CAIE
FP1
2012
June
Q11 EITHER
Challenging +1.8
Show that
$$\int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 + \mathrm { e } ^ { \pi } } { 2 }$$
Given that
$$I _ { n } = \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x$$
show that, for \(n \geqslant 2\),
$$I _ { n } = n ( n - 1 ) \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \cos ^ { 2 } x \sin ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$
and deduce that
$$\left( n ^ { 2 } + 1 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 } .$$
A curve has equation \(y = \mathrm { e } ^ { x } \sin ^ { 5 } x\). Find, in an exact form, the mean value of \(y\) over the interval \(0 \leqslant x \leqslant \pi\).
CAIE
FP1
2012
June
Q11 OR
Challenging +1.2
The position vectors of the points \(A , B , C , D\) are
$$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$
respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 .
Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).
\footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
}
CAIE
FP1
2012
June
Q11 OR
Challenging +1.2
The position vectors of the points \(A , B , C , D\) are
$$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$
respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 .
Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).
\footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
}
CAIE
FP1
2012
June
Q2
5 marks
Standard +0.3
2 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 1\) and \(u _ { r + 1 } = \frac { 3 u _ { r } - 2 } { 4 }\) for all \(r\). Prove by mathematical induction that \(u _ { n } = 4 \left( \frac { 3 } { 4 } \right) ^ { n } - 2\), for all positive integers \(n\).
CAIE
FP1
2012
June
Q5
9 marks
Standard +0.3
5 The matrix \(\mathbf { A }\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf { e }\). Prove that the matrix \(( \mathbf { A } + k \mathbf { I } )\), where \(k\) is a real constant and \(\mathbf { I }\) is the identity matrix, has an eigenvalue ( \(\lambda + k\) ) with corresponding eigenvector \(\mathbf { e }\).
The matrix \(\mathbf { B }\) is given by
$$\mathbf { B } = \left( \begin{array} { r r r }
2 & 2 & - 3 \\
2 & 2 & 3 \\
- 3 & 3 & 3
\end{array} \right) .$$
Two of the eigenvalues of \(\mathbf { B }\) are - 3 and 4 . Find corresponding eigenvectors.
Given that \(\left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\) is an eigenvector of \(\mathbf { B }\), find the corresponding eigenvalue.
Hence find the eigenvalues of \(\mathbf { C }\), where
$$\mathbf { C } = \left( \begin{array} { r r r }
- 1 & 2 & - 3 \\
2 & - 1 & 3 \\
- 3 & 3 & 0
\end{array} \right) ,$$
and state corresponding eigenvectors.
CAIE
FP1
2012
June
Q7
10 marks
Challenging +1.8
7 Expand \(\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }\) and, by substituting \(z = \cos \theta + \mathrm { i } \sin \theta\), find integers \(p , q , r , s\) such that
$$64 \sin ^ { 2 } \theta \cos ^ { 4 } \theta = p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta$$
Using the substitution \(x = 2 \cos \theta\), show that
$$\int _ { 1 } ^ { 2 } x ^ { 4 } \sqrt { } \left( 4 - x ^ { 2 } \right) \mathrm { d } x = \frac { 4 } { 3 } \pi + \sqrt { } 3$$
CAIE
FP1
2013
June
Q3
8 marks
Challenging +1.2
3 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 3 x + 4 = 0\) has roots \(\alpha , \beta , \gamma\). Given that \(c = \alpha + \beta + \gamma\), state the value of \(c\).
Use the substitution \(y = c - x\) to find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
Find a cubic equation whose roots are \(\frac { 1 } { \alpha + \beta } , \frac { 1 } { \beta + \gamma } , \frac { 1 } { \gamma + \alpha }\).
Hence evaluate \(\frac { 1 } { ( \alpha + \beta ) ^ { 2 } } + \frac { 1 } { ( \beta + \gamma ) ^ { 2 } } + \frac { 1 } { ( \gamma + \alpha ) ^ { 2 } }\).
CAIE
FP1
2013
June
Q4
8 marks
Challenging +1.2
4 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x\). Prove that, for every positive integer \(n\),
$$2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$$
Given that \(I _ { 1 } = \frac { 1 } { 4 } \pi\), find the exact value of \(I _ { 3 }\).
CAIE
FP1
2013
June
Q7
10 marks
Challenging +1.2
7 By considering the binomial expansion of \(\left( z - \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), express \(\sin ^ { 6 } \theta\) in the form
$$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$
where \(p , q , r\) and \(s\) are integers to be determined.
Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta\).