Questions PURE (178 questions)

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OCR PURE Q4
7 marks Moderate -0.8
4
  1. Simplify \(2 \binom { 6 } { - 3 } - 3 \binom { - 1 } { 2 }\).
  2. The vector \(\mathbf { a }\) is defined by \(\mathbf { a } = r \binom { 6 } { - 3 } + s \binom { - 1 } { 2 }\), where \(r\) and \(s\) are constants. Determine two pairs of values of \(r\) and \(s\) such that \(\mathbf { a }\) is parallel to the \(y\)-axis and \(| \mathbf { a } | = 3\).
OCR PURE Q5
8 marks Standard +0.3
5 The fuel consumption of a car, \(C\) miles per gallon, varies with the speed, \(v\) miles per hour. Jamal models the fuel consumption of his car by the formula \(C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 }\), for \(0 \leqslant v \leqslant 80\).
  1. Suggest a reason why Jamal has included an upper limit in his model.
  2. Determine the speed that gives the maximum fuel consumption. Amaya's car does more miles per gallon than Jamal's car. She proposes to model the fuel consumption of her car using a formula of the form \(C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 } + k\), for \(0 \leqslant v \leqslant 80\), where \(k\) is a positive constant.
  3. Give a reason why this model is not suitable.
  4. Suggest a different change to Jamal's formula which would give a more suitable model.
OCR PURE Q6
3 marks Easy -1.2
6 The power output, \(P\) watts, of a certain wind turbine is proportional to the cube of the wind speed \(v \mathrm {~ms} ^ { - 1 }\). When \(v = 3.6 , P = 50\).
Determine the wind speed that will give a power output of 225 watts.
OCR PURE Q7
3 marks Moderate -0.8
7 The relationship between the variables \(P\) and \(Q\) is modelled by the formula \(Q = a P ^ { n }\) where \(a\) and \(n\) are constants.
Some values of \(P\) and \(Q\) are obtained from an experiment.
  1. State appropriate quantities to plot so that the resulting points lie approximately in a straight line.
  2. Explain how to use such a graph to estimate the value of \(n\).
OCR PURE Q8
5 marks Standard +0.3
8
  1. Prove that the following statement is not true. $$p \text { is a positive integer } \Rightarrow 2 ^ { p } \geqslant p ^ { 2 }$$
  2. Prove that the following statement is true. \(m\) and \(n\) are consecutive positive odd numbers \(\Rightarrow m n + 1\) is the square of an even number
OCR PURE Q9
6 marks Standard +0.3
9 In this question you must show detailed reasoning.
Find the equation of the straight line with positive gradient that passes through \(( 0,2 )\) and is a tangent to the curve \(y = x ^ { 2 } - x + 6\).
OCR PURE Q10
6 marks Easy -1.8
10 Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town B .
  1. In the Printed Answer Booklet, complete the two-way frequency table.
    \multirow{2}{*}{}Town
    ABCTotal
    adult
    child
    Total
  2. One of the people is chosen at random.
    1. Find the probability that this person is an adult from town A .
    2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 . Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756 Jane used the same figures to write down the following sample of five student numbers.
      114, 287, 562, 81 and 817
    1. State, with a reason, which one of these samples is not random.
    2. Explain why Jane omitted the number 859 from her sample.
OCR PURE Q11
6 marks Moderate -0.5
11 A student is investigating changes in the number of residents in Local Authorities in the SouthEast Region between 2001 and 2011. The scatter diagram shows the number \(x\) of residents in these Local Authorities in the age group 8 to 9 in 2001 and the number \(y\) of residents in the same Local Authorities in the age group 18 to 19 in 2011.
[diagram]
  1. Suggest a reason why the student is comparing these two age groups in 2001 and 2011. The student notices that most of the data points are close to the line \(y = x\).
    1. Explain what this suggests about the residents in these Local Authorities.
    2. The student says that correlation does not imply causation, so there is no causal link between the values of \(x\) and the values of \(y\). Explain whether or not they are correct.
