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OCR Further Pure Core AS 2020 November Q1
6 marks Moderate -0.3
1 In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(- 77 - 36 \mathrm { i }\). \(2 \mathrm { P } , \mathrm { Q }\) and T are three transformations in 2-D.
P is a reflection in the \(x\)-axis. \(\mathbf { A }\) is the matrix that represents P .
  1. Write down the matrix \(\mathbf { A }\). Q is a shear in which the \(y\)-axis is invariant and the point \(\binom { 1 } { 0 }\) is transformed to the point \(\binom { 1 } { 2 }\). \(\mathbf { B }\) is the
    matrix that represents Q . matrix that represents Q.
  2. Find the matrix \(\mathbf { B }\). T is P followed by Q. C is the matrix that represents T.
  3. Determine the matrix \(\mathbf { C }\). \(L\) is the line whose equation is \(y = x\).
  4. Explain whether or not \(L\) is a line of invariant points under \(T\). An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
  5. Explain what the value of the determinant of \(\mathbf { C }\) means about
    • the area of \(N\) compared to the area of \(M\),
    • the orientation of \(N\) compared to the orientation of \(M\).
OCR Further Pure Core AS 2020 November Q3
12 marks Moderate -0.3
3 In this question you must show detailed reasoning. The complex number \(7 - 4 \mathrm { i }\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3 z - 4 z ^ { * }\)
    2. \(( z + 1 - 3 i ) ^ { 2 }\)
    3. \(\frac { z + 1 } { z - 1 }\)
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures.
  3. The complex number \(\omega\) is such that \(z \omega = \sqrt { 585 } ( \cos ( 0.5 ) + \mathrm { i } \sin ( 0.5 ) )\). Find the following.
    • \(| \omega |\)
    • \(\arg ( \omega )\), giving your answer correct to 3 significant figures
OCR Further Pure Core AS 2020 November Q5
7 marks Challenging +1.2
5 In this question you must show detailed reasoning. The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
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]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-5_807_872_443_239} \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 Pure Core Specimen Q15
8 marks Challenging +1.2
15 In this question you must show detailed reasoning. Show that $$\int _ { 0 } ^ { \frac { 2 } { 3 } } \operatorname { arsinh } 2 x \mathrm {~d} x = \frac { 2 } { 3 } \ln 3 - \frac { 1 } { 3 }$$
OCR MEI Further Mechanics Major 2019 June Q7
8 marks Challenging +1.2
7 In this question you must show detailed reasoning. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-04_503_885_1665_244} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 shows the curve with equation \(y = \frac { 2 } { 3 } \ln x\). The region R , shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0 , y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(y\)-coordinate of the centre of mass of the solid.
OCR MEI Further Mechanics Major 2022 June Q6
7 marks Standard +0.3
6 In this question the box should be modelled as a particle. A box of mass mkg is placed on a rough slope which makes an angle of \(\alpha\) with the horizontal.
  1. Show that the box is on the point of slipping if \(\mu = \tan \alpha\), where \(\mu\) is the coefficient of friction between the box and the slope. A box of mass 5 kg is pulled up a rough slope which makes an angle of \(15 ^ { \circ }\) with the horizontal. The box is subject to a constant frictional force of magnitude 3 N . The speed of the box increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point A on the slope to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point B on the slope with B higher up the slope than A . The distance AB is 10 m . \includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-05_535_876_957_255} The pulling force has constant magnitude P N and acts at a constant angle of \(25 ^ { \circ }\) above the slope, as shown in the diagram.
  2. Use the work-energy principle to determine the value of P .
OCR MEI Further Mechanics Major 2022 June Q13
17 marks Challenging +1.8
13 In this question take \(\boldsymbol { g = \mathbf { 1 0 }\).} A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 m . The other end of the string is attached to a fixed point O . The particle P rests in equilibrium at a point A with the string vertical.
  1. Show that the distance OA is 0.5 m . At time \(\mathrm { t } = 0 , \mathrm { P }\) is projected vertically downwards from A with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Throughout the subsequent motion, \(P\) experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) ).
  2. Given that the downward displacement of P from A at time t seconds is x metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 200 x = 0$$
  3. Verify that \(\mathrm { x } = \frac { 5 } { 56 } \mathrm { e } ^ { - 2 \mathrm { t } } \sin ( 14 \mathrm { t } )\).
  4. Determine whether the string becomes slack during the motion.
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.