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OCR MEI Paper 3 Specimen Q16
16 On a unit circle, the inscribed regular polygon with 12 edges gives a lower bound for \(\pi\), and the escribed regular polygon with 12 edges gives an upper bound for \(\pi\). Calculate the values of these bounds for \(\pi\), giving your answers:
  1. in surd form
  2. correct to 2 decimal places. {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 9 EA.
    OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
OCR MEI Paper 3 2022 June Q10
10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve \(\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }\) has a stationary point near \(x = 3\).
  • Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
  • Show that this stationary point is a maximum turning point.
OCR MEI Paper 3 2024 June Q3
3 In this question you must show detailed reasoning. The diagram shows the curve with equation \(y = x ^ { 5 }\) and the square \(O A B C\) where the points \(A , B\) and \(C\) have coordinates \(( 1,0 ) , ( 1,1 )\) and \(( 0,1 )\) respectively. The curve cuts the square into two parts.
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-04_658_780_1318_230} Show that the relationship between the areas of the two parts of the square is
\(\frac { \text { Area to left of curve } } { \text { Area below curve } } = 5\).
OCR MEI Paper 3 2024 June Q4
4 In this question you must show detailed reasoning. Determine the exact value of \(\frac { 1 } { \sqrt { 2 } + 1 } + \frac { 1 } { \sqrt { 3 } + \sqrt { 2 } } + \frac { 1 } { 2 + \sqrt { 3 } }\).
OCR MEI Paper 3 2024 June Q5
5 In this question you must show detailed reasoning. Using the substitution \(\mathrm { u } = \mathrm { x } + 1\), find the value of the positive integer \(c\) such that \(\int _ { \mathrm { c } } ^ { \mathrm { c } + 4 } \frac { \mathrm { x } } { ( \mathrm { x } + 1 ) ^ { 2 } } \mathrm { dx } = \ln 3 - \frac { 1 } { 3 }\).
OCR MEI Paper 3 2024 June Q6
6 In this question you must show detailed reasoning. Solve the equation \(\tan x - 3 \cot x = 2\) for values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
OCR MEI Paper 3 2021 November Q11
11 In this question you must show detailed reasoning. The diagram shows triangle ABC , with \(\mathrm { BC } = 8 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 45 ^ { \circ }\).
The point D on AC is such that \(\mathrm { DC } = 5 \mathrm {~cm}\) and \(\mathrm { BD } = 7 \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-7_684_553_1119_258} Determine the exact length of AB .
OCR MEI Paper 3 Specimen Q5
5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-05_878_1000_685_255} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Find the area of the part of the circle below the \(x\)-axis.
AQA Paper 3 2018 June Q2
2 A curve has equation \(y = x ^ { 5 } + 4 x ^ { 3 } + 7 x + q\) where \(q\) is a positive constant.
Find the gradient of the curve at the point where \(x = 0\)
Circle your answer.
0
4
7
\(q\)
AQA Paper 3 2018 June Q4
4
7
\(q\) 3 The line \(L\) has equation \(2 x + 3 y = 7\)
Which one of the following is perpendicular to \(L\) ?
Tick one box. $$\begin{aligned} & 2 x - 3 y = 7
& 3 x + 2 y = - 7
& 2 x + 3 y = - \frac { 1 } { 7 }
& 3 x - 2 y = 7 \end{aligned}$$ □
□
□
□ 4 Sketch the graph of \(y = | 2 x + a |\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes.
\includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-03_1001_1002_450_520}
AQA Paper 3 2018 June Q5
5 Show that, for small values of \(x\), the graph of \(y = 5 + 4 \sin \frac { x } { 2 } + 12 \tan \frac { x } { 3 }\) can be approximated by a straight line.
AQA Paper 3 2018 June Q7
7
\(q\) 3 The line \(L\) has equation \(2 x + 3 y = 7\)
Which one of the following is perpendicular to \(L\) ?
Tick one box. $$\begin{aligned} & 2 x - 3 y = 7
& 3 x + 2 y = - 7
& 2 x + 3 y = - \frac { 1 } { 7 }
& 3 x - 2 y = 7 \end{aligned}$$ □
□
□
□ 4 Sketch the graph of \(y = | 2 x + a |\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes.
\includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-03_1001_1002_450_520} 5 Show that, for small values of \(x\), the graph of \(y = 5 + 4 \sin \frac { x } { 2 } + 12 \tan \frac { x } { 3 }\) can be approximated by a straight line.
6 (b) Use the quotient rule to show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { x - 2 } { ( 2 x - 2 ) ^ { \frac { 3 } { 2 } } }\) 6 (a) State the maximum possible domain of f .
\(6 \quad\) A function f is defined by \(\mathrm { f } ( x ) = \frac { x } { \sqrt { 2 x - 2 } }\) $$\begin{gathered} \text { Do not write }
\text { outside the }
\text { box } \end{gathered}$$ 6 (a)
6 (c) Show that the graph of \(y = \mathrm { f } ( x )\) has exactly one point of inflection.
6 (d) Write down the values of \(x\) for which the graph of \(y = \mathrm { f } ( x )\) is convex.
7 (a) Given that \(\log _ { a } y = 2 \log _ { a } 7 + \log _ { a } 4 + \frac { 1 } { 2 }\), find \(y\) in terms of \(a\).
7 (b) When asked to solve the equation $$2 \log _ { a } x = \log _ { a } 9 - \log _ { a } 4$$ a student gives the following solution: $$\begin{aligned} & 2 \log _ { a } x = \log _ { a } 9 - \log _ { a } 4
& \Rightarrow 2 \log _ { a } x = \log _ { a } \frac { 9 } { 4 }
& \Rightarrow \log _ { a } x ^ { 2 } = \log _ { a } \frac { 9 } { 4 }
& \Rightarrow x ^ { 2 } = \frac { 9 } { 4 }
& \therefore x = \frac { 3 } { 2 } \text { or } - \frac { 3 } { 2 } \end{aligned}$$ Explain what is wrong with the student's solution.
AQA Paper 3 2018 June Q8
8
  1. Prove the identity \(\frac { \sin 2 x } { 1 + \tan ^ { 2 } x } \equiv 2 \sin x \cos ^ { 3 } x\) 8
  2. Hence find \(\int \frac { 4 \sin 4 \theta } { 1 + \tan ^ { 2 } 2 \theta } \mathrm {~d} \theta\)
AQA Paper 3 2018 June Q9
2 marks
9 Helen is creating a mosaic pattern by placing square tiles next to each other along a straight line.
\includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-10_364_1300_406_370} The area of each tile is half the area of the previous tile, and the sides of the largest tile have length \(w\) centimetres. 9
  1. Find, in terms of \(w\), the length of the sides of the second largest tile. 9
  2. Assume the tiles are in contact with adjacent tiles, but do not overlap.
    Show that, no matter how many tiles are in the pattern, the total length of the series of tiles will be less than \(3.5 w\).
    \(\mathbf { 9 }\) (c) Helen decides the pattern will look better if she leaves a 3 millimetre gap between adjacent tiles. Explain how you could refine the model used in part (b) to account for the 3 millimetre gap, and state how the total length of the series of tiles will be affected.
    [0pt] [2 marks]
AQA Paper 3 2018 June Q11
11 The table below shows the probability distribution for a discrete random variable \(X\).
\(\boldsymbol { x }\)12345
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)\(k\)\(2 k\)\(4 k\)\(2 k\)\(k\)
Find the value of \(k\). Circle your answer.
\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 10 }\)1
AQA Paper 3 2018 June Q14
14 A teacher in a college asks her mathematics students what other subjects they are studying. She finds that, of her 24 students:
12 study physics
8 study geography
4 study geography and physics
14
  1. A student is chosen at random from the class.
    Determine whether the event 'the student studies physics' and the event 'the student studies geography' are independent.
    14
  2. It is known that for the whole college:
    the probability of a student studying mathematics is \(\frac { 1 } { 5 }\)
    the probability of a student studying biology is \(\frac { 1 } { 6 }\)
    the probability of a student studying biology given that they study mathematics is \(\frac { 3 } { 8 }\)
    Calculate the probability that a student studies mathematics or biology or both.
AQA Paper 3 2018 June Q15
15 (e) State two necessary assumptions in context so that the distribution stated in part (a) is valid.
