Questions — OCR MEI Paper 3 (118 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 MEI Paper 3 2022 June Q12
12
  1. Show that \(\cos x = \sin \left( x + \frac { \pi } { 2 } \right)\).
  2. Hence show that \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) gives the approximation \(\cos x \approx \frac { \pi ^ { 2 } - 4 x ^ { 2 } } { \pi ^ { 2 } + x ^ { 2 } }\), as stated in line 31. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{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 OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
    OCR is part of Cambridge University Press \& Assessment, which is itself a department of the University of Cambridge.
OCR MEI Paper 3 2023 June Q1
1 In this question you must show detailed reasoning.
The obtuse angle \(\theta\) is such that \(\sin \theta = \frac { 2 } { \sqrt { 13 } }\).
Find the exact value of \(\cos \theta\).
OCR MEI Paper 3 2023 June Q2
2 The straight line \(y = 5 - 2 x\) is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-04_705_773_881_239}
  1. On the copy of the diagram in the Printed Answer Booklet, sketch the graph of \(y = | 5 - 2 x |\).
  2. Solve the inequality \(| 5 - 2 x | < 3\).
OCR MEI Paper 3 2023 June Q3
3 In this question you must show detailed reasoning.
Find the value of \(k\) such that \(\frac { 1 } { \sqrt { 5 } + \sqrt { 6 } } + \frac { 1 } { \sqrt { 6 } + \sqrt { 7 } } = \frac { k } { \sqrt { 5 } + \sqrt { 7 } }\).
OCR MEI Paper 3 2023 June Q4
4 In this question you must show detailed reasoning.
Find the coordinates of the points where the curve \(y = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6\) crosses the \(x\)-axis.
OCR MEI Paper 3 2023 June Q5
5 In this question you must show detailed reasoning.
This question is about the curve \(y = x ^ { 3 } - 5 x ^ { 2 } + 6 x\).
  1. Find the equation of the tangent, \(T\), to the curve at the point ( 0,0 ).
  2. Find the equation of the normal, \(N\), to the curve at the point ( 1,2 ).
  3. Find the coordinates of the point of intersection of \(T\) and \(N\).
OCR MEI Paper 3 2023 June Q6
6
  1. Quadrilateral KLMN has vertices \(\mathrm { K } ( - 4,1 ) , \mathrm { L } ( 5 , - 1 ) , \mathrm { M } ( 6,2 )\) and \(\mathrm { N } ( 2,5 )\), as shown in Fig. 6.1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_567_1004_404_319}
    \end{figure}
    1. Find the coordinates of the following midpoints.
      • P , the midpoint of KL
  2. Q, the midpoint of LM
  3. R, the midpoint of MN
  4. S, the midpoint of NK
    (ii) Verify that PQRS is a parallelogram.
  5. TVWX is a quadrilateral as shown in Fig. 6.2.
    6. Points A and B divide side TV into 3 equal parts. Points C and D divide side VW into 3 equal parts. Points E and F divide side WX into 3 equal parts. Points G and H divide side TX into 3 equal parts.
    \(\overrightarrow { \mathrm { TA } } = \mathbf { a } , \quad \overrightarrow { \mathrm { TH } } = \mathbf { b } , \quad \overrightarrow { \mathrm { VC } } = \mathbf { c }\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_577_671_1877_319}
    \end{figure} (i) Show that \(\overrightarrow { \mathrm { WX } } = k ( - \mathbf { a } + \mathbf { b } - \mathbf { c } )\), where \(k\) is a constant to be determined.
    (ii) Verify that AH is parallel to DE .
    (iii) Verify that BC is parallel to GF .
OCR MEI Paper 3 2023 June Q7
7 A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is \(r \mathrm {~cm}\) and the angle at the centre is \(\theta\) radians.
\includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-07_323_204_342_242} Determine the maximum possible area of the sector, showing that it is a maximum.
OCR MEI Paper 3 2023 June Q8
8 A circle with centre \(A\) and radius 8 cm and a circle with centre \(C\) and radius 12 cm intersect at points B and D . Quadrilateral \(A B C D\) has area \(60 \mathrm {~cm} ^ { 2 }\).
