Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/3 2013 November Q2
7 marks Moderate -0.3
The random variable \(X\) is defined as the difference (always positive or zero) between the scores when 2 ordinary dice are rolled.
  1. Copy and complete the probability distribution table for \(X\). [2]
    \(x\)012345
    P(\(X = x\))
  2. Find the expectation and variance of \(X\). [5]
Pre-U Pre-U 9794/3 2013 November Q3
5 marks Moderate -0.8
In a large examination room each candidate has just one electronic calculator.
  • \(G\) is the event that a randomly chosen candidate has a graphical calculator.
  • \(T\) is the event that a randomly chosen candidate has a 'Texio' brand calculator.
You are given the following probabilities. $$\text{P}(G) = 0.65 \quad \text{P}(T) = 0.4 \quad \text{P}(G \cap T) = 0.25$$
  1. Are the events \(G\) and \(T\) independent? Justify your answer with an appropriate calculation. [2]
  2. Find P(\(T | G\)) and explain, in the context of this question, what this probability represents. [3]
Pre-U Pre-U 9794/3 2013 November Q4
6 marks Moderate -0.8
As part of a study into the effects of alcohol, volunteers have their reaction times measured after they have consumed various fixed amounts of alcohol. For a random sample of 12 volunteers the following information was collected.
Units of alcohol consumed23344.55.5667889
Reaction time (seconds)12553.85.54.88.57.26.898
  1. Which is the independent variable in this experiment? [1]
  2. Find the least squares regression line of \(y\) (Reaction time) on \(x\) (Units of alcohol), and use it to estimate the reaction time of someone who has consumed 5 units of alcohol. [5]
Pre-U Pre-U 9794/3 2013 November Q5
8 marks Moderate -0.8
The table summarises 43 birth weights as recorded for babies born in a particular hospital during one week.
Birth weight (w kg)\(2.0 \leqslant w < 2.5\)\(2.5 \leqslant w < 3.0\)\(3.0 \leqslant w < 3.5\)\(3.5 \leqslant w < 4.0\)\(4.0 \leqslant w < 4.5\)
Frequency1691710
  1. State the type of skewness of the data. [1]
  2. Given that the lower quartile is 3.21 kg and the upper quartile is 3.96 kg, determine whether there are any babies whose birth weights might be regarded as outliers. [4]
  3. The mean birth weight was found to be 3.58 kg. However, it was discovered subsequently that the table includes the birth weight, 2.52 kg, of one baby that has been recorded twice. Find the mean birth weight after this error has been removed. [3]
Pre-U Pre-U 9794/3 2013 November Q6
9 marks Moderate -0.3
A company supplies tubs of coleslaw to a large supermarket chain. According to the labels on the tubs, each tub contains 300 grams of coleslaw. In practice the weights of coleslaw in the tubs are normally distributed with mean 305 grams and standard deviation 6 grams.
  1. Find the proportion of tubs that are underweight, according to the label. [3]
The supermarket chain requires that the proportion of underweight tubs should be reduced to 5%.
  1. If the standard deviation is kept at 6 grams, find the new mean weight needed to achieve the required reduction. [3]
  2. If the mean weight is kept at 305 grams, find the new standard deviation needed to achieve the required reduction. Explain why the company might prefer to adjust the standard deviation rather than the mean. [3]
Pre-U Pre-U 9794/3 2013 November Q7
6 marks Moderate -0.8
10 seconds after passing a warning signal, a train is travelling at 18 m s\(^{-1}\) and has gone 215 m beyond the signal. Find the acceleration (assumed to be constant) of the train during the 10 seconds and its velocity as it passed the signal. [6]
Pre-U Pre-U 9794/3 2013 November Q8
7 marks Moderate -0.3
A ball of mass 0.04 kg is released from rest at a height of 1 metre above a table. It rebounds to a height of 0.81 metre.
  1. Find the value of \(e\), the coefficient of restitution. [4]
  2. Find the impulse on the ball when it hits the table. [3]
Pre-U Pre-U 9794/3 2013 November Q9
9 marks Moderate -0.3
A tennis ball is served horizontally at a speed of 24 m s\(^{-1}\) from a height of 2.45 m above the ground.
