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Pre-U Pre-U 9795/1 2013 November Q6
8 marks Challenging +1.2
\(G\) is the set \(\{2, 4, 6, 8\}\), \(H\) is the set \(\{1, 5, 7, 11\}\) and \(\times_n\) denotes the operation of multiplication modulo \(n\).
  1. Construct the multiplication tables for \((G, \times_{10})\) and \((H, \times_{12})\). [2]
  2. By verifying the four group axioms, show that \(G\) and \(H\) are groups under their respective binary operations, and determine whether \(G\) and \(H\) are isomorphic. [6]
[You may assume that \(\times_n\) is associative.]
Pre-U Pre-U 9795/1 2013 November Q7
8 marks Standard +0.3
Relative to an origin \(O\), the points \(P\), \(Q\) and \(R\) have position vectors $$\mathbf{p} = \mathbf{i} + 2\mathbf{j} - 7\mathbf{k}, \quad \mathbf{q} = -3\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \mathbf{r} = 6\mathbf{i} + 4\mathbf{j} + \alpha\mathbf{k}$$ respectively.
  1. Determine \(\mathbf{p} \times \mathbf{q}\). [2]
  2. Deduce the value of \(\alpha\) for which
    1. \(OR\) is normal to the plane \(OPQ\), [1]
    2. the volume of tetrahedron \(OPQR\) is 50, [3]
    3. \(R\) lies in the plane \(OPQ\). [2]
Pre-U Pre-U 9795/1 2013 November Q8
10 marks Standard +0.8
  1. Determine \(x\) and \(y\) given that the complex number \(z = x + \text{i}y\) simultaneously satisfies $$|z - 1| = 1 \quad \text{and} \quad \arg(z + 1) = \frac{1}{6}\pi.$$ [4]
  2. On an Argand diagram, shade the region whose points satisfy $$1 \leqslant |z - 1| \leqslant 2 \quad \text{and} \quad \frac{1}{6}\pi \leqslant \arg(z + 1) \leqslant \frac{1}{4}\pi.$$ [6]
Pre-U Pre-U 9795/1 2013 November Q9
10 marks Challenging +1.2
  1. Show that there is exactly one value of \(k\) for which the system of equations \begin{align} kx + 2y + kz &= 4
    3x + 10y + 2z &= m
    (k - 1)x - 4y + z &= k \end{align} does not have a unique solution. [4]
  2. Given that the system of equations is consistent for this value of \(k\), find the value of \(m\). [4]
  3. Explain the geometrical significance of a non-unique solution to a \(3 \times 3\) system of linear equations. [2]
Pre-U Pre-U 9795/1 2013 November Q10
8 marks Standard +0.8
The roots of the equation \(x^4 - 2x^3 + 2x^2 + x - 3 = 0\) are \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Determine the values of
  1. \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\), [2]
  2. \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}\), [2]
  3. \(\alpha^3 + \beta^3 + \gamma^3 + \delta^3\). [4]
Pre-U Pre-U 9795/1 2013 November Q11
14 marks Standard +0.8
  1. Given that \(y = -4\) when \(x = 0\) and that $$\frac{dy}{dx} - y = e^{2x} + 3,$$ find the value of \(x\) for which \(y = 0\). [7]
  2. Find the general solution of $$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = e^{2x} + 3,$$ given that \(y = cx^2e^{2x} + d\) is a suitable form of particular integral. [7]
Pre-U Pre-U 9795/1 2013 November Q12
10 marks Challenging +1.3
    1. Use the method of differences to prove that $$\sum_{n=k}^N \frac{1}{n(n+1)} = \frac{1}{k} - \frac{1}{N+1}.$$ [4]
    2. Deduce the value of \(\sum_{n=k}^{\infty} \frac{1}{n(n+1)}\) and show that \(\sum_{n=k}^{\infty} \frac{1}{(n+1)^2} < \frac{1}{k}\). [3]
  2. Let \(S = \sum_{n=1}^{\infty} \frac{1}{n^2}\). Show that \(\frac{205}{144} < S < \frac{241}{144}\). [3]
Pre-U Pre-U 9795/1 2013 November Q13
24 marks Hard +2.3
  1. Let \(I_n = \int_0^{\alpha} \cosh^n x \, dx\) for integers \(n \geqslant 0\), where \(\alpha = \ln 2\).
    1. Prove that, for \(n \geqslant 2\), \(nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}\). [5]
    2. A curve has parametric equations \(x = 12 \sinh t + 4 \sinh^3 t\), \(y = 3 \cosh^4 t\), \(0 \leqslant t \leqslant \ln 2\). Find the length of the arc of this curve, giving your answer in the form \(a + b \ln 2\) for rational numbers \(a\) and \(b\). [8]
  2. The circle with equation \(x^2 + (y - R)^2 = r^2\), where \(r < R\), is rotated through one revolution about the \(x\)-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of \(\pi\), \(R\) and \(r\), the surface area of this torus. [11]
Pre-U Pre-U 9794/3 2014 June Q1
5 marks Easy -1.3
The masses, in kilograms, of 100 chickens on sale in a large supermarket were recorded as follows.
