Questions Paper 3 (350 questions)

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OCR MEI Paper 3 Specimen Q4
3 marks Moderate -0.5
4 Show that \(\sum _ { r = 1 } ^ { 4 } \ln \frac { r } { r + 1 } = - \ln 5\).
OCR MEI Paper 3 Specimen Q6
5 marks Standard +0.3
6 Fig. 6 shows the curve with equation \(y = x ^ { 4 } - 6 x ^ { 2 } + 4 x + 5\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-06_869_750_370_242} \captionsetup{labelformat=empty} \caption{Fig. 6
Find the coordinates of the points of inflection.}
\end{figure}
OCR MEI Paper 3 Specimen Q7
2 marks Moderate -0.8
7 By finding a counter example, disprove the following statement. If \(p\) and \(q\) are non-zero real numbers with \(p < q\), then \(\frac { 1 } { p } > \frac { 1 } { q }\).
OCR MEI Paper 3 Specimen Q8
8 marks Challenging +1.2
8 In Fig. 8, OAB is a thin bent rod, with \(\mathrm { OA } = 1 \mathrm {~m} , \mathrm { AB } = 2 \mathrm {~m}\) and angle \(\mathrm { OAB } = 120 ^ { \circ }\). Angles \(\theta , \phi\) and \(h\) are as shown in Fig. 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-07_949_949_429_214} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. Show that \(h = \sin \theta + 2 \sin \left( \theta + 60 ^ { \circ } \right)\). The rod is free to rotate about the origin so that \(\theta\) and \(\phi\) vary. You may assume that the result for \(h\) in part (a) holds for all values of \(\theta\).
  2. Find an angle \(\theta\) for which \(h = 0\).
OCR MEI Paper 3 Specimen Q9
7 marks Standard +0.3
9
  1. Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\) and \(R\) is positive and given in exact form. The function \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = \frac { 1 } { ( k + \cos \theta + 2 \sin \theta ) } , 0 \leq \theta \leq 2 \pi , k\) is a constant.
  2. The maximum value of \(\mathrm { f } ( \theta )\) is \(\frac { ( 3 + \sqrt { 5 } ) } { 4 }\). Find the value of \(k\).
OCR MEI Paper 3 Specimen Q10
10 marks Standard +0.3
10 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 4 x - 2\).
  1. Show that \(x = - 1\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Show that another root of \(\mathrm { f } ( x ) = 0\) lies between \(x = 1\) and \(x = 2\).
  3. Show that \(\mathrm { f } ( x ) = ( x + 1 ) \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } + a x + b\) and \(a\) and \(b\) are integers to be determined.
  4. Without further calculation, explain why \(\mathrm { g } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
  5. Use the Newton-Raphson formula to show that an iteration formula for finding roots of \(\mathrm { g } ( x ) = 0\) may be written $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$ Determine the root of \(\mathrm { g } ( x ) = 0\) which lies between \(x = 1\) and \(x = 2\) correct to 4 significant figures.
OCR MEI Paper 3 Specimen Q11
10 marks Challenging +1.8
11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).
    1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
    2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
  2. Would the result still hold with a larger domain? Give reasons for your answer.
OCR MEI Paper 3 Specimen Q12
1 marks Easy -2.5
12 Explain why the smaller regular hexagon in Fig. C1 has perimeter 6.
OCR MEI Paper 3 Specimen Q13
3 marks Moderate -0.8
13 Show that the larger regular hexagon in Fig. C1 has perimeter \(4 \sqrt { 3 }\).
OCR MEI Paper 3 Specimen Q14
3 marks Standard +0.8
14 Show that the two values of \(b\) given on line 36 are equivalent.
OCR MEI Paper 3 Specimen Q15
5 marks Challenging +1.2
15 Fig. 15 shows a unit circle and the escribed regular polygon with 12 edges. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-11_839_876_356_269} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure}
  1. Show that the perimeter of the polygon is \(24 \tan 15 ^ { \circ }\).
  2. Using the formula for \(\tan ( \theta - \phi )\) show that the perimeter of the polygon is \(48 - 24 \sqrt { 3 }\).
OCR MEI Paper 3 Specimen Q16
3 marks Challenging +1.2
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.
