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

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Pre-U Pre-U 9794/3 2012 June Q1
4 marks Easy -1.8
1 The heights in centimetres of 10 young women were measured and are given below. $$\begin{array} { l l l l l l l l l l } 140 & 145 & 162 & 174 & 153 & 167 & 147 & 151 & 148 & 156 \end{array}$$ Calculate the mean height of these women and show that the standard deviation is approximately 10 cm .
Pre-U Pre-U 9794/3 2012 June Q2
5 marks Moderate -0.8
2 A bag contains four black balls and one white ball. A man chooses a ball at random. If it is a black ball, he replaces it and chooses another at random. If he chooses the white ball, he stops.
  1. Name the probability distribution which models this situation.
  2. Calculate the probability that he will make exactly three attempts before he stops.
  3. Calculate the probability that he will make fewer than three attempts before he stops.
Pre-U Pre-U 9794/3 2012 June Q3
4 marks Easy -1.8
3 The lengths of snakes on a tropical island were measured and found to be normally distributed with a mean of 160 cm and a standard deviation of 6 cm . Find the probability that a randomly selected snake has a length of less than 170 cm .
Pre-U Pre-U 9794/3 2012 June Q4
6 marks Easy -1.2
4 In one department of a firm, four employees are selected for promotion from a staff of eighteen.
  1. In how many ways can four employees be selected? It is known that throughout the firm 5\% of those selected for promotion decline it.
  2. If 100 employees are randomly selected for promotion in the firm, calculate the number expected to decline promotion.
  3. If 20 employees are selected at random for promotion, use the binomial distribution to find the probability that fewer than five employees will decline promotion.
Pre-U Pre-U 9794/3 2012 June Q5
10 marks Moderate -0.8
5 In an archery competition, competitors are allowed up to three attempts to hit the bulls-eye. No one who succeeds may try again. \(45 \%\) of those entering the competition hit the bulls-eye first time. For those who fail to hit it the first time, \(60 \%\) of those attempting it for the second time succeed in hitting it. For those who fail twice, only \(15 \%\) of those attempting it for the third time succeed in hitting it. By drawing a tree diagram, or otherwise,
  1. find the probability that a randomly chosen competitor fails at all three attempts,
  2. find the probability that a randomly chosen competitor fails at the first attempt but succeeds at either the second or third attempt,
  3. find the probability that a randomly chosen competitor succeeds in hitting the bulls-eye,
  4. find the probability that a randomly chosen competitor requires exactly two attempts given that the competitor is successful.
Pre-U Pre-U 9794/3 2012 June Q6
11 marks Moderate -0.3
6 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) 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 \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.
  1. Construct a probability distribution table for \(X\).
  2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
  3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
  4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.
Pre-U Pre-U 9794/3 2012 June Q7
7 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-3_343_401_1439_872} The diagram shows two forces of magnitudes 10 N and 15 N acting in a horizontal plane on a particle \(P\).
  1. Find the component of the 15 N force which is parallel to the 10 N force.
  2. Write down the component of the 15 N force which is perpendicular to the 10 N force.
  3. Hence, or otherwise, calculate the magnitude and direction of the resultant force on \(P\).
Pre-U Pre-U 9794/3 2012 June Q8
4 marks Moderate -0.8
8 A crane lifts a crate of mass 20 kg using a light inextensible cable. The crate starts from rest and ascends 10 metres in 4 seconds during which time a constant tension of \(T \mathrm {~N}\) is applied in the cable. Find the value of \(T\).
Pre-U Pre-U 9794/3 2012 June Q9
6 marks Moderate -0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_430_565_260_790} The diagram shows a block of wood, weighing 100 N , at rest on a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 . A force of \(P \mathrm {~N}\) acts on the block up the slope.
  1. Find the maximum possible value of the friction acting on the block.
  2. Given that the block is on the point of moving up the slope, find \(P\).
  3. Given that the block is on the point of moving down the slope, find \(P\).
Pre-U Pre-U 9794/3 2012 June Q10
10 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_81_949_1283_598} Three particles \(A , B\) and \(C\), having masses \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) 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 \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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.