Mass (\(x\) kg)\(1.6 \leqslant x < 1.8\)\(1.8 \leqslant x < 2.0\)\(2.0 \leqslant x < 2.2\)\(2.2 \leqslant x < 2.4\)\(2.4 \leqslant x < 2.6\)
Number of chickens1627281811
Calculate estimates of the mean and standard deviation of the masses of these chickens. [5]
Pre-U Pre-U 9794/3 2014 June Q2
5 marks Moderate -0.8
\(A\) and \(B\) are two events. You are given that \(\mathrm{P}(A) = 0.6\), \(\mathrm{P}(B) = 0.5\) and \(\mathrm{P}(A \cup B) = 0.8\).
  1. Find \(\mathrm{P}(A \cap B)\). [2]
  2. Find \(\mathrm{P}(B | A)\). [2]
  3. Explain whether the events \(A\) and \(B\) are independent or not. [1]
Pre-U Pre-U 9794/3 2014 June Q3
6 marks Moderate -0.3
A discrete random variable \(X\) has the following probability distribution.
\(x\)12\(n\)7
\(\mathrm{P}(X = x)\)0.40.3\(p\)0.1
  1. Write down the value of \(p\). [1]
  2. Given that \(\mathrm{E}(X) = 2.5\), find \(n\). [2]
  3. Find \(\mathrm{Var}(X)\). [3]
Pre-U Pre-U 9794/3 2014 June Q4
6 marks Moderate -0.8
In a certain country 40% of the population have brown eyes. A random sample of 20 people is chosen from that population.
  1. Find the expected number of people in the sample who have brown eyes. [1]
  2. Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
  3. Find the probability that there are at least 8 people with brown eyes in the sample. [2]
Pre-U Pre-U 9794/3 2014 June Q5
7 marks Moderate -0.8
There are 15 students enrolled in a Maths club.
  1. In how many ways is it possible to choose 4 of the students to take part in a competition? [2]
There are 4 different medals to be allocated, at random, to the students in the Maths club.
  1. If there are no restrictions about how many medals a student may receive, in how many ways can the medals be allocated? [2]
  2. Find the probability that no student receives more than one medal. [3]
Pre-U Pre-U 9794/3 2014 June Q6
11 marks Standard +0.3
A machine is being used to manufacture ball bearings. The diameters of the ball bearings are normally distributed with mean 8.3 mm and standard deviation 0.20 mm.
  1. Find the probability that the diameter of a randomly chosen ball bearing lies between 8.1 mm and 8.5 mm. [5]
  2. Following an overhaul of the machine, it is now found that the diameters of 88% of ball bearings are less than 8.5 mm while 10% are less than 8.1 mm. Estimate the new mean and standard deviation of the diameters. [6]
Pre-U Pre-U 9794/3 2014 June Q7
5 marks Easy -1.2
A stone is projected vertically upwards from ground level at a speed of \(30\,\mathrm{m}\,\mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]
Pre-U Pre-U 9794/3 2014 June Q8
6 marks Moderate -0.8
A particle is being held in equilibrium by the following set of forces (in newtons). $$\mathbf{F}_1 = 5\mathbf{i} - 8\mathbf{j}, \quad \mathbf{F}_2 = -3\mathbf{i} - 4\mathbf{j}, \quad \mathbf{F}_3 = 6\mathbf{i} + 6\mathbf{j} \quad \text{and} \quad \mathbf{F}_4.$$
  1. Find \(\mathbf{F}_4\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [2]
  2. Hence find the magnitude and direction of \(\mathbf{F}_4\). [4]
Pre-U Pre-U 9794/3 2014 June Q9
7 marks Moderate -0.3
A particle of mass \(m\) is placed on a rough inclined plane. The plane makes an angle \(\theta\) with the horizontal. The coefficient of friction between the particle and the plane is \(\mu\) where \(\mu < \tan \theta\). The particle is released from rest and accelerates down the plane.
  1. Draw a fully labelled diagram to show the forces acting on the particle. [1]
  2. Find an expression in terms of \(g\), \(\theta\) and \(\mu\) for the acceleration of the particle. [5]
  3. Explain what would happen to the particle if \(\mu > \tan \theta\). [1]
Pre-U Pre-U 9794/3 2014 June Q10
10 marks Standard +0.3
A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a\,\mathrm{m}\,\mathrm{s}^{-2}\) is given by \(a = 12 - 6t\).
