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OCR Further Pure Core 1 2022 June Q9
5 marks Challenging +1.8
9 The cube roots of unity are represented on the Argand diagram below by the points \(A , B\) and \(C\). \includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-8_760_800_303_244} The points \(L , M\) and \(N\) are the midpoints of the line segments \(A B , B C\) and \(C A\) respectively. Determine a degree 6 polynomial equation with integer coefficients whose roots are the complex numbers represented by the points \(A , B , C , L , M\) and \(N\). \section*{END OF QUESTION PAPER}
OCR Further Pure Core 1 2023 June Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
Determine the value of \(\sum _ { r = 1 } ^ { 50 } r ^ { 2 } ( 16 - r )\).
OCR Further Pure Core 1 2023 June Q2
6 marks Standard +0.8
2 In this question you must show detailed reasoning.
The equation \(z ^ { 4 } + 4 z ^ { 3 } + 9 z ^ { 2 } + 10 z + 6 = 0\) has roots \(\alpha , \beta , \gamma\) and \(\delta\).
  1. Show that a quartic equation whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\) and \(\delta + 1\) is \(w ^ { 4 } + 3 w ^ { 2 } + 2 = 0\).
  2. Hence determine the exact roots of the equation \(z ^ { 4 } + 4 z ^ { 3 } + 9 z ^ { 2 } + 10 z + 6 = 0\).
OCR Further Pure Core 1 2023 June Q3
5 marks Standard +0.3
3
  1. Show that \(\frac { - 3 + \sqrt { 3 } \mathrm { i } } { 2 } = \sqrt { 3 } \mathrm { e } ^ { \frac { 5 } { 6 } \pi \mathrm { i } }\).
  2. Hence determine the exact roots of the equation \(z ^ { 5 } = \frac { 9 ( - 3 + \sqrt { 3 } \mathrm { i } ) } { 2 }\), giving the roots in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
OCR Further Pure Core 1 2023 June Q4
11 marks Standard +0.3
4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\)
  1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
  2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0 \\ 0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
    1. Find the exact area of the region \(R ^ { \prime }\).
    2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.
OCR Further Pure Core 1 2023 June Q5
6 marks Standard +0.8
5
  1. Find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + 5 y = 0\).
  2. Hence find the general solution of the differential equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + 5 y = x ( 4 - 5 x )\).
OCR Further Pure Core 1 2023 June Q7
11 marks Challenging +1.2
7 An engineer is modelling the motion of a particle \(P\) of mass 0.5 kg in a wind tunnel. \(P\) is modelled as travelling in a straight line. The point \(O\) is a fixed point within the wind tunnel. The displacement of \(P\) from \(O\) at time \(t\) seconds is \(x\) metres, for \(t \geqslant 0\). You are given that \(x \geqslant 0\) for all \(t \geqslant 0\) and that \(P\) does not reach the end of the wind tunnel.
If \(t \geqslant 0\), then \(P\) is subject to three forces which are modelled in the following way.
  • The first force has a magnitude of \(5 ( t + 1 ) \cosh t \mathrm {~N}\) and acts in the positive \(x\)-direction.
  • The second force has a magnitude of \(0.5 x \mathrm {~N}\) and acts towards \(O\).
  • The third force has a magnitude of \(\left| \frac { d x } { d t } \right| \mathrm { N }\) and acts in the direction of motion of the particle.
    1. The engineer applies the equation " \(F = m a\) " to the model of the motion of \(P\) and derives the following differential equation. \(5 ( t + 1 ) \operatorname { cosht } - 0.5 x + \frac { d x } { d t } = 0.5 \frac { d ^ { 2 } x } { d t ^ { 2 } }\)
      1. Explain the sign of the \(\frac { \mathrm { dx } } { \mathrm { dt } }\) term in the engineer's differential equation.
When \(t = 0\) the displacement of \(P\) is 6 m , and it is travelling towards \(O\) with a speed of \(5 \mathrm {~ms} ^ { - 1 }\).
