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CAIE FP1 2011 November Q1
1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).
CAIE FP1 2011 November Q2
2 The position vectors of points \(A , B , C\), relative to the origin \(O\), are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\), where $$\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { c } = 3 \mathbf { i } - \mathbf { j } - \mathbf { k }$$ Find \(\mathbf { a } \times \mathbf { b }\) and deduce the area of the triangle \(O A B\). Hence find the volume of the tetrahedron \(O A B C\), given that the volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.
CAIE FP1 2011 November Q3
3 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = 2 ^ { \frac { 1 } { 2 } n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$
CAIE FP1 2011 November Q4
4 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 4 & 2 & 5
6 & 7 & 5 & 8
9 & 9 & 9 & 9
15 & 16 & 14 & 17 \end{array} \right)$$ Find
  1. the rank of \(\mathbf { M }\) and a basis for the range space of T ,
  2. a basis for the null space of T .
CAIE FP1 2011 November Q5
5 The point \(P ( 2,1 )\) lies on the curve with equation $$x ^ { 3 } - 2 y ^ { 3 } = 3 x y$$ Find
  1. the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\),
  2. the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(P\).
CAIE FP1 2011 November Q6
6 Let \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } ( 1 - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 1\), $$( 3 + 2 n ) I _ { n } = 2 n I _ { n - 1 }$$ Hence find the exact value of \(I _ { 3 }\).
CAIE FP1 2011 November Q7
7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + p x + 1 } { x - 2 }\), where \(p\) is a constant. Given that \(C\) has two asymptotes, find the equation of each asymptote. Find the set of values of \(p\) for which \(C\) has two distinct turning points. Sketch \(C\) in the case \(p = - 1\). Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.
CAIE FP1 2011 November Q8
8 The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\). State the eigenvalues of the matrix \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & - 1 & 3
0 & 1 & 2
0 & 0 & 2 \end{array} \right) ,$$ and find corresponding eigenvectors. Show that \(\left( \begin{array} { l } 1
6
3 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { D }\), where $$\mathbf { D } = \left( \begin{array} { r r r } 1 & - 1 & 1
- 6 & - 3 & 4
- 9 & - 3 & 7 \end{array} \right) ,$$ and state the corresponding eigenvalue. Hence state an eigenvector of the matrix CD and give the corresponding eigenvalue.
CAIE FP1 2011 November Q9
9 The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant \ln 5\). Find
  1. the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \ln 5\),
  2. the arc length of \(C\),
  3. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2011 November Q10
10 The curve \(C\) has polar equation \(r = 3 + 2 \cos \theta\), for \(- \pi < \theta \leqslant \pi\). The straight line \(l\) has polar equation \(r \cos \theta = 2\). Sketch both \(C\) and \(l\) on a single diagram. Find the polar coordinates of the points of intersection of \(C\) and \(l\). The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\).
CAIE FP1 2011 November Q11 EITHER
Let \(\omega = \cos \frac { 1 } { 5 } \pi + \mathrm { i } \sin \frac { 1 } { 5 } \pi\). Show that \(\omega ^ { 5 } + 1 = 0\) and deduce that $$\omega ^ { 4 } - \omega ^ { 3 } + \omega ^ { 2 } - \omega = - 1$$ Show further that $$\omega - \omega ^ { 4 } = 2 \cos \frac { 1 } { 5 } \pi \quad \text { and } \quad \omega ^ { 3 } - \omega ^ { 2 } = 2 \cos \frac { 3 } { 5 } \pi$$ Hence find the values of $$\cos \frac { 1 } { 5 } \pi + \cos \frac { 3 } { 5 } \pi \quad \text { and } \quad \cos \frac { 1 } { 5 } \pi \cos \frac { 3 } { 5 } \pi$$ Find a quadratic equation having roots \(\cos \frac { 1 } { 5 } \pi\) and \(\cos \frac { 3 } { 5 } \pi\) and deduce the exact value of \(\cos \frac { 1 } { 5 } \pi\).
CAIE FP1 2011 November Q11 OR
Given that $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 \left( 1 + 4 x + 2 x ^ { 2 } \right) y = 8 x ^ { 2 }$$ and that \(x ^ { 2 } y = z\), show that $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} z } { \mathrm {~d} x } + 4 z = 8 x ^ { 2 }$$ Find the general solution for \(y\) in terms of \(x\). Describe the behaviour of \(y\) as \(x \rightarrow \infty\).
CAIE FP1 2012 November Q10
10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16
0 & 2 & 3
0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).
CAIE FP1 2012 November Q11 OR
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4
3 & 4 & 6 & 1
- 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,
  1. show that the dimension of \(R\) is 2 ,
  2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
  3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8
    7
    k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M x } = \left( \begin{array} { l } 8
    7
    k \end{array} \right)$$
CAIE FP1 2012 November Q1
1 Show that \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = \frac { 1 } { 6 } n ( 2 n + 1 ) ( 7 n + 1 )\).
CAIE FP1 2012 November Q2
2 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} a x + y + 2 z & = 0
3 x - 2 y & = 4
3 x - 4 y - 6 a z & = 14 \end{aligned}$$ has a unique solution.
CAIE FP1 2012 November Q3
3 Let \(S _ { N } = \frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { N } { ( N + 1 ) ! }\). Prove by mathematical induction that, for all positive integers \(N\), $$S _ { N } = 1 - \frac { 1 } { ( N + 1 ) ! }$$
CAIE FP1 2012 November Q4
4 The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } , 2 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }\) and \(2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\). Deduce, in either order, the exact value of
  1. the area of the triangle \(A B C\),
  2. the perpendicular distance from \(C\) to \(A B\).
CAIE FP1 2012 November Q5
5 The curve \(C\) has polar equation \(r = 1 + 2 \cos \theta\). Sketch the curve for \(- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi\). Find the area bounded by \(C\) and the half-lines \(\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi\).
CAIE FP1 2012 November Q6
6 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { 4 } t ^ { 4 } - \ln t$$ for \(1 \leqslant t \leqslant 2\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
CAIE FP1 2012 November Q7
7 A cubic equation has roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 4
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 14
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 34 \end{aligned}$$ Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\). Show that the cubic equation is $$x ^ { 3 } - 4 x ^ { 2 } + x + 6 = 0$$ and solve this equation.
CAIE FP1 2012 November Q8
8 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$ By considering \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, deduce the sum of the series $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$
CAIE FP1 2012 November Q9
9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 3 } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(- 1 < y < 3\). Find the coordinates of the turning points of \(C\). Sketch \(C\).
CAIE FP1 2012 November Q10
10 The curve \(C\) has equation \(x ^ { 3 } + y ^ { 3 } = 3 x y\), for \(x > 0\) and \(y > 0\). Find a relationship between \(x\) and \(y\) when \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\). Find the exact coordinates of the turning point of \(C\), and determine the nature of this turning point.
CAIE FP1 2012 November Q11
11 Show that \(\int x \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = - \frac { 1 } { 3 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } + c\), where \(c\) is a constant. Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), prove that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 }$$ Use the substitution \(x = \sin u\) to show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \pi$$ Find \(I _ { 4 }\).