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OCR FP2 2015 June Q9
11 marks Standard +0.8
9 The equation of a curve in polar coordinates is \(r = 2 \sin 3 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
  1. Sketch the curve.
  2. Find the area of the region enclosed by this curve.
  3. By expressing \(\sin 3 \theta\) in terms of \(\sin \theta\), show that a cartesian equation for the curve is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } y - 2 y ^ { 3 } .$$ \section*{END OF QUESTION PAPER}
OCR S1 2013 January Q1
7 marks Easy -1.2
1 When a four-sided spinner is spun, the number on which it lands is denoted by \(X\), where \(X\) is a random variable taking values \(2,4,6\) and 8 . The spinner is biased so that \(\mathrm { P } ( X = x ) = k x\), where \(k\) is a constant.
  1. Show that \(\mathrm { P } ( X = 6 ) = \frac { 3 } { 10 }\).
  2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Kathryn is allowed three attempts at a high jump. If she succeeds on any attempt, she does not jump again. The probability that she succeeds on her first attempt is \(\frac { 3 } { 4 }\). If she fails on her first attempt, the probability that she succeeds on her second attempt is \(\frac { 3 } { 8 }\). If she fails on her first two attempts, the probability that she succeeds on her third attempt is \(\frac { 3 } { 16 }\). Find the probability that she succeeds.
  4. Khaled is allowed two attempts to pass an examination. If he succeeds on his first attempt, he does not make a second attempt. The probability that he passes at the first attempt is 0.4 and the probability that he passes on either the first or second attempt is 0.58 . Find the probability that he passes on the second attempt, given that he failed on the first attempt.
OCR S1 2013 January Q3
12 marks Moderate -0.3
3 The Gross Domestic Product per Capita (GDP), \(x\) dollars, and the Infant Mortality Rate per thousand (IMR), \(y\), of 6 African countries were recorded and summarised as follows. $$n = 6 \quad \sum x = 7000 \quad \sum x ^ { 2 } = 8700000 \quad \sum y = 456 \quad \sum y ^ { 2 } = 36262 \quad \sum x y = 509900$$
  1. Calculate the equation of the regression line of \(y\) on \(x\) for these 6 countries. The original data were plotted on a scatter diagram and the regression line of \(y\) on \(x\) was drawn, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-3_721_1246_680_408}
  2. The GDP for another country, Tanzania, is 1300 dollars. Use the regression line in the diagram to estimate the IMR of Tanzania.
  3. The GDP for Nigeria is 2400 dollars. Give two reasons why the regression line is unlikely to give a reliable estimate for the IMR for Nigeria.
  4. The actual value of the IMR for Tanzania is 96. The data for Tanzania \(( x = 1300 , y = 96 )\) is now included with the original 6 countries. Calculate the value of the product moment correlation coefficient, \(r\), for all 7 countries.
  5. The IMR is now redefined as the infant mortality rate per hundred instead of per thousand, and the value of \(r\) is recalculated for all 7 countries. Without calculation state what effect, if any, this would have on the value of \(r\) found in part (iv).
OCR S1 2013 January Q4
10 marks Moderate -0.8
4
  1. How many different 3-digit numbers can be formed using the digits 1, 2 and 3 when
    (a) no repetitions are allowed,
    (b) any repetitions are allowed,
    (c) each digit may be included at most twice?
  2. How many different 4-digit numbers can be formed using the digits 1, 2 and 3 when each digit may be included at most twice?
OCR S1 2013 January Q5
10 marks Standard +0.3
5 A random variable \(X\) has the distribution \(\mathrm { B } \left( 5 , \frac { 1 } { 4 } \right)\).
  1. Find
    (a) \(\mathrm { E } ( X )\),
    (b) \(\mathrm { P } ( X = 2 )\).
  2. Two values of \(X\) are chosen at random. Find the probability that their sum is less than 2 .
  3. 10 values of \(X\) are chosen at random. Use an appropriate formula to find the probability that exactly 3 of these values are 2 s .
OCR S1 2013 January Q6
7 marks Moderate -0.3
6 The masses, \(x\) grams, of 800 apples are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{13d8d940-fd63-4b62-bd7a-aa7174f6af4b-4_592_1363_957_351}
  1. On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\).
  2. Find an estimate of the median mass of the apples.
  3. Two judges rank \(n\) competitors, where \(n\) is an even number. Judge 2 reverses each consecutive pair of ranks given by Judge 1, as shown.