AQA Paper 3 2018 June Q16
6 marks
16 A survey of 120 adults found that the volume, \(X\) litres per person, of carbonated drinks they consumed in a week had the following results: $$\sum x = 165.6 \quad \sum x ^ { 2 } = 261.8$$ 16
    1. Calculate the mean of \(X\).
      16
  2. (ii) Calculate the standard deviation of \(X\).
    16
  3. Assuming that \(X\) can be modelled by a normal distribution find
    16
    1. \(\mathrm { P } ( 0.5 < X < 1.5 )\)
      16
  5. (ii) \(\mathrm { P } ( X = 1 )\) 16
  6. Determine with a reason, whether a normal distribution is suitable to model this data. [2 marks]
    16
  7. It is known that the volume, \(Y\) litres per person, of energy drinks consumed in a week may be modelled by a normal distribution with standard deviation 0.21 Given that \(\mathrm { P } ( Y > 0.75 ) = 0.10\), find the value of \(\mu\), correct to three significant figures. [4 marks]
AQA Paper 3 2018 June Q17
17 Suzanne is a member of a sports club. For each sport she competes in, she wins half of the matches.
17
  1. After buying a new tennis racket Suzanne plays 10 matches and wins 7 of them.
    Investigate, at the \(10 \%\) level of significance, whether Suzanne's new racket has made a difference to the probability of her winning a match. 17
  2. After buying a new squash racket, Suzanne plays 20 matches. Find the minimum number of matches she must win for her to conclude, at the \(10 \%\) level of significance, that the new racket has improved her performance.
AQA Paper 3 2018 June Q18
18 In a region of England, the government decides to use an advertising campaign to encourage people to eat more healthily. Before the campaign, the mean consumption of chocolate per person per week was known to be 66.5 g , with a standard deviation of 21.2 g 18
  1. After the campaign, the first 750 available people from this region were surveyed to find out their average consumption of chocolate. 18
    1. State the sampling method used to collect the survey. 18
  3. (ii) Explain why this sample should not be used to conduct a hypothesis test.
    18
  4. A second sample of 750 people revealed that the mean consumption of chocolate per person per week was 65.4 g Investigate, at the \(10 \%\) level of significance, whether the advertising campaign has decreased the mean consumption of chocolate per person per week. Assume that an appropriate sampling method was used and that the consumption of chocolate is normally distributed with an unchanged standard deviation.
    \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-26_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-27_2492_1721_217_150}
    \includegraphics[max width=\textwidth, alt={}, center]{d9149857-5f94-4fa5-a6d8-550c0c07fefb-28_2496_1719_214_150}
AQA Paper 3 2019 June Q2
2 Find the value of \(\frac { 100 ! } { 98 ! \times 3 ! }\)
Circle your answer. $$\begin{array} { l l l l } \frac { 50 } { 147 } & 1650 & 3300 & 161700 \end{array}$$
AQA Paper 3 2019 June Q3
1 marks
3
Given \(u _ { 1 } = 1\), determine which one of the formulae below defines an increasing sequence for \(n \geq 1\) Circle your answer.
[0pt] [1 mark]
\(u _ { n + 1 } = 1 + \frac { 1 } { u _ { n } } \quad u _ { n } = 2 - 0.9 ^ { n - 1 } \quad u _ { n + 1 } = - 1 + 0.5 u _ { n } \quad u _ { n } = 0.9 ^ { n - 1 }\)
AQA Paper 3 2019 June Q4
4 Sketch the region defined by the inequalities $$y \leq ( 1 - 2 x ) ( x + 3 ) \text { and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\).
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-03_1000_1004_833_518}
AQA Paper 3 2019 June Q5
5 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 8 y = 264\)
\(A B\) is a chord of the circle. The angle at the centre of the circle, subtended by \(A B\), is 0.9 radians, as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-04_563_600_584_719} Find the area of the minor segment shaded on the diagram.
Give your answer to three significant figures.
AQA Paper 3 2019 June Q6
6 The three sides of a right-angled triangle have lengths \(a , b\) and \(c\), where \(a , b , c \in \mathbb { Z }\) 6
  1. State an example where \(a , b\) and \(c\) are all even.
    6
  2. Prove that it is not possible for all of \(a , b\) and \(c\) to be odd.