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Pre-U Pre-U 9794/3 2020 Specimen Q5
11 marks Moderate -0.3
James plays an arcade game. Each time he plays, he puts a £1 coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a £1 coin, James wins the game with a probability of 0.05 and the machine pays out ten £1 coins. The outcomes can be modelled by a random variable \(X\) representing the number of £1 coins gained at the end of a game.
  1. Construct a probability distribution table for \(X\). [2]
  2. Show that E(\(X\)) = -0.25 and find Var(\(X\)). [4]
James starts off with 10 £1 coins and decides to play exactly 10 games.
  1. Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
  2. Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q6
6 marks Easy -1.3
\includegraphics{figure_6} The diagram shows two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) acting at the origin \(O\) of rectangular coordinates \(Oxy\). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are -5 N and 7 N respectively.
  1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
  2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis. [4]
Pre-U Pre-U 9794/3 2020 Specimen Q7
6 marks Moderate -0.3
A particle travels along a straight line. Its velocity \(v\) ms\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).
  1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
  2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]
Pre-U Pre-U 9794/3 2020 Specimen Q8
6 marks Moderate -0.8
Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics{figure_8} The acceleration of the system is 0.3 ms\(^{-2}\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). [2]
Truck \(S\) is now subjected to an extra resistive force of 1800 N. The pulling force, \(P\), does not change.
  1. Calculate the new acceleration of the trucks. [2]
  2. Calculate the force in the coupling between the trucks. [2]
Pre-U Pre-U 9794/3 2020 Specimen Q9
10 marks Challenging +1.2
\includegraphics{figure_9} Three particles \(A\), \(B\) and \(C\), having masses of 1 kg, 2 kg and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed 14 ms\(^{-1}\). The coefficient of restitution between each pair of particles is 0.5.
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest. [4]
  2. Show that \(B\) reversed direction after the impact with \(C\). [3]
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q10
12 marks Challenging +1.8
\includegraphics{figure_10} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha < 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction R between \(A\) and the plane is mg(2\(\cos\alpha - \sin\alpha\)). [3]
  2. Show that R \(\geqslant \frac{1}{2}mg\sqrt{2}\). [3]
The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
  1. Show that \(0.5 < \tan\alpha < 1\). [3]
  2. Express \(\mu\) as a function of \(\tan\alpha\) and deduce its maximum value as \(\alpha\) varies. [3]
Pre-U Pre-U 9795/2 Specimen Q1
8 marks Challenging +1.8
A smooth sphere \(A\) of mass \(m\) is projected with speed \(u\) along a smooth horizontal surface and strikes a stationary smooth sphere \(B\) of equal radius but of mass \(M\). The direction of motion of \(A\) before the impact makes an acute angle \(\theta\) with the line of centres at the moment of contact. After the impact, the direction of motion of \(A\) is perpendicular to the initial direction of motion of \(A\). The coefficient of restitution between the two spheres is \(e\). Given that \(Me \geq m\), prove that $$\tan^2 \theta = \frac{Me - m}{m + M}.$$ [8]
Pre-U Pre-U 9795/2 Specimen Q2
9 marks Standard +0.8
One end of a light inextensible string of length \(l\) is attached to a fixed point \(O\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs at rest vertically below \(O\). The particle is then given a horizontal speed \(u\).
    1. Show that when \(OP\) has turned through an angle \(\theta\) the tension in the string is given by $$T = mg(3\cos \theta - 2) + \frac{mu^2}{l}$$ as long as the string remains taut. [5]
    2. Deduce that \(u^2 \geq 5gl\) in order for the particle to perform complete circles. [1]
    1. In the case \(u^2 = 3gl\), find the angle that \(OP\) makes with the downward vertical at \(O\) at the instant when the string becomes slack. [2]
    2. Describe the nature of the motion while the string is slack. [1]
Pre-U Pre-U 9795/2 Specimen Q3
11 marks Standard +0.8
A stone of mass \(m\) is projected vertically upwards with initial velocity \(u\). At time \(t\), the height risen above the point of projection is \(x\) and the resistance to motion is \(kv\) when the velocity of the stone is \(v\).
  1. Write down a first-order differential equation relating \(v\) and \(t\) and hence find \(t\) in terms of \(v\). [5]
  2. Write down a first-order differential equation relating \(v\) and \(x\) and hence find \(x\) in terms of \(v\). [6]
Pre-U Pre-U 9795/2 Specimen Q4
12 marks Challenging +1.2
A particle is projected with velocity \(V\) at an angle \(\alpha\) to the horizontal up a plane inclined at \(\beta\) to the horizontal, where \(\alpha > \beta\).
