]
    2. Calculate the probability that a member chosen at random likes Puccini given that this member likes Wagner. [2]
Pre-U Pre-U 9794/1 2011 June Q15
12 marks Standard +0.8
A firm produces chocolate bars whose weights are normally distributed with mean 120 g and standard deviation 6 g.
  1. Bars which weigh more than 114 g are sold at a profit of 15p per bar. The remaining bars are sold at no profit. Show that the expected profit per 100 bars is £12.62. [5]
  2. It is subsequently decided that bars which weigh more than \(x\) g should be sold at a profit of 20p per bar. Those which weigh \(x\) g or less are sold to employees at a profit of 3p per bar. The expected profit per 100 bars is £19.17. Find the value of \(x\). [7]
Pre-U Pre-U 9794/1 2011 June Q16
12 marks Standard +0.8
In a factory, computer chips are produced in large batches. A quality control procedure is used for each batch which requires a random sample of 8 chips to be tested. If no faulty chip is found, the batch is accepted. If two or more are faulty, the batch is rejected. If one is faulty, a further sample of 4 is selected and the batch is accepted if none of these is faulty. The probability of any chip being faulty is \(q\).
  1. Show that the probability of accepting a batch is \(p^8(1 + 8p^3 - 8p^4)\), where \(p = 1 - q\). [6]
  2. Find the expected number of chips sampled per batch, giving your answer in terms of \(p\). Hence show that when \(p = 0.75\), the expected number of chips sampled per batch is approximately 9. [6]
Pre-U Pre-U 9794/2 2011 June Q1
5 marks Easy -1.3
  1. Show that \(x = 4\) is a root of \(x^3 - 12x - 16 = 0\). [2]
  2. Hence completely factorise the expression \(x^3 - 12x - 16\). [3]
Pre-U Pre-U 9794/2 2011 June Q2
6 marks Easy -1.2
  1. Expand and simplify \((7 - 2\sqrt{3})^2\). [2]
  2. Show that $$\frac{\sqrt{125}}{2 + \sqrt{5}} = 25 - 10\sqrt{5}.$$ [4]
Pre-U Pre-U 9794/2 2011 June Q3
5 marks Moderate -0.8
Use integration by parts to find \(\int x \sin 3x \, dx\). [5]