4 \sqrt { 3 } & - 4
\end{array} \right)$$
- Determine
- the value of \(k\),
- the smallest value of \(\theta\)
A square \(S\) has vertices at the points with coordinates ( 0,0 ), ( \(a , - a\) ), ( \(2 a , 0\) ) and ( \(a , a\) ) where \(a\) is a constant.
The square \(S\) is transformed to the square \(S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
- Determine, in terms of \(a\), the area of \(S ^ { \prime }\)
- (a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos ^ { 2 } \left( \frac { x } { 3 } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\)
Give each term in simplest form. - Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
- Use the integration function on your calculator to evaluate
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
Give your answer to 5 decimal places.
- Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).
- The cubic equation
$$a x ^ { 3 } + b x ^ { 2 } - 19 x - b = 0$$
where \(a\) and \(b\) are constants, has roots \(\alpha , \beta\) and \(\gamma\)
The cubic equation
$$w ^ { 3 } - 9 w ^ { 2 } - 97 w + c = 0$$
where \(c\) is a constant, has roots \(( 4 \alpha - 1 ) , ( 4 \beta - 1 )\) and \(( 4 \gamma - 1 )\)
Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\).
- \(\mathbf { A }\) is a 2 by 2 matrix and \(\mathbf { B }\) is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate - \(\mathbf { A B }\)
- \(\mathbf { A } + \mathbf { B }\)
(ii) Given that
$$\left( \begin{array} { r r r }
- 5 & 3 & 1
a & 0 & 0
b & a & b
\end{array} \right) \left( \begin{array} { r r r }
0 & 5 & 0
2 & 12 & - 1
- 1 & - 11 & 3
\end{array} \right) = \lambda \mathbf { I }$$
where \(a\), \(b\) and \(\lambda\) are constants, - determine
- the value of \(\lambda\)
- the value of \(a\)
- the value of \(b\)
- Hence deduce the inverse of the matrix \(\left( \begin{array} { r r r } - 5 & 3 & 1
a & 0 & 0
b & a & b \end{array} \right)\)
(iii) Given that
$$\mathbf { M } = \left( \begin{array} { c c c }
1 & 1 & 1
0 & \sin \theta & \cos \theta
0 & \cos 2 \theta & \sin 2 \theta
\end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$
determine the values of \(\theta\) for which the matrix \(\mathbf { M }\) is singular.