Questions — OCR (4619 questions)

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OCR Pure 1 2018 March Q4
4
  1. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided in the Printed Answer Booklet.
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of intersection of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).
OCR Pure 1 2018 March Q5
5 An ice cream seller expects that the number of sales will increase by the same amount every week from May onwards. 150 ice creams are sold in Week 1 and 166 ice creams are sold in Week 2. The ice cream seller makes a profit of \(\pounds 1.25\) for each ice cream sold.
  1. Find the expected profit in Week 10.
  2. In which week will the total expected profits first exceed \(\pounds 5000\) ?
  3. Give two reasons why this model may not be appropriate.
OCR Pure 1 2018 March Q6
6 Prove by contradiction that \(\sqrt { 7 }\) is irrational.
OCR Pure 1 2018 March Q7
7 Two lifeboat stations, \(P\) and \(Q\), are situated on the coastline with \(Q\) being due south of \(P\). A stationary ship is at sea, at a distance of 4.8 km from \(P\) and a distance of 2.2 km from \(Q\). The ship is on a bearing of \(155 ^ { \circ }\) from \(P\).
  1. Find any possible bearings of the ship from \(Q\).
  2. Find the shortest distance from the ship to the line \(P Q\).
  3. Give a reason why the actual distance from the ship to the coastline may be different to your answer to part (ii).
OCR Pure 1 2018 March Q8
8
  1. Given that \(y = \sec x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  2. In this question you must show detailed reasoning. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } ( \sec 2 x + \tan 2 x ) ^ { 2 } \mathrm {~d} x\).
OCR Pure 1 2018 March Q9
9
  1. Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
  2. Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
  3. State the set of values for which the expansion in part (ii) is valid.
OCR Pure 1 2018 March Q10
10 In this question you must show detailed reasoning.
Show that the curve with equation \(x ^ { 2 } - 4 x y + 8 y ^ { 3 } - 4 = 0\) has exactly one stationary point.
OCR Pure 1 2018 March Q11
11 The height, in metres, of the sea at a coastal town during a day may be modelled by the function $$\mathrm { f } ( t ) = 1.7 + 0.8 \sin ( 30 t ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
  1. (a) Find the maximum height of the sea as given by this model.
    (b) Find the time of day at which this maximum height first occurs.
  2. Determine the time when, according to this model, the height of the sea will first be 1.2 m . The height, in metres, at a different coastal town during a day may be modelled by the function $$\mathrm { g } ( t ) = a + b \sin ( c t + d ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
  3. It is given that at this different coastal town the maximum height of the sea is 3.1 m , and this height occurs at 0500 and 1700. The minimum height of the sea is 0.7 m , and this height occurs at 1100 and 2300 . Find the values of the constants \(a , b , c\) and \(d\).
  4. It is instead given that the maximum height of the sea actually occurs at 0500 and 1709 . State, with a reason, how this will affect the value of \(c\) found in part (iii).
    \includegraphics[max width=\textwidth, alt={}, center]{74a37bca-0b28-4c48-bd21-a9304f31b8f8-6_563_568_322_751} The diagram shows the curve \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\) for \(x \geqslant 0\).
  5. Use the substitution \(u ^ { 2 } = x + 1\) to find \(\int \mathrm { e } ^ { \sqrt { x + 1 } } \mathrm {~d} x\).
  6. Make \(x\) the subject of the equation \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\).
  7. Hence show that \(\int _ { \mathrm { e } } ^ { \mathrm { e } ^ { 4 } } \left( ( \ln y ) ^ { 2 } - 1 \right) \mathrm { d } y = 9 \mathrm { e } ^ { 4 }\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA
OCR Stats 1 2018 March Q1
1 Part of the graph of \(y = \mathrm { f } ( x )\) is shown below, where \(\mathrm { f } ( x )\) is a cubic polynomial.
\includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-04_681_679_475_694}
  1. Find \(\mathrm { f } ( - 1 )\).
  2. Write down three linear factors of \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d\).
  3. Show that \(a = - 2\).
  4. Find \(b , c\) and \(d\).
OCR Stats 1 2018 March Q2
2 Angela makes the following claim. \begin{displayquote} " \(n\) is an odd positive integer greater than \(1 \Rightarrow 2 ^ { n } - 1\) is prime" \end{displayquote} Prove that Angela's claim is false.
OCR Stats 1 2018 March Q3
3 On a particular voyage, a ship sails 500 km at a constant speed of \(v \mathrm {~km} / \mathrm { h }\). The cost for the voyage is \(\pounds R\) per hour. The total cost of the voyage is \(\pounds T\).
  1. Show that \(T = \frac { 500 R } { v }\). The running cost is modelled by the following formula. $$R = 270 + \frac { v ^ { 3 } } { 200 }$$ The ship's owner wishes to sail at a speed that will minimise the total cost for the voyage. It is given that the graph of \(T\) against \(v\) has exactly one stationary point, which is a minimum.
