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 \(l _ { 2 }\).
- Find the position vector of \(P\).
- 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 \(l _ { 1 }\). - 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 \(4 x ^ { 2 } + 4 x + 17 = 0\). - Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
- \(A , B\) 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 \(\gamma\). - 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 \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
- 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 - (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 \(\left( \begin{array} { c } x
6
2 \end{array} \right)\). - 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