(ii) Without attempting to solve the differential equation, find the acceleration of \(P\) when \(t = 0\). Let the particular solution to the differential equation in part (a) be a function f such that \(\mathrm { x } = \mathrm { f } ( \mathrm { t } )\) for \(t \geqslant 0\). The particular solution to the differential equation can be expressed as a Maclaurin series.
    1. Show that the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t\) is \(6 - 5 t\).
    2. Use your answer to part (a)(ii) to show that the term in \(t ^ { 2 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(- 3 t ^ { 2 }\).
    3. By differentiating the differential equation in part (a) with respect to \(t\), show that the term in \(t ^ { 3 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(0.5 t ^ { 3 }\). You are given that the complete Maclaurin series for the function f is valid for all values of \(t \geqslant 0\).
      After 0.25 seconds \(P\) has travelled 1.43 m towards the origin.
    1. By using the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\), evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 0.25\).
    2. Explain why it might not be sensible to use the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\) to evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 10\).
    •
    OCR Further Pure Core 1 2023 June Q8
    15 marks Challenging +1.2
    8 The points \(P , Q\) and \(R\) have coordinates \(( 0,2,3 ) , ( 2,0,1 )\) and \(( 1,3,0 )\) respectively.
    The acute angle between the line segments \(P Q\) and \(P R\) is \(\theta\).
    1. Show that \(\sin \theta = \frac { 2 } { 11 } \sqrt { 22 }\). The triangle \(P Q R\) lies in the plane \(\Pi\).
    2. Determine an equation for \(\Pi\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\), where \(a , b , c\) and \(d\) are integers. The point \(S\) has coordinates \(( 5,3 , - 1 )\).
    3. By finding the shortest distance between \(S\) and the plane \(\Pi\), show that the volume of the tetrahedron \(P Q R S\) is \(\frac { 14 } { 3 }\).
      [0pt] [The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height] The tetrahedron \(P Q R S\) is transformed to the tetrahedron \(\mathrm { P } ^ { \prime } \mathrm { Q } ^ { \prime } \mathrm { R } ^ { \prime } \mathrm { S } ^ { \prime }\) by a rotation about the \(y\)-axis.
      The \(x\)-coordinate of \(S ^ { \prime }\) is \(2 \sqrt { 2 }\).
    4. By using the matrix for a rotation by angle \(\theta\) about the \(y\)-axis, as given in the Formulae Booklet, determine in exact form the possible coordinates of \(R ^ { \prime }\).
    OCR Further Pure Core 1 2024 June Q1
    3 marks Moderate -0.5
    1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).
    OCR Further Pure Core 1 2024 June Q2
    10 marks Standard +0.3
    2 The locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \left\{ z : 0 \leqslant \arg ( z + i ) \leqslant \frac { 1 } { 4 } \pi \right\}\).
    1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
    2. Determine whether the complex number \(1.2 + 0.8\) is is \(C _ { 1 }\). The locus \(C _ { 2 }\) is the set of complex numbers represented by the interior of the circle with radius 2 and centre 3 . The locus \(C _ { 2 }\) is illustrated on the Argand diagram below. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-2_698_920_1009_239}
    3. Use set notation to define \(C _ { 2 }\).
    4. Determine whether the complex number \(1.2 + 0.8\) is in \(C _ { 2 }\).
    OCR Further Pure Core 1 2024 June Q3
    8 marks Standard +0.8
    3 A transformation T is represented by the matrix \(\mathbf { N } = \left( \begin{array} { l l l } a & 4 & 2 \\ 5 & 1 & 0 \\ 3 & 6 & 3 \end{array} \right)\), where \(a\) is a constant.
    1. Find \(\mathbf { N } ^ { 2 }\) in terms of \(a\).
    2. Find det \(\mathbf { N }\) in terms of \(a\). The value of \(a\) is 13 to the nearest integer.
      A shape \(S _ { 1 }\) has volume 11.6 to 1 decimal place. Shape \(S _ { 1 }\) is mapped to shape \(S _ { 2 }\) by the transformation T . A student claims that the volume of \(S _ { 2 }\) is less than 400 .
