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 \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } \theta \mathrm {~d} \theta\).
CAIE
FP1
2013
June
Q11 OR
Standard +0.8
The points \(A , B , C\) and \(D\) have coordinates as follows:
$$A ( 2,1 , - 2 ) , \quad B ( 4,1 , - 1 ) , \quad C ( 3 , - 2 , - 1 ) \quad \text { and } \quad D ( 3,6,2 ) .$$
The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(C\). Find a cartesian equation of \(\Pi _ { 1 }\).
Find the area of triangle \(A B C\) and hence, or otherwise, find the volume of the tetrahedron \(A B C D\).
[0pt]
[The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.]
The plane \(\Pi _ { 2 }\) passes through the points \(A , B\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
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CAIE
FP1
2013
June
Q2
6 marks
Challenging +1.2
2 The roots of the equation \(x ^ { 4 } - 4 x ^ { 2 } + 3 x - 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\); the sum \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). By using the relation \(y = x ^ { 2 }\), or otherwise, show that \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\) and \(\delta ^ { 2 }\) are the roots of the equation
$$y ^ { 4 } - 8 y ^ { 3 } + 12 y ^ { 2 } + 7 y + 4 = 0$$
State the value of \(S _ { 2 }\) and hence show that
$$S _ { 8 } = 8 S _ { 6 } - 12 S _ { 4 } - 72 .$$
CAIE
FP1
2013
June
Q8
11 marks
Challenging +1.2
8 The curve \(C\) has parametric equations \(x = \frac { 3 } { 2 } t ^ { 2 } , y = t ^ { 3 }\), for \(0 \leqslant t \leqslant 2\). Find the arc length of \(C\).
Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 6\).