CAIE
FP1
2010
November
Q6
6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r r }
1 & 2 & - 1 & \alpha
2 & 3 & - 1 & 0
2 & 1 & 2 & - 2
0 & 1 & - 3 & - 2
\end{array} \right)$$
Given that the dimension of the range space of T is 4 , show that \(\alpha \neq 1\).
It is now given that \(\alpha = 1\). Show that the vectors
$$\left( \begin{array} { l }
1
2
2
0
\end{array} \right) , \quad \left( \begin{array} { l }
2
3
1
1
\end{array} \right) \quad \text { and } \quad \left( \begin{array} { r }
- 1
- 1
2
- 3
\end{array} \right)$$
form a basis for the range space of T .
Given also that the vector \(\left( \begin{array} { c } p
1
1
q \end{array} \right)\) is in the range space of T , find a condition satisfied by \(p\) and \(q\).
CAIE
FP1
2010
November
Q7
7 The roots of the equation \(x ^ { 3 } + 4 x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = \frac { 1 } { 1 + x }\) to show that the equation \(6 y ^ { 3 } - 7 y ^ { 2 } + 3 y - 1 = 0\) has roots \(\frac { 1 } { \alpha + 1 } , \frac { 1 } { \beta + 1 }\) and \(\frac { 1 } { \gamma + 1 }\).
For the cases \(n = 1\) and \(n = 2\), find the value of
$$\frac { 1 } { ( \alpha + 1 ) ^ { n } } + \frac { 1 } { ( \beta + 1 ) ^ { n } } + \frac { 1 } { ( \gamma + 1 ) ^ { n } }$$
Deduce the value of \(\frac { 1 } { ( \alpha + 1 ) ^ { 3 } } + \frac { 1 } { ( \beta + 1 ) ^ { 3 } } + \frac { 1 } { ( \gamma + 1 ) ^ { 3 } }\).
Hence show that \(\frac { ( \beta + 1 ) ( \gamma + 1 ) } { ( \alpha + 1 ) ^ { 2 } } + \frac { ( \gamma + 1 ) ( \alpha + 1 ) } { ( \beta + 1 ) ^ { 2 } } + \frac { ( \alpha + 1 ) ( \beta + 1 ) } { ( \gamma + 1 ) ^ { 2 } } = \frac { 73 } { 36 }\).
CAIE
FP1
2010
November
Q10
10 By using de Moivre's theorem to express \(\sin 5 \theta\) and \(\cos 5 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\), show that
$$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$
where \(t = \tan \theta\).
Show that the roots of the equation \(x ^ { 4 } - 10 x ^ { 2 } + 5 = 0\) are \(\tan \left( \frac { 1 } { 5 } n \pi \right)\) for \(n = 1,2,3,4\).
By considering the product of the roots of this equation, find the exact value of \(\tan \left( \frac { 1 } { 5 } \pi \right) \tan \left( \frac { 2 } { 5 } \pi \right)\).
CAIE
FP1
2010
November
Q12 OR
The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )\). Obtain a cartesian equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = d\).
The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }\) and is parallel to \(3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }\), where \(a\) and \(c\) are positive constants. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\frac { 15 } { \sqrt { } 6 }\) and that the acute angle between \(l\) and \(\Pi _ { 1 }\) is \(\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)\), find the values of \(a\) and \(c\).
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CAIE
FP1
2011
November
Q8
8 The curve \(C\) has polar equation \(r = 1 + \sin \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\).
The area of the region enclosed by the initial line, the half-line \(\theta = \frac { 1 } { 2 } \pi\), and the part of \(C\) for which \(\theta\) is positive, is denoted by \(A _ { 1 }\). The area of the region enclosed by the initial line, and the part of \(C\) for which \(\theta\) is negative, is denoted by \(A _ { 2 }\). Find the ratio \(A _ { 1 } : A _ { 2 }\), giving your answer correct to 1 decimal place.
CAIE
FP1
2011
November
Q10
10 A curve \(C\) has equation
$$y = \frac { 5 \left( x ^ { 2 } - x - 2 \right) } { x ^ { 2 } + 5 x + 10 }$$
Find the coordinates of the points of intersection of \(C\) with the axes.
Show that, for all real values of \(x , - 1 \leqslant y \leqslant 15\).
Sketch \(C\), stating the coordinates of any turning points and the equation of the horizontal asymptote.
[0pt]
[Question 11 is printed on the next page.]
CAIE
FP1
2011
November
Q11 OR
Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where
$$\mathbf { A } = \left( \begin{array} { r r r }
1 & 1 & 2
0 & 2 & 2
- 1 & 1 & 3
\end{array} \right)$$
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is defined by \(\mathbf { x } \mapsto \mathbf { A x }\). Let \(\mathbf { e } , \mathbf { f }\) be two linearly independent eigenvectors of \(\mathbf { A }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively, and let \(\Pi\) be the plane, through the origin, containing \(\mathbf { e }\) and \(\mathbf { f }\). By considering the parametric equation of \(\Pi\), show that all points of \(\Pi\) are mapped by T onto points of \(\Pi\).
Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.