CAIE
FP1
2013
November
Q4
7 marks
Standard +0.3
4 A curve has parametric equations
$$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$
Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$
except for certain values of \(\theta\), which should be stated.
Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(\theta = \frac { 1 } { 4 } \pi\).
CAIE
FP1
2013
November
Q8
11 marks
Standard +0.3
8 The points \(A , B , C\) have position vectors
$$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$
respectively, relative to the origin \(O\). Find a cartesian equation of the plane \(A B C\).
The point \(D\) has position vector \(6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\). Find the coordinates of \(E\), the point of intersection of the line \(O D\) with the plane \(A B C\).
Find the acute angle between the line \(E D\) and the plane \(A B C\).
CAIE
FP1
2013
November
Q9
11 marks
Challenging +1.2
9 Prove by mathematical induction that, for every positive integer \(n\),
$$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$
Express \(\sin ^ { 5 } \theta\) in the form \(p \sin 5 \theta + q \sin 3 \theta + r \sin \theta\), where \(p , q\) and \(r\) are rational numbers to be determined.
CAIE
FP1
2013
November
Q11 EITHER
Challenging +1.2
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
Standard +0.3
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
7 marks
Standard +0.8
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
10 marks
Challenging +1.8
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
11 marks
Challenging +1.2
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
$$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$