  2. Show that \(B\) reverses direction after an impact with \(C\).
  3. Find the distance between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2012 June Q11
13 marks Standard +0.3
11 A particle \(P\) of mass 2 kg can move along a line of greatest slope on the smooth surface of a wedge which is fixed to the ground. The sloping face \(O A\) of the wedge has length 10 metres and is inclined at \(30 ^ { \circ }\) to the horizontal (see Fig. 1). \(P\) is fired up the slope from the lowest point \(O\), with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_295_1529_484_310} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the time taken for \(P\) to reach \(A\) and show that the speed of \(P\) at \(A\) is \(10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(P\) has reached \(A\) it becomes a projectile (see Fig. 2). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_424_1533_1123_306} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Find the total horizontal distance travelled by \(P\) from \(O\) when it hits the ground.
Pre-U Pre-U 9794/3 2013 June Q1
4 marks Easy -1.3
1 Pupils at a certain school carried out a survey of traffic passing the school during a two-hour period one morning. One pupil recorded the number of people in each of the first 100 cars. Her results were as follows.
Number of people12345
Number of cars482614102
Find the mean and the standard deviation of the number of people per car in her sample.
Pre-U Pre-U 9794/3 2013 June Q2
4 marks Moderate -0.8
2 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }\) and \(\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 }\).
Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\mathrm { P } ( B )\).
Pre-U Pre-U 9794/3 2013 June Q3
12 marks Moderate -0.8
3 At a local athletics club, data on the ages of the members and their times to run a 10 km course are recorded. For a random sample of 25 club members aged between 20 and 60, their ages ( \(x\) years) and times ( \(y\) minutes) are summarised as follows. $$n = 25 \quad \Sigma x = 1002 \quad \Sigma x ^ { 2 } = 43508 \quad \Sigma y = 1865 \quad \Sigma y ^ { 2 } = 142749 \quad \Sigma x y = 77532$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Show that the equation of the least squares regression line of \(y\) on \(x\) is \(y = 0.83 x + 41.28\), where the coefficients are given correct to 2 decimal places.
  3. Use the equation given in part (ii) to estimate the time taken by someone who is
    1. 50 years old,
    2. 65 years old. Comment on the validity of each of these estimates.
Pre-U Pre-U 9794/3 2013 June Q4
10 marks Standard +0.3
4 A tomato grower grows just one variety of tomatoes. The weights of these tomatoes are found to be normally distributed with a mean of 85.1 grams and a standard deviation of 3.4 grams.
  1. Find the probability that a randomly chosen tomato of this variety weighs less than 80 grams.
  2. The grower puts the tomatoes in packs of 6 . Find the probability that, in a randomly chosen pack of 6 , at most one tomato weighs less than 80 grams.
  3. The grower supplies consignments of 250 packs of these tomatoes to a retailer. For a randomly chosen consignment, find the expected number of packs having more than one tomato weighing less than 80 grams.
Pre-U Pre-U 9794/3 2013 June Q5
10 marks Standard +0.3
5 A game is played with cards, each of which has a single digit printed on it. Eleanor has 7 cards with the digits \(1,1,2,3,4,5,6\) on them.
  1. How many different 7-digit numbers can be made by arranging Eleanor's cards?
  2. Eleanor is going to select 5 of the 7 cards and use them to form a 5 -digit number. How many different 5-digit numbers are possible?
Pre-U Pre-U 9794/3 2013 June Q6
13 marks Moderate -0.3
6 A particle travels along a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 6 t ^ { 2 } + 8 t \text { for } 0 \leqslant t \leqslant 4$$ When \(t = 0\) the particle is at rest at the point \(P\).
  1. Find the times (other than \(t = 0\) ) when the particle is at rest. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
  2. Find the acceleration of the particle when \(t = 2\).
  3. Find an expression for the displacement of the particle from \(P\) after \(t\) seconds. Hence state its displacement from \(P\) when \(t = 2\) and find its average speed between \(t = 0\) and \(t = 2\).
Pre-U Pre-U 9794/3 2013 June Q7
8 marks Standard +0.3
7 A particle \(A\) of mass \(4 m\), on a smooth horizontal plane, is moving with speed \(u\) directly towards another particle \(B\), of mass \(2 m\), which is at rest. The coefficient of restitution between the two particles is \(e\).
  1. Show that, after the collision, the velocity of \(A\) is \(\frac { 1 } { 3 } ( 2 - e ) u\) and find the velocity of \(B\).
