CAIE
FP1
2013
June
Q3
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
Q7
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
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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