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OCR Further Statistics AS Specimen Q5
7 marks Moderate -0.3
  1. The random variable \(X\) has the distribution \(\operatorname { Geo } ( 0.6 )\).
    (a) Find \(\mathrm { P } ( X \geq 8 )\).
    (b) Find the value of \(\mathrm { E } ( X )\).
    (c) Find the value of \(\operatorname { Var } ( X )\).
  2. The random variable \(Y\) has the distribution \(\operatorname { Geo } ( p )\). It is given that \(\mathrm { P } ( Y < 4 ) = 0.986\) correct to 3 significant figures. Use an algebraic method to find the value of \(p\). Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
  3. State these two assumptions.
  4. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\operatorname { Po } ( 0.8 )\).
  5. (a) Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house.
    (b) Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house.
  6. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house.
  7. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition.
OCR Further Mechanics AS 2019 June Q5
14 marks Standard +0.8
  1. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
  2. Show that
    1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
    2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
  3. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
  4. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero.
OCR Further Pure Core 2 2019 June Q6
6 marks Standard +0.8
  1. Show that the motion of the particle can be modelled by the differential equation \end{itemize} $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when \(t = 0\).
  2. Find \(v\) in terms of \(t\).
  3. Find the velocity of the particle when \(t = 2\). When \(t = 2\) the force acting in the positive \(x\)-direction is replaced by a constant force of magnitude \(\frac { 1 } { 2 } \mathrm {~N}\) in the same direction.
  4. Refine the differential equation given in part (a) to model the motion for \(t \geqslant 2\).
  5. Use the refined model from part (d) to find an exact expression for \(v\) in terms of \(t\) for \(t \geqslant 2\). \(6 \quad A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after 0.2 seconds at point \(B\) whose displacement is 0.2 m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x \mathrm {~m}\).
  6. On the axes provided in the Printed Answer Booklet, sketch a graph of \(x\) against \(t\) for \(0 \leqslant t \leqslant 0.4\).
  7. Find the displacement of \(P\) from \(M\) at 0.75 seconds after release.
OCR Further Pure Core 2 2019 June Q9
11 marks Challenging +1.2
  1. Find the exact area enclosed by the curve.
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\).
OCR Further Pure Core 2 2022 June Q8
7 marks Challenging +1.2
  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 Further Pure Core 2 Specimen Q7
7 marks Challenging +1.2
  1. Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2 x \sin 4 x\) up to and including the term in \(x ^ { 3 }\).
  2. Hence find, in exact surd form, an approximation to the least positive root of the equation \(2 \sin x \sin 2 x \sin 4 x = x\).
OCR Further Mechanics 2024 June Q7
14 marks Standard +0.8
  1. Show that \(B\) 's motion can be modelled by the differential equation \(\frac { 1 } { \mathrm { v } } \frac { \mathrm { dv } } { \mathrm { dx } } = - 4\).
    1. Solve the differential equation in part (a) to find the particular solution for \(v\) in terms of \(x\) and \(u\).
    2. By considering the behaviour of \(v\) as \(x \longrightarrow \infty\) describe one feature of the model that is not realistic. At the instant when \(B\) reaches the point \(A\), where \(\mathrm { x } = \mathrm { X }\), its speed is \(V \mathrm {~ms} ^ { - 1 }\). The work done by the resistance as \(B\) moves from \(O\) to \(A\) is denoted by \(W \mathrm {~J}\).
    1. Use the formula \(\mathrm { W } = \int \mathrm { F } \mathrm { dx }\) to determine an expression for \(W\) in terms of \(X\) and \(u\).
    2. Explain the relevance of the sign of your answer in part (c)(i).
    3. By writing your answer to part (c)(i) in terms of \(V\) and \(u\) show how the quantity \(W\) relates to the energy of \(B\).
OCR MEI Further Pure Core AS 2023 June Q7
10 marks Standard +0.3
  1. By expanding \(( \sqrt { 3 } + \mathrm { i } ) ^ { 5 }\), express \(z ^ { 5 }\) in the form \(\mathrm { a } +\) bi where \(a\) and \(b\) are real and exact.
    1. Express \(z\) in modulus-argument form.
    2. Hence find \(z ^ { 5 }\) in modulus-argument form.
    3. Use this result to verify your answers to part (a).
OCR MEI Further Mechanics B AS Specimen Q5
7 marks Standard +0.8
  1. Find an expression for the stiffness of the spring, \(k \mathrm { Nm } ^ { - 1 }\), in terms of \(m , h\) and \(g\). The particle is pushed down a further distance from the equilibrium position and released from rest. At time \(t\) seconds, the displacement of the particle from the equilibrium position of the system is \(y \mathrm {~m}\) in the downward direction, as shown in Fig. 5.3. You are given that \(| y | \leq h\).
