Questions — OCR AS Pure (21 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
OCR AS Pure 2017 Specimen Q1
1 Given that \(\mathrm { f } ( x ) = 6 x ^ { 3 } - 5 x\), find
  1. \(\mathrm { f } ^ { \prime } ( x )\),
  2. \(f ^ { \prime \prime } ( 2 )\).
OCR AS Pure 2017 Specimen Q2
2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.
  1. Find the coordinates of the centre of the circle.
  2. Find the equation of the tangent to the circle at the point \(B\).
OCR AS Pure 2017 Specimen Q3
3 The points \(P , Q\) and \(R\) have coordinates \(( - 1,6 ) , ( 2,10 )\) and \(( 11,1 )\) respectively. Find the angle \(P R Q\).
OCR AS Pure 2017 Specimen Q4
4 The curve \(y = 2 x ^ { 3 } + 3 x ^ { 2 } - k x + 4\) has a stationary point where \(x = 2\).
  1. Determine the value of the constant \(k\).
  2. Determine whether this stationary point is a maximum or a minimum point.
OCR AS Pure 2017 Specimen Q5
5
  1. Find \(\int \left( x ^ { 3 } - 6 x \right) \mathrm { d } x\).
    1. Find \(\int \left( \frac { 4 } { x ^ { 2 } } - 1 \right) \mathrm { d } x\).
    2. The diagram shows part of the curve \(y = \frac { 4 } { x ^ { 2 } } - 1\).
      \includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-04_707_1283_708_415} The curve crosses the \(x\)-axis at \(( 2,0 )\).
      The shaded region is bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = 5\). Calculate the area of the shaded region.
OCR AS Pure 2017 Specimen Q6
6 In this question you must show detailed reasoning. The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } + 4 x ^ { 2 } + 7 x - 5\).
  1. Show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Hence solve the equation \(4 \sin ^ { 3 } \theta + 4 \sin ^ { 2 } \theta + 7 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).
OCR AS Pure 2017 Specimen Q7
7
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\).
  2. Hence, or otherwise, solve \(2 x ^ { 2 } - x - 3 < 0\).
  3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of k .
OCR AS Pure 2017 Specimen Q8
8 A club secretary wishes to survey a sample of members of his club. He uses all members present at a particular meeting as his sample.
  1. Explain why this sample is likely to be biased. Later the secretary decides to choose a random sample of members.
    The club has 253 members and the secretary numbers the members from 1 to 253 . He then generates random 3-digit numbers on his calculator. The first six random numbers generated are 156, 965, 248, 156, 073 and 181. The secretary uses each number, where possible, as the number of a member in the sample.
  2. Find possible numbers for the first four members in the sample.
OCR AS Pure 2017 Specimen Q9
9 The probability distribution of a random variable \(X\) is given in the table.
\(x\)123
\(\mathrm { P } ( X = x )\)0.60.30.1
Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.
OCR AS Pure 2017 Specimen Q10
10
  1. Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
    Give your answer in ascending powers of \(x\).
  2. Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
  3. A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
    The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.
OCR AS Pure 2017 Specimen Q11
3 marks
11 The scatter diagram below shows data taken from the 2011 UK census for each of the Local Authorities in the North East and North West regions.
The scatter diagram shows the total population of the Local Authority and the proportion of its workforce that travel to work by bus, minibus or coach.
\includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-07_938_1136_664_260}
  1. Samuel suggests that, with a few exceptions, the data points in the diagram show that Local Authorities with larger populations generally have higher proportions of workers travelling by bus, minibus or coach. On the diagram in the Printed Answer Booklet draw a ring around each of the data points that Samuel might regard as an exception.
  2. Jasper suggests that it is possible to separate these Local Authorities into more than one group with different relationships between population and proportion travelling to work by bus, minibus or coach. Discuss Jasper's suggestion, referring to the data and to how differences between the Local Authorities could explain the patterns seen in the diagram.
    [0pt] [3]
OCR AS Pure 2017 Specimen Q12
12 It is known that under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, 450 patients with this disease were selected and the number, \(X\), that experienced side effects within one year was noted. It was found that 51 of the 450 patients experienced side effects within one year.
  1. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
  2. It was later discovered that all 450 patients selected for the trial were treated in the same hospital. Comment on the validity of the model used in part (a).