  1. Show that the time of flight is \(\frac{2V \sin(\alpha - \beta)}{g \cos \beta}\). [3]
  2. Show that the range on the inclined plane is \(\frac{2V^2 \sin(\alpha - \beta) \cos \alpha}{g \cos^2 \beta}\). [4]
  3. If the particle strikes the plane at right angles, prove that \(\tan \alpha = \cot \beta + 2 \tan \beta\). [5]
Pre-U Pre-U 9795/2 Specimen Q5
8 marks Moderate -0.3
A girl can paddle her canoe at \(5 \text{ m s}^{-1}\) in still water. She wishes to cross a river which is \(100 \text{ m}\) wide and flowing at \(8 \text{ m s}^{-1}\).
    1. Write down the angle to the river bank at which the boat must head, in order to cross the river in the least possible time. [1]
    2. Find the acute angle to the river bank at which the boat must head, in order to cross the river by the shortest route. [4]
  2. Calculate the times taken for each of the two cases in part (i). [3]
Pre-U Pre-U 9795/2 Specimen Q6
12 marks Challenging +1.8
A light elastic string of natural length \(2a\) and modulus of elasticity \(\lambda\) is stretched between two points \(A\) and \(B\), which are \(3a\) apart on a smooth horizontal table. A particle of mass \(m\) is attached to the mid-point of the string, pulled aside to \(A\) and released.
  1. Prove that, while one part of the string is taut and the other part is slack, the particle is describing simple harmonic motion. [2]
  2. Find the speed of the particle when the slack part of the string becomes taut. [2]
  3. Prove that the total time for the particle to reach the mid-point of the string for the first time is $$\sqrt{\frac{ma}{\lambda}} \left( \frac{\pi}{3} + \frac{1}{\sqrt{2}} \sin^{-1} \frac{1}{\sqrt{7}} \right).$$ [8]
Pre-U Pre-U 9795/2 Specimen Q7
6 marks Challenging +1.2
The length \(M\) of male snakes of a certain species may be regarded as a normal random variable with mean \(0.45\) metres and standard deviation \(0.06\) metres. The length \(F\) of female snakes of the same species may be regarded as a normal random variable with mean \(0.55\) metres and standard deviation \(0.08\) metres. Assuming that \(M\) and \(F\) are independent, find the probability that a randomly chosen male snake of this species is more than three-quarters of the length of a randomly chosen female snake of this species. [6]
Pre-U Pre-U 9795/2 Specimen Q8
9 marks Standard +0.3
  1. The random variable \(X\) is such that \(\text{E}(X) = a\theta + b\), where \(a\) and \(b\) are constants and \(\theta\) is a parameter. Show that \(\frac{X - b}{a}\) is an unbiased estimator of \(\theta\). [2]
  2. The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{8}(\theta + 4 - x) & \theta \leq x \leq \theta + 4, \\ 0 & \text{otherwise}. \end{cases}$$ Find \(\text{E}(X)\) and hence find an unbiased estimator of \(\theta\). [7]
Pre-U Pre-U 9795/2 Specimen Q9
10 marks Standard +0.3
A certain type of fossil occurs at a mean rate of \(0.5\) per square metre at a particular location.
  1. State an assumption that must be made so that the above situation can be modelled by a Poisson distribution. [1]
  2. Find the probability of at least 7 of these fossils occurring in an area of \(10 \text{ m}^2\). [2]
  3. Given that at least 4 such fossils have occurred in an area of \(5 \text{ m}^2\), find the probability that there will be more than 6 found in this area of \(5 \text{ m}^2\). [3]
  4. Find the least area that must be searched in order that the probability of finding at least one fossil of this type is greater than \(0.999\). Give your answer to the nearest square metre. [4]
Pre-U Pre-U 9795/2 Specimen Q10
10 marks Standard +0.3
A biased tetrahedral die has faces numbered \(1\) to \(4\). The random variable \(X\) is the number on the face of the die which is in contact with the table after the die has been thrown. It is known, for this die, that \(\text{P}(X = x) = kx\) where \(k\) is a constant.
