Questions — OCR MEI Further Pure Core AS (71 questions)

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OCR MEI Further Pure Core AS 2021 November Q3
3 Three planes have the following equations. $$\begin{aligned} 2 x - 3 y + z & = - 3
x - 4 y + 2 z & = 1
- 3 x - 2 y + 3 z & = 14 \end{aligned}$$
    1. Write the system of equations in matrix form.
    2. Hence find the point of intersection of the planes.
  2. In this question you must show detailed reasoning. Find the acute angle between the planes \(2 x - 3 y + z = - 3\) and \(x - 4 y + 2 z = 1\).
OCR MEI Further Pure Core AS 2021 November Q4
4 Anika thinks that, for two square matrices \(\mathbf { A }\) and \(\mathbf { B }\), the inverse of \(\mathbf { A B }\) is \(\mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }\). Her attempted proof of this is as follows. $$\begin{aligned} ( \mathbf { A B } ) \left( \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \right) & = \mathbf { A } \left( \mathbf { B A } ^ { - 1 } \right) \mathbf { B } ^ { - 1 }
& = \mathbf { A } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) \mathbf { B } ^ { - 1 }
& = \left( \mathbf { A } \mathbf { A } ^ { - 1 } \right) \left( \mathbf { B B } ^ { - 1 } \right)
& = \mathbf { I } \times \mathbf { I }
& = \mathbf { I }
\text { Hence } ( \mathbf { A B } ) ^ { - 1 } & = \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \end{aligned}$$
  1. Explain the error in Anika's working.
  2. State the correct inverse of the matrix \(\mathbf { A B }\) and amend Anika's working to prove this.
OCR MEI Further Pure Core AS 2021 November Q5
5 Prove by induction that \(\sum _ { r = 1 } ^ { n } r \times 2 ^ { r - 1 } = 1 + ( n - 1 ) 2 ^ { n }\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2021 November Q6
6 A transformation T of the plane has associated matrix \(\mathbf { M } = \left( \begin{array} { c c } 1 & \lambda + 1
\lambda - 1 & - 1 \end{array} \right)\), where \(\lambda\) is a non-zero
constant.
    1. Show that T reverses orientation.
    2. State, in terms of \(\lambda\), the area scale factor of T .
    1. Show that \(\mathbf { M } ^ { 2 } - \lambda ^ { 2 } \mathbf { I } = \mathbf { 0 }\).
    2. Hence specify the transformation equivalent to two applications of T .
  3. In the case where \(\lambda = 1 , \mathrm {~T}\) is equivalent to a transformation S followed by a reflection in the \(x\)-axis.
    1. Determine the matrix associated with S .
    2. Hence describe the transformation S .
OCR MEI Further Pure Core AS 2021 November Q7
7
    1. Find the modulus and argument of \(z _ { 1 }\), where \(z _ { 1 } = 1 + \mathrm { i }\).
    2. Given that \(\left| z _ { 2 } \right| = 2\) and \(\arg \left( z _ { 2 } \right) = \frac { 1 } { 6 } \pi\), express \(z _ { 2 }\) in a + bi form, where \(a\) and \(b\) are exact real numbers.
  2. Using these results, find the exact value of \(\sin \frac { 5 } { 12 } \pi\), giving the answer in the form \(\frac { \sqrt { m } + \sqrt { n } } { p }\), where \(m , n\) and \(p\) are integers.
OCR MEI Further Pure Core AS 2021 November Q9
9
  1. On a single Argand diagram, sketch the loci defined by
    • \(\arg ( z - 2 ) = \frac { 3 } { 4 } \pi\),
    • \(\quad | z | = | z + 2 - i |\).
    • In this question you must show detailed reasoning.
    The point of intersection of the two loci in part (a) represents the complex number \(w\). Find \(w\), giving your answer in exact form. \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS Specimen Q1
1 The complex number \(z _ { 1 }\) is \(1 + \mathrm { i }\) and the complex number \(z _ { 2 }\) has modulus 4 and argument \(\frac { \pi } { 3 }\).
  1. Express \(z _ { 2 }\) in the form \(a + b \mathrm { i }\), giving \(a\) and \(b\) in exact form.
  2. Express \(\frac { z _ { 2 } } { z _ { 1 } }\) in the form \(c + d i\), giving \(c\) and \(d\) in exact form.
  3. Describe fully the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2
    0 & 1 \end{array} \right)\).
