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OCR H240/02 2021 November Q5
8 marks Standard +0.3
5 In this question you must show detailed reasoning. Points \(A , B\) and \(C\) have coordinates \(( 0,6 ) , ( 7,5 )\) and \(( 6 , - 2 )\) respectively.
  1. Find an equation of the perpendicular bisector of \(A B\).
  2. Hence, or otherwise, find an equation of the circle that passes through points \(A , B\) and \(C\).
OCR H240/03 2018 June Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:
  • the length of the flower bed is to be 3 m longer than the width,
  • the length of the flower bed must be at least 14.5 m ,
  • the area of the flower bed must be less than \(180 \mathrm {~m} ^ { 2 }\).
The width of the flower bed is \(x \mathrm {~m}\).
By writing down and solving appropriate inequalities in \(x\), determine the set of possible values for the width of the flower bed.
OCR H240/03 2020 November Q6
11 marks Challenging +1.2
6 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-06_495_800_312_244}
The diagram shows the curve with equation \(4 x y = 2 \left( x ^ { 2 } + 4 y ^ { 2 } \right) - 9 x\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 4 y - 9 } { 4 x - 16 y }\). At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
  2. Show that the distance \(P Q\) is \(k \sqrt { 5 }\), where \(k\) is a rational number to be determined.
OCR H240/03 2022 June Q6
8 marks Standard +0.3
6 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{e69f8d73-764e-4f13-a126-faec02c4ad08-06_611_1344_306_242}
The diagram shows the curves \(y = \sqrt { 2 x + 9 }\) and \(y = 4 \mathrm { e } ^ { - 2 x } - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined.
OCR H240/03 2022 June Q7
8 marks Standard +0.8
7 In this question you must show detailed reasoning.
  1. Show that the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\) can be expressed in the form $$m \tan ^ { 2 } \theta - 4 \tan \theta + ( m + 3 ) = 0 .$$
  2. It is given that there is only one value of \(\theta\), for \(0 < \theta < \pi\), satisfying the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\). Given also that \(m\) is a negative integer, find this value of \(\theta\), correct to \(\mathbf { 3 }\) significant figures.
OCR H240/03 2023 June Q12
13 marks Standard +0.3
12 In this question you should take the acceleration due to gravity to be \(10 \mathrm {~ms ^ { - 2 }\).}
\includegraphics[max width=\textwidth, alt={}]{977ffad6-2440-46bf-9f17-0f30817d2ddf-09_410_1344_324_244}
A small ball \(P\) is projected from a point \(A\) with speed \(39 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\) and \(\cos \theta = \frac { 12 } { 13 }\). Point \(A\) is 20 m vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.
  1. Find the maximum height of \(P\) above the ground during its motion. The time taken for \(P\) to travel from \(A\) to \(C\) is \(T\) seconds.
  2. Determine the value of \(T\).
  3. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geqslant 0\), the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(Q\) is given by \(v = k t ^ { 3 } + 6 t ^ { 2 } + \frac { 3 } { 2 } t\),
    where \(k\) is a positive constant.
  4. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision. \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-10_607_803_303_246} The diagram shows a small block \(B\), of mass 2 kg , and a particle \(P\), of mass 4 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60 ^ { \circ }\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.
  5. Determine, in terms of \(g\), the tension in the string. When \(P\) is moving at \(2 \mathrm {~ms} ^ { - 1 }\) the string breaks. In the 0.5 seconds after the string breaks \(P\) moves 1.9 m down the plane.
  6. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures.
OCR PURE 2018 May Q8
6 marks Standard +0.8
8 In this question you must show detailed reasoning. The lines \(y = \frac { 1 } { 2 } x\) and \(y = - \frac { 1 } { 2 } x\) are tangents to a circle at \(( 2,1 )\) and \(( - 2,1 )\) respectively. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
OCR PURE 2020 October Q11
4 marks Challenging +1.2
11 In this question you must show detailed reasoning. A biased four-sided spinner has edges numbered \(1,2,3,4\). When the spinner is spun, the probability that it will land on the edge numbered \(X\) is given by \(P ( X = x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 10 } x & x = 1,2,3,4 , \\ 0 & \text { otherwise } . \end{cases}\)
  1. Draw a table showing the probability distribution of \(X\). The spinner is spun three times and the value of \(X\) is noted each time.
  2. Find the probability that the third value of \(X\) is greater than the sum of the first two values of \(X\).
OCR PURE 2023 May Q5
6 marks Standard +0.3
5 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = x ^ { 3 } - 4 x\). \includegraphics[max width=\textwidth, alt={}, center]{e42b1a99-c3ca-4ce1-becd-cd346aec757e-05_499_695_404_251} Determine the total area enclosed by the curve and the \(x\)-axis.