  3. Some of these Local Authorities contain universities.
    1. On the diagram in the Printed Answer Booklet, circle three points that are likely to represent Local Authorities containing universities.
    2. Give a reason for your choice of points in part (c)(i). Assume that the proportion of residents in age group 8 to 9 in 2001 was roughly the same in each Local Authority in the South-East. The Local Authority in this region with the largest population is Medway.
  4. On the diagram in the Printed Answer Booklet, label clearly with the letter \(M\) the point that corresponds to Medway.
OCR PURE Q12
7 marks Moderate -0.3
12 The variable \(X\) has the distribution \(\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)\). The probabilities \(\mathrm { P } ( X = r )\) for \(r = 0\) to 50 are given by the terms of the expansion of \(( a + b ) ^ { n }\) for specific values of \(a , b\) and \(n\).
  1. State the values of \(a\), \(b\) and \(n\). A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the \(5 \%\) significance level.
  2. Write down suitable hypotheses for the test.
  3. Determine the rejection region for the test, showing the values of any relevant probabilities.
OCR PURE Q13
7 marks Challenging +1.2
13
  1. The probability distribution of a random variable \(X\) is shown in the table, where \(p\) is a constant.
    \(x\)0123
    \(P ( X = x )\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 4 }\)\(p\)\(3 p\)
    Two values of \(X\) are chosen at random. Determine the probability that their product is greater than their sum.
  2. A random variable \(Y\) takes \(n\) values, each of which is equally likely. Two values, \(Y _ { 1 }\) and \(Y _ { 2 }\), of \(Y\) are chosen at random. It is given that \(\mathrm { P } \left( Y _ { 1 } = Y _ { 2 } \right) = 0.02\).
    Find \(\mathrm { P } \left( Y _ { 1 } > Y _ { 2 } \right)\).
OCR PURE Q1
2 marks Moderate -0.8
1 Find the term in \(x ^ { 3 }\) in the binomial expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR PURE Q2
5 marks Moderate -0.8
2 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 5 x ^ { 3 } - 4 x ^ { 2 } + a x - 2\), where \(a\) is a constant. You are given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  1. Find the value of \(a\).
  2. Find all the factors of \(\mathrm { f } ( x )\).
OCR PURE Q3
7 marks Standard +0.3
3 The diagram in the Printed Answer Booklet shows part of the graph of \(y = x ^ { 2 } - 4 x + 3\).
  1. It is required to solve the equation \(x ^ { 2 } - 3 x + 1 = 0\) graphically by drawing a straight line with equation \(y = m x + c\) on the diagram, where \(m\) and \(c\) are constants. Find the values of \(m\) and \(c\).
  2. Use the graph to find approximate values of the roots of the equation \(x ^ { 2 } - 3 x + 1 = 0\).
  3. By shading, or otherwise, indicate clearly the regions where all of the following inequalities are satisfied. You should use the values of \(m\) and \(c\) found in part (a). \(x \geqslant 0\) \(x \leqslant 4\) \(y \leqslant x ^ { 2 } - 4 x + 3\) \(y \geqslant m x + c\)
OCR PURE Q4
8 marks Moderate -0.3
4 In this question you must show detailed reasoning. Solve the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. \(2 \tan x + 1 = 4\)
  2. \(5 \sin x - 1 = 2 \cos ^ { 2 } x\)
OCR PURE Q5
8 marks Moderate -0.8
5 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 3 x\). The curve passes through the point (6, 20).
  1. Determine the equation of the curve.
  2. Hence determine \(\int _ { 1 } ^ { p } y \mathrm {~d} x\) in terms of the constant \(p\).
OCR PURE Q6
11 marks Standard +0.3
6 During some research the size, \(P\), of a population of insects, at time \(t\) months after the start of the research, is modelled by the following formula. \(P = 100 \mathrm { e } ^ { t }\)
  1. Use this model to answer the following.
    1. Find the value of \(P\) when \(t = 4\).
    2. Find the value of \(t\) when the population is 9000 .
  2. It is suspected that a more appropriate model would be the following formula. \(P = k a ^ { t }\) where \(k\) and \(a\) are constants.
