Questions — OCR MEI Further Pure Core AS (73 questions)

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OCR MEI Further Pure Core AS 2020 November Q1
3 marks Moderate -0.3
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
7 marks Challenging +1.2
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 2019 June Q8
11 marks Standard +0.3
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
6 marks Standard +0.8
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.
OCR MEI Further Pure Core AS 2018 June Q1
4 marks Moderate -0.8
The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 1 & 3 \end{pmatrix}.$$ Calculate all possible products formed from two of these three matrices. [4]
OCR MEI Further Pure Core AS 2018 June Q2
3 marks Moderate -0.8
Find, to the nearest degree, the angle between the vectors \(\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ -3 \end{pmatrix}\). [3]
OCR MEI Further Pure Core AS 2018 June Q3
5 marks Moderate -0.8
Find real numbers \(a\) and \(b\) such that \((a - 3i)(5 - i) = b - 17i\). [5]
OCR MEI Further Pure Core AS 2018 June Q4
5 marks Moderate -0.3
Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]
OCR MEI Further Pure Core AS 2018 June Q5
7 marks Standard +0.3
A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \((2, 3)\) under this transformation. [2]
  2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]
OCR MEI Further Pure Core AS 2018 June Q6
4 marks Standard +0.3
Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]
OCR MEI Further Pure Core AS 2018 June Q7
9 marks Standard +0.8
  1. Express \(\frac{1}{2r-1} - \frac{1}{2r+1}\) as a single fraction. [2]
  2. Find how many terms of the series $$\frac{2}{1 \times 3} + \frac{2}{3 \times 5} + \frac{2}{5 \times 7} + \ldots + \frac{2}{(2r-1)(2r+1)} + \ldots$$ are needed for the sum to exceed \(0.999999\). [7]
OCR MEI Further Pure Core AS 2018 June Q8
6 marks Standard +0.3
Prove by induction that \(\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}\) for all positive integers \(n\). [6]
OCR MEI Further Pure Core AS 2018 June Q9
9 marks Standard +0.3
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}
  1. Find, in modulus-argument form, the complex number represented by the point P. [2]
  2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    lie within this region. [4]
OCR MEI Further Pure Core AS 2018 June Q10
8 marks Standard +0.3
Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.
  1. Investigate the arrangement of the planes:
    [6]
  2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]
OCR MEI Further Pure Core AS Specimen Q1
4 marks Moderate -0.8
The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).
  1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
  2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]
OCR MEI Further Pure Core AS Specimen Q2
4 marks Moderate -0.8
  1. Describe fully the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). [2]
  2. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). Explaining your reasoning, find the area of the image of the triangle following this transformation. [2]
OCR MEI Further Pure Core AS Specimen Q3
4 marks Moderate -0.3
  1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. [2] \includegraphics{figure_3}
  2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg(z - 2i) = \frac{\pi}{4}\). [2]
OCR MEI Further Pure Core AS Specimen Q4
6 marks Standard +0.3
  1. Find the coordinates of the point where the following three planes intersect. Give your answers in terms of \(a\). $$x - 2y - z = 6$$ $$3x + y + 5z = -4$$ $$-4x + 2y - 3z = a$$ [4]
  2. Determine whether the intersection of the three planes could be on the \(z\)-axis. [2]
OCR MEI Further Pure Core AS Specimen Q5
7 marks Challenging +1.2
The cubic equation \(x^3 - 4x^2 + px + q = 0\) has roots \(\alpha\), \(\frac{2}{\alpha}\) and \(\alpha + \frac{2}{\alpha}\). Find
  • the values of the roots of the equation,
  • the value of \(p\).
[7]
OCR MEI Further Pure Core AS Specimen Q6
5 marks Standard +0.8
  1. Show that, when \(n = 5\), \(\sum_{r=n+1}^{2n} r^2 = 330\). [1]
  2. Find, in terms of \(n\), a fully factorised expression for \(\sum_{r=n+1}^{2n} r^2\). [4]
OCR MEI Further Pure Core AS Specimen Q7
7 marks Moderate -0.3
The plane \(\Pi\) has equation \(3x - 5y + z = 9\).
  1. Show that \(\Pi\) contains
    and
    [4]
  2. Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through \((4,1,2)\). [3]
OCR MEI Further Pure Core AS Specimen Q8
9 marks Challenging +1.8
In this question you must show detailed reasoning.
  1. Explain why all cubic equations with real coefficients have at least one real root. [2]
  2. Points representing the three roots of the equation \(z^3 + 9z^2 + 27z + 35 = 0\) are plotted on an Argand diagram. Find the exact area of the triangle which has these three points as its vertices. [7]
OCR MEI Further Pure Core AS Specimen Q9
14 marks Challenging +1.2
You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).
  1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
  2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]
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 \(n\).
  1. Explain geometrically why this claim is true. [2]
  2. Verify algebraically that this claim is true. [3]