    3. Comment on the student's claim.
    OCR Further Pure Core 1 2024 June Q4
    3 marks Challenging +1.2
    4 In this question you must show detailed reasoning.
    The equation \(2 x ^ { 3 } + 3 x ^ { 2 } + 6 x - 3 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
    Determine a cubic equation with integer coefficients that has roots \(\alpha ^ { 2 } \beta \gamma , \alpha \beta ^ { 2 } \gamma\) and \(\alpha \beta \gamma ^ { 2 }\).
    OCR Further Pure Core 1 2024 June Q5
    5 marks Standard +0.8
    5 Express \(\frac { 12 x ^ { 3 } } { ( 2 x + 1 ) \left( 2 x ^ { 2 } + 1 \right) }\) using partial fractions.
    OCR Further Pure Core 1 2024 June Q6
    4 marks Challenging +1.2
    6 In this question you must show detailed reasoning.
    Determine the exact value of \(\int _ { 9 } ^ { \infty } \frac { 18 } { x ^ { 2 } \sqrt { x } } \mathrm {~d} x\).
    OCR Further Pure Core 1 2024 June Q7
    9 marks Standard +0.8
    7
    1. By using the definitions of \(\cosh u\) and \(\sinh u\) in terms of \(\mathrm { e } ^ { u }\) and \(\mathrm { e } ^ { - u }\), show that \(\sinh 2 u \equiv 2 \sinh u \cosh u\). The equation of a curve, \(C\), is \(\mathrm { y } = 16 \cosh \mathrm { x } - \sinh 2 \mathrm { x }\).
    2. Show that there is only one solution to the equation \(\frac { d ^ { 2 } y } { d x ^ { 2 } } = 0\) You are now given that \(C\) has exactly one point of inflection.
    3. Use your answer to part (b) to determine the exact coordinates of this point of inflection. Give your answer in a logarithmic form where appropriate.
    OCR Further Pure Core 1 2024 June Q8
    5 marks Standard +0.3
    8 Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geqslant 0\).
    OCR Further Pure Core 1 2024 June Q9
    6 marks Standard +0.8
    9
    1. Find the Maclaurin series of \(( \ln ( 1 + x ) ) ^ { 2 }\) up to and including the term in \(x ^ { 4 }\). The diagram below shows parts of the graphs of the curves with equations \(y = ( \ln ( 1 + x ) ) ^ { 2 }\) and \(y = 2 x ^ { 3 }\). The curves intersect at the origin, \(O\), and at the point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-4_663_906_831_248} \section*{(b) In this question you must show detailed reasoning.} Use your answer to part (a) to determine an approximation for the value of the \(x\)-coordinate of \(A\). Give your answer to \(\mathbf { 2 }\) decimal places.
    OCR Further Pure Core 1 2024 June Q10
    10 marks Challenging +1.2
    10 A particle \(B\), of mass 3 kg , moves in a straight line and has velocity \(v \mathrm {~ms} ^ { - 1 }\).
    At time \(t\) seconds, where \(0 \leqslant t < \frac { 1 } { 4 } \pi\), a variable force of \(- ( 15 \sin 4 \mathrm { t } + 6 \mathrm { v } \tan 2 \mathrm { t } )\) Newtons is applied to \(B\). There are no other forces acting on \(B\). Initially, when \(t = 0 , B\) has velocity \(4.5 \mathrm {~ms} ^ { - 1 }\). The motion of \(B\) can be modelled by the differential equation \(\frac { d v } { d t } + P ( t ) v = Q ( t )\) where \(P ( t )\) and \(\mathrm { Q } ( \mathrm { t } )\) are functions of \(t\).