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OCR MEI Paper 3 2022 June Q10
5 marks Standard +0.3
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
4 marks Standard +0.8
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
2 marks Moderate -0.8
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
6 marks Standard +0.3
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
5 marks Standard +0.3
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
5 marks Challenging +1.2
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 marks Standard +0.3
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.
Edexcel Paper 3 2018 June Q1
5 marks Moderate -0.8
  1. Helen believes that the random variable \(C\), representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.
    1. Write down the probability distribution for \(C\).
    2. Using this model, find the probability that cloud cover is less than 50\%
    Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than \(50 \%\) was 0.315
  2. Comment on the suitability of Helen's model in the light of this information.
  3. Suggest an appropriate refinement to Helen's model.
Edexcel Paper 3 2018 June Q2
7 marks Standard +0.3
  1. Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, \(\pounds w\), and the average weekly temperature, \(t ^ { \circ } \mathrm { C }\), for 8 weeks during the summer.
    The product moment correlation coefficient for these data is - 0.915
    1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.
    2. Suggest a possible reason for this correlation.
    Tessa suggests that a linear regression model could be used to model these data.
  2. State, giving a reason, whether or not the correlation coefficient is consistent with Tessa's suggestion.
  3. State, giving a reason, which variable would be the explanatory variable. Tessa calculated the linear regression equation as \(w = 10755 - 171 t\)
  4. Give an interpretation of the gradient of this regression equation.
Edexcel Paper 3 2018 June Q3
11 marks Moderate -0.3
  1. In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable \(H\) represents the number of times the dart hits the target in the first 10 throws.
Peta models \(H\) as \(\mathrm { B } ( 10,0.1 )\)
  1. State two assumptions Peta needs to make to use her model.
  2. Using Peta's model, find \(\mathrm { P } ( H \geqslant 4 )\) For each child the random variable \(F\) represents the number of the throw on which the dart first hits the target. Using Peta's assumptions about this experiment,
  3. find \(\mathrm { P } ( F = 5 )\) Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models \(\mathrm { P } ( F = n )\) as $$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$ where \(\alpha\) is a constant.
  4. Find the value of \(\alpha\)
  5. Using Thomas' model, find \(\mathrm { P } ( F = 5 )\)
  6. Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.
Edexcel Paper 3 2018 June Q4
13 marks Easy -1.3
  1. Charlie is studying the time it takes members of his company to travel to the office. He stands by the door to the office from 0840 to 0850 one morning and asks workers, as they arrive, how long their journey was.
    1. State the sampling method Charlie used.
    2. State and briefly describe an alternative method of non-random sampling Charlie could have used to obtain a sample of 40 workers.
    Taruni decided to ask every member of the company the time, \(x\) minutes, it takes them to travel to the office.
  2. State the data selection process Taruni used. Taruni's results are summarised by the box plot and summary statistics below. \includegraphics[max width=\textwidth, alt={}, center]{65e4b254-fb7b-45c2-9702-32f034018193-10_378_1349_1050_367} $$n = 95 \quad \sum x = 4133 \quad \sum x ^ { 2 } = 202294$$
  3. Write down the interquartile range for these data.
  4. Calculate the mean and the standard deviation for these data.
  5. State, giving a reason, whether you would recommend using the mean and standard deviation or the median and interquartile range to describe these data. Rana and David both work for the company and have both moved house since Taruni collected her data. Rana's journey to work has changed from 75 minutes to 35 minutes and David's journey to work has changed from 60 minutes to 33 minutes. Taruni drew her box plot again and only had to change two values.
  6. Explain which two values Taruni must have changed and whether each of these values has increased or decreased.
Edexcel Paper 3 2018 June Q5
14 marks Challenging +1.2
  1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.
Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.
  1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
  2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
  3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
  4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.
Edexcel Paper 3 2018 June Q6
6 marks Standard +0.3
6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves in the \(x - y\) plane in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = t ^ { - \frac { 1 } { 2 } } \mathbf { i } - 4 \mathbf { j }$$ When \(t = 1 , P\) is at the point \(A\) and when \(t = 4 , P\) is at the point \(B\).
Find the exact distance \(A B\).