  2. Show that the motion of the particle is modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { g y } { h } = 0\).
  3. Find an expression for the period of the motion of the particle.
  4. Would the model for the motion of the particle be valid for large values of \(m\) ? Justify your answer.
OCR MEI Further Pure Core 2020 November Q10
7 marks Standard +0.3
  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 Q7
11 marks Challenging +1.2
  1. Use the Maclaurin series for \(\ln ( 1 + x )\) up to the term in \(x ^ { 3 }\) to obtain an approximation to \(\ln 1.5\).
  2. (A) Find the error in the approximation in part (i).
    (B) Explain why the Maclaurin series in part (i), with \(x = 2\), should not be used to find an approximation to \(\ln 3\).
  3. Find a cubic approximation to \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
  4. (A) Use the approximation in part (iii) to find approximations to
    • ln 1.5 and
    • \(\quad \ln 3\).
      (B) Comment on your answers to part (iv) (A).
OCR MEI Further Pure Core Specimen Q14
18 marks Challenging +1.2
  1. Starting with the result $$\mathrm { e } ^ { \mathrm { i } \theta } = \cos \theta + \mathrm { i } \sin \theta$$ show that
    (A) \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta\) (B) \(\cos \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { - \mathrm { i } \theta } \right)\).
  2. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity
  3. Using the result in part (i) (B), obtain the values of the constants \(P , Q , R\) and \(S\) in the identity
  4. Show that \(\cos \frac { \pi } { 12 } = \left( \frac { 26 + 15 \sqrt { 3 } } { 64 } \right) ^ { \frac { 1 } { 6 } }\).
  5. Using the result in part (i) (A), obtain the values of the constants \(a , b , c\) and \(d\) in the identity $$\cos 6 \theta \equiv a \cos ^ { 6 } \theta + b \cos ^ { 4 } \theta + c \cos ^ { 2 } \theta + d$$ $$\cos ^ { 6 } \theta \equiv P \cos 6 \theta + Q \cos 4 \theta + R \cos 2 \theta + S$$
OCR MEI Further Mechanics Major 2023 June Q9
12 marks Challenging +1.8
  1. Determine the following, in either order.
    • The components of the velocity of P , parallel and perpendicular to the plane, immediately before P hits the plane at A .
    • The distance OA.
    After P hits the plane at A it continues to move away from O . Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.
  2. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. \includegraphics[max width=\textwidth, alt={}, center]{41b1f65b-8806-4183-81a1-0276691e203c-08_615_759_251_244} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O . The point A lies on the rim of the bowl such that AO makes an angle of \(30 ^ { \circ }\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A , with speed \(u\), where \(\mathrm { u } > \sqrt { \frac { \mathrm { gr } } { 2 } }\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A . The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.
  3. Determine, in terms of \(m , g , u , r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B . The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4 m g\).
  4. Determine, in terms of \(r\), the vertical distance of B above the floor. It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.
  5. Show that \(\mathrm { u } ^ { 2 } > 2 \mathrm { gr }\).
OCR MEI Further Mechanics Major 2020 November Q9
10 marks Standard +0.8
  1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4 a\).
  2. Determine the coefficient of friction between the bar and the ring. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-07_850_835_258_255} \captionsetup{labelformat=empty} \caption{Fig. 10}
    \end{figure} Fig. 10 shows a small bead P of mass \(m\) which is threaded on a smooth thin wire. The wire is in the form of a circle of radius \(a\) and centre O . The wire is fixed in a vertical plane. The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by \(\theta\).
  3. Determine the value of \(\theta\) when the magnitude of the reaction of the wire on the bead is \(\frac { 7 } { 2 } m g\).
  4. Show that the angular velocity of P when OP makes an angle \(\theta\) with the downward vertical is given by \(k \sqrt { \frac { g } { a } } \cos \left( \frac { \theta } { 2 } \right)\), stating the value of the constant \(k\).
  5. Hence determine, in terms of \(g\) and \(a\), the angular acceleration of P when \(\theta\) takes the value found in part (a).
