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OCR Further Pure Core 2 2022 June Q6
10 marks Challenging +1.2
6 A particle, \(P\), positioned at the origin, \(O\), is projected with a certain velocity along the \(x\)-axis. \(P\) is then acted on by a single force which varies in such a way that \(P\) moves backwards and forwards along the \(x\)-axis. When the time after projection is \(t\) seconds, the displacement of \(P\) from the origin is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~ms} ^ { - 1 }\). The motion of \(P\) is modelled using the differential equation \(\ddot { x } + \omega ^ { 2 } x = 0\), where \(\omega\) rads \(^ { - 1 }\) is a positive constant.
  1. Write down the general solution of this differential equation. \(D\) is the point where \(x = d\) for some positive constant, \(d\). When \(P\) reaches \(D\) it comes to instantaneous rest.
  2. Using the answer to part (a), determine expressions, in terms of \(\omega\), \(d\) and \(t\) only, for the following quantities
    The quantity \(z\) is defined by \(z = \frac { 1 } { v }\).
  3. Using part (c), determine an expression for \(\mathrm { Z } _ { \mathrm { m } }\), the mean value of z with respect to the displacement, as \(P\) moves directly from \(O\) to \(D\). One measure of the validity of the model is consideration of the value of \(\mathrm { z } _ { \mathrm { m } }\). If \(\mathrm { z } _ { \mathrm { m } }\) exceeds 8 then the model is considered to be valid. The value of \(d\) is measured as 0.25 to 2 significant figures. The value of \(\omega\) is measured as \(0.75 \pm 0.02\).
  4. Determine what can be inferred about the validity of the model from the given information.
  5. Find, according to the model, the least possible value of the velocity with which \(P\) was initially projected. Give your answer to \(\mathbf { 2 }\) significant figures.
OCR Further Pure Core 2 2022 June Q7
13 marks Standard +0.8
7 You are given that \(a\) is a parameter which can take only real values.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c r } 2 & 4 & - 6 \\ - 3 & 10 - 4 a & 9 \\ 7 & 4 & 4 \end{array} \right)\).
  1. Find an expression for the determinant of \(\mathbf { A }\) in terms of \(a\). You are given the following system of equations in \(x , y\) and \(z\). $$\begin{array} { r r } 2 x + & 4 y - 6 z = \\ - 3 x + & ( 10 - 4 a ) y + 9 z = \\ 7 x + & 4 y + 4 z = \\ 7 x + & 11 \end{array}$$ The system can be written in the form \(\mathbf { A } \left( \begin{array} { c } \mathrm { x } \\ \mathrm { y } \\ \mathrm { z } \end{array} \right) = \left( \begin{array} { r } 6 \\ - 9 \\ 11 \end{array} \right)\).
    1. In the case where \(\mathbf { A }\) is not singular, solve the given system of equations by using \(\mathbf { A } ^ { - 1 }\).
    2. In the case where \(\mathbf { A }\) is singular describe the configuration of the planes whose equations are the three equations of the system. The transformation represented by \(\mathbf { A }\) is denoted by T .
      A 3-D object of volume \(| 5 a - 20 |\) is transformed by T to a 3-D image.
    1. Determine the range of values of \(a\) for which the orientation of the image is the reverse of the orientation of the object.
    2. Determine the range of values of \(a\) for which the volume of the image is less than the volume of the object.
OCR Further Pure Core 2 2022 June Q10
8 marks Challenging +1.8
10 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2 , - 1\) ) and ( \(13,10,9\) ) respectively.
  • The plane \(\Pi _ { A }\) contains \(A\) and the plane \(\Pi _ { B }\) contains \(B\).
  • The planes \(\Pi _ { A }\) and \(\Pi _ { B }\) are parallel.
  • The \(x\) and \(y\) components of any normal to plane \(\Pi _ { A }\) are equal.
  • The shortest distance between \(\Pi _ { A }\) and \(\Pi _ { B }\) is 2 .
There are two possible solution planes for \(\Pi _ { A }\) which satisfy the above conditions.
Determine the acute angle between these two possible solution planes.
OCR Further Pure Core 2 2023 June Q1
8 marks Easy -1.3
1
  1. The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l l l } 1 & 0 & - 2 & 2 \\ 4 & 2 & - 2 & 3 \end{array} \right)\).
    1. Write down the dimensions of \(\mathbf { P }\).
    2. Write down the transpose of \(\mathbf { P }\).
  2. The matrices \(\mathbf { Q } , \mathbf { R }\) and \(\mathbf { S }\) are given by \(\mathbf { Q } = \left( \begin{array} { l l } 1 & 2 \end{array} \right) , \mathbf { R } = \left( \begin{array} { r r } 3 & - 4 \\ 2 & 3 \end{array} \right)\) and \(\mathbf { S } = \left( \begin{array} { l l } 3 & - 2 \end{array} \right)\). Write down the sum of the two of these matrices which are conformable for addition.
  3. The dimensions of matrix \(\mathbf { A }\) are 4 by 5. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are conformable for multiplication so that the matrix \(\mathbf { C } = \mathbf { B A }\) can be formed. The matrix \(\mathbf { C }\) has 6 rows.
