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

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Pre-U Pre-U 9794/3 2017 June Q7
9 marks Moderate -0.8
7 A building 33.8 m high stands on horizontal ground. A particle is projected horizontally from the top of the building and hits the ground 31.2 m away.
  1. Find the initial speed of the particle.
  2. Find the magnitude and direction of the velocity of the particle when it hits the ground.
Pre-U Pre-U 9794/3 2017 June Q8
6 marks Standard +0.3
8 An object of weight 16 N is supported in equilibrium by a force of \(P \mathrm {~N}\) at \(30 ^ { \circ }\) to the vertical and by another of 10 N at \(\theta ^ { \circ }\) to the vertical as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85c5c346-8eb5-47ea-b94e-80b1a0038ce1-4_549_483_397_831}
  1. Draw a triangle to show that the forces acting on the object are in equilibrium.
  2. Find the two possible values of \(\theta\) and the corresponding values of \(P\).
Pre-U Pre-U 9794/3 2017 June Q9
8 marks Moderate -0.8
9 A particle moves along a straight line such that its displacement from \(O\), a fixed point on the line, is \(x\). The particle travels from rest from the point \(P\), where \(x = 2\), to the point \(Q\), where \(x = 5.6\). All distances are in metres. Two models for the motion of the particle are proposed.
  1. In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from \(P\) to \(Q\). Find the velocity of the particle when it reaches \(Q\).
  2. In Model 2, the velocity after \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)\).
    1. Write down the values of \(t\) when \(v = 0\).
    2. Show that \(x = 5.6\) when \(t = 18\).
    3. The particle represents a fragile instrument that is being moved from \(P\) to \(Q\) across a laboratory. Explain why Model 2 might be more appropriate than Model 1.
Pre-U Pre-U 9794/3 2017 June Q10
5 marks Moderate -0.8
10 A cyclist travelling at a steady speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) passes a bus which is at rest at a bus stop. 5 seconds later the bus sets off following the cyclist and accelerating at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\). How soon after setting off does the bus catch up with the cyclist? How fast is the bus going at this time? {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9794/3 2018 June Q2
9 marks Moderate -0.3
2 A teacher is monitoring the progress of students. The length of time, \(x\) hours, spent revising in a given week is compared to the score, \(y\), achieved in an assessment at the end of the week. The scatter diagram for a random sample of 8 students is shown below. \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-2_866_967_715_589} The data are summarised as \(\Sigma x = 24.6 , \Sigma y = 404 , \Sigma x ^ { 2 } = 105.56 , \Sigma y ^ { 2 } = 20820\) and \(\Sigma x y = 1350.2\).
  1. Find the equation of the least squares regression line of \(y\) on \(x\).
  2. Calculate the product moment correlation coefficient for the data.
  3. A ninth student, Jane, revises for 1.5 hours.
    1. Estimate her score in the assessment.
    2. Comment on the reliability of this estimate.
Pre-U Pre-U 9794/3 2018 June Q3
5 marks Easy -1.2
3 John plays a game with two unbiased coins. John tosses the coins. If he gets two heads he wins \(\pounds 1\). If he gets two tails he wins 20 p. If he gets one head and one tail he wins nothing. Let \(X\) be the random variable for the amount of money, in pence, John wins per game.
  1. Construct a probability distribution table for \(X\).
  2. Calculate \(\mathrm { E } ( X )\).
  3. John pays \(s\) pence to play the game. State the values of \(s\) for which John should expect to make a loss.
Pre-U Pre-U 9794/3 2018 June Q4
6 marks Moderate -0.3
4 On a particular day at a busy international airport, 75\% of the scheduled flights depart on time. A random sample of 16 flights is chosen.
  1. Find the expected number of flights that depart on time.
  2. For these 16 flights, find the probability that fewer than 14 flights depart on time.
  3. For these 16 flights, the probability that at least \(k\) flights depart on time is greater than 0.9 . Find the largest possible value of \(k\).
