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

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OCR MEI Further Pure Core AS 2018 June Q1
1 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows: $$\mathbf { A } = \left( \begin{array} { l } 1
OCR MEI Further Pure Core AS 2018 June Q3
3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices. 2 Find, to the nearest degree, the angle between the vectors \(\left( \begin{array} { r } 1
0
- 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 2
3
- 3 \end{array} \right)\). 3 Find real numbers \(a\) and \(b\) such that \(( a - 3 i ) ( 5 - i ) = b - 17 i\).
OCR MEI Further Pure Core AS 2018 June Q4
4 Find a cubic equation with real coefficients, two of whose roots are \(2 - \mathrm { i }\) and 3.
OCR MEI Further Pure Core AS 2018 June Q5
5 A transformation of the \(x - y\) plane is represented by the matrix \(\left( \begin{array} { r r } \cos \theta & 2 \sin \theta
2 \sin \theta & - \cos \theta \end{array} \right)\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \(( 2,3 )\) under this transformation.
  2. You are given that this image is the point ( \(a , 0\) ). Find the value of \(a\).
OCR MEI Further Pure Core AS 2018 June Q6
6 Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0
4 & - 1 \end{array} \right)\).
OCR MEI Further Pure Core AS 2018 June Q7
7
  1. Express \(\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 }\) as a single fraction.
  2. Find how many terms of the series $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 3 \times 5 } + \frac { 2 } { 5 \times 7 } + \ldots + \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) } + \ldots$$ are needed for the sum to exceed 0.999999.
OCR MEI Further Pure Core AS 2018 June Q8
8 Prove by induction that \(\left( \begin{array} { l l } 1 & 1
0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 2 ^ { n } - 1
0 & 2 ^ { n } \end{array} \right)\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2018 June Q9
9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9ef04b56-c6e5-46ea-a485-fe872932e9d8-3_549_520_397_751} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find, in modulus-argument form, the complex number represented by the point P .
  2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    • \(3 + 5 \mathrm { i }\)
    • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\)
      lie within this region.
OCR MEI Further Pure Core AS 2018 June Q10
10 Three planes have equations $$\begin{aligned} - x + 2 y + z & = 0
2 x - y - z & = 0
x + y & = a \end{aligned}$$ where \(a\) is a constant.
  1. Investigate the arrangement of the planes:
    • when \(a = 0\);
    • when \(a \neq 0\).
    • 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.
OCR MEI Further Pure Core AS 2019 June Q1
1 In this question you must show detailed reasoning.
Find \(\sum _ { r = 1 } ^ { 100 } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\), giving your answer correct to 4 decimal places.
OCR MEI Further Pure Core AS 2019 June Q2
2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).
OCR MEI Further Pure Core AS 2019 June Q3
3 In this question you must show detailed reasoning.
\(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1
- 1 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A B }\).
  2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1
    0 & 1 \end{array} \right)\), find \(\mathbf { B }\).
OCR MEI Further Pure Core AS 2019 June Q4
4
  1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3
    - 1 & 1 & 2
    - 2 & 1 & 2 \end{array} \right)\).
  2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19
    - x + y + 2 z & = 4
    - 2 x + y + 2 z & = k \end{aligned}$$
  3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).
OCR MEI Further Pure Core AS 2019 June Q5
5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).
OCR MEI Further Pure Core AS 2019 June Q6
6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k
1 & \lambda - k \end{array} \right)\), and \(\lambda\)
and \(k\) are real constants. and \(k\) are real constants.
  1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
    1. Find the range of possible values of \(k\).
    2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
  2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
  3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
    Find the possible values of \(\lambda\).
OCR MEI Further Pure Core AS 2019 June Q7
7
  1. Sketch on a single Argand diagram
    1. the set of points for which \(| z - 1 - 3 i | = 3\),
    2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
  2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
OCR MEI Further Pure Core AS 2022 June Q1
1
    1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7
      2 x - 4 y - 3 z & = - 5
      - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
    2. Hence solve the equations.
  2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1
    2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.
OCR MEI Further Pure Core AS 2022 June Q2
2
  1. Show that the vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) is parallel to the plane \(2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10\).
  2. Determine the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x - y - 3 z = 3\).
OCR MEI Further Pure Core AS 2022 June Q3
3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
OCR MEI Further Pure Core AS 2022 June Q5
5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\).
\includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242} State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.
OCR MEI Further Pure Core AS 2022 June Q6
6
  1. Using standard summation formulae, show that \(\sum _ { r = 1 } ^ { n } r ( r + 2 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 7 )\).
  2. Use induction to prove the result in part (a).
OCR MEI Further Pure Core AS 2022 June Q7
7 On an Argand diagram, the point A represents the complex number \(z\) with modulus 2 and argument \(\frac { 1 } { 3 } \pi\). The point B represents \(\frac { 1 } { z }\).
  1. Sketch an Argand diagram showing the origin O and the points A and B .
  2. The point C is such that OACB is a parallelogram. C represents the complex number \(w\). Determine each of the following.
    • The modulus of \(w\), giving your answer in exact form.
    • The argument of \(w\), giving your answer correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Further Pure Core AS 2022 June Q8
8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta
\sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).
  1. Show that T leaves areas unchanged for all values of \(\theta\).
  2. Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T . The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2
    0 & 1 \end{array} \right)\).
    1. Find \(\mathbf { M N } ^ { - 1 }\).
    2. Hence describe fully a sequence of two transformations of the plane that is equivalent to T . \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR MEI Further Pure Core AS 2023 June Q1
1 The transformation R of the plane is reflection in the line \(x = 0\).
  1. Write down the matrix \(\mathbf { M }\) associated with R .
  2. Find \(\mathbf { M } ^ { 2 }\).
  3. Interpret the result of part (b) in terms of the transformation \(R\).
OCR MEI Further Pure Core AS 2023 June Q2
2 In this question you must show detailed reasoning.
The equation \(\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\).
Find \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha }\) in terms of \(k\), simplifying your answer.