  2. Hence write down their velocities in the case when \(e = \frac { 1 } { 2 }\). Particle \(B\) now collides directly with a third particle \(C\), of mass \(m\), which is at rest. The coefficient of restitution in both collisions is \(\frac { 1 } { 2 }\).
  3. Use your answers to part (ii) to find the velocities of \(A , B\) and \(C\) after the second collision has taken place.
  4. Explain briefly whether any further collisions take place.
Pre-U Pre-U 9794/3 2013 June Q8
10 marks Standard +0.3
8 A particle is projected from a point \(O\) with initial speed \(U\) at an angle \(\theta\) above the horizontal. At time \(t\) after projection the position of the particle is \(( x , y )\) relative to horizontal and vertical axes through \(O\).
  1. Write down expressions for \(x\) and \(y\) at time \(t\). Hence derive the cartesian equation of the trajectory of the particle.
  2. A player in a cricket match throws the ball with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to another player who is 45 metres away. Assume that the players throw and catch the ball at the same height above the ground. Show that there are two possible trajectories and find their respective angles of projection. [4]
  3. Describe briefly one advantage of each trajectory.
Pre-U Pre-U 9794/3 2013 June Q9
9 marks Standard +0.3
9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}
  1. Copy and complete the diagram to show all the forces acting on the particle.
  2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
  3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.
Pre-U Pre-U 9794/3 2015 June Q1
5 marks Moderate -0.8
1 The information below summarises the percentages of males unemployed ( \(x\) ) and the percentages of females unemployed ( \(y\) ) in 10 different locations in the UK. $$n = 10 \quad \Sigma x = 87.6 \quad \Sigma x ^ { 2 } = 804.34 \quad \Sigma y = 76.4 \quad \Sigma y ^ { 2 } = 596 \quad \Sigma x y = 684.02$$ Find the product-moment correlation coefficient for these data.
Pre-U Pre-U 9794/3 2015 June Q2
6 marks Standard +0.3
2 Jill is collecting picture cards given away in packets of a particular brand of breakfast cereal. There are five different cards in the complete set. Each packet contains one card which is equally likely to be any of the five cards in the set.
  1. Find the probability that Jill has a complete set of cards from the first five packets that she buys.
  2. At some point Jill needs just one more card to complete the set. Let \(X\) be the random variable that represents the number of additional packets that Jill will need to buy in order to complete the set.
    1. Write down the distribution of \(X\).
    2. State the expected number of additional packets that Jill will need to buy.
    3. Find the probability that Jill will need to buy at least 3 additional packets in order to complete the set.
Pre-U Pre-U 9794/3 2015 June Q3
4 marks Moderate -0.5
3 Jack's journey time, in minutes, to work each morning is modelled by the normal distribution \(\mathrm { N } \left( 43.2,6.3 ^ { 2 } \right)\).
  1. If Jack leaves home at 0810 , find the probability that he arrives at work by 0900 .
  2. Find the time by which Jack should leave home in order to be at least \(95 \%\) certain that he arrives at work by 0900 .
Pre-U Pre-U 9794/3 2015 June Q4
9 marks Moderate -0.3
4 At a sixth form college, the student council has 16 members made up as follows. There are 3 male and 3 female students from Year 12, and 6 male and 4 female students from Year 13. Two members of the council are chosen at random to represent the college at conference. Find the probability that the 2 members chosen are
  1. the same sex,
  2. the same sex and from the same year,
  3. from the same year given that they are the same sex.
Pre-U Pre-U 9794/3 2015 June Q5
12 marks Standard +0.3
5 A garden centre grows a particular variety of plant for sale. They sow 3 seeds in each pot and there are 6 pots in a tray. The probability that a seed germinates is 0.7 , independently of any other seeds.
  1. State the probability distribution of the number of seeds in a pot that germinate.
  2. Find the probability that, in a randomly chosen pot,
    1. exactly 2 seeds germinate,
    2. at least 1 seed germinates. After the seeds have germinated and become seedlings, some are removed (and discarded) so that there remains at most 1 seedling per pot.
    3. Write out the probability distribution of the number of seedlings per pot that remain.
    4. Find the probability that there is a seedling in every one of the 6 pots in a randomly chosen tray.