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OCR MEI S1 2009 January Q1
7 marks Easy -1.2
1 A supermarket chain buys a batch of 10000 scratchcard draw tickets for sale in its stores. 50 of these tickets have a \(\pounds 10\) prize, 20 of them have a \(\pounds 100\) prize, one of them has a \(\pounds 5000\) prize and all of the rest have no prize. This information is summarised in the frequency table below.
Prize money\(\pounds 0\)\(\pounds 10\)\(\pounds 100\)\(\pounds 5000\)
Frequency992950201
  1. Find the mean and standard deviation of the prize money per ticket.
  2. I buy two of these tickets at random. Find the probability that I win either two \(\pounds 10\) prizes or two \(\pounds 100\) prizes.
OCR MEI S1 2009 January Q2
5 marks Moderate -0.8
2 Thomas has six tiles, each with a different letter of his name on it.
  1. Thomas arranges these letters in a random order. Find the probability that he arranges them in the correct order to spell his name.
  2. On another occasion, Thomas picks three of the six letters at random. Find the probability that he picks the letters T, O and M (in any order).
OCR MEI S1 2009 January Q3
8 marks Easy -1.3
3 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(p\)0.10.050.050.25
  1. Find the value of \(p\).
  2. Find the expectation and variance of \(X\).
  3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.
OCR MEI S1 2009 January Q4
8 marks Moderate -0.8
4 A pottery manufacturer makes teapots in batches of 50. On average 3\% of teapots are faulty.
  1. Find the probability that in a batch of 50 there is
    (A) exactly one faulty teapot,
    (B) more than one faulty teapot.
  2. The manufacturer produces 240 batches of 50 teapots during one month. Find the expected number of batches which contain exactly one faulty teapot.
OCR MEI S1 2009 January Q5
8 marks Moderate -0.8
5 Each day Anna drives to work.
  • \(R\) is the event that it is raining.
  • \(L\) is the event that Anna arrives at work late.
You are given that \(\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25\) and \(\mathrm { P } ( R \cap L ) = 0.2\).
  1. Determine whether the events \(R\) and \(L\) are independent.
  2. Draw a Venn diagram showing the events \(R\) and \(L\). Fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( L \mid R )\). State what this probability represents.
OCR MEI S1 2009 January Q6
17 marks Easy -1.2
6 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
  3. Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
  4. Copy and complete the frequency table below for these data.
    Temperature
    \(( t\) degrees Celsius \()\)
    		\(3.0 \leqslant t \leqslant 3.4\)\(3.4 < t \leqslant 3.8\)\(3.8 < t \leqslant 4.2\)\(4.2 < t \leqslant 4.6\)\(4.6 < t \leqslant 5.0\)
    Frequency243157
  5. Use your table to calculate an estimate of the mean.
  6. The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.
OCR MEI S1 2009 January Q7
19 marks Standard +0.3
7 An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.
  1. Find the probability that
    (A) exactly 8 of these orders are delivered within 24 hours,
    (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
  2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
  3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
OCR MEI S1 2016 June Q1
7 marks Moderate -0.8
1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows.
250
26058
2779
28145
29002
3077
316
32047
3333
Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams
  1. Find the median and interquartile range of the data.
  2. Determine whether there are any outliers.
OCR MEI S1 2016 June Q2
7 marks Moderate -0.8
2 In a hockey league, each team plays every other team 3 times. The probabilities that Team A wins, draws and loses to Team B are given below.
  • \(\mathrm { P } (\) Wins \() = 0.5\)
  • \(\mathrm { P } (\) Draws \() = 0.3\)
  • \(\mathrm { P } (\) Loses \() = 0.2\)
The outcomes of the 3 matches are independent.
  1. Find the probability that Team A does not lose in any of the 3 matches.
  2. Find the probability that Team A either wins all 3 matches or draws all 3 matches or loses all 3 matches.
  3. Find the probability that, in the 3 matches, exactly two of the outcomes, 'Wins', 'Draws' and 'Loses' occur for Team A.
OCR MEI S1 2016 June Q3
6 marks Easy -1.3
3
  1. There are 5 runners in a race. How many different finishing orders are possible? [You should assume that there are no 'dead heats', where two runners are given the same position.] For the remainder of this question you should assume that all finishing orders are equally likely.