OCR PURE 2023 May Q8
7 marks Challenging +1.2
8 In this question you must show detailed reasoning. A circle has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 12 = 0\). Two tangents to this circle pass through the point \(( 0,1 )\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same.
Find the angle between these two tangents.
OCR PURE 2018 May Q8
9 marks Standard +0.3
8 In this question you must show detailed reasoning. The diagram shows part of the graph of \(y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }\). The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 8\) and \(x = a\), where \(a > 8\). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of \(a\).
OCR PURE 2019 May Q1
3 marks Easy -1.2
1 In this question you must show detailed reasoning. Solve the equation \(x ( 3 - \sqrt { 5 } ) = 24\), giving your answer in the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
OCR PURE 2020 October Q3
7 marks Moderate -0.8
3 In this question you must show detailed reasoning. Find the equation of the normal to the curve \(y = 4 \sqrt { x } - 3 x + 1\) at the point on the curve where \(x = 4\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR PURE 2020 October Q4
9 marks Standard +0.3
4 In this question you must show detailed reasoning. The cubic polynomial \(6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
  1. Use the factor theorem to show that \(k = - 37\).
  2. Using this value of \(k\), factorise \(\mathrm { f } ( x )\) completely.
    1. Hence find the three values of \(t\) satisfying the equation \(6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0\).
    2. Express the sum of the three values found in part (c)(i) as a single logarithm.
OCR PURE 2020 October Q6
6 marks Moderate -0.3
6 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{7fc02f90-8f8b-4153-bba1-dc0807124e96-4_650_661_1765_242}
The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.
OCR PURE 2022 June Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning. Solve the equation \(3 x + 1 = 4 \sqrt { x }\).
OCR PURE 2023 May Q2
4 marks Moderate -0.3
2 In this question you must show detailed reasoning. Solve the equation \(x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x\). Give your answer in the form \(a \sqrt { 5 } + b\), where \(a\) and \(b\) are integers to be determined.
OCR PURE 2023 May Q8
7 marks Moderate -0.3
8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).
OCR MEI Paper 2 2018 June Q12
5 marks Standard +0.3
12 You must show detailed reasoning in this question. In the summer of 2017 in England a large number of candidates sat GCSE examinations in both mathematics and English. 56\% of these candidates achieved at least level 4 in mathematics and \(80 \%\) of these candidates achieved at least level 4 in English. 14\% of these candidates did not achieve at least level 4 in either mathematics or English. Determine whether achieving level 4 or above in English and achieving level 4 or above in mathematics were independent events.
OCR MEI Paper 2 2019 June Q15
6 marks Challenging +1.2
15 You must show detailed reasoning in this question. The screenshot in Fig. 15 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_387_954_1599_260} \captionsetup{labelformat=empty} \caption{Fig. 15}
\end{figure} The distribution is symmetrical about the line \(x = 35\) and there is a point of inflection at \(x = 31\).
Fifty independent readings of \(X\) are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .
OCR MEI Paper 2 2023 June Q15
7 marks Standard +0.3
15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.
OCR MEI Paper 2 2023 June Q17
6 marks Standard +0.8
17 In this question you must show detailed reasoning. Solve the equation \(2 \sin x + \sec x = 4 \cos x\), where \(- \pi < x < \pi\).
OCR MEI Paper 2 2024 June Q16
12 marks Standard +0.3
16 In this question you must show detailed reasoning. Find the particular solution of the differential equation $$\frac { d y } { d x } = \frac { 9 y } { ( x - 1 ) ( x + 2 ) }$$ given that \(x = 2\) when \(y = 16\). \section*{END OF QUESTION PAPER}
OCR MEI Paper 2 2020 November Q10
9 marks Standard +0.3
10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$
  1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
  2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]
    \(n\)01234567
    \(\mathrm { x } _ { \mathrm { n } }\)10.9585090.9500840.9482610.947860.9477720.9477530.947748
    \captionsetup{labelformat=empty} \caption{Fig. 10.1}
    \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]
    \(x\)\(y\)
    0.9477475\(- 7.79967 \mathrm { E } - 07\)
    0.9477485\(- 2.90821 \mathrm { E } - 06\)
    \(x\)\(y\)
    0.947745\(4.54066 \mathrm { E } - 06\)
    0.947755\(- 1.67417 \mathrm { E } - 05\)
    \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{table}
  3. Consider the information in Fig. 10.1 and Fig. 10.2.
    • Write 4.54066E-06 in standard mathematical notation.
    • State the value of the root as accurately as you can, justifying your answer.
OCR MEI Paper 2 2020 November Q14
8 marks Challenging +1.2
14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-16_647_898_404_233} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.