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OCR MEI M1 Q2
18 marks Standard +0.3
2 In this question the value of \(g\) should be taken as \(10 \mathrm {~m \mathrm {~s} ^ { 2 }\).} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { 1 }\) vertically and \(20 \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{362d5995-bd39-4b07-b6a4-63eb1dd3e69d-2_461_1114_464_505} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that, \(t\) seconds after projection, the height in metres of each particle above its point of projection is \(10 t - 5 t ^ { 2 }\).
  2. Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
  3. Calculate the horizontal distance travelled by B before reaching the ground.
  4. Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
  5. Verify that the particles collide 0.75 seconds after A is projected.
OCR M3 2006 January Q10
Standard +0.8
10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes
  • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  • Answer all the questions.
  • Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
  • The acceleration due to gravity is denoted by \(\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
  • You are permitted to use a graphical calculator in this paper.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total number of marks for this paper is 72.
  • Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
  • You are reminded of the need for clear presentation in your answers.
OCR FS1 AS 2021 June Q1
8 marks Moderate -0.8
1 A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is 0.1 . The first weekday in September on which he receives a delivery of books to review is the \(X\) th weekday of September.
  1. State an assumption needed for \(X\) to be well modelled by a geometric distribution.
  2. Find \(\mathrm { P } ( X = 11 )\).
  3. Find \(\mathrm { P } ( X \leqslant 8 )\).
  4. Find \(\operatorname { Var } ( X )\).
  5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review.
OCR FS1 AS 2021 June Q2
8 marks Standard +0.3
2 The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(\mathrm { P } ( W = w )\)0.250.36\(x\)\(x ^ { 2 }\)
  1. Show that \(\operatorname { Var } ( W ) = 0.8571\).
  2. Find \(\operatorname { Var } ( 3 W + 6 )\). In this question you must show detailed reasoning.
    The random variable \(T\) has a binomial distribution. It is known that \(\mathrm { E } ( T ) = 5.625\) and the standard deviation of \(T\) is 1.875 . Find the values of the parameters of the distribution.
OCR FS1 AS 2021 June Q4
9 marks Standard +0.8
4 The table shows the results of a random sample drawn from a population which is thought to have the distribution \(\mathrm { U } ( 20 )\). \end{table}
OCR FP1 AS 2017 December Q1
4 marks Moderate -0.3
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } - 3 & 3 & 2 \\ 5 & - 4 & - 3 \\ - 1 & 1 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { - 1 }\).
  2. Solve the simultaneous equations $$\begin{aligned} - 3 x + 3 y + 2 z & = 12 a \\ 5 x - 4 y - 3 z & = - 6 \\ - x + y + z & = 7 \end{aligned}$$ giving your solution in terms of \(a\).
OCR FP1 AS 2017 December Q2
9 marks Standard +0.3
2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - ( 3 + 2 \mathrm { i } ) | = 2\) and \(\arg ( z - ( 3 + 2 \mathrm { i } ) ) = \frac { 5 \pi } { 6 }\) respectively.
  1. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single Argand diagram.
  2. Find, in surd form, the number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z - ( 3 + 2 i ) | \leqslant 2 \text { and } \frac { 5 \pi } { 6 } \leqslant \arg ( z - ( 3 + 2 i ) ) \leqslant \pi$$
OCR FP1 AS 2017 December Q3
8 marks Standard +0.3
3 Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
OCR FP1 AS 2017 December Q5
7 marks Standard +0.8
5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the position vector of \(P\).
  2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
  3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
    The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
  4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
  5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
    The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
  7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\).
OCR FP1 AS 2017 December Q6
5 marks
6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\).
OCR FP1 AS 2017 December Q7
7 marks Standard +0.8
7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s \\ t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears.
OCR FP1 AS 2017 December Q10
Standard +0.3
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the position vector of \(P\).
  2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
  3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
    The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
  4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
  5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
    The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
  7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\). 6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\). 7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s \\ t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears. 8
  8. (a) Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2 \\ 5 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { c } x \\ 6 \\ 2 \end{array} \right)\).
    (b) Find the shortest possible vector of the form \(\left( \begin{array} { l } 1 \\ a \\ b \end{array} \right)\) which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2 \\ 5 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { c } x \\ 6 \\ 2 \end{array} \right)\).
  9. Vector \(\mathbf { v }\) is perpendicular to both \(\left( \begin{array} { c } - 1 \\ 1 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { c } 1 \\ p \\ p ^ { 2 } \end{array} \right)\) where \(p\) is a real number. Show that it is impossible for \(\mathbf { v }\) to be perpendicular to the vector \(\left( \begin{array} { c } 1 \\ 1 \\ p - 1 \end{array} \right)\). \section*{OCR} Oxford Cambridge and RSA
OCR FS1 AS 2017 December Q1
8 marks Moderate -0.3
1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that \(5 \%\) of letters receive a favourable reply.