    1. Write down the number of columns that \(\mathbf { C }\) has.
    2. Write down the dimensions of \(\mathbf { B }\).
    3. Explain whether the matrix \(\mathbf { A B }\) can be formed.
  4. Find the value of \(c\) for which \(\left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5 \\ 10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3 \\ 6 & 10 \end{array} \right)\).
OCR Further Pure Core 2 2023 June Q2
7 marks Standard +0.3
2 In this question you must show detailed reasoning.
  1. Write the complex number \(- 24 + 7 \mathrm { i }\) in modulus-argument form.
  2. Solve the simultaneous equations given below, giving your answers in cartesian form. $$\begin{aligned} i z + 3 w & = - 7 i \\ - 6 z + 5 i w & = 3 + 13 i \end{aligned}$$
OCR Further Pure Core 2 2023 June Q3
6 marks Standard +0.3
3
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } u } \left( \sinh ^ { - 1 } u \right) = \frac { 1 } { \sqrt { u ^ { 2 } + 1 } }\).
  2. Find the equation of the normal to the graph of \(\mathrm { y } = \sinh ^ { - 1 } 2 \mathrm { x }\) at the point where \(x = \sqrt { 6 }\). Give your answer in the form \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) where \(m\) and \(c\) are given in exact, non-hyperbolic form.
OCR Further Pure Core 2 2023 June Q6
8 marks Standard +0.8
6 The equation of the plane \(\Pi\) is \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right)\).
  1. Find the acute angle between \(\Pi\) and the plane with equation \(\mathbf { r } . \left( \begin{array} { l } 2 \\ 0 \\ 3 \end{array} \right) = 4\). The point \(A\) has coordinates ( \(9 , - 7,20\) ).
    The point \(F\) is the point of intersection between \(\Pi\) and the perpendicular from \(A\) to \(\Pi\).
  2. Determine the coordinates of \(F\).
OCR Further Pure Core 2 2023 June Q7
8 marks Challenging +1.8
7 In this question you must show detailed reasoning.
  1. Show that $$\sum _ { r = 1 } ^ { n } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } } = \frac { a } { n + 1 } + b + c \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers whose values are to be determined. You are given that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\) exists and is equal to \(\frac { 1 } { 6 } \pi ^ { 2 }\).
  2. Show that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } }\) exists and is equal to \(( \pi - 1 ) ( \pi + 1 )\).
OCR Further Pure Core 2 2023 June Q8
11 marks Challenging +1.8
8 A surge in the current, \(I\) units, through an electrical component at a time, \(t\) seconds, is to be modelled. The surge starts when \(t = 0\) and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, \(I\) is modelled by the following differential equation. \(\left( 2 t - t ^ { 2 } \right) \frac { d l } { d t } = \left( 2 t - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - 2 ( t - 1 ) l\)
  1. By using an integrating factor show that, according to the model, while the surge is occurring, \(I\) is given by \(\mathrm { I } = \left( 2 \mathrm { t } - \mathrm { t } ^ { 2 } \right) \left( \sin ^ { - 1 } ( \mathrm { t } - 1 ) + 5 \right)\). The surge lasts until there is again no current through the component.
  2. Determine the length of time that the surge lasts according to the model.
  3. Determine, according to the model, the rate of increase of the current at the start of the surge. Give your answer in an exact form.
OCR Further Pure Core 2 2023 June Q9
9 marks Challenging +1.2
9 A function is defined by \(y = f ( t )\) where \(f ( t ) = \ln ( 1 + a t )\) and \(a\) is a constant.
  1. By considering \(\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }\) and \(\frac { d ^ { 4 } y } { d t ^ { 4 } }\), make a conjecture for a general formula for \(\frac { d ^ { n } y } { d t ^ { n } }\) in terms of \(n\) and \(a\) for any integer \(n \geqslant 1\).
  2. Use induction to prove the formula conjectured in part (a).
  3. In the case where \(\mathrm { f } ( t ) = \ln ( 1 + 2 t )\), find the rate at which the \(6 ^ { \text {th } }\) derivative of \(\mathrm { f } ( t )\) is varying when \(t = \frac { 3 } { 2 }\).
OCR Further Pure Core 2 2020 November Q1
5 marks Moderate -0.3
1 In this question you must show detailed reasoning.
Solve the equation \(4 z ^ { 2 } - 20 z + 169 = 0\). Give your answers in modulus-argument form.
OCR Further Pure Core 2 2020 November Q2
6 marks Challenging +1.3
2 In this question you must show detailed reasoning.
The roots of the equation \(3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).
  1. Find a cubic equation with integer coefficients whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
  2. Find the exact value of \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).
OCR Further Pure Core 2 2020 November Q3
6 marks Standard +0.8
3 In this question you must show detailed reasoning.
  1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
  2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).
OCR Further Pure Core 2 2020 November Q4
9 marks Standard +0.3
4 The equations of two intersecting lines \(l _ { 1 }\) and \(l _ { 2 }\) are \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ a \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 3 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 7 \\ 9 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right)\) where \(a\) is a constant.
The equation of the plane \(\Pi\) is
r. \(\left( \begin{array} { l } 1 \\ 5 \\ 3 \end{array} \right) = - 14\). \(l _ { 1 }\) and \(\Pi\) intersect at \(Q\). \(l _ { 2 }\) and \(\Pi\) intersect at \(R\).
  1. Verify that the coordinates of \(R\) are (13, 3, -14).
  2. Determine the exact value of the length of \(Q R\).
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.