  2. The runners are denoted by \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\). Find the probability that they either finish in the order ABCDE or in the order EDCBA.
  3. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in that order.
  4. Find the probability that the first 3 runners to finish are \(\mathrm { A } , \mathrm { B }\) and C , in any order.
OCR MEI S1 2016 June Q4
8 marks Moderate -0.3
4 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = \frac { k } { r ( r - 1 ) } \text { for } r = 2,3,4,5,6 .$$
  1. Show that the value of \(k\) is 1.2 . Using this value of \(k\), show the probability distribution of \(X\) in a table.
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR MEI S1 2016 June Q5
8 marks Easy -1.3
5 Measurements of sunshine and rainfall are made each day at a particular weather station. For a randomly chosen day,
  • \(R\) is the event that at least 1 mm of rainfall is recorded,
  • \(S\) is the event that at least 1 hour of sunshine is recorded.
You are given that \(\mathrm { P } ( R ) = 0.28 , \mathrm { P } ( S ) = 0.87\) and \(\mathrm { P } ( R \cup S ) = 0.94\).
  1. Find \(\mathrm { P } ( R \cap S )\).
  2. Draw a Venn diagram showing the events \(R\) and \(S\), and fill in the probability corresponding to each of the four regions of your diagram.
  3. Find \(\mathrm { P } ( R \mid S )\) and state what this probability represents in this context.
OCR MEI S1 2016 June Q6
18 marks Moderate -0.8
6 An online store has a total of 930 different types of women's running shoe on sale. The prices in pounds of the types of women's running shoe are summarised in the table below.
Price \(( \pounds x )\)\(10 \leqslant x \leqslant 40\)\(40 < x \leqslant 50\)\(50 < x \leqslant 60\)\(60 < x \leqslant 80\)\(80 < x \leqslant 200\)
Frequency147109182317175
  1. Calculate estimates of the mean and standard deviation of the shoe prices.
  2. Calculate an estimate of the percentage of types of shoe that cost at least \(\pounds 100\).
  3. Draw a histogram to illustrate the data. The corresponding histogram below shows the prices in pounds of the 990 types of men's running shoe on sale at the same online store. \includegraphics[max width=\textwidth, alt={}, center]{aff0c5b2-011b-49a0-bf05-6d905f890eba-4_643_1192_340_440}
  4. State the type of skewness shown by the histogram for men's running shoes.
  5. Martin is investigating the percentage of types of shoe on sale at the store that cost more than \(\pounds 100\). He believes that this percentage is greater for men's shoes than for women's shoes. Estimate the percentage for men's shoes and comment on whether you can be certain which percentage is higher.
  6. You are given that the mean and standard deviation of the prices of men's running shoes are \(\pounds 68.83\) and \(\pounds 42.93\) respectively. Compare the central tendency and variation of the prices of men's and women's running shoes at the store.
OCR MEI S1 2016 June Q7
18 marks Moderate -0.3
7 To withdraw money from a cash machine, the user has to enter a 4-digit PIN (personal identification number). There are several thousand possible 4-digit PINs, but a survey found that \(10 \%\) of cash machine users use the PIN '1234'.
  1. 16 cash machine users are selected at random.
    (A) Find the probability that exactly 3 of them use 1234 as their PIN.
    (B) Find the probability that at least 3 of them use 1234 as their PIN.
    (C) Find the expected number of them who use 1234 as their PIN. An advertising campaign aims to reduce the number of people who use 1234 as their PIN. A hypothesis test is to be carried out to investigate whether the advertising campaign has been successful.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. A random sample of 20 cash machine users is selected.
    (A) Explain why the test could not be carried out at the \(10 \%\) significance level.
    (B) The test is to be carried out at the \(k \%\) significance level. State the lowest integer value of \(k\) for which the test could result in the rejection of the null hypothesis.