  4. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2
    0 & 1 \end{array} \right)\). Explaining your reasoning, find the area of the image of the triangle following this transformation.
OCR MEI Further Pure Core AS Specimen Q3
3
  1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7728fdf9-2000-4265-a0cb-f34a6561c2ca-2_917_825_1334_699} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg ( z - 2 \mathrm { i } ) = \frac { \pi } { 4 }\).
OCR MEI Further Pure Core AS Specimen Q4
4
  1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$\begin{aligned} x - 2 y - z & = 6
    3 x + y + 5 z & = - 4
    - 4 x + 2 y - 3 z & = a \end{aligned}$$
  2. Determine whether the intersection of the three planes could be on the \(z\)-axis.
OCR MEI Further Pure Core AS Specimen Q5
5 The cubic equation \(x ^ { 3 } - 4 x ^ { 2 } + p x + q = 0\) has roots \(\alpha , \frac { 2 } { \alpha }\) and \(\alpha + \frac { 2 } { \alpha }\). Find
  • the values of the roots of the equation,
  • the value of \(p\).
OCR MEI Further Pure Core AS Specimen Q6
6
  1. Show that, when \(n = 5 , \sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = 330\).
  2. Find, in terms of \(n\), a fully factorised expression for \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 }\).
OCR MEI Further Pure Core AS Specimen Q7
7 The plane \(\Pi\) has equation \(3 x - 5 y + z = 9\).
  1. Show that \(\Pi\) contains
    • the point (4,1,2)
      and
    • the vector \(\left( \begin{array} { l } 1
      1
      2 \end{array} \right)\).
    • Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through ( \(4,1,2\) ).
OCR MEI Further Pure Core AS Specimen Q9
9 You are given that matrix \(\mathbf { M } = \left( \begin{array} { l l } - 3 & 8
- 2 & 5 \end{array} \right)\).
  1. Prove that, for all positive integers \(n , \mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 - 4 n & 8 n
    - 2 n & 1 + 4 n \end{array} \right)\).
  2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf { M }\). It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf { M } ^ { n }\), for any positive integer \(n\).
  3. Explain geometrically why this claim is true.
  4. Verify algebraically that this claim is true. \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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OCR MEI Further Pure Core AS 2023 June Q7
  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 Pure Core AS 2024 June Q4
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 \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).
OCR MEI Further Pure Core AS 2024 June Q9
9 In this question you must show detailed reasoning. Find a vector \(\mathbf { v }\) which has the following properties.
  • It is a unit vector.
  • It is parallel to the plane \(2 x + 2 y + z = 10\).
  • It makes an angle of \(45 ^ { \circ }\) with the normal to the plane \(\mathrm { x } + \mathrm { z } = 5\).
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OCR MEI Further Pure Core AS 2020 November Q1
1 In this question you must show detailed reasoning. Find \(\sum _ { r = 2 } ^ { 50 } \left( \frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \right)\), expressing the answer as an exact fraction.
OCR MEI Further Pure Core AS 2021 November Q8
8 In this question you must show detailed reasoning. The equation \(\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0\) has roots \(\alpha , \beta\) and \(\frac { \alpha } { \beta }\). Given that \(\alpha > 0\), find, in any order,
  • the roots of the equation,
  • the value of \(k\).
OCR MEI Further Pure Core AS Specimen Q8
8 In this question you must show detailed reasoning.
  1. Explain why all cubic equations with real coefficients have at least one real root.
  2. Points representing the three roots of the equation \(z ^ { 3 } + 9 z ^ { 2 } + 27 z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices.
OCR MEI Further Pure Core AS 2019 June Q8
8 In this question you must show detailed reasoning. You are given that i is a root of the equation \(z ^ { 4 } - 2 z ^ { 3 } + 3 z ^ { 2 } + a z + b = 0\), where \(a\) and \(b\) are real constants.
  1. Show that \(a = - 2\) and \(b = 2\).
  2. Find the other roots of this equation.
OCR MEI Further Pure Core AS 2022 June Q4
4 In this question you must show detailed reasoning. The equation \(z ^ { 3 } + 2 z ^ { 2 } + k z + 3 = 0\), where \(k\) is a constant, has roots \(\alpha , \frac { 1 } { \alpha }\) and \(\beta\).
Determine the roots in exact form.