Questions — OCR MEI (4456 questions)

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OCR MEI Paper 3 2021 November Q4
3 marks Moderate -0.8
4 The diagram shows points \(A\) and \(B\) on the curve \(y = \left( \frac { x } { 4 } \right) ^ { - x }\).
The \(x\)-coordinate of A is 1 and the \(x\)-coordinate of B is 1.1 . \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-4_522_707_1758_278}
  1. Find the gradient of chord AB . Give your answer correct to 2 decimal places.
  2. Give the \(x\)-coordinate of a point C on the curve such that the gradient of chord AC is a better approximation to the gradient of the tangent to the curve at A .
OCR MEI Paper 3 2021 November Q5
8 marks Moderate -0.8
5
  1. The diagram shows the curve \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-5_574_682_315_328} On the axes in the Printed Answer Booklet, sketch graphs of
    1. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(x\),
    2. \(\frac { \mathrm { dy } } { \mathrm { dx } }\) against \(y\).
  2. Wolves were introduced to Yellowstone National Park in 1995. The population of wolves, \(y\), is modelled by the equation \(y = A e ^ { k t }\),
    where \(A\) and \(k\) are constants and \(t\) is the number of years after 1995.
    1. Give a reason why this model might be suitable for the population of wolves.
    2. When \(t = 0 , y = 21\) and when \(t = 1 , y = 51\). Find values of \(A\) and \(k\) consistent with the data.
    3. Give a reason why the model will not be a good predictor of wolf populations many years after 1995.
OCR MEI Paper 3 2021 November Q6
4 marks Moderate -0.8
6 In this question you must show detailed reasoning.
Show that \(\sum _ { r = 1 } ^ { 3 } \frac { 1 } { \sqrt { r + 1 } + \sqrt { r } } = 1\).
OCR MEI Paper 3 2021 November Q7
3 marks Moderate -0.3
7 Determine \(\int x \cos 2 x \mathrm {~d} x\).
OCR MEI Paper 3 2021 November Q8
3 marks Challenging +1.2
8 For a particular value of \(a\), the curve \(\mathrm { y } = \frac { \mathrm { a } } { \mathrm { x } ^ { 2 } }\) passes through the point \(( 3,1 )\).
Find the coordinates of all the other points on the curve where both the \(x\)-coordinate and the \(y\)-coordinate are integers.
OCR MEI Paper 3 2021 November Q9
11 marks Standard +0.3
9 The diagram shows the curve \(\mathrm { y } = 3 - \sqrt { \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-6_810_1008_1155_283}
  1. Draw the line \(\mathrm { y } = 5 \mathrm { x } - 1\) on the copy of the diagram in the Printed Answer Booklet.
  2. In this question you must show detailed reasoning. Determine the exact area of the region bounded by the curve \(y = 3 - \sqrt { x }\), the lines \(y = 5 x - 1\) and \(x = 4\) and the \(x\)-axis.
OCR MEI Paper 3 2021 November Q10
9 marks Standard +0.3
10
  1. Express \(\frac { 1 } { ( 4 x + 1 ) ( x + 1 ) }\) in partial fractions.
  2. A curve passes through the point \(( 0,2 )\) and satisfies the differential equation \(\frac { d y } { d x } = \frac { y } { ( 4 x + 1 ) ( x + 1 ) }\),
    for \(x > - \frac { 1 } { 4 }\).
    Show by integration that \(\mathrm { y } = \mathrm { A } \left( \frac { 4 \mathrm { x } + 1 } { \mathrm { x } + 1 } \right) ^ { \mathrm { B } }\) where \(A\) and \(B\) are constants to be determined.
OCR MEI Paper 3 2021 November Q12
3 marks Moderate -0.5
12 Show that \(\beta = \arctan \left( \frac { 1 } { 3 } \right)\), as given in line 15 .
OCR MEI Paper 3 2021 November Q13
3 marks Moderate -0.8
13
  1. Use triangle ABE in Fig. C 2 to show that \(\arctan x + \arctan \left( \frac { 1 } { x } \right) = \frac { \pi } { 2 }\), as given in line 29 .
  2. Sketch the graph of \(\mathrm { y } = \arctan \mathrm { x }\).
  3. What property of the arctan function ensures that \(\mathrm { y } > \frac { 1 } { \mathrm { x } } \Rightarrow \arctan y > \arctan \left( \frac { 1 } { \mathrm { x } } \right)\), as given in line 30 ?
OCR MEI Paper 3 2021 November Q14
5 marks Challenging +1.2
14
  1. Show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) \Rightarrow \arctan \left( \frac { 1 } { 2 } \right) + \arctan \left( \frac { 1 } { 3 } \right) = \arctan 1 .$$
  2. Use the arctan addition formula in line 23 to show that $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) , \text { as given in line } 39 .$$
OCR MEI Paper 3 2021 November Q15
4 marks Challenging +1.8
15 Prove that \(\arctan 1 + \arctan 2 + \arctan 3 = \pi\), as given in line 41 . \section*{END OF QUESTION PAPER} \section*{OCR
Oxford Cambridge and RSA}
OCR MEI Paper 3 Specimen Q1
2 marks Easy -1.8
1 Express \(\frac { 2 } { x - 1 } + \frac { 5 } { 2 x + 1 }\) as a single fraction.
OCR MEI Paper 3 Specimen Q2
4 marks Moderate -0.5
2 Find the first four terms of the binomial expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.
OCR MEI Paper 3 Specimen Q3
4 marks Easy -1.2
3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.
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 \(48 - 24 \sqrt { 3 }\).