  1. Determine the value of \(k\) and state the moment generating function of \(X\). [3]
  2. Hence find \(\text{E}(X)\) and \(\text{Var}(X)\). [7]
Pre-U Pre-U 9795/2 Specimen Q11
12 marks Standard +0.3
  1. State briefly the conditions under which the binomial distribution \(\text{B}(n, p)\) may be approximated by a normal distribution. [2]
  2. A multiple-choice test has \(50\) questions. Each question has four possible answers. A student passes the test if answering \(36\%\) or more of the questions correctly. Using a suitable distributional approximation, estimate the probability that a student who selects answers to all the questions randomly will pass the test. [5]
  3. A test similar to that in part (ii) has \(N\) questions instead of \(50\) questions. Estimate the least value of \(N\) so that the probability that a student gets \(36\%\) or more of the questions correct, by selecting answers to all questions randomly, will be less than \(0.01\). (A continuity correction is not required in this part of the question.) [5]
Pre-U Pre-U 9795/2 Specimen Q12
13 marks Standard +0.8
The length of time, in years, that a salesman keeps his company car may be modelled by the continuous random variable \(T\). The probability density function of \(T\) is given by $$f(t) = \begin{cases} \frac{2}{5}t e^{-\frac{t^2}{5}} & t \geq 0, \\ 0 & \text{otherwise}. \end{cases}$$
  1. Sketch the graph of \(f(t)\). [2]
  2. Find the cumulative distribution function \(F(t)\) and hence find the median value of \(T\). [3]
  3. Find the probability that \(T\) is greater than the modal value of \(T\). [5]
  4. The probability that a randomly chosen salesman keeps his car longer than \(N\) years is \(0.05\). Find the value of \(N\) correct to \(3\) significant figures. [3]
Pre-U Pre-U 9794/2 Specimen Q1
4 marks Moderate -0.3
  1. Show that \(\binom{n}{n-2} = \frac{n(n-1)}{2}\), where the positive integer \(n\) satisfies \(n \geqslant 2\). [1]
  2. Solve the equation \(\binom{2n+1}{2n-1} - 2 \times \binom{n}{n-2} = 24\). [3]
Pre-U Pre-U 9794/2 Specimen Q2
4 marks Standard +0.3
Solve the simultaneous equations $$x - 2y = 5,$$ $$\frac{4}{x} - \frac{2}{y} = 5.$$ [4]
Pre-U Pre-U 9794/2 Specimen Q3
5 marks Standard +0.8
The equation of a curve is \(y = x^{\frac{3}{2}} \ln x\). Find the exact coordinates of the stationary point on the curve. [5]
Pre-U Pre-U 9794/2 Specimen Q4
7 marks Standard +0.3
A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).
  1. Determine the area of the smaller segment. [4]
The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.
  1. Find the equation of the line in its final position. [3]
Pre-U Pre-U 9794/2 Specimen Q5
9 marks Standard +0.3
  1. Divide the quartic \(2x^4 - 5x^3 + 4x^2 + 2x - 3\) by the quadratic \(x^2 + x - 2\), identifying the quotient and the remainder. [4]
    1. Show that \((x - 1)\) is a factor of \(nx^{n+1} - (n + 1)x^n + 1\), where \(n\) is a positive integer. [1]
    2. Hence, or otherwise, find all the roots of \(3x^4 - 4x^3 + 1 = 0\). [4]
Pre-U Pre-U 9794/2 Specimen Q6
10 marks Standard +0.8
  1. Express \(y^3 - 3y - 2\) in terms of \(x\), where \(x = y + 1\). [1]
  2. Hence express $$\frac{2y + 5}{y^3 - 3y - 2}$$ in partial fractions. [5]
  3. Find the exact value of $$\int_0^1 \frac{2y + 5}{y^3 - 3y - 2} dy.$$ [4]
Pre-U Pre-U 9794/2 Specimen Q7
12 marks Moderate -0.3
  1. Given that \(\cos \theta = \frac{7}{25}\), where \(\frac{3}{2}\pi < \theta < 2\pi\), determine the exact values of
    1. \(\sin \theta\), [3]
    2. \(\sin(\frac{1}{2}\theta)\), [3]
    3. \(\sec(\frac{1}{2}\theta)\). [1]
    1. Express \(4 \cos x - 3 \sin x\) in the form \(A \cos(x + \alpha)\), where \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [2]
    2. Hence find the greatest and least values of \(4 \cos x - 3 \sin x\) for \(0 \leqslant x \leqslant \pi\). [3]