  1. Bill sends a letter to each of 40 potential sponsors. Assuming that the number \(N\) of favourable responses can be modelled by a binomial distribution, find the mean and variance of \(N\).
  2. Gill sends one letter at a time to potential sponsors. \(L\) is the number of letters she sends, up to and including the first letter that receives a favourable response.
    (a) State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
    (b) Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a \(90 \%\) chance of receiving at least one favourable reply.
OCR FS1 AS 2017 December Q2
7 marks Moderate -0.3
2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.
  1. All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.
  2. 4 cards are chosen at random. Find the probability that at least three consonants ( \(\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }\) ) are on the cards chosen.
OCR FS1 AS 2017 December Q3
7 marks Standard +0.3
3 Over a long period Jenny counts the number of trolleys used at her local supermarket between 10 am and 10.20 am each day. She finds that the mean number of trolleys used between these times on a weekday is 40.00. You should assume that the use of trolleys occurs randomly, independently of one another, and at a constant average rate.
  1. Calculate the probability that, on a randomly chosen weekday, the number of trolleys used between these times is between 32 and 50 inclusive.
  2. Write down an expression for the probability that, on a randomly chosen weekday, exactly 5 trolleys are used during a time period of \(t\) minutes between 10 am and 10.20 am. Jenny carries out this process for seven consecutive days. She finds that the mean number of trolleys used between 10 am and 10.20 am is 35.14 and the variance is 91.55 .
  3. Explain why this suggests that the distribution of the number of trolleys used between these times on these seven consecutive days is not well modelled by a Poisson distribution.
  4. Give a reason why it might not be appropriate to apply the Poisson model to the total number of trolleys used between these times on seven consecutive days.
OCR FS1 AS 2017 December Q4
10 marks Standard +0.3
4 The discrete random variable \(X\) has the distribution \(\mathrm { U } ( n )\).
  1. Use the results \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) and \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) to show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\). It is given that \(\mathrm { E } ( X ) = 13\).
  2. Find the value of \(n\).
  3. Find \(\mathrm { P } ( X < 7.5 )\). It is given that \(\mathrm { E } ( a X + b ) = 10\) and \(\operatorname { Var } ( a X + b ) = 117\), where \(a\) and \(b\) are positive.
  4. Calculate the value of \(a\) and the value of \(b\).
OCR FS1 AS 2017 December Q5
8 marks Moderate -0.5
5 A shop manager recorded the maximum daytime temperature \(T ^ { \circ } \mathrm { C }\) and the number \(C\) of ice creams sold on 9 summer days. The results are given in the table and illustrated in the scatter diagram.
\(T\)172125262727293030
\(C\)211620383237353942
\includegraphics[max width=\textwidth, alt={}]{64d7ed6d-fadd-4c59-afb0-97d1788ba369-3_661_1189_1320_431}
$$n = 9 , \Sigma t = 232 , \Sigma c = 280 , \Sigma t ^ { 2 } = 6130 , \Sigma c ^ { 2 } = 9444 , \Sigma t c = 7489$$
  1. State, with a reason, whether one of the variables \(C\) or \(T\) is likely to be dependent upon the other.
  2. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
  3. State with a reason what the value of \(r\) would have been if the temperature had been measured in \({ } ^ { \circ } \mathrm { F }\) rather than \({ } ^ { \circ } \mathrm { C }\).
  4. Calculate the equation of the least squares regression line of \(c\) on \(t\).
  5. The regression line is drawn on the copy of the scatter diagram in the Printed Answer Booklet. Use this diagram to explain what is meant by "least squares".
OCR FS1 AS 2017 December Q6
9 marks Standard +0.3
6 Arlosh, Sarah and Desi are investigating the ratings given to six different films by two critics.
  1. Arlosh calculates Spearman's rank correlation coefficient \(r _ { s }\) for the critics' ratings. He calculates that \(\Sigma d ^ { 2 } = 72\). Show that this value must be incorrect.
  2. Arlosh checks his working with Sarah, whose answer \(r _ { s } = \frac { 29 } { 35 }\) is correct. Find the correct value of \(\Sigma d ^ { 2 }\).
  3. Carry out an appropriate two-tailed significance test of the value of \(r _ { s }\) at the \(5 \%\) significance level, stating your hypotheses clearly. Each critic gives a score out of 100 to each film. Desi uses these scores to calculate Pearson's product-moment correlation coefficient. She carries out a two-tailed significance test of this value at the \(5 \%\) significance level.
