Questions PURE (137 questions)

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OCR PURE Q6
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
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
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
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.
  1. 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.
  2. 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
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.
    • The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
    • The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.
    • Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011.
    • In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011.
OCR PURE Q11
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
1
  1. Prove that \(\cos x + \sin x \tan x \equiv \frac { 1 } { \cos x }\) (where \(x \neq \frac { 1 } { 2 } n \pi\) for any odd integer \(n\) ).
  2. Solve the equation \(2 \sin ^ { 2 } x = \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
    1. Show that \(B\) lies on \(A C\).
    2. Find the ratio \(A B : B C\).
  3. The diagram shows the line \(x + y = 6\) and the point \(P ( 2,4 )\) that lies on the line. A copy of the diagram is given in the Printed Answer Booklet.
    \includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-03_741_720_1080_324} The distinct point \(Q\) also lies on the line \(x + y = 6\) and is such that \(| \overrightarrow { O Q } | = | \overrightarrow { O P } |\), where \(O\) is the origin. Find the magnitude and direction of the vector \(\overrightarrow { P Q }\).
  4. The point \(R\) is such that \(\overrightarrow { P R }\) is perpendicular to \(\overrightarrow { O P }\) and \(| \overrightarrow { P R } | = \frac { 1 } { 2 } | \overrightarrow { O P } |\). Write down the position vectors of the two possible positions of the point \(R\).
OCR PURE Q3
3 The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a quadratic function of \(x\).
A copy of the diagram is given in the Printed Answer Booklet.
\includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-04_903_926_406_242}
  1. On the copy of the diagram in the Printed Answer Booklet, draw a possible graph of the gradient function \(y = \mathrm { f } ^ { \prime } ( x )\).
  2. State the gradient of the graph of \(y = \mathrm { f } ^ { \prime \prime } ( x )\).
OCR PURE Q4
4 A curve has equation \(y = \mathrm { e } ^ { 3 x }\).
  1. Determine the value of \(x\) when \(y = 10\).
  2. Determine the gradient of the tangent to the curve at the point where \(x = 2\).
OCR PURE Q6
6
  1. Determine the two real roots of the equation \(8 x ^ { 6 } + 7 x ^ { 3 } - 1 = 0\).
  2. Determine the coordinates of the stationary points on the curve \(y = 8 x ^ { 7 } + \frac { 49 } { 4 } x ^ { 4 } - 7 x\).
  3. For each of the stationary points, use the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine whether it is a maximum or a minimum.
OCR PURE Q7
7
  1. Two real numbers are denoted by \(a\) and \(b\).
    1. Write down expressions for the following.
      • The mean of the squares of \(a\) and \(b\)
  2. The square of the mean of \(a\) and \(b\)
    (ii) Prove that the mean of the squares of \(a\) and \(b\) is greater than or equal to the square of their mean.
  3. You are given that the result in part (a)(ii) is true for any two or more real numbers.
  4. Explain what this result shows about the variance of a set of data.
OCR PURE Q9
9 In a survey, 50 people were asked whether they had passed A-level English and whether they had passed A-level Mathematics. The numbers of people in various categories are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-07_540_808_598_255}
  1. A person is chosen at random from the 50 people. Find the probability that this person has passed A-level Mathematics.
  2. Two people are chosen at random, without replacement, from those who have passed A-level in at least one of the two subjects. Determine the probability that both of these people have passed A-level Mathematics.
OCR PURE Q10
10 The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-08_734_1693_342_178}
  1. Calculate the number of boulders with masses between 60 and 65 kg .
    1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample.
    2. Explain why your answers are only estimates.
  2. Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution.
  3. Give one advantage of using a histogram rather than a pie chart in this context.
OCR PURE Q11
11 Casey and Riley attend a large school. They are discussing the music preferences of the students at their school. Casey believes that the favourite band of 75\% of the students is Blue Rocking. Riley believes that the true figure is greater than 75\%. They plan to carry out a hypothesis test at the \(5 \%\) significance level, using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.75\) and \(\mathrm { H } _ { 1 } : p > 0.75\). They choose a random sample of 60 students from the school, and note the number, \(X\), who say that their favourite band is Blue Rocking. They find that \(X = 50\).
  1. Assuming the null hypothesis to be true, Riley correctly calculates that \(\mathrm { P } ( X = 50 ) = 0.0407\), correct to 3 significant figures. Riley says that, because this value is less than 0.05 , the null hypothesis should be rejected.
