OCR MEI Paper 3 (Paper 3) 2023 June

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 \(\cos \theta\).

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 \(| 5 - 2 x | < 3\).

3 In this question you must show detailed reasoning.
Find the value of \(k\) such that \(\frac { 1 } { \sqrt { 5 } + \sqrt { 6 } } + \frac { 1 } { \sqrt { 6 } + \sqrt { 7 } } = \frac { k } { \sqrt { 5 } + \sqrt { 7 } }\).

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.

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 \(N\).

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
      • Q, the midpoint of LM
      • R, the midpoint of MN
      • S, the midpoint of NK
      • Verify that PQRS is a parallelogram.
      • TVWX is a quadrilateral as shown in Fig. 6.2.
      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}
      1. Show that \(\overrightarrow { \mathrm { WX } } = k ( - \mathbf { a } + \mathbf { b } - \mathbf { c } )\), where \(k\) is a constant to be determined.
      2. Verify that AH is parallel to DE .
      3. Verify that BC is parallel to GF .

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.

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.

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.
4. 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.

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.

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 \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).

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 .

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 \(\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }\).

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.

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.