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

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OCR MEI Further Pure Core AS 2023 June Q3
3 In this question you must show detailed reasoning.
The function \(\mathrm { f } ( \mathrm { z } )\) is given by \(\mathrm { f } ( \mathrm { z } ) = 2 \mathrm { z } ^ { 3 } - 7 \mathrm { z } ^ { 2 } + 16 \mathrm { z } - 15\).
By first evaluating \(\mathrm { f } \left( \frac { 3 } { 2 } \right)\), find the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\).
OCR MEI Further Pure Core AS 2023 June Q4
4 You are given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { ar } + \mathrm { b } ) = \mathrm { n } ^ { 2 }\) for all \(n\), where \(a\) and \(b\) are constants.
By finding \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { ar } + \mathrm { b } )\) in terms of \(a , b\) and \(n\), determine the values of \(a\) and \(b\).
OCR MEI Further Pure Core AS 2023 June Q5
5 The Argand diagram below shows the points representing 1 and \(z\), where \(| z | = 2\).
\includegraphics[max width=\textwidth, alt={}, center]{26cec6f9-78a7-4f0b-969a-13ad02510c25-3_577_595_312_242} Mark the points representing the following complex numbers on the copy of the diagram in the Printed Answer Booklet, labelling them clearly.
  • \(\mathrm { Z } ^ { * }\)
  • \(\frac { 1 } { z }\)
  • \(1 + z\)
  • iz
OCR MEI Further Pure Core AS 2023 June Q6
6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1
- 1 & 0 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
  2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
  3. Prove your conjecture.
OCR MEI Further Pure Core AS 2023 June Q8
8 The equations of three planes are $$\begin{array} { r } 2 x + y + 3 z = 3
3 x - y - 2 z = 2
- 4 x + 3 y + 7 z = k \end{array}$$ where \(k\) is a constant.
  1. By considering a suitable determinant, show that the planes do not meet at a single point.
  2. Given that the planes form a sheaf, determine the value of \(k\).
OCR MEI Further Pure Core AS 2023 June Q9
9 A transformation T of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { c c } k + 1 & - 1
1 & k \end{array} \right)\), where \(k\) is a
constant. constant. Show that, for all values of \(k , \mathrm {~T}\) has no invariant lines through the origin.
OCR MEI Further Pure Core AS 2023 June Q10
10 The plane P has normal vector \(2 \mathbf { i } + a \mathbf { j } - \mathbf { k }\), where \(a\) is a positive constant, and the point ( \(3 , - 1,1\) ) lies in P . The plane \(\mathrm { x } - \mathrm { z } = 3\) makes an angle of \(45 ^ { \circ }\) with P . Find the cartesian equation of P . \section*{END OF QUESTION PAPER}
OCR MEI Further Pure Core AS 2024 June Q1
1 The quadratic equation \(\mathrm { x } ^ { 2 } + \mathrm { ax } + \mathrm { b } = 0\), where \(a\) and \(b\) are real constants, has a root 2-3.
  1. Write down the other root.
  2. Hence or otherwise determine the values of \(a\) and \(b\).
OCR MEI Further Pure Core AS 2024 June Q2
2 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 1 & a
- 1 & 2 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 2 & 0
1 & - 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { r r } - 1 & 0
2 & 1 \end{array} \right)\), where \(a\) is
a constant. a constant.
  1. By multiplying out the matrices on both sides of the equation, verify that \(\mathbf { A } ( \mathbf { B C } ) = ( \mathbf { A B } ) \mathbf { C }\).
  2. State the property of matrix multiplication illustrated by this result.
OCR MEI Further Pure Core AS 2024 June Q3
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
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
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
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
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
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
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 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
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
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 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
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
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
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
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
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 \(3 - 2 \beta\).