    1. Show that, using this model, the graph of \(\log _ { 10 } P\) against \(t\) would be a straight line. Some observations of \(t\) and \(P\) gave the following results.
      \(t\)12345
      \(P\)1005001800700019000
      \(\log _ { 10 } P\)2.002.703.263.854.28
    2. On the grid in the Printed Answer Booklet, draw a line of best fit for the data points \(\left( t , \log _ { 10 } P \right)\) given in the table.
    3. Hence estimate the values of \(k\) and \(a\).
OCR PURE Q7
8 marks Standard +0.3
7
  1. In this question you must show detailed reasoning. Find the range of values of the constant \(m\) for which the simultaneous equations \(y = m x\) and \(x ^ { 2 } + y ^ { 2 } - 6 x - 2 y + 5 = 0\) have real solutions.
  2. Give a geometrical interpretation of the solution in the case where \(m = 2\).
OCR PURE Q8
4 marks Easy -1.8
8 A random sample of 10 students from a college was chosen. They were asked how much time, \(x\) hours, they spent studying, and how much money, \(\pounds y\), they earned, in a typical week during term time. The results are shown in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{c4cc2cd8-46bf-448f-b223-92378984bfde-5_544_741_555_242}
  1. Comment on the relationship shown by the diagram between hours spent studying and money earned, during term time, by these 10 students. The coordinates of the points in the diagram are \(( 18,23 ) , ( 20,21 ) , ( 23,20 ) , ( 25,19 ) , ( 25,21 )\), \(( 27,18 ) , ( 32,16 ) , ( 38,17 ) , ( 40,16 )\) and \(( 41,23 )\).
  2. Find the mean and standard deviation of the number of hours spent per week studying during term time by these 10 students.
OCR PURE Q9
10 marks Standard +0.3
9 Last year, market research showed that \(8 \%\) of adults living in a certain town used a particular local coffee shop. Following an advertising campaign, it was expected that this proportion would increase. In order to test whether this had happened, a random sample of 150 adults in the town was chosen. The random variable \(X\) denotes the number of these 150 adults who said that they used the local coffee shop.
    1. Assuming that the proportion of adults using the local coffee shop is unchanged from the previous year, state a suitable binomial distribution with which to model the variable \(X\).
    2. The probabilities given by this model are the terms of the binomial expansion of an expression of the form \(( a + b ) ^ { n }\). Write down this expression, using appropriate values of \(a , b\) and \(n\). It was found that 18 of these 150 adults said that they use the local coffee shop.
  2. Test, at the 5\% significance level, whether the proportion of adults in the town who use the local coffee shop has increased. It was later discovered by a statistician that the random sample of 150 adults had been chosen from shoppers in the town on a Friday and a Saturday.
  3. Explain why this suggests that the assumptions made when using a binomial model for \(X\) may not be valid in this context.