OCR MEI Further Mechanics Major 2021 November Q10
13 marks Challenging +1.2
  1. Determine the magnitude of the normal reaction of the wire on P in terms of \(m , g , a , u\) and \(\theta\), when P is between B and C . P collides with a fixed barrier at C . The coefficient of restitution between P and the fixed barrier is \(e\). After this collision P moves back towards B . On the straight portion BA , the motion of P is resisted by a constant horizontal force \(F\).
  2. Show that P will reach A if $$F b \leqslant \frac { 1 } { 2 } m \left[ e ^ { 2 } u ^ { 2 } + k \left( 1 - e ^ { 2 } \right) g a \right] ,$$ where \(k\) is an integer to be determined.
OCR MEI Further Statistics Major 2022 June Q6
11 marks Standard +0.3
  1. Determine a 95\% confidence interval for the mean weight of liquid paraffin in a tub.
  2. Explain whether the confidence interval supports the researcher's belief.
  3. Explain why the sample has to be random in order to construct the confidence interval.
    [0pt]
  4. A 95\% confidence interval for the mean weight in grams of another ingredient in the skin cream is [1.202, 1.398]. This confidence interval is based on a large sample and the unbiased estimate of the population variance calculated from the sample is 0.25 . Find each of the following.
    • The mean of the sample
    • The size of the sample
OCR MEI Further Extra Pure 2020 November Q5
8 marks Standard +0.3
  1. Show that \(\mathbf { f }\) is also an eigenvector of \(\mathbf { A }\).
  2. State the eigenvalue associated with \(\mathbf { f }\). You are now given that \(\mathbf { A }\) represents a reflection in 3-D space.
  3. Explain the significance of \(\mathbf { e }\) and \(\mathbf { f }\) in relation to the transformation that \(\mathbf { A }\) represents.
  4. State the cartesian equation of the plane of reflection of the transformation represented by \(\mathbf { A }\).
WJEC Unit 1 Specimen Q4
5 marks Moderate -0.8
  1. Identify the statement which is false. Find a counter example to show that this statement is in fact false.
  2. Identify the statement which is true. Give a proof to show that this statement is in fact true. \item Figure 1 shows a sketch of the graph of \(y = f ( x )\). The graph has a minimum point at \(( - 3 , - 4 )\) and intersects the \(x\)-axis at the points \(( - 8,0 )\) and \(( 2,0 )\). \end{enumerate} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-3_540_992_422_518} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  3. Sketch the graph of \(y = f ( x + 3 )\), indicating the coordinates of the stationary point and the coordinates of the points of intersection of the graph with the \(x\)-axis.
  4. Figure 2 shows a sketch of the graph having one of the following equations with an appropriate value of either \(p , q\) or \(r\). \(y = f ( p x )\), where \(p\) is a constant \(y = f ( x ) + q\), where \(q\) is a constant \(y = r f ( x )\), where \(r\) is a constant \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-3_513_1072_1683_587} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Write down the equation of the graph sketched in Figure 2, together with the value of the corresponding constant.
WJEC Further Unit 1 2018 June Q2
6 marks Standard +0.3
  1. Find the values of \(\alpha , \beta\), and \(\gamma\).
  2. Find the cubic equation with roots \(3 \alpha , 3 \beta , 3 \gamma\). Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), where \(a , b , c , d\) are constants to be determined.
WJEC Further Unit 3 2018 June Q2
10 marks Standard +0.3
  1. Calculate the distance \(A P\) when \(P\) is instantaneously at rest for the first time, giving your answer correct to 2 decimal places.
  2. Estimate the distance \(A P\) when \(P\) is instantaneously at rest for the second time and clearly state one assumption that you have made in making your estimate. \item The position vector \(\mathbf { x }\) metres at time \(t\) seconds of an object of mass 3 kg may be modelled by \end{enumerate} $$\mathbf { x } = 3 \sin t \mathbf { i } - 4 \cos 2 t \mathbf { j } + 5 \sin t \mathbf { k }$$
  3. Find an expression for the velocity vector \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) at time \(t\) seconds and determine the least value of \(t\) when the object is instantaneously at rest.
