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OCR MEI S1 Q1
8 marks Moderate -0.8
1 A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
  • Reducing car use
  • Insulating their homes
  • Avoiding air travel
The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-1_699_1085_849_569} One of the citizens is selected at random.
  1. Find the probability that this citizen
    (A) has avoided air travel,
    (B) has used at least two of the three methods.
  2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. Three of the citizens are selected at random.
  3. Find the probability that none of them have avoided air travel.
OCR MEI S1 Q2
6 marks Standard +0.8
2 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.
  1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
  2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
  3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.
OCR MEI S1 Q3
18 marks Moderate -0.3
3 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.
  1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
  2. Find the probability that
    (A) all three of these trains are on time,
    (B) just one of these three trains is on time,
    (C) the 1200 train is on time.
  3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time.
OCR MEI S1 Q4
8 marks Moderate -0.8
4 In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random.
  • \(G\) is the event that this person goes to the gym.
  • \(R\) is the event that this person goes running.
You are given that \(\mathrm { P } ( G ) = 0.24 , \mathrm { P } ( R ) = 0.13\) and \(\mathrm { P } ( G \cap R ) = 0.06\).
  1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram.
  2. Determine whether the events \(G\) and \(R\) are independent.
  3. Find \(\mathrm { P } ( R \mid G )\).
OCR MEI S1 Q5
3 marks Easy -1.2
5 My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.
  1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.
  2. If I now guess my number, find the probability that I guess it correctly.
OCR MEI S1 Q6
6 marks Moderate -0.8
6 Whitefly, blight and mosaic virus are three problems which can affect tomato plants. 100 tomato plants are examined for these problems. The numbers of plants with each type of problem are shown in the Venn diagram. 47 of the plants have none of the problems. \includegraphics[max width=\textwidth, alt={}, center]{e54eba7c-d862-435a-acdd-27df6ede5fab-3_654_804_1262_699}
  1. One of the 100 plants is selected at random. Find the probability that this plant has
    (A) at most one of the problems,
    (B) exactly two of the problems.
  2. Three of the 100 plants are selected at random. Find the probability that all of them have at least one of the problems.
OCR C1 Q5
6 marks Moderate -0.3
5
  1. Solve the simultaneous equations $$y = x ^ { 2 } - 3 x + 2 , \quad y = 3 x - 7 .$$
  2. What can you deduce from the solution to part (i) about the graphs of \(y = x ^ { 2 } - 3 x + 2\) and \(y = 3 x - 7\) ?
  3. Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 3 x + 2\) at the point ( 3,2 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
OCR C1 Q7
13 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.
  1. Calculate the radius of the circle and the coordinates of the centre.
  2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
  3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).
OCR MEI S1 Q1
18 marks Easy -1.2
1 Laura frequently flies to business meetings and often finds that her flights are delayed. A flight may be delayed due to technical problems, weather problems or congestion problems, with probabilities \(0.2,0.15\) and 0.1 respectively. The tree diagram shows this information. \includegraphics[max width=\textwidth, alt={}, center]{10679ff3-494d-4f4e-a38a-0832faa91690-1_605_1650_534_284}
  1. Write down the values of the probabilities \(a , b\) and \(c\) shown in the tree diagram. One of Laura's flights is selected at random.
  2. Find the probability that Laura's flight is not delayed and hence write down the probability that it is delayed.
  3. Find the probability that Laura's flight is delayed due to just one of the three problems.
  4. Given that Laura's flight is delayed, find the probability that the delay is due to just one of the three problems.
  5. Given that Laura's flight has no technical problems, find the probability that it is delayed.
  6. In a particular year, Laura has 110 flights. Find the expected number of flights that are delayed.
OCR MEI S1 Q2
8 marks Moderate -0.8
2 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 Q3
8 marks Moderate -0.8
3 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.
  • \(W\) is the event that this person speaks Welsh.
  • \(C\) is the event that this person is a child.
You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).
  1. Determine whether the events \(W\) and \(C\) are independent.
  2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
  3. Find \(\mathrm { P } ( W \mid C )\).
  4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults. [1]
OCR C1 2005 January Q1
6 marks Easy -1.8
1
  1. Express \(11 ^ { - 2 }\) as a fraction.
  2. Evaluate \(100 ^ { \frac { 3 } { 2 } }\).
  3. Express \(\sqrt { 50 } + \frac { 6 } { \sqrt { 3 } }\) in the form \(a \sqrt { } 2 + b \sqrt { } 3\), where \(a\) and \(b\) are integers.
