Questions — OCR MEI Paper 3 (118 questions)

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OCR MEI Paper 3 2024 June Q14
14 Substitute appropriate values of \(t _ { 1 }\) and \(t _ { 2 }\) to verify that the expression \(t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } + t _ { 1 } t _ { 2 } + \frac { 1 } { 2 }\) gives the correct value for the \(y\)-coordinate of the point of intersection of the normals at the points A and B in Fig. C2.
OCR MEI Paper 3 2024 June Q15
15
  1. Show that, for the curve \(y = a x ^ { 2 } + b x + c\), the equation of the tangent at the point with \(x\)-coordinate \(t\) is \(\mathrm { y } = ( 2 \mathrm { at } + \mathrm { b } ) \mathrm { x } - \mathrm { at } ^ { 2 } + \mathrm { c }\).
  2. Hence show that for the curve with equation \(y = a x ^ { 2 } + b x + c\), the tangents at two points, \(P\) and Q , on the curve cross at a point which has \(x\)-coordinate equal to the mean of the \(x\)-coordinates of points P and Q , as given in lines 11 to 14 .
OCR MEI Paper 3 2024 June Q16
16 Show that the expression \(a \left( \frac { x _ { P } + x _ { Q } } { 2 } \right) ^ { 2 } + b \left( \frac { x _ { P } + x _ { Q } } { 2 } \right) + c - a \left( \frac { x _ { P } - x _ { Q } } { 2 } \right) ^ { 2 }\) is equivalent to \(a x _ { P } x _ { Q } + b \left( \frac { x _ { P } + x _ { Q } } { 2 } \right) + c\), as given in lines 15 and 16 .
OCR MEI Paper 3 2024 June Q17
17 Show that, for the curve \(y = x ^ { 2 }\), the equation of the normal at the point \(\left( t , t ^ { 2 } \right)\) is \(y = - \frac { x } { 2 t } + t ^ { 2 } + \frac { 1 } { 2 }\), as given in line 27.
OCR MEI Paper 3 2024 June Q18
18 A student is investigating the intersection points of tangents to the curve \(y = 6 x ^ { 2 } - 7 x + 1\). She uses software to draw tangents at pairs of points with \(x\)-coordinates differing by 5 . Find the equation of the curve that all the intersection points lie on.
OCR MEI Paper 3 2020 November Q1
1 Find the value of \(\sum _ { r = 1 } ^ { 5 } 2 ^ { r } ( r - 1 )\).
OCR MEI Paper 3 2020 November Q2
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 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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
7 Determine \(\int x \cos 2 x \mathrm {~d} x\).
OCR MEI Paper 3 2021 November Q8
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.