  4. Explain with a reason whether you would expect the conclusion of Desi's test to be the same as the result of the test in part (iii).
OCR FS1 AS 2017 December Q7
11 marks Standard +0.3
7 Josh is investigating whether sticking pins into a map at random, while blindfolded, provides a random sample of regions of the map. Josh divides the map into 49 squares of equal size and asks each of 98 friends to stick a pin into the map at random, while blindfolded. He then notes the number of pins in each square. To analyse the results he groups the squares as shown in the diagram.
DDDDDDD
DCCCCCD
DCBBBCD
DCBABCD
DCBBBCD
DCCCCCD
DDDDDDD
The results are summarised in the table.
RegionABCD
Number of squares181624
Number of pins6213338
  1. Test at the 10\% significance level whether the use of pins in this way provides a random sample of regions of the map.
  2. What can be deduced from considering the different contributions to the test statistic? \section*{OCR} \section*{Oxford Cambridge and RSA}
OCR FM1 AS 2017 December Q1
4 marks Moderate -0.8
1 A climber of mass 65 kg climbs from the bottom to the top of a vertical cliff which is 78 m in height. The climb takes 90 minutes so the velocity of the climber can be neglected.
  1. Calculate the work done by the climber in climbing the cliff.
  2. Calculate the average power generated by the climber in climbing the cliff.
OCR FM1 AS 2017 December Q2
4 marks Moderate -0.8
2 The universal law of gravitation states that \(F = \frac { G m _ { 1 } m _ { 2 } } { r ^ { 2 } }\) where \(F\) is the magnitude of the force between two objects of masses \(m _ { 1 }\) and \(m _ { 2 }\) which are a distance \(r\) apart and \(G\) is a constant. Find the dimensions of \(G\).
OCR FM1 AS 2017 December Q3
8 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-2_473_298_1037_884} A particle \(P\) of mass 1.5 kg is attached to one end of a light inextensible string of length 2.4 m . The other end of the string is attached to a fixed point \(O\). The particle is initially at rest directly below \(O\). A horizontal impulse of magnitude 9.3 Ns is applied to \(P\). In the subsequent motion the string remains taut and makes an angle of \(\theta\) radians with the downwards vertical at \(O\), as shown in the diagram.
  1. Find the speed of \(P\) when \(\theta = \frac { 1 } { 6 } \pi\).
  2. Determine whether \(P\) will reach the same horizontal level as \(O\).
OCR FM1 AS 2017 December Q4
9 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-3_216_1219_255_415} \(A\) and \(B\) are two long straight parallel horizontal sections of railway track. An engine on track \(A\) is attached to a carriage of mass 6000 kg on track \(B\) by a light inextensible chain which remains horizontal and taut in the ensuing motion. The chain is 13 m in length and the points of attachment on the engine and carriage are a perpendicular distance of 5 m apart. The engine and carriage start at rest and then the engine accelerates uniformly to a speed of \(5.6 \mathrm {~ms} ^ { - 1 }\) while travelling 250 m . It is assumed that any resistance to motion can be ignored.
  1. Find the work done on the carriage by the tension in the chain.
  2. Find the magnitude of the tension in the chain. The mass of the engine is 10000 kg .
  3. At a point further along the track the engine and the carriage are moving at a speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) and the power of the engine is 68 kW . Find the acceleration of the engine at this instant.
OCR FM1 AS 2017 December Q5
13 marks Standard +0.3
5 Two discs, \(A\) and \(B\), have masses 1.4 kg and 2.1 kg respectively. They are sliding towards each other in the same straight line across a large sheet of horizontal ice. Immediately before the collision \(A\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision \(A\) 's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Explain why it is impossible for \(A\) to be travelling in the same direction after the collision as it was before the collision.
  2. Find the velocity of \(B\) immediately after the collision.
  3. Calculate the coefficient of restitution between \(A\) and \(B\).
  4. State what your answer to part (iii) means about the kinetic energy of the system. The discs are made from the same material. The discs will be damaged if subjected to an impulse of magnitude greater than 6.5 Ns .
  5. Determine whether \(B\) will be damaged as a result of the collision.
  6. Explain why \(A\) will be damaged if, and only if, \(B\) is damaged.
OCR FM1 AS 2017 December Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-4_547_597_251_735} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\) which is 1.8 m above a smooth horizontal table. The particle moves on the table in a circular path at constant speed with the string taut (see diagram). The particle has a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its angular velocity is \(0.625 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Show that the radius of the circular path is 0.8 m .
  2. Find the magnitude of the normal contact force between the particle and the table. The speed is changed to \(v \mathrm {~ms} ^ { - 1 }\). At this speed the particle is just about to lose contact with the table.
  3. Find the value of \(v\).