OCR AS Pure 2017 Specimen Q1
1
  1. The diagram below shows the graph of \(y = \mathrm { f } ( x )\).
    \includegraphics[max width=\textwidth, alt={}, center]{54bdddcb-c1ef-4b60-af6c-ac944cae29fe-03_793_1470_571_356}
    1. On the diagram in the Printed Answer Booklet draw the graph of \(y = \mathrm { f } ( x + 3 )\).
    2. Describe fully the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = - \mathrm { f } ( x )\).
  2. The point \(( 2,3 )\) lies on the graph of \(y = \mathrm { g } ( x )\). State the coordinates of its image when \(y = \mathrm { g } ( x )\) is transformed to
    1. \(\quad y = 4 \mathrm {~g} ( x )\)
    2. \(\quad y = \mathrm { g } ( 4 x )\).
OCR AS Pure 2017 Specimen Q2
2 In this question you must show detailed reasoning. Solve the equation \(2 \cos ^ { 2 } x = 2 - \sin x\) for \(0 ^ { \circ } \leq x \leq 180 ^ { \circ }\).
OCR AS Pure 2017 Specimen Q3
3 The number of members of a social networking site is modelled by \(m = 150 \mathrm { e } ^ { 2 t }\), where \(m\) is the number of members and \(t\) is time in weeks after the launch of the site.
  1. State what this model implies about the relationship between \(m\) and the rate of change of \(m\).
  2. What is the significance of the integer 150 in the model?
  3. Find the week in which the model predicts that the number of members first exceeds 60000 .
  4. The social networking site only expects to attract 60000 members. Suggest how the model could be refined to take account of this.
OCR AS Pure 2017 Specimen Q4
4 The points \(A , B\) and \(C\) have position vectors \(\binom { - 2 } { 1 } , \binom { 2 } { 5 }\) and \(\binom { 6 } { 3 }\) respectively. \(M\) is the midpoint of \(B C\).
  1. Find the position vector of the point \(D\) such that \(\overrightarrow { B C } = \overrightarrow { A D }\).
  2. Find the magnitude of \(\overrightarrow { A M }\).
OCR AS Pure 2017 Specimen Q5
5 A doctors' surgery starts a campaign to reduce missed appointments.
The number of missed appointments for each of the first five weeks after the start of the campaign is shown below.
Number of weeks after
the start \(( x )\)
12345
Number of missed
appointments \(( y )\)
235149995938
This data could be modelled by an equation of the form \(y = p q ^ { x }\) where \(p\) and \(q\) are constants.
  1. Show that this relationship may be expressed in the form \(\log _ { 10 } y = m x + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). The diagram below shows \(\log _ { 10 } y\) plotted against \(x\), for the given data.
    \includegraphics[max width=\textwidth, alt={}, center]{54bdddcb-c1ef-4b60-af6c-ac944cae29fe-05_737_1668_1233_258}
  2. Estimate the values of \(p\) and \(q\).
  3. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable.
  4. A student suggests that, for any prime number between 20 and 40, when its digits are squared and then added, the sum is an odd number. For example, 23 has digits 2 and 3 which gives \(2 ^ { 2 } + 3 ^ { 2 } = 13\), which is odd. Show by counter example that this suggestion is false.
  5. Prove that the sum of the squares of any three consecutive positive integers cannot be divided by 3 .
OCR AS Pure 2017 Specimen Q7
7 Differentiate \(\mathrm { f } ( x ) = x ^ { 4 }\) from first principles.
OCR AS Pure 2017 Specimen Q8
8 A curve has equation \(y = k x ^ { \frac { 3 } { 2 } }\) where \(k\) is a constant.
The point \(P\) on the curve has \(x\)-coordinate 4.
The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(P Q\) is the radius of a circle centre \(P\). Show that \(k = \frac { 1 } { 2 }\).
Find the equation of the circle.
OCR AS Pure 2017 Specimen Q9
9 The diagram below shows the velocity-time graph of a car moving along a straight road, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t \mathrm {~s}\) after it passes through the point \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{54bdddcb-c1ef-4b60-af6c-ac944cae29fe-07_689_1557_520_310}
  1. Calculate the acceleration of the car at \(t = 6\).
  2. Jasmit says "The distance travelled by the car during the first 20 seconds of the car's motion is more than five times its displacement from \(A\) after the first 20 seconds of the car's motion". Give evidence to support Jasmit's statement.
OCR AS Pure 2017 Specimen Q10
10 A student is attempting to model the flight of a boomerang.
She throws the boomerang from a fixed point \(O\) and catches it when it returns to \(O\).
She suggests the model for the displacement, \(s\) metres, after \(t\) seconds in given by
\(s = 9 t ^ { 2 } - \frac { 3 } { 2 } t ^ { 3 } , 0 \leq t \leq 6\). For this model,
  1. determine what happens at \(t = 6\),
  2. find the greatest displacement of the boomerang from \(O\),
  3. find the velocity of the boomerang 1 second before the student catches it,
  4. find the acceleration of the boomerang 1 second before the student catches it.