  2. Find the speed that gives the minimum value of \(T\).
  3. Find the minimum value of the total cost.
OCR Stats 1 2018 March Q4
4 The diagram shows part of the graph of \(y = \cos x\), where \(x\) is measured in radians.
\includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-05_609_846_294_607}
  1. Use the copy of this diagram in the Printed Answer Booklet to find an approximate solution to the equation \(x = \cos x\).
  2. Use an iterative method to find the solution to the equation \(x = \cos x\) correct to 3 significant figures. You should show your first, second and last two iterations, writing down all the figures on your calculator.
OCR Stats 1 2018 March Q5
5 Points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1
2
3 \end{array} \right) , \left( \begin{array} { c } 2
- 1
5 \end{array} \right)\) and \(\left( \begin{array} { c } - 4
0
3 \end{array} \right)\) respectively.
  1. Find the exact distance between the midpoint of \(A B\) and the midpoint of \(B C\). Point \(D\) has position vector \(\left( \begin{array} { c } x
    - 6
    z \end{array} \right)\) and the line \(C D\) is parallel to the line \(A B\).
  2. Find all the possible pairs of \(x\) and \(z\).
OCR Stats 1 2018 March Q6
6 In this question you must show detailed reasoning.
  1. Use the formula for \(\tan ( A - B )\) to show that \(\tan \frac { \pi } { 12 } = 2 - \sqrt { 3 }\).
  2. Solve the equation \(2 \sqrt { 3 } \sin 3 A - 2 \cos 3 A = 1\) for \(0 ^ { \circ } \leqslant A < 180 ^ { \circ }\).
OCR Stats 1 2018 March Q7
7 A tank is shaped as a cuboid. The base has dimensions 10 cm by 10 cm . Initially the tank is empty. Water flows into the tank at \(25 \mathrm {~cm} ^ { 3 }\) per second. Water also leaks out of the tank at \(4 h ^ { 2 } \mathrm {~cm} ^ { 3 }\) per second, where \(h \mathrm {~cm}\) is the depth of the water after \(t\) seconds. Find the time taken for the water to reach a depth of 2 cm .
OCR Stats 1 2018 March Q8
8 The masses, \(X\) grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and \(70 \%\) of the tomatoes have masses greater than 53.0 g .
  1. Find the percentage of tomatoes with masses greater than 59.0 g .
  2. Find the percentage of tomatoes with masses greater than 65.0 g .
  3. Given that \(\mathrm { P } ( a < X < 50 ) = 0.1\), find \(a\).
OCR Stats 1 2018 March Q9
9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.
  1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
  2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
  3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
  4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.
OCR Stats 1 2018 March Q10
7 marks
10 A researcher is investigating the actual lengths of time that patients spend at their appointments with the doctors at a certain clinic. There are 12 doctors at the clinic, and each doctor has 24 appointments per day. The researcher plans to choose a sample of 24 appointments on a particular day.
  1. The researcher considers the following two methods for choosing the sample. Method A: Choose a random sample of 24 appointments from the 288 on that day.
    Method B: Choose one doctor's 1st and 2nd appointments. Choose another doctor's 3rd and 4th appointments and so on until the last doctor's 23rd and 24th appointments. For each of A and B state a disadvantage of using this method. Appointments are scheduled to last 10 minutes. The researcher suspects that the actual times that patients spend are more than 10 minutes on average. To test this suspicion, he uses method A , and takes a random sample of 24 appointments. He notes the actual time spent for each appointment and carries out a hypothesis test at the \(1 \%\) significance level.
  2. Explain why a 1-tail test is appropriate. The population mean of the actual times that patients spend at their appointments is denoted by \(\mu\) minutes.
  3. Assuming that \(\mu = 10\), state the probability that the conclusion of the test will be that \(\mu\) is not greater than 10 . The actual lengths of time, in minutes, that patients spend for their appointments may be assumed to have a normal distribution with standard deviation 3.4.
    [0pt]
  4. Given that the total length of time spent for the 24 appointments is 285 minutes, carry out the test. [7]
  5. In part (iv) it was necessary to use the fact that the sample mean is normally distributed. Give a reason why you know that this is true in this case.
OCR Stats 1 2018 March Q11
11 The scatter diagram shows data, taken from the pre-release data set, for several Local Authorities in one region of the UK in 2011. The diagram shows, for each Local Authority, the number of workers who drove to work, and the number of workers who walked to work. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{2011} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-08_483_956_479_557}
\end{figure}
  1. Four students calculated the value of Pearson's product-moment correlation coefficient for the data in the diagram. Their answers were \(0.913,0.124 , - 0.913\) and - 0.124 . One of these values is correct. Without calculation state, with a reason, which is the correct value.
  2. Sanjay makes the following statement.
    "The diagram shows that, in any Local Authority, if there are a large number of people who drive to work there will be a large number who walk to work." Give a reason why this statement is incorrect.