Determine the two possible values for the length AC.
OCR MEI Paper 3 2023 June Q9
9 A small country started using solar panels to produce electrical energy in the year 2000. Electricity production is measured in megawatt hours (MWh). For the period from 2000 to 2009, the annual electrical energy produced using solar panels can be modelled by the equation \(\mathrm { P } = 0.3 \mathrm { e } ^ { 0.5 \mathrm { t } }\), where \(P\) is the annual amount of electricity produced in MWh and \(t\) is the time in years after the year 2000.
  1. According to this model, find the amount of electricity produced using solar panels in each of the following years.
    1. 2000
    2. 2009
  2. Give a reason why the model is unlikely to be suitable for predicting the annual amount of electricity produced using solar panels in the year 2025. An alternative model is suggested; the curve representing this model is shown in Fig. 9. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 9} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-08_702_1587_1265_230}
    \end{figure}
  3. Explain how the graph shows that the alternative model gives a value for the amount of electricity produced in 2009 that is consistent with the original model.
    1. On the axes given in the Printed Answer Booklet, sketch the gradient function of the model shown in Fig. 9.
    2. State approximately the value of \(t\) at the point of inflection in Fig. 9.
    3. Interpret the significance of the point of inflection in the context of the model.
  5. State approximately the long term value of the annual amount of electricity produced using solar panels according to the model represented in Fig. 9.
OCR MEI Paper 3 2023 June Q10
10
  1. You are given that \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = x ^ { 6 } + 3 x ^ { 4 } y ^ { 2 } + 3 x ^ { 2 } y ^ { 4 } + y ^ { 6 }\).
    Hence, or otherwise, prove that \(\sin ^ { 6 } \theta + \cos ^ { 6 } \theta = 1 - \frac { 3 } { 4 } \sin ^ { 2 } 2 \theta\) for all values of \(\theta\).
  2. Use the result from part (a) to determine the minimum value of \(\sin ^ { 6 } \theta + \cos ^ { 6 } \theta\). The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.
OCR MEI Paper 3 2023 June Q11
11
  1. Evaluate \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).
  2. Show that Euler's approximate formula, as given in line 13, gives the exact value of \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).
OCR MEI Paper 3 2023 June Q12
12 With the aid of a suitable diagram, show that the three triangles referred to in line 26 have the areas given in line 27 .
OCR MEI Paper 3 2023 June Q13
13 Prove that Euler's approximate formula, as given in line 13, when applied to \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 2 }\) gives exactly \(\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }\).
OCR MEI Paper 3 2023 June Q14
14 Show that the expression given in line 33 simplifies to \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { \mathrm { r } } \approx \ln \mathrm { n } + \frac { 13 } { 24 } + \frac { 6 \mathrm { n } + 5 } { 12 \mathrm { n } ( \mathrm { n } + 1 ) }\), as given in line 34.
OCR MEI Paper 3 2023 June Q15
15 The expression given in line 34 is used to calculate \(\sum _ { r = 1 } ^ { 6 } \frac { 1 } { r }\).
Show that the error in the result is less than \(1.5 \%\) of the true value.
OCR MEI Paper 3 2024 June Q1
1 Solve the inequality \(\frac { x } { 5 } > 6 - x\).
OCR MEI Paper 3 2024 June Q2
2
  1. The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \sqrt { 1 + 2 x } \text { for } x \geqslant - \frac { 1 } { 2 }$$ Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of this inverse function.
  2. Explain why \(\mathrm { g } ( x ) = 1 + x ^ { 2 }\), with domain all real numbers, has no inverse function.
OCR MEI Paper 3 2024 June Q7
7 Prove that \(\sin 8 \theta \tan 4 \theta + \cos 8 \theta = 1\).
OCR MEI Paper 3 2024 June Q8
8 In this question you must show detailed reasoning.
  1. Express \(\cos x + \sqrt { 3 } \sin x\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the values of \(R\) and \(\alpha\) in exact form.
