Questions — Pre-U Pre-U 9794/3 (125 questions)

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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 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 9794/3 2014 June Q1
5 marks Easy -1.8
The masses, in kilograms, of 100 chickens on sale in a large supermarket were recorded as follows.
Mass (\(x\) kg)\(1.6 \leq x < 1.8\)\(1.8 \leq x < 2.0\)\(2.0 \leq x < 2.2\)\(2.2 \leq x < 2.4\)\(2.4 \leq x < 2.6\)
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 Q7
5 marks Moderate -0.8
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 Q10
10 marks Moderate -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 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 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 Moderate -0.8
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 Q10
10 marks Moderate -0.8
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 2016 June Q1
4 marks Moderate -0.8
The following data refer to the annual rate of inflation and the annual percentage pay increase measured on 10 randomly chosen occasions.
Inflation rate (\%)0.91.21.61.51.73.04.13.72.84.2
Pay increase (\%)4.84.73.84.45.65.52.40.40.61.7
Show that, for these data, the product moment correlation coefficient between the rate of inflation and the annual pay increase is \(-0.679\), correct to 3 significant figures. [4]
Pre-U Pre-U 9794/3 2016 June Q2
8 marks Moderate -0.8
The weights of pineapples on sale at a wholesaler are normally distributed with mean \(1.349\) kg and standard deviation \(0.236\) kg. Before going on sale the pineapples are classified as 'Small', 'Medium', 'Large' and 'Extra Large'.
  1. A pineapple is classified as 'Small' if it weighs less than \(1.100\) kg. Find the probability that a randomly chosen pineapple will be classified as 'Small'. [5]
  2. \(10\%\) of pineapples are classified as 'Extra Large'. Find the minimum weight required for a pineapple to be classified as 'Extra Large'. [3]
Pre-U Pre-U 9794/3 2016 June Q3
11 marks Moderate -0.3
Chris plays for his local hockey club. In his first 20 games for the club, the mean number of goals per game he has scored is \(0.7\), with a standard deviation of \(0.9\). In the next 5 games he scores \(0, 1, 0, 2, 1\) goals.
  1. Find the mean and standard deviation for the number of goals per game Chris has scored in all 25 games. [7]
  2. A sponsor pays Chris £65 each time he plays for the club and a further £25 for each goal he scores. Find the mean and standard deviation of the amount per game he earns from the sponsor for all 25 games. [4]
Pre-U Pre-U 9794/3 2016 June Q4
8 marks Moderate -0.3
A certain type of sweet is made in a variety of colours. \(20\%\) of the sweets made are blue. Sweets of the various colours are thoroughly mixed before being put into packets.
  1. In a packet that contains 10 sweets, find the probability that the packet contains
    1. at most 3 blue sweets, [1]
    2. exactly 3 blue sweets, [2]
    3. at least 1 blue sweet. [2]
  2. What is the smallest number of sweets that a packet should contain in order to be at least \(95\%\) certain of having at least 1 blue sweet? [3]
Pre-U Pre-U 9794/3 2016 June Q5
4 marks Moderate -0.3
The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]
Pre-U Pre-U 9794/3 2016 June Q6
5 marks Moderate -0.8
\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\). Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]