OCR FP1 AS (Further Pure 1 AS) 2017 December

1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } - 3 & 3 & 2
5 & - 4 & - 3
- 1 & 1 & 1 \end{array} \right)\).

1. Find \(\mathbf { A } ^ { - 1 }\).
2. Solve the simultaneous equations $$\begin{aligned} - 3 x + 3 y + 2 z & = 12 a
   5 x - 4 y - 3 z & = - 6
   - x + y + z & = 7 \end{aligned}$$ giving your solution in terms of \(a\).

2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - ( 3 + 2 \mathrm { i } ) | = 2\) and \(\arg ( z - ( 3 + 2 \mathrm { i } ) ) = \frac { 5 \pi } { 6 }\) respectively.

1. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single Argand diagram.
2. Find, in surd form, the number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Indicate, by shading, the region of the Argand diagram for which $$| z - ( 3 + 2 i ) | \leqslant 2 \text { and } \frac { 5 \pi } { 6 } \leqslant \arg ( z - ( 3 + 2 i ) ) \leqslant \pi$$

3 Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2

5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
   \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
   The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
   The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\).

6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\).

7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s
t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears.

10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
   \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
   The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
   The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\). 6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\). 7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s
   t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears. 8
8. (a) Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2
   5
   1 \end{array} \right)\) and \(\left( \begin{array} { c } x
   6
   2 \end{array} \right)\).
   (b) Find the shortest possible vector of the form \(\left( \begin{array} { l } 1
   a
   b \end{array} \right)\) which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2
   5
   1 \end{array} \right)\) and \(\left( \begin{array} { c } x
   6
   2 \end{array} \right)\).
9. Vector \(\mathbf { v }\) is perpendicular to both \(\left( \begin{array} { c } - 1
   1
   1 \end{array} \right)\) and \(\left( \begin{array} { c } 1
   p
   p ^ { 2 } \end{array} \right)\) where \(p\) is a real number. Show that it is impossible for \(\mathbf { v }\) to be perpendicular to the vector \(\left( \begin{array} { c } 1
   1
   p - 1 \end{array} \right)\). \section*{OCR} Oxford Cambridge and RSA