Questions — OCR Further Pure Core 2 (129 questions)

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OCR Further Pure Core 2 2020 November Q5
7 marks Standard +0.3
5 A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geqslant 0 , Q\) can be modelled by the differential equation \(\frac { d ^ { 2 } Q } { d t ^ { 2 } } - 2 \frac { d Q } { d t } - 15 Q = 0\). Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.
  1. Determine the charge on the capacitor when \(t = 0.5\).
  2. Determine the finite limit of \(Q\) as \(t\) tends to infinity.
OCR Further Pure Core 2 2020 November Q6
6 marks Challenging +1.8
6 The equation of a curve in polar coordinates is \(r = \ln ( 1 + \sin \theta )\) for \(\alpha \leqslant \theta \leqslant \beta\) where \(\alpha\) and \(\beta\) are non-negative angles. The curve consists of a single closed loop through the pole.
  1. By solving the equation \(r = 0\), determine the smallest possible values of \(\alpha\) and \(\beta\).
  2. Find the area enclosed by the curve, giving your answer to 4 significant figures.
  3. Hence, by considering the value of \(r\) at \(\theta = \frac { \alpha + \beta } { 2 }\), show that the loop is not circular.
OCR Further Pure Core 2 2020 November Q7
6 marks Challenging +1.2
7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4 \\ - 0.8 & 1.8 \end{array} \right)\).
  1. Find \(\operatorname { det } \mathbf { A }\). The matrix A represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
  2. By considering the determinants of these transformations, determine the scale factor of the stretch.
  3. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
  4. Find the angle of rotation.
OCR Further Pure Core 2 2020 November Q9
11 marks Challenging +1.2
9 Two thin poles, \(O A\) and \(B C\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \(( 0,3 ) , ( 5,0 )\) and \(( 2,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{c07ba83a-75fa-42dc-9bfd-6fc2f9226a23-5_805_1554_452_258} It is required to find the height of pole \(B C\) by modelling the shape of the curve that the chain forms.
Jofra models the curve using the equation \(\mathrm { y } = \mathrm { k } \cosh ( \mathrm { ax } - \mathrm { b } ) - 1\) where \(k , a\) and \(b\) are positive constants.
  1. Determine the value of \(k\).
  2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. Holly models the curve using the equation \(y = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 3\).
  3. Write down the coordinates of the point, \(( u , v )\) where \(u\) and \(v\) are both non-zero, at which the two models will agree.
  4. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(B C\) by 3.32 m to 3 significant figures.
OCR Further Pure Core 2 2020 November Q10
10 marks Standard +0.8
10 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).
    1. Determine \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
    3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
  2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.
OCR Further Pure Core 2 2021 November Q1
3 marks Standard +0.8
1 Two matrices, \(\mathbf { A }\) and \(\mathbf { B }\), are given by \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & - 1 \\ 2 & - 3 & 1 \\ a & 1 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 6 & 3 & - 4 \\ - 1 & 6 & - 4 \\ 8 & - 8 & - 1 \end{array} \right)\) where \(a\) is a constant. Find the value of \(a\) for which \(\mathbf { A B } = \mathbf { B A }\).
OCR Further Pure Core 2 2021 November Q3
9 marks Standard +0.3
3 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 2 \end{array} \right)\).
The plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 5 \\ - 3 \end{array} \right) = 4\).
  1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(\Pi\).
  2. Find the acute angle between \(l _ { 1 }\) and \(\Pi\). \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 1\). \(l _ { 2 }\) is the line with the following properties.
OCR Further Pure Core 2 2021 November Q4
3 marks Moderate -0.8
4 In this question you must show detailed reasoning.
Determine the value of \(\sum _ { r = 1 } ^ { 100 } ( 2 r + 3 ) ^ { 2 }\).
OCR Further Pure Core 2 2021 November Q5
8 marks Standard +0.8
5 In this question you must show detailed reasoning.
  1. Using the definition of \(\cosh x\) in terms of exponentials, show that \(\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1\).
