CAIE
FP1
2013
November
Q11 EITHER
Let \(I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { n } \mathrm {~d} x\). Show that, for all integers \(n\),
$$( 2 n + 1 ) I _ { n } = 2 n I _ { n - 1 } + 2 ^ { n }$$
Evaluate \(I _ { 0 }\) and hence find \(I _ { 3 }\).
Given that \(I _ { - 1 } = \frac { 1 } { 4 } \pi\), find \(I _ { - 3 }\).
CAIE
FP1
2013
November
Q11 OR
The vector \(\mathbf { e }\) is an eigenvector of each of the \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Justifying your answer, state an eigenvalue of \(\mathbf { A } + \mathbf { B }\).
The matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
6 & - 1 & - 6
1 & 0 & - 2
3 & - 1 & - 3
\end{array} \right)$$
has eigenvectors \(\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)\). Find the corresponding eigenvalues.
The matrix \(\mathbf { B }\), where
$$\mathbf { B } = \left( \begin{array} { r r r }
8 & - 2 & - 8
2 & 0 & - 4
4 & - 2 & - 4
\end{array} \right) ,$$
also has eigenvectors \(\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)\), for which \(- 2,2,4\), respectively, are corresponding eigenvalues. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \mathbf { A } + \mathbf { B } - 5 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. State the eigenvalues of \(\mathbf { M }\).
Find matrices \(\mathbf { R }\) and \(\mathbf { S }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } ^ { 5 } = \mathbf { R D S }\).
[0pt]
[You should show clearly all the elements of the matrices \(\mathbf { R } , \mathbf { S }\) and \(\mathbf { D }\).]
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CAIE
FP1
2014
November
Q3
3 It is given that \(u _ { r } = r \times r !\) for \(r = 1,2,3 , \ldots\). Let \(S _ { n } = u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots + u _ { n }\). Write down the values of
$$2 ! - S _ { 1 } , \quad 3 ! - S _ { 2 } , \quad 4 ! - S _ { 3 } , \quad 5 ! - S _ { 4 }$$
Conjecture a formula for \(S _ { n }\).
Prove, by mathematical induction, a formula for \(S _ { n }\), for all positive integers \(n\).
CAIE
FP1
2014
November
Q7
7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x\). Show that, for all positive integers \(n\),
$$I _ { n } = n I _ { n - 1 } - 1$$
Find the exact value of \(I _ { 4 }\).
By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }\) in the interval \(0 \leqslant x \leqslant 1\), show that
$$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$
CAIE
FP1
2014
November
Q8
8 A circle has polar equation \(r = a\), for \(0 \leqslant \theta < 2 \pi\), and a cardioid has polar equation \(r = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram.
Write down the polar coordinates of the points of intersection of the circle and the cardioid.
Show that the area of the region that is both inside the circle and inside the cardioid is
$$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$
CAIE
FP1
2016
November
Q2
2 Find the cubic equation with roots \(\alpha , \beta\) and \(\gamma\) such that
$$\begin{aligned}
\alpha + \beta + \gamma & = 3
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30
\end{aligned}$$
giving your answer in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers to be found.
CAIE
FP1
2016
November
Q3
3 Find a matrix \(\mathbf { A }\) whose eigenvalues are \(- 1,1,2\) and for which corresponding eigenvectors are
$$\left( \begin{array} { l }
1
0
0
\end{array} \right) , \quad \left( \begin{array} { l }
1
1
0
\end{array} \right) , \quad \left( \begin{array} { l }
0
1
1
\end{array} \right) ,$$
respectively.
CAIE
FP1
2016
November
Q8
8 A curve \(C\) has equation \(x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0\). Show that, at stationary points on \(C , x = - 2 y\).
Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the stationary points.
CAIE
FP1
2016
November
Q9
9 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\).
Given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), prove that, for \(n > 1\),
$$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
By first using the substitution \(x = \cos ^ { - 1 } u\), find the value of
$$\int _ { 0 } ^ { 1 } \left( \cos ^ { - 1 } u \right) ^ { 3 } \mathrm {~d} u$$
giving your answer in an exact form.