    Explain why this statement is incorrect.
  2. Carry out the test.
    1. State which mathematical model is used in the calculation in part (b), including the value(s) of any parameter(s).
    2. The random sample was chosen without replacement. Explain whether this invalidates the model used in part (b).
OCR PURE Q12
12 This question deals with information about the populations of Local Authorities (LAs) in the North of England, taken from the 2011 census. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 1} \includegraphics[alt={},max width=\textwidth]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-10_437_903_450_109}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 2} \includegraphics[alt={},max width=\textwidth]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-10_423_905_466_1046}
\end{figure} Fig. 1 and Fig. 2 both show strong correlation, but of two different kinds.
  1. For each diagram, use a single word to describe the kind of correlation shown.
  2. For each diagram, suggest a reason, in context, why the correlation is of the particular kind described in part (a). Fig. 3 is the same as Fig. 2 but with the point \(A\) marked.
    Fig. 4 shows information about the same LAs as Fig. 2 and Fig. 3. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 3} \includegraphics[alt={},max width=\textwidth]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-10_417_904_1674_95}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 4} \includegraphics[alt={},max width=\textwidth]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-10_449_887_1644_1080}
    \end{figure}
  3. Point \(A\) in Fig. 3 and point \(B\) in Fig. 4 represent the same LA. Explain how you can tell that this LA has a large population. \section*{END OF QUESTION PAPER}
OCR PURE Q1
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
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 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
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 \(p > 0\).
OCR PURE Q5
5 In this question you must show detailed reasoning.
The line \(x + 5 y = k\) is a tangent to the curve \(x ^ { 2 } - 4 y = 10\). Find the value of the constant \(k\).
OCR PURE Q6
6 A pan of water is heated until it reaches \(100 ^ { \circ } \mathrm { C }\). Once the water reaches \(100 ^ { \circ } \mathrm { C }\), the heat is switched off and the temperature \(T ^ { \circ } \mathrm { C }\) of the water decreases. The temperature of the water is modelled by the equation $$T = 25 + a \mathrm { e } ^ { - k t }$$ where \(t\) denotes the time, in minutes, after the heat is switched off and \(a\) and \(k\) are positive constants.
  1. Write down the value of \(a\).
  2. Explain what the value of 25 represents in the equation \(T = 25 + a \mathrm { e } ^ { - k t }\). When the heat is switched off, the initial rate of decrease of the temperature of the water is \(15 ^ { \circ } \mathrm { C }\) per minute.
  3. Calculate the value of \(k\).
  4. Find the time taken for the temperature of the water to drop from \(100 ^ { \circ } \mathrm { C }\) to \(45 ^ { \circ } \mathrm { C }\).
  5. A second pan of water is heated, but the heat is turned off when the water is at a temperature of less than \(100 ^ { \circ } \mathrm { C }\). Suggest how the equation for the temperature as the water cools would be modified by this.
OCR PURE Q7
7
  1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
  2. Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.
OCR PURE Q9
9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).
OCR PURE Q10
10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}
  1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
  2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.
OCR PURE Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-7_127_1147_260_459} A particle \(P\) is moving along a straight line with constant acceleration. Initially the particle is at \(O\). After 9 s , \(P\) is at a point \(A\), where \(O A = 18 \mathrm {~m}\) (see diagram) and the velocity of \(P\) at \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { O A }\).
  1. (a) Show that the initial speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
    (b) Find the acceleration of \(P\).
    \(B\) is a point on the line such that \(O B = 10 \mathrm {~m}\), as shown in the diagram.
  2. Show that \(P\) is never at point \(B\). A second particle \(Q\) moves along the same straight line, but has variable acceleration. Initially \(Q\) is at \(O\), and the displacement of \(Q\) from \(O\) at time \(t\) seconds is given by $$x = a t ^ { 3 } + b t ^ { 2 } + c t$$ where \(a\), \(b\) and \(c\) are constants.
    It is given that
    • the velocity and acceleration of \(Q\) at the point \(O\) are the same as those of \(P\) at \(O\),
    • \(\quad Q\) reaches the point \(A\) when \(t = 6\).
    • Find the velocity of \(Q\) at \(A\).
    \section*{OCR} \section*{Oxford Cambridge and RSA}