Pre-U Pre-U 9794/3 2018 June Q5
9 marks Standard +0.3
5 A soft drinks company has an automated bottling machine that fills 500 ml bottles with soft drink. The contents of the bottles are measured during a check on the machine. In the check, \(5 \%\) of the bottles contain more than 500 ml and \(2.5 \%\) contain less than 495 ml . It is given that the amount of drink dispensed per bottle is normally distributed.
  1. Find the mean and standard deviation of the amount of drink dispensed per bottle, giving your answers to 4 significant figures.
  2. It is subsequently found that the measurements of volume made in the checking process are all 3 ml below their true value. Using a corrected distribution, find the probability that a bottle chosen at random contains more than 500 ml of the drink.
Pre-U Pre-U 9794/3 2018 June Q6
12 marks Moderate -0.3
6 A volleyball squad has 11 players. A volleyball team consists of 6 players.
  1. Find the total number of different teams that could be chosen from the squad. The squad has 5 women and 6 men.
  2. Find the total number of different teams that contain at least 3 women. The squad includes a man and a woman who are married to one another.
  3. It is given that the team chosen has exactly 3 women and all such teams are equally likely to be chosen. Calculate the probability that a team chosen includes the married couple.
Pre-U Pre-U 9794/3 2018 June Q7
5 marks Moderate -0.8
7 A particle is projected with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle 0.4 s after projection.
Pre-U Pre-U 9794/3 2018 June Q8
7 marks Easy -1.2
8 A small ball is thrown vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point 5 m above the ground. Assuming air resistance is negligible, find
  1. the greatest height above the ground reached by the ball,
  2. the time taken for the ball to reach the ground.
Pre-U Pre-U 9794/3 2018 June Q9
5 marks Moderate -0.8
9 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal table. \(P\) collides directly with a stationary particle \(Q\) of mass 0.5 kg . This collision reverses the direction of motion of \(P\). Immediately after the collision the speed of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the value of \(m\),
  2. the coefficient of restitution between the two particles.
Pre-U Pre-U 9794/3 2018 June Q10
7 marks Standard +0.3
10 A particle \(P\) moves in a straight line starting from \(O\). At time \(t\) seconds after leaving \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 5 + 1.5 t - 0.125 t ^ { 3 }\).
  1. Find the displacement of \(P\) between the times \(t = 1\) and \(t = 4\).
  2. Find the time at which the velocity of \(P\) is a maximum, justifying your answer.
Pre-U Pre-U 9794/3 2018 June Q11
13 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-4_285_700_1043_721} Three forces are acting on a particle \(A\) as shown in the diagram. The forces act in the same plane and the particle is in equilibrium.
  1. Evaluate \(P\) and \(\theta\). The 8 N force is removed.
  2. State the direction of the instantaneous acceleration of \(A\).
Pre-U Pre-U 9794/3 2018 June Q12
12 marks Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635} The diagram shows a block \(B\) of mass 2 kg and a particle \(A\) of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley \(C\) at the edge of the plane and \(A\) hangs in equilibrium 1.2 m above horizontal ground. The part of the string between \(B\) and \(C\) is parallel to a line of greatest slope of the plane. \(B\) is released and begins to move up the plane.
  1. Show that the acceleration of \(A\) is \(3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures, and find the tension in the string.
  2. When \(A\) reaches the ground it remains there. Given that \(B\) does not reach \(C\) in the subsequent motion, find the total time that \(B\) is moving up the plane.
Pre-U Pre-U 9794/3 2013 November Q1
5 marks Easy -1.2
  1. Given that \(X \sim \text{Geo}\left(\frac{1}{6}\right)\), write down the values of E(\(X\)) and Var(\(X\)). [2]
  2. \(Y \sim \text{B}(n, p)\). Given that E(\(Y\)) = 4 and Var(\(Y\)) = \(\frac{8}{3}\), find the values of \(n\) and \(p\). [3]
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]