  2. Hence solve the equation \(\cos x = \sqrt { 3 } ( 1 - \sin x )\) for values of \(x\) in the interval \(- \pi \leqslant x \leqslant \pi\). Give the roots of this equation in exact form.
OCR MEI Paper 3 2024 June Q9
9 This question is about the equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } - x - \frac { 1 } { 3 x - 2 }\).
Fig. 9.1 shows the curve \(y = f ( x )\).
Fig. 9.1
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-06_940_929_518_239}
  1. Show, by calculation, that the equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
  2. Fig. 9.2 shows part of a spreadsheet being used to find a root of the equation. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 9.2}
    AB
    1\(x\)\(f ( x )\)
    21.53.1625
    31.250.619977679
    41.125- 0.250466087
    5
    \end{table} Write down a suitable number to use as the next value of \(x\) in the spreadsheet.
  3. Determine a root of the equation \(\mathrm { f } ( x ) = 0\). Give your answer correct to \(\mathbf { 1 }\) decimal place.
  4. Fig. 9.3 shows a similar spreadsheet being used to search for another root of \(\mathrm { f } ( x ) = 0\). \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 9.3}
    AB
    1xf(x)
    200.5
    31-1
    40.51.5625
    50.75-4.4336
    60.64.5296
    70.7-10.4599
    80.6519.5285
    90.675-40.4674
    100.662579.5301
    110.66875-160.4687
    10
    \end{table}
    1. Explain why it looks from rows 2 and 3 of the spreadsheet as if there is a root between 0 and 1.
    2. Explain why this process will not find a root between 0 and 1 .
OCR MEI Paper 3 2024 June Q10
10 The diagram below shows the curve \(y = f ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-07_942_679_1500_242} Sketch the graph of the gradient function, \(y = f ^ { \prime } ( x )\), on the copy of the diagram in the Printed Answer Booklet.
OCR MEI Paper 3 2024 June Q11
11 Fig. 11.1 shows the curve with equation \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) where \(\mathrm { g } ( x ) = x \sin x + \cos x\) and the curve of the gradient function \(\mathrm { y } = \mathrm { g } ^ { \prime } ( \mathrm { x } )\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 11.1} \includegraphics[alt={},max width=\textwidth]{60e1e785-c34b-48ef-a63f-13a25fee186e-08_1136_1196_459_246}
\end{figure}
  1. Show that the \(x\)-coordinates of the points on the curve \(y = g ( x )\) where the gradient is 1 satisfy the equation \(\frac { 1 } { x } - \cos x = 0\). Fig. 11.2 shows part of the curve with equation \(y = \frac { 1 } { x } - \cos x\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Fig. 11.2} \includegraphics[alt={},max width=\textwidth]{60e1e785-c34b-48ef-a63f-13a25fee186e-09_678_1363_424_239}
    \end{figure}
  2. Use the Newton-Raphson method with a suitable starting value to find the smallest positive \(x\)-coordinate of a point on the curve \(y = x \sin x + \cos x\) where the gradient is 1 . You should write down at least the following.
    • The iteration you use
    • The starting value
    • The solution correct to \(\mathbf { 4 }\) decimal places
    • Explain why \(x _ { 1 } = 3\) is not a suitable starting value for the Newton-Raphson method in part (b).
OCR MEI Paper 3 2024 June Q12
12 The diagram shows the curve with parametric equations
\(x = \sin 2 \theta + 2 , y = 2 \cos \theta + \cos 2 \theta\), for \(0 \leqslant \theta < 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-10_771_673_397_239}
  1. In this question you must show detailed reasoning. Determine the exact coordinates of all the stationary points on the curve.
  2. Write down the equation of the line of symmetry of the curve.
OCR MEI Paper 3 2024 June Q13
13 Substitute appropriate values of \(t _ { 1 }\) and \(t _ { 2 }\) to verify that \(t _ { 1 } t _ { 2 }\) gives the correct value for the \(y\)-coordinate of the point of intersection of the tangents at the points A and B in Fig. \(\mathbf { C 1 . }\)