    1. Find the functions \(\mathrm { P } ( \mathrm { t } )\) and \(\mathrm { Q } ( \mathrm { t } )\).
    2. Using an integrating factor, determine the first time at which \(B\) is stationary according to the model.
    OCR Further Pure Core 1 2024 June Q11
    7 marks Standard +0.8
    11 A 3-D coordinate system, whose units are metres, is set up to model a construction site. The construction site contains four vertical poles \(P _ { 1 } , P _ { 2 } , P _ { 3 }\) and \(P _ { 4 }\). The floor of the construction site is modelled as lying in the \(x - y\) plane and the poles are modelled as vertical line segments. One end of each pole lies on the floor of the construction site, and the other end of each pole is modelled by the points \(( 0,0,18 ) , ( 12,14,20 ) , ( 0,11,7 )\) and \(( 18,2,16 )\) respectively. A wire, \(S\), runs from the top of \(P _ { 1 }\) to the top of \(P _ { 2 }\). A second wire, \(T\), runs from the top of \(P _ { 3 }\) to the top of \(P _ { 4 }\). The wires are modelled by straight lines segments. The layout of the construction site is illustrated on the diagram below which is not drawn to scale. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-5_707_871_696_242} A vector equation of the line segment that represents the wire \(S\) is given by \(\mathbf { r } = \left( \begin{array} { c } 0 \\ 0 \\ 18 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 7 \\ 1 \end{array} \right) , 0 \leqslant \lambda \leqslant 2\).
    1. Find, in the same form, a vector equation of the line segment that represents the wire \(T\). The components of the direction vector should be integers whose only positive common factor is 1 . For the construction site to be considered safe, it must pass two tests.
      Test 1: The wires \(S\) and \(T\) need to be at least 5 metres apart at all positions on \(S\) and \(T\).
    2. By using an appropriate formula, determine whether the construction site passes Test 1. A security camera is placed at a point \(Q\) on wire \(S\). Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of \(P _ { 3 }\) must be at least 19 m .
    3. Determine whether it is possible to find point \(Q\) on \(S\) such that the construction site passes Test 2.
    OCR Further Pure Core 1 2024 June Q12
    7 marks Challenging +1.8
    12 For any positive parameter \(k\), the curve \(C _ { k }\) is defined by the polar equation \(\mathrm { r } = \mathrm { k } ( \cos \theta + 1 ) + \frac { 10 } { \mathrm { k } } , 0 \leqslant \theta \leqslant 2 \pi\).
    For each value of \(k\) the curve is a single, closed loop with no self-intersections. The diagram shows \(C _ { 10.5 }\) for the purpose of illustration. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-6_558_723_550_242} Each curve, \(C _ { k }\), encloses a certain area, \(A _ { k }\).
    You are given that there is a single minimum value of \(A _ { k }\).
    Determine, in an exact form, the value of \(k\) for which \(C _ { k }\) encloses this minimum area.
    OCR Further Pure Core 1 2020 November Q1
    2 marks Easy -1.2
    1 Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).
    OCR Further Pure Core 1 2020 November Q2
    3 marks Standard +0.8
    2 Find an expression for \(1 \times 2 ^ { 2 } + 2 \times 3 ^ { 2 } + 3 \times 4 ^ { 2 } + \ldots + n ( n + 1 ) ^ { 2 }\) in terms of \(n\). Give your answer in fully factorised form.
    OCR Further Pure Core 1 2020 November Q3
    5 marks Moderate -0.3
    3 You are given the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 0 \end{array} \right)\).
    1. Find \(\mathbf { A } ^ { 4 }\).
    2. Describe the transformation that \(\mathbf { A }\) represents. The matrix \(\mathbf { B }\) represents a reflection in the plane \(x = 0\).
    3. Write down the matrix \(\mathbf { B }\). The point \(P\) has coordinates (2, 3, 4). The point \(P ^ { \prime }\) is the image of \(P\) under the transformation represented by \(\mathbf { B }\).
    4. Find the coordinates of \(P ^ { \prime }\).
    OCR Further Pure Core 1 2020 November Q4
    4 marks Moderate -0.3
    4 In this question you must show detailed reasoning.
    1. Determine the square roots of 25 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(0 \leqslant \theta < 2 \pi\).
    2. Illustrate the number 25i and its square roots on an Argand diagram.
    OCR Further Pure Core 1 2020 November Q5
    5 marks Standard +0.8
    5 By expanding \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 3 }\), where \(z = e ^ { \mathrm { i } \theta }\), show that \(4 \cos ^ { 3 } 2 \theta = \cos 6 \theta + 3 \cos 2 \theta\).