    Competitor\(C _ { 1 }\)\(C _ { 2 }\)\(C _ { 3 }\)\(C _ { 4 }\)\(C _ { 5 }\)\(C _ { 6 }\)\(\ldots \ldots\)\(C _ { n - 1 }\)\(C _ { n }\)
    Judge 1 rank123456\(\ldots \ldots\)\(n - 1\)\(n\)
    Judge 2 rank214365\(\ldots \ldots\)\(n\)\(n - 1\)
    Given that the value of Spearman's coefficient of rank correlation is \(\frac { 63 } { 65 }\), find \(n\).
  4. An experiment produced some data from a bivariate distribution. The product moment correlation coefficient is denoted by \(r\), and Spearman's rank correlation coefficient is denoted by \(r _ { s }\).
    (a) Explain whether the statement $$r = 1 \Rightarrow r _ { s } = 1$$ is true or false.
    (b) Use a diagram to explain whether the statement $$r \neq 1 \Rightarrow r _ { s } \neq 1$$ is true or false. 8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  5. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  6. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR S1 2013 January Q8
13 marks Standard +0.8
8 Sandra makes repeated, independent attempts to hit a target. On each attempt, the probability that she succeeds is 0.1 .
  1. Find the probability that
    (a) the first time she succeeds is on her 5th attempt,
    (b) the first time she succeeds is after her 5th attempt,
    (c) the second time she succeeds is before her 4th attempt. Jill also makes repeated attempts to hit the target. Each attempt of either Jill or Sandra is independent. Each time that Jill attempts to hit the target, the probability that she succeeds is 0.2 . Sandra and Jill take turns attempting to hit the target, with Sandra going first.
  2. Find the probability that the first person to hit the target is Sandra, on her
    (a) 2nd attempt,
    (b) 10th attempt. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
    7
OCR S1 2013 June Q1
7 marks Easy -1.2
1 The lengths, in centimetres, of 18 snakes are given below. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 24 & 62 & 20 & 65 & 27 & 67 & 69 & 32 & 40 & 53 & 55 & 47 & 33 & 45 & 55 & 56 & 49 & 58 \end{array}$$
  1. Draw an ordered stem-and-leaf diagram for the data.
  2. Find the mean and median of the lengths of the snakes.
  3. It was found that one of the lengths had been measured incorrectly. After this length was corrected, the median increased by 1 cm . Give two possibilities for the incorrect length and give a corrected value in each case.
OCR S1 2013 June Q2
7 marks Moderate -0.3
2
  1. The table shows the times, in minutes, spent by five students revising for a test, and the grades that they achieved in the test.
    StudentAnnBillCazDenEd
    Time revising0603510045
    GradeCDEBA
    Calculate Spearman's rank correlation coefficient.
  2. The table below shows the ranks given by two judges to four competitors.
    CompetitorPQRS
    Judge 1 rank1234
    Judge 2 rank3214
    Spearman's rank correlation coefficient for these ranks is denoted by \(r _ { s }\). With the same set of ranks for Judge 1, write down a different set of ranks for Judge 2 which gives the same value of \(r _ { s }\). There is no need to find the value of \(r _ { s }\).
OCR S1 2013 June Q3
10 marks Moderate -0.5
3 The probability distribution of a random variable \(X\) is shown.
\(x\)1357
\(\mathrm { P } ( X = x )\)0.40.30.20.1
  1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. Three independent values of \(X\), denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), are chosen. Given that \(X _ { 1 } + X _ { 2 } + X _ { 3 } = 19\), write down all the possible sets of values for \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and hence find \(\mathrm { P } \left( X _ { 1 } = 7 \right)\).
  3. 11 independent values of \(X\) are chosen. Use an appropriate formula to find the probability that exactly 4 of these values are 5 s .
OCR S1 2013 June Q4
6 marks Moderate -0.8
4 At a stall in a fair, contestants have to estimate the mass of a cake. A group of 10 people made estimates, \(m \mathrm {~kg}\), and for each person the value of \(( m - 5 )\) was recorded. The mean and standard deviation of \(( m - 5 )\) were found to be 0.74 and 0.13 respectively.
  1. Write down the mean and standard deviation of \(m\). The mean and standard deviation of the estimates made by another group of 15 people were found to be 5.6 kg and 0.19 kg respectively.
  2. Calculate the mean of all 25 estimates.
  3. Fiona claims that if a group's estimates are more consistent, they are likely to be more accurate. Given that the true mass of the cake is 5.65 kg , comment on this claim.
OCR S1 2013 June Q5
9 marks Moderate -0.8
5 The table shows some of the values of the seasonally adjusted Unemployment Rate (UR), \(x \%\), and the Consumer Price Index (CPI), \(y \%\), in the United Kingdom from April 2008 to July 2010.
