CAIE
FP1
2011
November
Q7
7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + p x + 1 } { x - 2 }\), where \(p\) is a constant. Given that \(C\) has two asymptotes, find the equation of each asymptote.
Find the set of values of \(p\) for which \(C\) has two distinct turning points.
Sketch \(C\) in the case \(p = - 1\). Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.
CAIE
FP1
2011
November
Q8
8 The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\).
State the eigenvalues of the matrix \(\mathbf { C }\), where
$$\mathbf { C } = \left( \begin{array} { r r r }
- 1 & - 1 & 3
0 & 1 & 2
0 & 0 & 2
\end{array} \right) ,$$
and find corresponding eigenvectors.
Show that \(\left( \begin{array} { l } 1
6
3 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { D }\), where
$$\mathbf { D } = \left( \begin{array} { r r r }
1 & - 1 & 1
- 6 & - 3 & 4
- 9 & - 3 & 7
\end{array} \right) ,$$
and state the corresponding eigenvalue.
Hence state an eigenvector of the matrix CD and give the corresponding eigenvalue.
CAIE
FP1
2012
November
Q11
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
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
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 \(\tan \phi = \frac { 4 } { 3 }\).