  4. A new random sample of 60 cash machine users is selected. It is found that 2 of them use 1234 as their PIN. You are given that, if \(X \sim \mathrm {~B} ( 60,0.1 )\), then (to 4 decimal places) $$\mathrm { P } ( X = 2 ) = 0.0393 , \quad \mathrm { P } ( X < 2 ) = 0.0138 , \quad \mathrm { P } ( X \leqslant 2 ) = 0.0530 .$$ Using the same hypotheses as in part (ii), carry out the test at the \(5 \%\) significance level. \section*{END OF QUESTION PAPER}
OCR M1 2014 June Q1
7 marks Moderate -0.3
1 A particle \(P\) is projected vertically downwards with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) which is 5 m above horizontal ground.
  1. Find the speed of \(P\) immediately before it strikes the ground. After striking the ground, \(P\) rebounds and moves vertically upwards and 0.87 s after leaving the ground \(P\) passes through \(A\).
  2. Calculate the speed of \(P\) immediately after it leaves the ground. It is given that the mass of \(P\) is 0.2 kg .
  3. Calculate the change in the momentum of \(P\) as a result of its collision with the ground.
OCR M1 2014 June Q2
7 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes \(2.5 \mathrm {~N} , F \mathrm {~N}\) and 2.4 N act on the particle on bearings \(\theta ^ { \circ } , 180 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram). The particle is in equilibrium.
  1. Find \(\theta\) and \(F\). The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
  2. Find the magnitude and direction of the acceleration of the particle.
OCR M1 2014 June Q3
8 marks Moderate -0.8
3 A particle \(P\) travels in a straight line. The velocity of \(P\) at time \(t\) seconds after it passes through a fixed point \(A\) is given by \(\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }\). Find
  1. the velocity of \(P\) when it passes through \(A\),
  2. the displacement of \(P\) from \(A\) when \(t = 1.5\),
  3. the velocity of \(P\) when it has acceleration \(6 \mathrm {~ms} ^ { - 2 }\).
OCR M1 2014 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_136_824_260_623} Particles \(P\) and \(Q\) are moving towards each other with constant speeds \(4 \mathrm {~ms} ^ { - 1 }\) and \(2 \mathrm {~ms} ^ { - 1 }\) along the same straight line on a smooth horizontal surface (see diagram). \(P\) has mass 0.2 kg and \(Q\) has mass 0.3 kg . The two particles collide.
  1. Show that \(Q\) must change its direction of motion in the collision.
  2. Given that \(P\) and \(Q\) move with equal speed after the collision, calculate both possible values for their speed after they collide.
OCR M1 2014 June Q5
12 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_652_1675_959_187} A particle \(P\) can move in a straight line on a horizontal surface. At time \(t\) seconds the displacement of \(P\) from a fixed point \(A\) on the line is \(x \mathrm {~m}\). The diagram shows the \(( t , x )\) graph for \(P\). In the interval \(0 \leqslant t \leqslant 10\), either the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\), or \(P\) is at rest.
  1. Show by calculation that \(T = 1.75\).
  2. State the velocity of \(P\) when
    1. \(t = 2\),
    2. \(t = 8\),
    3. \(t = 9\).
    4. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 10\). For \(t > 10\), the displacement of \(P\) from \(A\) is given by \(x = 20 t - t ^ { 2 } - 96\).
    5. Calculate the value of \(t\), where \(t > 10\), for which the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
OCR M1 2014 June Q6
14 marks Moderate -0.3
6 A particle \(P\) of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on \(P\), and \(P\) is in limiting equilibrium.
  1. Calculate the coefficient of friction between \(P\) and the surface.
  2. Find the magnitude and direction of the contact force exerted by the surface on \(P\).
  3. \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-4_190_579_580_598} The initial 3 N force continues to act on \(P\) in its original direction. An additional force of magnitude \(T \mathrm {~N}\), acting in the same vertical plane as the 3 N force, is now applied to \(P\) at an angle of \(\theta ^ { \circ }\) above the horizontal (see diagram). \(P\) is again in limiting equilibrium.
    1. Given that \(\theta = 0\), find \(T\).
    2. Given instead that \(\theta = 30\), calculate \(T\).
OCR M1 2014 June Q7
16 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(M\) is the mid-point of \(A B\). Two particles \(P\) and \(Q\), joined by a taut light inextensible string, are placed on the plane at \(A\) and \(M\) respectively. The particles are simultaneously projected with speed \(0.6 \mathrm {~ms} ^ { - 1 }\) down the line of greatest slope (see diagram). The particles move down the plane with acceleration \(0.9 \mathrm {~ms} ^ { - 2 }\). At the instant 2 s after projection, \(P\) is at \(M\) and \(Q\) is at \(B\). The particle \(Q\) subsequently remains at rest at \(B\).