OCR PURE Q10
6 marks Easy -1.8
10 The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK.
Geography codeLocal authorityWork mainly at or from homeUnderground, metro, light rail or tramBus, minibus or coachDriving a car or vanPassenger in a car or vanOn foot
E06000047County Durham0.74\%0.05\%-1.50\%4.58\%-2.99\%-0.97\%
E06000005Darlington0.26\%-0.01\%-3.25\%3.06\%-1.28\%0.99\%
E08000020Gateshead-0.01\%-0.01\%-2.28\%4.62\%-2.35\%-0.18\%
E06000001Hartlepool0.03\%-0.04\%-1.62\%4.80\%-2.38\%-0.26\%
E06000002Middlesbrough-0.34\%-0.01\%-2.32\%2.19\%-1.33\%0.67\%
E08000021Newcastle upon Tyne0.10\%-0.23\%-0.67\%-0.48\%-1.51\%1.75\%
E08000022North Tyneside0.05\%0.54\%-1.18\%3.30\%-2.21\%-0.60\%
E06000048Northumberland1.39\%-0.08\%-0.95\%3.50\%-2.37\%-1.44\%
E06000003Redcar and Cleveland-0.02\%-0.01\%-2.09\%4.20\%-2.06\%-0.49\%
E08000023South Tyneside-0.36\%2.03\%-3.05\%4.50\%-2.41\%-0.51\%
E06000004Stockton-on-Tees0.14\%0.03\%-2.02\%3.52\%-2.01\%-0.15\%
E08000024Sunderland0.17\%1.48\%-3.11\%4.89\%-2.21\%-0.52\%
Increase in percentage of employees travelling to work by various methods
The first two digits of the Geography code give the type of each of the LAs:
06: Unitary authority
07: Non-metropolitan district
08: Metropolitan borough
  1. In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
  2. Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.
OCR PURE Q11
6 marks Standard +0.3
11 Alex models the number of goals that a local team will score in any match as follows.
Number of goals01234
More
than 4
Probability\(\frac { 3 } { 25 }\)\(\frac { 1 } { 5 }\)\(\frac { 8 } { 25 }\)\(\frac { 7 } { 25 }\)\(\frac { 2 } { 25 }\)0
The number of goals scored in any match is independent of the number of goals scored in any other match.
  1. Alex chooses 3 matches at random. Use the model to determine the probability of each of the following.
    1. The team will score a total of exactly 1 goal in the 3 matches.
    2. The numbers of goals scored in the first 2 of the 3 matches will be equal, but the number of goals scored in the 3rd match will be different. During the first 10 matches this season, the team scores a total of 31 goals.
  2. Without carrying out a formal test, explain briefly whether this casts doubt on the validity of Alex's model. \section*{END OF QUESTION PAPER}
OCR PURE Q1
6 marks Standard +0.3
1 In triangle \(A B C , A B = 20 \mathrm {~cm}\) and angle \(B = 45 ^ { \circ }\).
  1. Given that \(A C = 16 \mathrm {~cm}\), find the two possible values for angle \(C\), correct to 1 decimal place.
  2. Given instead that the area of the triangle is \(75 \sqrt { 2 } \mathrm {~cm} ^ { 2 }\), find \(B C\).
OCR PURE Q2
4 marks Moderate -0.8
2
  1. The curve \(y = \frac { 2 } { 3 + x }\) is translated by four units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
  2. Describe fully the single transformation that transforms the curve \(y = \frac { 2 } { 3 + x }\) to \(y = \frac { 5 } { 3 + x }\).
OCR PURE Q3
3 marks Standard +0.3
3 In each of the following cases choose one of the statements $$P \Rightarrow Q \quad P \Leftarrow Q \quad P \Leftrightarrow Q$$ to describe the relationship between \(P\) and \(Q\).
  1. \(P : y = 3 x ^ { 5 } - 4 x ^ { 2 } + 12 x\) \(Q : \frac { \mathrm { d } y } { \mathrm {~d} x } = 15 x ^ { 4 } - 8 x + 12\)
  2. \(\quad P : x ^ { 5 } - 32 = 0\) where \(x\) is real \(Q : x = 2\)
  3. \(\quad P : \ln y < 0\) \(Q : y < 1\)
OCR PURE Q4
6 marks Moderate -0.3
4
  1. Express \(4 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. State the number of real roots of the equation \(4 x ^ { 2 } - 12 x + 11 = 0\).
  3. Explain fully how the value of \(r\) is related to the number of real roots of the equation \(p ( x + q ) ^ { 2 } + r = 0\) where \(p , q\) and \(r\) are real constants and \(p > 0\).