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OCR MEI Further Pure Core AS 2024 June Q3
6 marks Standard +0.3
3
  1. Using standard summation formulae, write down an expression in terms of \(n\) for \(\sum _ { r = 1 } ^ { 2 n } r ^ { 3 }\).
  2. Hence show that \(\sum _ { \mathrm { r } = \mathrm { n } + 1 } ^ { 2 \mathrm { n } } \mathrm { r } ^ { 3 } = \frac { 1 } { 4 } \mathrm { n } ^ { 2 } ( \mathrm { an } + \mathrm { b } ) ( \mathrm { cn } + \mathrm { d } )\), where \(a , b , c\) and \(d\) are integers to be determined.
OCR MEI Further Pure Core AS 2024 June Q5
6 marks Moderate -0.5
5
  1. Find the volume scale factor of the transformation with associated matrix \(\left( \begin{array} { r r r } 1 & 2 & 0 \\ 0 & 3 & - 1 \\ - 1 & 0 & 2 \end{array} \right)\).
  2. The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf { P }\) and \(\mathbf { Q }\) respectively.
    1. Write down an expression for the associated matrix of the combined transformation S followed by T. The determinant of \(\mathbf { P }\) is 3 and \(\mathbf { Q } = \left( \begin{array} { r r } k & 3 \\ - 1 & 2 \end{array} \right)\), where \(k\) is a constant.
    2. Given that this combined transformation preserves both orientation and area, determine the value of \(k\).
OCR MEI Further Pure Core AS 2024 June Q6
9 marks Standard +0.8
6 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 4 & - 9 \\ 1 & - 2 \end{array} \right)\).
  1. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 + 3 n & - 9 n \\ n & 1 - 3 n \end{array} \right)\) for all positive integers \(n\).
  2. A student thinks that this formula, when \(n = 0\) and \(n = - 1\), gives the identity matrix and the inverse matrix \(\mathbf { M } ^ { - 1 }\) respectively. Determine whether the student is correct.
OCR MEI Further Pure Core AS 2024 June Q7
6 marks Standard +0.8
7 Three planes have equations $$\begin{array} { r } x + 2 y - 3 z = 0 \\ - x + 3 y - 2 z = 0 \\ x - 2 y + k z = k \end{array}$$ where \(k\) is a constant.
  1. For the case \(k = 0\), the origin lies on all three planes. Use a determinant to explain whether there are any other points that lie on all three planes in this case.
  2. You are now given that \(k = 1\).
    1. Show that there are no points that lie on all three planes.
    2. Describe the geometrical arrangement of the three planes.
OCR MEI Further Pure Core AS 2024 June Q8
9 marks Standard +0.8
8 In an Argand diagram, the point P representing the complex number \(w\) lies on the locus defined by \(\left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\}\). You are given that \(\operatorname { Re } ( w ) = 1\).
  1. Find \(w\). The point P also lies on the locus defined by \(\{ \mathrm { z } : | \mathrm { z } + 3 - 9 \mathrm { i } | = \mathrm { k } \}\), where \(k\) is a constant.
  2. Find the complex number represented by the other point of intersection of the loci defined by $$\{ z : | z + 3 - 9 i | = k \} \text { and } \left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\} .$$
OCR MEI Further Pure Core AS 2020 November Q2
4 marks Moderate -0.8
2 Fig. 2 shows two complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) represented on an Argand diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{55a4a9f1-ed86-44bb-8759-dfee0b66f56d-2_985_997_781_239} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. On the copy of Fig. 2 in the Printed Answer Booklet, mark points representing each of the following complex numbers.
    • \(\mathrm { Z } _ { 1 } { } ^ { * }\)
    • \(z _ { 2 } - z _ { 1 }\)
    • In this question you must show detailed reasoning.
    In the case where \(z _ { 1 } = 1 + 2 \mathrm { i }\) and \(z _ { 2 } = 3 + \mathrm { i }\), find \(\frac { z _ { 2 } - z _ { 1 } } { z _ { 1 } ^ { * } }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
OCR MEI Further Pure Core AS 2020 November Q3
7 marks Standard +0.3
3 In this question you must show detailed reasoning.
The roots of the equation \(x ^ { 2 } - 2 x + 4 = 0\) are \(\alpha\) and \(\beta\).
  1. Find \(\alpha\) and \(\beta\) in modulus-argument form.
  2. Hence or otherwise show that \(\alpha\) and \(\beta\) are both roots of \(x ^ { 3 } + \lambda = 0\), where \(\lambda\) is a real constant to be determined.