  1. Find the distance \(A B\). The plane is rough between \(A\) and \(M\), but smooth between \(M\) and \(B\).
  2. Calculate the speed of \(P\) when it reaches \(B\). \(P\) has mass 0.4 kg and \(Q\) has mass 0.3 kg .
  3. By considering the motion of \(Q\), calculate the tension in the string while both particles are moving down the plane.
  4. Calculate the coefficient of friction between \(P\) and the plane between \(A\) and \(M\). \section*{END OF QUESTION PAPER}
OCR MEI FP3 2011 June Q1
24 marks Challenging +1.2
1 The points \(\mathrm { A } ( 2 , - 1,3 ) , \mathrm { B } ( - 2 , - 7,7 )\) and \(\mathrm { C } ( 7,5,1 )\) are three vertices of a tetrahedron ABCD .
The plane ABD has equation \(x + 4 y + 7 z = 19\).
The plane ACD has equation \(2 x - y + 2 z = 11\).
  1. Find the shortest distance from \(B\) to the plane \(A C D\).
  2. Find an equation for the line AD .
  3. Find the shortest distance from C to the line AD .
  4. Find the shortest distance between the lines \(A D\) and \(B C\).
  5. Given that the tetrahedron ABCD has volume 20, find the coordinates of the two possible positions for the vertex \(D\).
OCR MEI FP3 2011 June Q2
24 marks Challenging +1.8
2 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x\).
  1. Sketch the section of \(S\) given by \(y = - 3\), and sketch the section of \(S\) given by \(x = - 6\). Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
  2. From your sketches in part (i), deduce that \(( - 6 , - 3 , - 324 )\) is a stationary point on \(S\), and state the nature of this stationary point.
  3. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\), and hence find the coordinates of the other three stationary points on \(S\).
  4. Show that there are exactly two values of \(k\) for which the plane with equation $$120 x - z = k$$ is a tangent plane to \(S\), and find these values of \(k\).
OCR MEI FP3 2011 June Q3
24 marks Challenging +1.8
3
    1. Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\). The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
    2. Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
    3. Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
    1. Show that the normal to the ellipse at the point with parameter \(\theta\) has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
    2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
    3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
      (A) at the point \(( 2,0 )\),
      (B) at the point \(( 0,1 )\).
OCR MEI FP3 2011 June Q4
24 marks Challenging +1.8
4
  1. Show that the set \(G = \{ 1,3,4,5,9 \}\), under the binary operation of multiplication modulo 11 , is a group. You may assume associativity.
  2. Explain why any two groups of order 5 must be isomorphic to each other. The set \(H = \left\{ 1 , \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 4 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 6 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 8 } { 5 } \pi \mathrm { j } } \right\}\) is a group under the binary operation of multiplication of complex numbers.
  3. Specify an isomorphism between the groups \(G\) and \(H\). The set \(K\) consists of the 25 ordered pairs \(( x , y )\), where \(x\) and \(y\) are elements of \(G\). The set \(K\) is a group under the binary operation defined by $$\left( x _ { 1 } , y _ { 1 } \right) \left( x _ { 2 } , y _ { 2 } \right) = \left( x _ { 1 } x _ { 2 } , y _ { 1 } y _ { 2 } \right)$$ where the multiplications are carried out modulo 11 ; for example, \(( 3,5 ) ( 4,4 ) = ( 1,9 )\).
  4. Write down the identity element of \(K\), and find the inverse of the element \(( 9,3 )\).
  5. Explain why \(( x , y ) ^ { 5 } = ( 1,1 )\) for every element \(( x , y )\) in \(K\).
  6. Deduce that all the elements of \(K\), except for one, have order 5. State which is the exceptional element.
  7. A subgroup of \(K\) has order 5 and contains the element (9, 3). List the elements of this subgroup.
  8. Determine how many subgroups of \(K\) there are with order 5 .