  4. Write down the momentum vector at time \(t\) seconds.
  5. Find, in vector form, an expression for the force acting on the object at time \(t\) seconds.
WJEC Further Unit 3 Specimen Q2
12 marks Standard +0.8
  1. Show that the speed of \(P\) when it first begins to move in a circle is \(\sqrt { 3 g }\).
  2. In the subsequent motion, when the string first makes an angle of \(45 ^ { \circ }\) with the downwards vertical,
    1. calculate the speed \(v\) of \(P\),
    2. determine the tension in the string. \item At time \(t = 0 \mathrm {~s}\), the position vector of an object \(A\) is \(\mathbf { i } \mathrm { m }\) and the position vector of another object \(B\) is \(3 \mathbf { i } \mathrm {~m}\). The constant velocity vector of \(A\) is \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } \mathrm {~ms} ^ { - 1 }\) and the constant velocity vector of \(B\) is \(\mathbf { i } + 3 \mathbf { j } - 5 \mathbf { k } \mathrm {~ms} ^ { - 1 }\). Determine the value of \(t\) when \(A\) and \(B\) are closest together and find the least distance between \(A\) and \(B\). \item Relative to a fixed origin \(O\), the position vector \(\mathbf { r } \mathrm { m }\) at time \(t \mathrm {~s}\) of a particle \(P\), of mass 0.4 kg , is given by \end{enumerate} $$\mathbf { r } = \mathrm { e } ^ { 2 t } \mathbf { i } + \sin ( 2 t ) \mathbf { j } + \cos ( 2 t ) \mathbf { k }$$
  3. Show that the velocity vector \(\mathbf { v }\) and the position vector \(\mathbf { r }\) are never perpendicular to each other.
  4. Given that the speed of \(P\) at time \(t\) is \(v _ { \mathrm { ms } } ^ { - 1 }\), show that $$v ^ { 2 } = 4 \mathrm { e } ^ { 4 t } + 4$$
  5. Find the kinetic energy of \(P\) at time \(t\).
  6. Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\).
  7. Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\).
OCR Further Pure Core 1 2023 June Q9
14 marks Challenging +1.8
9 In this question you must show detailed reasoning.}
  1. Use de Moivre's theorem to determine constants \(A\), \(B\) and \(C\) such that $$\sin ^ { 4 } \theta \equiv A \cos 4 \theta + B \cos 2 \theta + C .$$ The function f is defined by \(\mathrm { f } ( x ) = \sin \left( 4 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) - 8 \sin \left( 2 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) \right) + 12 \sin ^ { - 1 } \left( x ^ { \frac { 1 } { 5 } } \right) , \quad x \in \mathbb { R } , 0 \leqslant x < 1\).
  2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 5 \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\). \includegraphics[max width=\textwidth, alt={}, center]{478c66d2-16a0-41ef-9444-25cfcd47d11d-7_894_842_1000_260} The diagram shows the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 1 - x ^ { \frac { 2 } { 5 } } } }\) for \(0 \leqslant x < 1\) and the asymptote \(x = 1\). The region \(R\) is the unbounded region between the curve, the \(x\)-axis, the line \(x = 0\) and the line \(x = 1\). You are given that the area of \(R\) is finite.
  3. Determine the exact area of \(R\).
OCR Further Pure Core 1 2023 June Q6
4 marks Standard +0.8
6 In this question you must show detailed reasoning.} The power output, \(p\) watts, of a machine at time \(t\) hours after it is switched on can be modelled by the equation \(\mathrm { p } = 20 - 20 \tanh ( 1.44 \mathrm { t } )\) for \(t \geqslant 0\). Determine, according to the model, the mean power output of the machine over the first half hour after it is switched on. Give your answer correct to \(\mathbf { 2 }\) decimal places.
OCR D2 2007 January Q6
12 marks Moderate -0.5
6 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a maximin problem.
StageStateActionWorkingMaximin
\multirow{2}{*}{1}0044
1033
\multirow{6}{*}{2}00\(\min ( 6,4 ) = 4\)\multirow{2}{*}{}
1\(\min ( 2,3 ) = 2\)
\multirow{2}{*}{1}0\(\min ( 2,4 ) =\)\multirow{2}{*}{}
1\(\min ( 4,3 ) =\)
\multirow{2}{*}{2}0min(2,\multirow{2}{*}{}
1min(3,
\multirow{3}{*}{3}\multirow{3}{*}{0}0min(5,\multirow{3}{*}{}
1\(\min ( 5\),
2\(\min ( 2\),
  1. Complete the last two columns of the table in the insert.
  2. State the maximin value and write down the maximin route.
OCR MEI Further Pure Core AS 2024 June Q4
7 marks Standard +0.8
4 In this question you must show detailed reasoning. The roots of the cubic equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\) are \(\alpha , \beta\) and \(\gamma\).
  1. Find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { 2 } ( \alpha - 1 ) , \frac { 1 } { 2 } ( \beta - 1 )\) and \(\frac { 1 } { 2 } ( \gamma - 1 )\).
  2. Hence or otherwise solve the equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).