CAIE
FP1
2012
November
Q11
13 marks
Challenging +1.2
11 Show that \(\int x \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = - \frac { 1 } { 3 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } + c\), where \(c\) is a constant.
Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), prove that, for \(n \geqslant 2\),
$$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 }$$
Use the substitution \(x = \sin u\) to show that
$$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \pi$$
Find \(I _ { 4 }\).
CAIE
FP1
2012
November
Q12 EITHER
Standard +0.8
The vector \(\mathbf { e }\) is an eigenvector of each of the \(n \times n\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Prove that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with eigenvalue \(\lambda \mu\).
It is given that the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & 2 & 2 \\
- 2 & - 2 & - 2 \\
1 & 2 & 2
\end{array} \right) ,$$
has eigenvectors \(\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 0 \\ - 1 \end{array} \right)\). Find the corresponding eigenvalues.
Given that 2 is also an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
The matrix \(\mathbf { B }\), where
$$\mathbf { B } = \left( \begin{array} { r r r }
- 1 & 2 & 2 \\
2 & 2 & 2 \\
- 3 & - 6 & - 6
\end{array} \right) ,$$
has the same eigenvectors as \(\mathbf { A }\). Given that \(\mathbf { A B } = \mathbf { C }\), find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that
$$\mathbf { P } ^ { - 1 } \mathbf { C } ^ { 2 } \mathbf { P } = \mathbf { D }$$
CAIE
FP1
2012
November
Q12 OR
Challenging +1.2
Obtain the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 75 \cos 2 t$$
Given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\), find \(x\) in terms of \(t\).
Show that, for large positive values of \(t\) and for any initial conditions,
$$x \approx 5 \cos ( 2 t - \phi ) ,$$
where the constant \(\phi\) is such that \(\tan \phi = \frac { 4 } { 3 }\).
CAIE
FP1
2013
November
Q3
7 marks
Standard +0.3
3 It is given that
$$S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r } = 2 n ^ { 2 } + n$$
Write down the values of \(S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }\). Express \(u _ { r }\) in terms of \(r\), justifying your answer.
Find
$$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } .$$
CAIE
FP1
2013
November
Q6
9 marks
Challenging +1.2
6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where
$$\mathbf { M } = \left( \begin{array} { r r r r }
1 & - 3 & - 1 & 2 \\
4 & - 10 & 0 & 2 \\
1 & - 1 & 3 & - 4 \\
5 & - 12 & 1 & 1
\end{array} \right)$$
Find, in either order, the rank of \(\mathbf { M }\) and a basis for the null space \(K\) of T .
Evaluate
$$\mathbf { M } \left( \begin{array} { r }
1 \\
- 2 \\
- 3 \\
- 4
\end{array} \right)$$
and hence show that every solution of
$$\mathbf { M x } = \left( \begin{array} { r }
2 \\
16 \\
10 \\
22
\end{array} \right)$$
has the form
$$\mathbf { x } = \left( \begin{array} { r }
1 \\
- 2 \\
- 3 \\
- 4
\end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$
where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\).