Questions — OCR MEI (4301 questions)

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OCR MEI Paper 3 2023 June Q5
8 marks Moderate -0.8
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
10 marks Standard +0.3
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
6 marks Standard +0.8
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
7 marks Challenging +1.2
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
8 marks Moderate -0.3
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
6 marks Standard +0.8
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
3 marks Easy -1.2
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
3 marks Standard +0.8
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
4 marks Challenging +1.2
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
3 marks Standard +0.8
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
2 marks Challenging +1.2
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
2 marks Easy -1.8
1 Solve the inequality \(\frac { x } { 5 } > 6 - x\).
OCR MEI Paper 3 2024 June Q2
4 marks Moderate -0.8
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
3 marks Challenging +1.2
7 Prove that \(\sin 8 \theta \tan 4 \theta + \cos 8 \theta = 1\).
OCR MEI Paper 3 2024 June Q8
8 marks Standard +0.3
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
6 marks Standard +0.3
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
3 marks Standard +0.3
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
8 marks Standard +0.3
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
9 marks Standard +0.8
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
1 marks Moderate -0.5
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. \(\mathbf { C 1 . }\)
OCR MEI Paper 3 2024 June Q14
1 marks Moderate -0.5
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
6 marks Standard +0.3
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
2 marks Moderate -0.5
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
3 marks Easy -1.2
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
2 marks Challenging +1.2
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.