OCR MEI Further Pure Core AS 2020 November Q4
4 marks Moderate -0.3
4 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r r } 0 & - 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\).
    1. Calculate \(\operatorname { det } \mathbf { M }\).
    2. State two geometrical consequences of this value for the transformation associated with \(\mathbf { M }\).
  2. Describe fully the transformation associated with \(\mathbf { M }\).
OCR MEI Further Pure Core AS 2020 November Q5
6 marks Standard +0.3
5 You are given that \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 n + 4\).
Prove by induction that \(u _ { n } = n ^ { 2 } + 3 n + 1\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2020 November Q6
8 marks Moderate -0.3
6 The matrices \(\mathbf { M }\) and \(\mathbf { N }\) are \(\left( \begin{array} { l l } \lambda & 2 \\ 2 & \lambda \end{array} \right)\) and \(\left( \begin{array} { c c } \mu & 1 \\ 1 & \mu \end{array} \right)\) respectively, where \(\lambda\) and \(\mu\) are constants.
  1. Investigate whether \(\mathbf { M }\) and \(\mathbf { N }\) are commutative under multiplication.
  2. You are now given that \(\mathbf { M N } = \mathbf { I }\).
    1. Write down a relationship between \(\operatorname { det } \mathbf { M }\) and \(\operatorname { det } \mathbf { N }\).
    2. Given that \(\lambda > 0\), find the exact values of \(\lambda\) and \(\mu\).
    3. Hence verify your answer to part (i).
OCR MEI Further Pure Core AS 2020 November Q7
7 marks Standard +0.8
7 In the quartic equation \(2 x ^ { 4 } - 20 x ^ { 3 } + a x ^ { 2 } + b x + 250 = 0\), the coefficients \(a\) and \(b\) are real. One root of the equation is \(2 + \mathrm { i }\). Find the other roots.
OCR MEI Further Pure Core AS 2020 November Q8
7 marks Moderate -0.3
8
  1. The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 0 & - 1 \\ - 1 & 0 \end{array} \right)\).
    1. Find \(\mathbf { M } ^ { 2 }\).
    2. Write down the transformation represented by \(\mathbf { M }\).
    3. Hence state the geometrical significance of the result of part (i).
  2. The matrix \(\mathbf { N }\) is \(\left( \begin{array} { c c } k + 1 & 0 \\ k & k + 2 \end{array} \right)\), where \(k\) is a constant. Using determinants, investigate whether \(\mathbf { N }\) can represent a reflection.
OCR MEI Further Pure Core AS 2020 November Q9
7 marks Standard +0.3
9 Three planes have equations \(k x + y - 2 z = 0\) \(2 x + 3 y - 6 z = - 5\) \(3 x - 2 y + 5 z = 1\) where \(k\) is a constant. Investigate the arrangement of the planes for each of the following cases. If in either case the planes meet at a unique point, find the coordinates of that point.
  1. \(k = - 1\)
  2. \(k = \frac { 2 } { 3 }\)
OCR MEI Further Pure Core AS 2020 November Q10
7 marks Challenging +1.2
10 A vector \(\mathbf { v }\) has magnitude 1 unit. The angle between \(\mathbf { v }\) and the positive \(z\)-axis is \(60 ^ { \circ }\), and \(\mathbf { v }\) is parallel to the plane \(x - 2 y = 0\). Given that \(\mathbf { v } = a \mathbf { i } + b \mathbf { j } + c \mathbf { k }\), where \(a , b\) and \(c\) are all positive, find \(\mathbf { v }\). \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS 2021 November Q1
3 marks Moderate -0.8
1 Using standard summation formulae, find \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - 3 r \right)\), giving your answer in fully factorised form.
OCR MEI Further Pure Core AS 2021 November Q2
3 marks Standard +0.3
2 The equation \(3 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\alpha\) and \(\beta\).
Find an equation with integer coefficients whose roots are \(3 - 2 \alpha\) and \(3 - 2 \beta\).
OCR MEI Further Pure Core AS 2021 November Q3
7 marks Standard +0.3
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
5 marks Moderate -0.3
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 marks Standard +0.3
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
12 marks Standard +0.8
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
9 marks Challenging +1.2
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 marks Challenging +1.2
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
4 marks Moderate -0.8
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
4 marks Standard +0.3
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
6 marks Standard +0.3
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.