  1. Show that it will clear the net at a point where the net is 1 m high and 12 m from the server. [5]
  2. How far beyond the net will it land? [4]
Pre-U Pre-U 9794/3 2013 November Q10
5 marks Standard +0.3
A parcel \(P\) of weight 50 N is being held in equilibrium by two light, inextensible strings \(AP\) and \(BP\). The string \(AP\) is attached to a wall at \(A\), and string \(BP\) passes over a smooth pulley which is at the same height as \(A\), as shown in the diagram. When the tension in \(BP\) is 40 N, the strings are at right angles to each other. \includegraphics{figure_10}
  1. Find the tension in string \(AP\). [4]
  2. Explain why the parcel can never be in equilibrium with both strings horizontal. [1]
Pre-U Pre-U 9794/3 2013 November Q11
13 marks Standard +0.3
Two particles, \(A\) and \(B\), each of mass 1 kg are connected by a light inextensible string. Particle \(A\) is at rest on a slope inclined at 30° to the horizontal. The string passes over a small smooth pulley at the top of the slope and particle \(B\) hangs freely, as shown in the diagram. \includegraphics{figure_11}
    1. In the case when the slope is smooth, draw a fully labelled diagram to show the forces acting on the particles. Hence find the acceleration of the particles and the tension in the string. [7]
    2. Write down the direction of the resultant force exerted by the string on the pulley. [1]
  2. In fact the contact between particle \(A\) and the slope is rough. The coefficient of friction between \(A\) and the slope is \(\mu\). The system is in equilibrium. Find the set of possible values of \(\mu\). [5]
Pre-U Pre-U 9795/1 2013 November Q1
4 marks Moderate -0.8
For real values of \(t\), the non-singular matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A}^{-1} = \begin{pmatrix} t & 5 \\ 2 & 8 \end{pmatrix} \quad \text{and} \quad \mathbf{B}^{-1} = \begin{pmatrix} 2 & -t \\ 3 & -1 \end{pmatrix}.$$
  1. Determine the values which \(t\) cannot take. [2]
  2. Without finding either \(\mathbf{A}\) or \(\mathbf{B}\), determine \((\mathbf{AB})^{-1}\) in terms of \(t\). [2]
Pre-U Pre-U 9795/1 2013 November Q2
5 marks Standard +0.3
Use de Moivre's theorem to express \(\cos 3\theta\) in terms of powers of \(\cos \theta\) only, and deduce the identity \(\cos 6x \equiv \cos 2x(2\cos 4x - 1)\). [5]
Pre-U Pre-U 9795/1 2013 November Q3
7 marks Standard +0.3
The curve \(C\) has equation \(y = \frac{2x}{x^2 + 1}\).
  1. Write down the equation of the asymptote of \(C\) and the coordinates of any points where \(C\) meets the coordinate axes. [2]
  2. Show that the curve meets the line \(y = k\) if and only if \(-1 \leqslant k \leqslant 1\). Deduce the coordinates of the turning points of the curve. [5]
[Note: You are NOT required to sketch \(C\).]
Pre-U Pre-U 9795/1 2013 November Q4
4 marks Standard +0.8
Let \(f(n) = 2(5^{n-1} + 1)\) for integers \(n = 1, 2, 3, \ldots\).
  1. Prove that, if \(f(n)\) is divisible by 8, then \(f(n + 1)\) is also divisible by 8. [3]
  2. Explain why this result does not imply that the statement '\(f(n)\) is divisible by 8 for all positive integers \(n\)' follows by mathematical induction. [1]
Pre-U Pre-U 9795/1 2013 November Q5
8 marks Challenging +1.2
The curve \(S\) has polar equation \(r = 1 + \sin \theta + \sin^2 \theta\) for \(0 \leqslant \theta < 2\pi\).
  1. Determine the polar coordinates of the points on \(S\) where \(\frac{dr}{d\theta} = 0\). [5]
  2. Sketch \(S\). [3]
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]