Questions — OCR MEI (4301 questions)

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OCR MEI Paper 3 2020 November Q1
2 marks Easy -1.2
1 Find the value of \(\sum _ { r = 1 } ^ { 5 } 2 ^ { r } ( r - 1 )\).
OCR MEI Paper 3 2020 November Q2
4 marks Moderate -0.8
2 The graph of \(y = | 1 - x | - 2\) is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-04_625_1102_794_242} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Determine the set of values of \(x\) for which \(| 1 - x | > 2\).
OCR MEI Paper 3 2020 November Q3
3 marks Moderate -0.8
3 A particular phone battery will last 10 hours when it is first used. Every time it is recharged, it will only last \(98 \%\) of its previous time. Find the maximum total length of use for the battery.
OCR MEI Paper 3 2020 November Q4
3 marks Standard +0.8
4 Fig. 4 shows the regular octagon ABCDEFGH . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-05_689_696_301_239} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} \(\overrightarrow { \mathrm { AB } } = \mathbf { i } , \overrightarrow { \mathrm { CD } } = \mathbf { j }\), where \(\mathbf { i }\) is a unit vector parallel to the \(x\)-axis and \(\mathbf { j }\) is a unit vector parallel to the \(y\)-axis. Find an exact expression for \(\overrightarrow { \mathrm { BC } }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
OCR MEI Paper 3 2020 November Q5
11 marks Standard +0.3
5 Fig. 5 shows part of the curve \(y = \operatorname { cosec } x\) together with the \(x\) - and \(y\)-axes. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-06_732_625_317_244} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. For the section of the curve which is shown in Fig. 5, write down
    1. the equations of the two vertical asymptotes,
    2. the coordinates of the minimum point.
  2. Show that the equation \(x = \operatorname { cosec } x\) has a root which lies between \(x = 1\) and \(x = 2\).
  3. Use the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\), with \(x _ { 0 } = 1\), to find
    1. the values of \(x _ { 1 }\) and \(x _ { 2 }\), correct to 5 decimal places,
    2. this root of the equation, correct to 3 decimal places.
  4. There is another root of \(x = \operatorname { cosec } x\) which lies between \(x = 2\) and \(x = 3\). Determine whether the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\) with \(x _ { 0 } = 2.5\) converges to this root.
  5. Sketch the staircase or cobweb diagram for the iteration, starting with \(x _ { 0 } = 2.5\), on the diagram in the Printed Answer Booklet.
OCR MEI Paper 3 2020 November Q6
12 marks Moderate -0.3
6
    1. Write down the derivative of \(\mathrm { e } ^ { \mathrm { kx } }\), where \(k\) is a constant.
    2. A business has been running since 2009. They sell maths revision resources online. Give a reason why an exponential growth model might be suitable for the annual profits for the business. Fig. 6 shows the relationship between the annual profits of the business in thousands of pounds ( \(y\) ) and the time in years after \(2009 ( x )\). The graph of lny plotted against \(x\) is approximately a straight line. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-07_1052_1157_751_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
      \end{figure}
  2. Show that the straight line is consistent with a model of the form \(\mathbf { y } = \mathrm { Ae } ^ { \mathrm { kx } }\), where \(A\) and \(k\) are constants.
  3. Estimate the values of \(A\) and \(k\).
  4. Use the model to predict the profit in the year 2020.
  5. How reliable do you expect the prediction in part (d) to be? Justify your answer.
OCR MEI Paper 3 2020 November Q7
9 marks Standard +0.8
7
  1. Express \(\frac { 1 } { x } + \frac { 1 } { A - x }\) as a single fraction. The population of fish in a lake is modelled by the differential equation
    \(\frac { d x } { d t } = \frac { x ( 400 - x ) } { 400 }\)
    where \(x\) is the number of fish and \(t\) is the time in years.
    When \(t = 0 , x = 100\).
  2. In this question you must show detailed reasoning. Find the number of fish in the lake when \(t = 10\), as predicted by the model.
OCR MEI Paper 3 2020 November Q8
16 marks Standard +0.3
8
  1. The curve \(y = \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } }\) is shown in Fig. 8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-08_495_1058_1105_315} \captionsetup{labelformat=empty} \caption{Fig. 8}
    \end{figure}
    1. Show that \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = \frac { 20 x ^ { 2 } - 4 } { \left( 1 + x ^ { 2 } \right) ^ { 4 } }\).
    2. In this question you must show detailed reasoning. Find the set of values of \(x\) for which the curve is concave downwards.
  2. Use the substitution \(x = \tan \theta\) to find the exact value of \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } d x\). Answer all the questions.
    Section B (15 marks) 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 2020 November Q9
3 marks Standard +0.3
9
  1. Show that if \(a = 1\) and \(b > 1\) then \(\mathrm { a } ^ { \mathrm { b } } < \mathrm { b } ^ { \mathrm { a } }\).
  2. Find integer values of \(a\) and \(b\) with \(b > a > 1\) and \(\mathrm { a } ^ { \mathrm { b } }\) not greater than \(\mathrm { b } ^ { \mathrm { a } }\) (a counter example to the conjecture given in lines 7-8).
OCR MEI Paper 3 2020 November Q10
2 marks Easy -1.2
10 In this question you must show detailed reasoning.
Show that \(\int _ { \mathrm { e } } ^ { \pi } \frac { 1 } { x } \mathrm {~d} x = \ln \pi - 1\) as given in line 37.
OCR MEI Paper 3 2020 November Q11
2 marks Easy -2.5
11 Show that \(\mathrm { e } ^ { x }\) is an increasing function for all values of \(x\), as stated in line 39 .
OCR MEI Paper 3 2020 November Q12
8 marks Standard +0.8
12
  1. Show that the only stationary point on the curve \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } }\) occurs where \(x = \mathrm { e }\), as given in line 45.
  2. Show that the stationary point is a maximum.
  3. It follows from part (b) that, for any positive number \(a\) with \(a \neq \mathrm { e }\),
    \(\frac { \ln \mathrm { e } } { \mathrm { e } } > \frac { \ln a } { a }\).
    Use this fact to show that \(\mathrm { e } ^ { a } > a ^ { \mathrm { e } }\).
OCR MEI Paper 3 2021 November Q1
5 marks Easy -1.3
1
  1. Express \(x ^ { 2 } + 8 x + 2\) in the form \(( x + a ) ^ { 2 } + b\).
  2. Write down the coordinates of the turning point of the curve \(y = x ^ { 2 } + 8 x + 2\).
  3. State the transformation(s) which map(s) the curve \(y = x ^ { 2 }\) onto the curve \(y = x ^ { 2 } + 8 x + 2\).
OCR MEI Paper 3 2021 November Q2
2 marks Moderate -0.8
2 Solve the equation \(\sin 2 x = 0.3\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). Give your answer(s) correct to \(\mathbf { 1 }\) decimal place.
OCR MEI Paper 3 2021 November Q3
7 marks Moderate -0.8
3
  1. Determine, in terms of \(k\), the coordinates of the point where the lines with the following equations intersect. $$\begin{array} { r } x + y = k \\ 2 x - y = 1 \end{array}$$
  2. Determine, in terms of \(k\), the coordinates of the points where the line \(\mathrm { x } + \mathrm { y } = \mathrm { k }\) crosses the curve \(y = x ^ { 2 } + k\).
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 $$\arctan \left( \frac { 1 } { n + 1 } \right) + \arctan \left( \frac { 1 } { n ^ { 2 } + n + 1 } \right) = \arctan \left( \frac { 1 } { n } \right) , \text { as given in line } 39 .$$