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OCR Further Pure Core 1 2022 June Q1
6 marks Standard +0.8
1 In this question you must show detailed reasoning.
  1. Show that \(\cosh ( 2 \ln 3 ) = \frac { 41 } { 9 }\). The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \sqrt { \operatorname { sinhx } }\), the \(x\)-axis and the line with equation \(x = 2 \ln 3\) (see diagram). The units of the axes are centimetres. \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-2_652_668_740_242} A manufacturer produces bell-shaped chocolate pieces. Each piece is modelled as being the shape of the solid formed by rotating \(R\) completely about the \(x\)-axis.
  2. Determine, according to the model, the exact volume of one chocolate piece.
OCR Further Pure Core 2 2021 November Q2
8 marks Moderate -0.3
2 In this question you must show detailed reasoning. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 3 - 7 \mathrm { i }\) and \(z _ { 2 } = 2 + 4 \mathrm { i }\).
  1. Express each of the following as exact numbers in the form \(a + b \mathrm { i }\).
    1. \(3 z _ { 1 } + 4 z _ { 2 }\)
    2. \(z _ { 1 } z _ { 2 }\)
    3. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
  2. Write \(z _ { 1 }\) in modulus-argument form giving the modulus in exact form and the argument correct to \(\mathbf { 3 }\) significant figures.
OCR Further Statistics 2022 June Q3
8 marks Standard +0.8
3 In this question you must show detailed reasoning. A discrete random variable \(V\) has the following probability distribution, where \(p\) and \(q\) are constants.
\(v\)0123
\(\mathrm { P } ( \mathrm { V } = \mathrm { v } )\)\(p\)\(q\)0.120.2
It is given that \(\mathrm { E } ( V ) = \operatorname { Var } ( V )\). Determine the value of \(p\) and the value of \(q\).
OCR Further Pure Core AS 2022 June Q3
6 marks Challenging +1.2
3 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).
OCR Further Pure Core AS 2024 June Q9
8 marks Challenging +1.8
9 In this question you must show detailed reasoning. You are given that \(a\) is a real root of the equation \(x ^ { 4 } + x ^ { 3 } + 3 x ^ { 2 } - 5 x = 0\).
You are also given that \(a + 2 + 3 \mathrm { i }\) is one root of the equation \(z ^ { 4 } - 2 ( 1 + a ) z ^ { 3 } + ( 21 a - 10 ) z ^ { 2 } + ( 86 - 80 a ) z + ( 285 a - 195 ) = 0\). Determine all possible values of \(z\).
OCR Further Mechanics 2022 June Q5
9 marks Challenging +1.8
5 In this question you must show detailed reasoning. The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = 4\) and the curve with equation \(\mathrm { y } = \frac { 15 } { \sqrt { \mathrm { x } ^ { 2 } + 9 } }\) is occupied by a uniform lamina. The centre of mass of the lamina is at the point \(G ( \bar { x } , \bar { y } )\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-4_944_954_598_228}
  1. Show that \(\bar { x } = \frac { 2 } { \ln 3 }\).
  2. Determine the value of \(\bar { y }\). Give your answer correct to \(\mathbf { 3 }\) significant figures. \(P\) is the point on the curved edge of the lamina where \(x = 3\). The lamina is freely suspended from \(P\) and hangs in equilibrium in a vertical plane.
  3. Determine the acute angle that the longest straight edge of the lamina makes with the vertical.
OCR Further Additional Pure 2019 June Q8
11 marks Hard +2.3
8 In this question you must show detailed reasoning.
  1. Prove that \(2 ( p - 2 ) ^ { p - 2 } \equiv - 1 ( \bmod p )\), where \(p\) is an odd prime.
  2. Find two odd prime factors of the number \(N = 2 \times 34 ^ { 34 } - 2 ^ { 15 }\). \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR Further Additional Pure 2021 November Q1
3 marks Moderate -0.8
1 In this question you must show detailed reasoning. Express the number \(\mathbf { 4 1 7 2 3 } _ { 10 }\) in hexadecimal (base 16).
OCR MEI Further Mechanics B AS 2021 November Q3
8 marks Standard +0.8
3 In this question you must show detailed reasoning. [In this question you may use the formula: Volume of cone \(= \frac { 1 } { 3 } \times\) base area × height.]