  2. Solve the equation \(\cosh 2 x = 3 \cosh x + 1\), giving all your answers in exact logarithmic form.
OCR Further Pure Core 2 2021 November Q6
6 marks Standard +0.8
6 In this question you must show detailed reasoning.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).
  1. Define the transformation represented by \(\mathbf { A }\).
  2. Show that the area of any object shape is invariant under the transformation represented by \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r l } 7 & 2 \\ 21 & 7 \end{array} \right)\). You are given that \(\mathbf { B }\) represents the transformation which is the result of applying the following three transformations in the given order.
OCR Further Pure Core 2 2021 November Q7
10 marks Challenging +1.3
7 In this question you must show detailed reasoning.
  1. Find the values of \(A , B\) and \(C\) for which \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \equiv A + \frac { B x + C } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\).
  2. Hence express \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\) using partial fractions.
  3. Using your answer to part (b), determine \(\int _ { 0 } ^ { 2 } \frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \mathrm {~d} x\) expressing your answer in the form \(a + \ln b + c \pi\) where \(a\) is an integer, and \(b\) and \(c\) are both rational.
OCR Further Pure Core 2 2021 November Q8
16 marks Challenging +1.2
8 A particle \(P\) of mass 2 kg can only move along the straight line segment \(O A\), where \(O A\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(O A\) is 0.9 m . When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). \(P\) is subject to a force of magnitude \(4 \mathrm { e } ^ { - 2 t } \mathrm {~N}\) in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2 , v = 0.5\) and the resultant force on \(P\) is 0 N .
  1. Show that, according to the model, \(\frac { d v } { d t } + v = 2 e ^ { - 2 t }\).
  2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\).
  3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\) 's speed must reach a maximum value for some \(t > 0\).
  4. Determine the maximum speed considered in part (c).
  5. Determine the greatest value of \(t\) for which the model is valid.
OCR Further Pure Core 2 2021 November Q9
6 marks Challenging +1.2
9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 2 \end{array} \right)\).
  1. By considering \(\mathbf { A } , \mathbf { A } ^ { 2 } , \mathbf { A } ^ { 3 }\) and \(\mathbf { A } ^ { 4 }\) make a conjecture about the form of the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\) for \(n \geqslant 1\).
  2. Use induction to prove the conjecture made in part (a).
OCR Further Pure Core 2 2021 November Q10
6 marks Challenging +1.2
10 In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
  2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).
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 $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ 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 \(x \mathrm {~m}\).
    1. On the axes provided in the Printed Answer Booklet, sketch a graph of \(x\) against \(t\) for \(0 \leqslant t \leqslant 0.4\).
    2. 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 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 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 Further Pure Core 2 2023 June Q4
4 marks Challenging +1.2
4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm . \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.
OCR Further Pure Core 2 2023 June Q5
7 marks Challenging +1.2
5 In this question you must show detailed reasoning.
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that \(\sinh 2 x \equiv 2 \sinh x \cosh x\).
  2. Solve the equation \(15 \sinh x + 16 \cosh x - 6 \sinh 2 x = 20\), giving all your answers in logarithmic form.
OCR Further Pure Core 2 2023 June Q10
7 marks Challenging +1.2
10 In this question you must show detailed reasoning. A region, \(R\), of the floor of an art gallery is to be painted for the purposes of an art installation. A suitable polar coordinate system is set up on the floor of the gallery with units in metres and radians. \(R\) is modelled as being the region enclosed by two curves, \(C _ { 1 }\) and \(C _ { 2 }\). The polar equations of \(C _ { 1 }\) and \(C _ { 2 }\) are $$\begin{array} { l l } C _ { 1 } : r = 5 , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \\ C _ { 2 } : r = 3 \cosh \theta , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \end{array}$$ Both curves are shown in the diagram, with \(R\) indicated. \includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-6_1481_821_836_251} The gallery must buy tins of paint to paint \(R\). Each tin of paint can cover an area of \(0.5 \mathrm {~m} ^ { 2 }\).
Determine the smallest number of tins of paint that the gallery must buy in order to be able to paint \(R\) completely.
OCR Further Pure Core 2 2020 November Q8
9 marks Standard +0.8
8 In this question you must show detailed reasoning. The complex number \(- 4 + i \sqrt { 48 }\) is denoted by \(z\).
  1. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
  2. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.