CAIE
FP1
2015
June
Q3
7 marks
3 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } > 5\) and \(a _ { n + 1 } = \frac { 4 a _ { n } } { 5 } + \frac { 5 } { a _ { n } }\) for every positive integer \(n\).
Prove by mathematical induction that \(a _ { n } > 5\) for every positive integer \(n\).
Prove also that \(a _ { n } > a _ { n + 1 }\) for every positive integer \(n\).
CAIE
FP1
2015
June
Q5
9 marks
Standard +0.8
5 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations
$$\begin{array} { l l }
C _ { 1 } : & r = \frac { 1 } { \sqrt { 2 } } , \quad \text { for } 0 \leqslant \theta < 2 \pi \\
C _ { 2 } : & r = \sqrt { } \left( \sin \frac { 1 } { 2 } \theta \right) , \quad \text { for } 0 \leqslant \theta \leqslant \pi
\end{array}$$
Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
Find the exact value of the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the half-line \(\theta = 0\).
CAIE
FP1
2015
June
Q6
9 marks
Standard +0.3
6 A curve has equation \(x ^ { 2 } - 6 x y + 25 y ^ { 2 } = 16\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at the point \(( 3,1 )\).
By finding the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 3,1 )\), determine the nature of this turning point.
CAIE
FP1
2015
June
Q8
11 marks
Challenging +1.8
8 By considering \(\sum _ { r = 1 } ^ { n } z ^ { 2 r - 1 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin \theta \neq 0\),
$$\sum _ { r = 1 } ^ { n } \sin ( 2 r - 1 ) \theta = \frac { \sin ^ { 2 } n \theta } { \sin \theta }$$
Deduce that
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) \cos \left[ \frac { ( 2 r - 1 ) \pi } { 2 n } \right] = - \operatorname { cosec } \left( \frac { \pi } { 2 n } \right) \cot \left( \frac { \pi } { 2 n } \right)$$
CAIE
FP1
2015
June
Q9
11 marks
Challenging +1.2
9 The curve \(C\) has parametric equations
$$x = 4 t + 2 t ^ { \frac { 3 } { 2 } } , \quad y = 4 t - 2 t ^ { \frac { 3 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 4$$
Find the arc length of \(C\), giving your answer correct to 3 significant figures.
Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 32\).
CAIE
FP1
2015
June
Q10
12 marks
Challenging +1.2
10 The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r }
2 & 2 & - 3 \\
2 & 2 & 3 \\
- 3 & 3 & 3
\end{array} \right)$$
The matrix \(\mathbf { A }\) has an eigenvector \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\). Find the corresponding eigenvalue.
The matrix \(\mathbf { A }\) also has eigenvalues 4 and 6. Find corresponding eigenvectors.
Hence find a matrix \(\mathbf { P }\) such that \(\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\), where \(\mathbf { D }\) is a diagonal matrix which is to be determined.
The matrix \(\mathbf { B }\) is such that \(\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }\), where
$$\mathbf { Q } = \left( \begin{array} { r r r }
4 & 11 & 5 \\
1 & 4 & 2 \\
1 & 2 & 1
\end{array} \right)$$
By using the expression \(\mathbf { P D P } ^ { - 1 }\) for \(\mathbf { A }\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf { B }\).
[0pt]
[Question 11 is printed on the next page.]
CAIE
FP1
2016
June
Q1
4 marks
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
9 marks
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
11 marks
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
12 marks
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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