  3. Rosie makes the following statement.
    "The diagram must be wrong because it shows good positive correlation. If there are more people driving to work, there will be fewer people walking to work, so there would be negative correlation." Explain briefly why Rosie's statement is incorrect.
  4. The diagram shows a fairly close relationship between the two variables. One point on the diagram represents a Local Authority where this relationship is less strong than for the others. On the diagram in the Printed Answer Booklet, label this point A.
  5. Given that the point A represents a metropolitan borough, suggest a reason why the relationship is less strong for this Local Authority than for the others in the region. The scatter diagram below shows the corresponding data for the same region in 2001. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{2001} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-09_481_885_388_591}
    \end{figure}
  6. (a) State a change that has taken place in the metropolitan borough represented by the point A between 2001 and 2011.
    (b) Suggest a possible reason for this change.
OCR Stats 1 2018 March Q12
12 Rob has two six-sided dice, each with sides numbered 1, 2, 3, 4, 5, 6.
One dice is fair. The other dice is biased, with probabilities as shown in the table.
Biased die
\(y\)123456
\(\mathrm { P } ( Y = y )\)0.30.250.20.140.10.01
Rob throws each dice once and notes the two scores, \(X\) on the fair dice and \(Y\) on the biased dice. He then calculates the value of the variable \(S\) which is defined as follows.
  • If \(X \leqslant 3\), then \(S = X + 2 Y\).
  • If \(X > 3\), then \(S = X + Y\).
    1. (a) Draw up a sample space diagram showing all the possible outcomes and the corresponding values of \(S\).
      (b) On your diagram, circle the four cells where the value \(S = 10\) occurs.
    2. Explain the mistake in the following calculation.
$$\mathrm { P } ( S = 10 ) = \frac { \text { Number of outcomes giving } S = 10 } { \text { Total number of outcomes } } = \frac { 4 } { 36 } = \frac { 1 } { 9 } .$$
  • Find the correct value of \(\mathrm { P } ( S = 10 )\).
  • Given that \(S = 10\), find the probability that the score on one of the dice is 4 .
  • The events " \(X = 1\) or 2 " and " \(S = n\) " are mutually exclusive. Given that \(\mathrm { P } ( S = n ) \neq 0\), find the value of \(n\).
    •
    OCR Mechanics 1 2018 March Q1
    1 Show in a sketch the region of the \(x - y\) plane within which all three of the following inequalities are satisfied. $$3 y \geqslant 4 x \quad y - x \leqslant 1 \quad y \geqslant ( x - 1 ) ^ { 2 }$$ You should indicate the region for which the inequalities hold by labelling the region \(R\).
    OCR Mechanics 1 2018 March Q2
    2 The first term of a geometric progression is 12 and the second term is 9 .
    1. Find the fifth term. The sum of the first \(N\) terms is denoted by \(S _ { N }\) and the sum to infinity is denoted by \(S _ { \infty }\). It is given that the difference between \(S _ { \infty }\) and \(S _ { N }\) is at most 0.0096 .
    2. Show that \(\left( \frac { 3 } { 4 } \right) ^ { N } \leqslant 0.0002\).
    3. Use logarithms to find the smallest possible value of \(N\).
    OCR Mechanics 1 2018 March Q3
    3 A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm { e } ^ { 3 x } - 5\). Give details of these transformations.
    OCR Mechanics 1 2018 March Q4
    4 A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t ^ { 2 } , \quad y = t ^ { 3 }$$
    1. Show that the equation of the tangent at the point with parameter \(t\) is $$2 y = 3 t x - t ^ { 3 } .$$
    2. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A \left( \frac { 19 } { 12 } , - \frac { 15 } { 8 } \right)\) and it meets the \(x\)-axis at the point \(B\).
      Find the area of triangle \(O A B\), where \(O\) is the origin.
    OCR Mechanics 1 2018 March Q5
    5 In this question you must show detailed reasoning.
    \includegraphics[max width=\textwidth, alt={}]{467d7747-6a07-40ea-bb47-41ea3117f233-5_392_1102_319_466}
    The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 3 x \right) \mathrm { e } ^ { - x }$$ The diagram shows the curve \(y = \mathrm { f } ( x )\).
    1. Find the range of f.
    2. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm { g } ( x ) = \left( 2 x ^ { 2 } - 3 x \right) \mathrm { e } ^ { - x } .$$ Given that g is a one-one function, state the least possible value of \(k\).
    3. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis.
    4. Determine the values of \(p\) and \(q\) for which $$x ^ { 2 } - 6 x + 10 \equiv ( x - p ) ^ { 2 } + q .$$
    5. Use the substitution \(x - p = \tan u\), where \(p\) takes the value found in part (i), to evaluate $$\int _ { 3 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 6 x + 10 } \mathrm {~d} x .$$
    6. Determine the value of $$\int _ { 3 } ^ { 4 } \frac { x } { x ^ { 2 } - 6 x + 10 } \mathrm {~d} x$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants to be determined.