The region between the line \(\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }\), where \(a > 0\), the \(x\)-axis and the \(y\)-axis is rotated about the \(y\)-axis to form a uniform right circular cone C with base radius \(a\).
  1. Show that the centre of mass of C is \(\frac { 3 } { 4 } a\) from its base. The cone C is fixed on top of a uniform cube, B , of edge length \(2 a\), so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} The centre of mass of the combined shape lies on the boundary of C and B .
    The density of \(B\) is not equal to the density of \(C\).
  2. Determine the exact value of \(\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }\).
    [0pt] [3]
OCR MEI Further Mechanics B AS Specimen Q6
12 marks Standard +0.8
6 In this question you must show detailed reasoning. As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_725_449_539_751} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure}
  1. Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
  2. Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\). Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure}
  3. Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.
OCR MEI Further Statistics A AS Specimen Q3
10 marks Standard +0.3
3 In this question you must show detailed reasoning. A student is investigating what people think about organic food. She wishes to see if there is any difference between the opinions of females and males. She takes a random sample of 100 people and asks each of them if they think that organic food is better for their health than non-organic food. She will use the data to conduct a hypothesis test. The table below shows the opinions of these 100 people.
\cline { 3 - 4 } \multicolumn{2}{c|}{}Sex
\cline { 3 - 4 } \multicolumn{2}{c|}{}FemaleMale
\multirow{2}{*}{
Opinion on
organic food
}		Organic better3518
\cline { 2 - 4 }Not better2225
  1. Explain why the student should use a random sample.
  2. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a person's sex and their opinion on organic food. Show your calculations.
OCR MEI Further Pure Core 2019 June Q4
3 marks Challenging +1.2
4 In this question you must show detailed reasoning. Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{01a574f1-f6f6-40f5-baa5-535c36269731-2_501_670_1329_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
OCR MEI Further Pure Core 2019 June Q8
8 marks Standard +0.3
8 In this question you must show detailed reasoning. The roots of the equation \(x ^ { 3 } - x ^ { 2 } + k x - 2 = 0\) are \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
  1. Evaluate, in exact form, the roots of the equation.
  2. Find \(k\).
OCR MEI Further Pure Core 2019 June Q10
8 marks Standard +0.8
10 In this question you must show detailed reasoning.
  1. You are given that \(- 1 + \mathrm { i }\) is a root of the equation \(z ^ { 3 } = a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. Find \(a\) and \(b\).
  2. Find all the roots of the equation in part (a), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r\) and \(\theta\) are exact.
  3. Chris says "the complex roots of a polynomial equation come in complex conjugate pairs". Explain why this does not apply to the polynomial equation in part (a).
OCR MEI Further Pure Core 2019 June Q15
8 marks Challenging +1.2
15 In this question you must show detailed reasoning. Show that \(\int _ { \frac { 3 } { 4 } } ^ { \frac { 3 } { 2 } } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 4 x + 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { 3 + \sqrt { 5 } } { 2 } \right)\).
OCR MEI Further Pure Core 2023 June Q15
5 marks Standard +0.3
15 In this question you must show detailed reasoning. Evaluate \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 1 + 2 x - x ^ { 2 } } } d x\), giving your answer in terms of \(\pi\).
OCR MEI Further Pure Core 2024 June Q16
6 marks Challenging +1.2
16 In this question you must show detailed reasoning. Show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } } \mathrm { dx } = \ln \left( \frac { \mathrm { a } + \mathrm { b } \sqrt { 3 } } { \mathrm { c } } \right)\), where \(a , b\) and \(c\) are integers to be determined.
OCR MEI Further Pure Core 2020 November Q11
8 marks Standard +0.8
11 In this question you must show detailed reasoning. In Fig. 11, the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }\) and FA are \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L respectively. \begin{figure}[h]
[diagram]
\captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
OCR MEI Further Statistics Minor 2020 November Q3
8 marks Standard +0.3
3 In this question you must show detailed reasoning. In a survey into pet ownership, one of the questions was 'Do you own either a cat or a dog (or both)?'. A total of 121 people took part in the survey and you should assume that they form a random sample of people in a particular town. The results, classified by the age of the person being surveyed, are shown in Table 3. \begin{table}[h]
\multirow{2}{*}{}Ownership of cat or dog
Does ownDoes not own
\multirow{2}{*}{Age}Over 45 years3829
Under 45 years2331
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table} Carry out a test at the 10\% significance level to investigate whether, for people in this town, there is any association between age and ownership of a cat or dog.