OCR C1 2005 January Q2
4 marks Moderate -0.8
2 Given that \(2 x ^ { 2 } - 12 x + p = q ( x - r ) ^ { 2 } + 10\) for all values of \(x\), find the constants \(p , q\) and \(r\).
OCR C1 2005 January Q3
4 marks Easy -1.2
3
  1. The curve \(y = 5 \sqrt { } x\) is transformed by a stretch, scale factor \(\frac { 1 } { 2 }\), parallel to the \(x\)-axis. Find the equation of the curve after it has been transformed.
  2. Describe the single transformation which transforms the curve \(y = 5 \sqrt { } x\) to the curve \(y = ( 5 \sqrt { } x ) - 3\).
OCR C1 2005 January Q4
5 marks Moderate -0.5
4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$
OCR C1 2005 January Q5
7 marks Easy -1.2
5 On separate diagrams,
  1. sketch the curve \(y = \frac { 1 } { x }\),
  2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
  3. sketch the curve \(y = - \sqrt { } x\).
OCR C1 2005 January Q6
7 marks Moderate -0.8
6
  1. Calculate the discriminant of \(- 2 x ^ { 2 } + 7 x + 3\) and hence state the number of real roots of the equation \(- 2 x ^ { 2 } + 7 x + 3 = 0\).
  2. The quadratic equation \(2 x ^ { 2 } + ( p + 1 ) x + 8 = 0\) has equal roots. Find the possible values of \(p\).
OCR C1 2005 January Q7
9 marks Easy -1.3
7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:
  1. \(y = \frac { 1 } { 2 } x ^ { 4 } - 3 x\),
  2. \(y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )\),
  3. \(y = \sqrt [ 5 ] { x }\).
OCR C1 2005 January Q8
8 marks Moderate -0.8
8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.
  1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
  2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
  3. By solving the inequalities in parts (i) and (ii), determine the set of possible values of \(x\).
OCR C1 2005 January Q9
9 marks Moderate -0.3
9
  1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
  2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
  3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
  4. Deduce the value of \(k\).
OCR C1 2005 January Q10
13 marks Moderate -0.8
10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.
  1. Calculate the gradient of \(D E\).
  2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
  3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
  4. Calculate the length of \(D F\).
  5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).
OCR MEI S1 Q1
16 marks Moderate -0.3
1 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining \(1 \%\) overseas. Two people are selected at random. You may use the tree diagram below in answering this question. \includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-1_944_1118_626_547}
  1. Find the probability that
    (A) both of these people were born in the rest of the UK,
    (B) at least one of these people was born in England,
    (C) neither of these people was born overseas.
  2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
  3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
    (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.
OCR MEI S1 Q2
8 marks Moderate -0.8
2 Steve is going on holiday. The probability that he is delayed on his outward flight is 0.3 . The probability that he is delayed on his return flight is 0.2 , independently of whether or not he is delayed on the outward flight.
  1. Find the probability that Steve is delayed on his outward flight but not on his return flight.
  2. Find the probability that he is delayed on at least one of the two flights.
  3. Given that he is delayed on at least one flight, find the probability that he is delayed on both flights.
OCR MEI S1 Q3
8 marks Standard +0.8
3 Sophie and James are having a tennis competition. The winner of the competition is the first to win 2 matches in a row. If the competition has not been decided after 5 matches, then the player who has won more matches is declared the winner of the competition. For example, the following sequences are two ways in which Sophie could win the competition. ( \(\mathbf { S }\) represents a match won by Sophie; \(\mathbf { J }\) represents a match won by James.) \section*{SJSS SJSJS}
  1. Explain why the sequence \(\mathbf { S S J }\) is not possible.
  2. Write down the other three possible sequences in which Sophie wins the competition.
  3. The probability that Sophie wins a match is 0.7 . Find the probability that she wins the competition in no more than 4 matches.
OCR MEI S1 Q4
8 marks Easy -1.2
4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram. \includegraphics[max width=\textwidth, alt={}, center]{b56ccabe-0e51-4555-b550-78ba347f69bb-3_803_804_520_717} One of the residents is selected at random.
  1. Find the probability that this resident recycles
    (A) at least one of the materials,
    (B) exactly one of the materials.
  2. Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
  3. Find the probability that exactly one of them recycles kitchen waste.