DateApril 2008July 2008October 2008January 2009April 2009July 2009October 2009January 2010April 2010July 2010
UR, \(x \%\)5.25.76.16.87.57.87.87.97.87.7
CPI, \(y \%\)3.04.44.53.02.31.81.53.53.73.1
These data are summarised below. $$n = 10 \quad \sum x = 70.3 \quad \sum x ^ { 2 } = 503.45 \quad \sum y = 30.8 \quad \sum y ^ { 2 } = 103.94 \quad \sum x y = 211.9$$
  1. Calculate the product moment correlation coefficient, \(r\), for the data, showing that \(- 0.6 < r < - 0.5\).
  2. Karen says "The negative value of \(r\) shows that when the Unemployment Rate increases, it causes the Consumer Price Index to decrease." Give a criticism of this statement.
  3. (a) Calculate the equation of the regression line of \(x\) on \(y\).
    (b) Use your equation to estimate the value of the Unemployment Rate in a month when the Consumer Price Index is 4.0\%.
OCR S1 2013 June Q6
7 marks Easy -1.2
6 The diagram shows five cards, each with a letter on it. \includegraphics[max width=\textwidth, alt={}, center]{d06430a6-7957-4313-beea-bb320fadb282-4_113_743_315_662} The letters A and E are vowels; the letters B, C and D are consonants.
  1. Two of the five cards are chosen at random, without replacement. Find the probability that they both have vowels on them.
  2. The two cards are replaced. Now three of the five cards are chosen at random, without replacement. Find the probability that they include exactly one card with a vowel on it.
  3. The three cards are replaced. Now four of the five cards are chosen at random without replacement. Find the probability that they include the card with the letter B on it.
OCR S1 2013 June Q7
11 marks Standard +0.3
7 In a factory, an inspector checks a random sample of 30 mugs from a large batch and notes the number, \(X\), which are defective. He then deals with the batch as follows.
  • If \(X < 2\), the batch is accepted.
  • If \(X > 2\), the batch is rejected.
  • If \(X = 2\), the inspector selects another random sample of only 15 mugs from the batch. If this second sample contains 1 or more defective mugs, the batch is rejected. Otherwise the batch is accepted.
It is given that \(5 \%\) of mugs are defective.
  1. (a) Find the probability that the batch is rejected after just the first sample is checked.
    (b) Show that the probability that the batch is rejected is 0.327 , correct to 3 significant figures.
  2. Batches are checked one after another. Find the probability that the first batch to be rejected is either the 4th or the 5th batch that is checked.
  3. A bag contains 12 black discs, 10 white discs and 5 green discs. Three discs are drawn at random from the bag, without replacement. Find the probability that all three discs are of different colours.
  4. A bag contains 30 red discs and 20 blue discs. A second bag contains 50 discs, each of which is either red or blue. A disc is drawn at random from each bag. The probability that these two discs are of different colours is 0.54 . Find the number of red discs that were in the second bag at the start.
OCR S1 2013 June Q9
8 marks Standard +0.3
9 A game is played with a token on a board with a grid printed on it. The token starts at the point \(( 0,0 )\) and moves in steps. Each step is either 1 unit in the positive \(x\)-direction with probability 0.8 , or 1 unit in the positive \(y\)-direction with probability 0.2 . The token stops when it reaches a point with a \(y\)-coordinate of 1 . It is given that the token stops at \(( X , 1 )\).
  1. (a) Find the probability that \(X = 10\).
    (b) Find the probability that \(X < 10\).
  2. Find the expected number of steps taken by the token.
  3. Hence, write down the value of \(\mathrm { E } ( X )\).
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
    (a) \(t = 2\),
    (b) \(t = 8\),
    (c) \(t = 9\).
  3. 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 \(x = 20 t - t ^ { 2 } - 96\).
  4. 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.
    (a) Given that \(\theta = 0\), find \(T\).
    (b) 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 M2 2007 January Q1
3 marks Moderate -0.3
1 A uniform solid cylinder has height 20 cm and diameter 12 cm . It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cylinder topples when the angle of inclination is \(\alpha\). Find \(\alpha\).
OCR M2 2007 January Q2
4 marks Moderate -0.3
2 Two smooth spheres \(A\) and \(B\), of equal radius and of masses 0.2 kg and 0.1 kg respectively, are free to move on a smooth horizontal table. \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with \(B\), which is stationary. The collision is perfectly elastic. Calculate the speed of \(A\) after the impact. [4]
OCR M2 2007 January Q3
8 marks Standard +0.3
3 A small sphere of mass 0.2 kg is projected vertically downwards with speed \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 40 m above horizontal ground. It hits the ground and rebounds vertically upwards, coming to instantaneous rest at its initial point of projection. Ignoring air resistance, calculate
  1. the coefficient of restitution between the sphere and the ground,
  2. the magnitude of the impulse which the ground exerts on the sphere.