OCR MEI Further Statistics Major 2020 November Q8
10 marks Standard +0.3
8 In this question you must show detailed reasoning. On the manufacturer's website, it is claimed that the average daily electricity consumption of a particular model of fridge is 1.25 kWh (kilowatt hours). A researcher at a consumer organisation decides to check this figure. A random sample of 40 fridges is selected. Summary statistics for the electricity consumption \(x \mathrm { kWh }\) of these fridges, measured over a period of 24 hours, are as follows. \(\Sigma x = 51.92 \quad \Sigma x ^ { 2 } = 70.57\) Carry out a test at the \(5 \%\) significance level to investigate the validity of the claim on the website.
[0pt] [10]
OCR Further Pure Core 2 2019 June Q3
5 marks Standard +0.3
3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }
OCR Further Pure Core 2 2022 June Q3
6 marks Challenging +1.2
3 In this question you must show detailed reasoning. The roots of the equation \(4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
OCR Further Pure Core 2 2022 June Q9
9 marks Challenging +1.2
9 In this question you must show detailed reasoning.
  1. Show that \(\operatorname { Re } \left( \mathrm { e } ^ { \mathrm { Ai } \theta } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right) ^ { 4 } \right) = a \cos 4 \theta \cos ^ { 4 } \theta\), where \(a\) is an integer to be determined.
  2. Hence show that \(\cos \frac { 1 } { 12 } \pi = \frac { 1 } { 2 } \sqrt [ 4 ] { \mathrm { b } + \mathrm { c } \sqrt { 3 } }\), where \(b\) and \(c\) are integers to be determined.
OCR D2 2007 June Q4
16 marks Moderate -0.5
4 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a minimax problem.
StageStateA ctionWorkingM inimax
\multirow{3}{*}{1}0044
1033
2022
\multirow{9}{*}{2}\multirow{3}{*}{0}0\(\max ( 6,4 ) = 6\)\multirow{3}{*}{3}
1\(\max ( 2,3 ) = 3\)
2\(\max ( 3,2 ) = 3\)
\multirow{3}{*}{1}0\(\max ( 2,4 ) =\)\multirow{3}{*}{}
1\(\max ( 4,3 ) =\)
2\(\max ( 5,2 ) =\)
\multirow{3}{*}{2}0max(2,\multirow{3}{*}{}
1max(3,
2max(4,
\multirow{3}{*}{3}\multirow{3}{*}{0}0max(5,\multirow{3}{*}{}
1max(5,
2max(2,
  1. On the insert, complete the last two columns of the table.
  2. State the minimax value and write down the minimax route.
  3. Complete the diagram on the insert to show the network that is represented by the table.
OCR D2 2007 June Q5
13 marks Moderate -0.5
5 Answer this question on the insert provided. The network represents a system of pipes through which fluid can flow from a source, S , to a sink, T . \includegraphics[max width=\textwidth, alt={}, center]{09d4aacd-026b-4d81-a826-3d3f29f9c105-5_1310_1301_447_424} The arrows are labelled to show excess capacities and potential backflows (how much more and how much less could flow in each pipe). The excess capacities and potential backflows are measured in litres per second. Currently the flow is 6 litres per second, all flowing along a single route through the system.
  1. Write down the route of the 6 litres per second that is flowing from \(S\) to \(T\).
  2. What is the capacity of the pipe AG and in which direction can fluid flow along this pipe?
  3. Calculate the capacity of the \(\operatorname { cut } \mathrm { X } = \{ \mathrm { S } , \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \} , \mathrm { Y } = \{ \mathrm { F } , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { T } \}\).
  4. Describe how a further 7 litres per second can flow from S to T and update the labels on the arrows to show your flow. Explain how you know that this is the maximum flow. {}