  1. Find an expression in terms of \(t\) for its velocity \(v\,\mathrm{m}\,\mathrm{s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
  2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
  3. Find the velocity of \(P\) when it returns to \(O\). [3]
Pre-U Pre-U 9794/3 2014 June Q11
12 marks Standard +0.3
A light inextensible string passes over a smooth fixed pulley. Particles of mass 0.2 kg and 0.3 kg are attached to opposite ends of the string, so that the parts of the string not in contact with the pulley are vertical. The system is released from rest with the string taut.
  1. Find the acceleration of the particles and the tension in the string. [6]
When the heavier particle has fallen 2.25 m it hits the ground and is brought to rest (and the string goes slack).
  1. Find the speed with which it hits the ground. [2]
  2. Find the magnitude of the impulse of the ground on the particle. [2]
  3. If the impact between the particle and the ground lasts for 0.005 seconds, find the constant force that would be needed to bring the particle to rest. [2]
Pre-U Pre-U 9795/2 2014 June Q1
8 marks Standard +0.3
A machine is selecting independently and at random long rods and short rods. The length of the long rods, \(X\) cm, is normally distributed with mean 25 cm and variance 3 cm\(^2\) and the length of the short rods, \(Y\) cm, is normally distributed with mean 15 cm and variance 2 cm\(^2\). Assume that \(X\) and \(Y\) are independent random variables.
  1. One long rod and one short rod are chosen at random. Find the probability that the difference in the lengths, \(X - Y\), is between 8 cm and 11 cm. [4]
  2. Two long rods and two short rods are chosen at random and are assembled into an approximately rectangular frame. Find the probability that the perimeter of the resulting frame is more than 75 cm. [4]
Pre-U Pre-U 9795/2 2014 June Q2
8 marks Challenging +1.2
The mean of a random sample of \(n\) observations drawn from a normal distribution with mean \(\mu\) and variance \(\sigma^2\) is denoted by \(\bar{X}\). It is given that P(\(\mu - 0.5\sigma < \bar{X} < \mu + 0.5\sigma\)) > 0.95.
  1. Find the smallest possible value of \(n\). [5]
  2. With this value of \(n\), find P(\(\bar{X} > \mu - 0.1\sigma\)). [3]
Pre-U Pre-U 9795/2 2014 June Q3
8 marks Standard +0.8
A random sample of 400 seabirds is taken from a colony, ringed, and returned, unharmed, to the colony. After a suitable period of time has elapsed, a second random sample of 400 seabirds is taken, and 20 of this second sample are found to be ringed. You may assume that the probability that a seabird is captured is independent of whether or not it has been ringed and that the colony remains unchanged at the time of the second sampling.
  1. Estimate the number of seabirds in the colony. [1]
  2. Find a 98% confidence interval for the proportion of seabirds in the colony which are ringed. [5]
  3. Deduce a 98% confidence interval for the number of seabirds in the colony. [2]
Pre-U Pre-U 9795/2 2014 June Q4
10 marks Challenging +1.2
The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} 3e^{-x} & 0 \leq x \leq k, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(e^{-k} = \frac{2}{3}\). [2]
  2. Show that the moment generating function of \(X\) is given by \(M_X(t) = \frac{3}{1-t}\left(1 - \frac{2}{3}e^{kt}\right)\). [4]
  3. By expanding \(M_X(t)\) as a power series in \(t\), up to and including the term in \(t^2\), show that $$M_X(t) = 1 + (1 - 2k)t + (1 - 2k - k^2)t^2 + \ldots.$$ [3] [You may use the standard series for \((1-t)^{-1}\) and \(e^{kt}\) without proof.]
  4. Deduce that the exact value of E\((X)\) is \(1 - 2\ln\left(\frac{2}{3}\right)\). [1]
Pre-U Pre-U 9795/2 2014 June Q5
13 marks Standard +0.3
  1. The discrete random variable \(X\) has a Poisson distribution with mean \(\lambda\). Use the probability generating function for \(X\) to show that both the mean and the variance have the value \(\lambda\). [5]
  2. The number of eggs laid by a certain insect has a Poisson distribution with variance 250. Find, using a suitable approximation, the probability that between 230 and 260 (inclusive) eggs are laid. [5]
  3. An insect lays 250 eggs. The probability that any egg that is laid survives to maturity is 0.1. Use a suitable approximation to find the probability that more than 30 eggs survive to maturity. [3]
Pre-U Pre-U 9795/2 2014 June Q6
13 marks Challenging +1.2
The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{4}{\pi(1+x^2)} & 0 \leq x \leq 1, \\ 0 & \text{otherwise.} \end{cases}$$
  1. Verify that the median value of \(X\) lies between 0.41 and 0.42. [3]
  2. Show that E\((X) = \frac{2}{\pi}\ln 2\). [2]
  3. Find Var\((X)\). [5]
  4. Given that \(\tan\frac{1}{8}\pi = \sqrt{2} - 1\), find the exact value of P(\(X > \frac{1}{4}\sqrt{3}|X > \sqrt{2} - 1\)). [3]