OCR MEI Further Pure Core AS (Further Pure Core AS) 2024 June

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 \(b\).

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.

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.

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 \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).

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 \(k\).

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.

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.

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 $$\{ z : | z + 3 - 9 i | = k \} \text { and } \left\{ z : \arg ( z - 7 ) = \frac { 3 } { 4 } \pi \right\} .$$

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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