CAIE
FP1
2016
June
Q1
Standard +0.8
1 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 7 = 0\) are \(\alpha , \beta\) and \(\gamma\). Using the substitution \(y = 1 + \frac { 1 } { x }\), or otherwise, find the cubic equation whose roots are \(1 + \frac { 1 } { \alpha } , 1 + \frac { 1 } { \beta }\) and \(1 + \frac { 1 } { \gamma }\), giving your answer in the form \(a y ^ { 3 } + b y ^ { 2 } + c y + d = 0\), where \(a , b , c\) and \(d\) are constants to be found.
CAIE
FP1
2016
June
Q6
Challenging +1.8
6 Use de Moivre's theorem to express \(\cot 7 \theta\) in terms of \(\cot \theta\).
Use the equation \(\cot 7 \theta = 0\) to show that the roots of the equation
$$x ^ { 6 } - 21 x ^ { 4 } + 35 x ^ { 2 } - 7 = 0$$
are \(\cot \left( \frac { 1 } { 14 } k \pi \right)\) for \(k = 1,3,5,9,11,13\), and deduce that
$$\cot ^ { 2 } \left( \frac { 1 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 3 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 5 } { 14 } \pi \right) = 7$$
CAIE
FP1
2016
June
Q8
Standard +0.8
8 Find a cartesian equation of the plane \(\Pi _ { 1 }\) passing through the points with coordinates \(( 2 , - 1,3 )\), \(( 4,2 , - 5 )\) and \(( - 1,3 , - 2 )\).
The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - y + 2 z = 5\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
Find a vector equation of the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE
FP1
2016
June
Q10
Standard +0.8
10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
- 2 & 1 & - 1 \\
0 & - 1 & 2 \\
0 & 0 & 1
\end{array} \right)$$
and find corresponding eigenvectors.
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), and hence find the matrix \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
[0pt]
[Question 11 is printed on the next page.]
CAIE
FP1
2016
June
Q15
Challenging +1.2
15
33
66
81
\end{array} \right)$$
has the form \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }\), where \(\lambda\) and \(\mu\) are scalars and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\).
Hence obtain a solution \(\mathbf { x } ^ { \prime }\) of ( \(*\) ) such that the sum of the components \(\mathbf { x } ^ { \prime }\) is 6 and the sum of the squares of the components of \(\mathbf { x } ^ { \prime }\) is 26 .
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CAIE
FP1
2017
June
Q7
Challenging +1.2
7 By finding a cubic equation whose roots are \(\alpha , \beta\) and \(\gamma\), solve the set of simultaneous equations
$$\begin{aligned}
\alpha + \beta + \gamma & = - 1 , \\
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 29 , \\
\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - 1 .
\end{aligned}$$