CAIE FP1 (Further Pure Mathematics 1) 2011 November

1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).

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.

3 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = 2 ^ { \frac { 1 } { 2 } n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$

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 .

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 \(P\).

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 \(I _ { 3 }\).

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.

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.

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.

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 \(R\).

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 \(\cos \frac { 1 } { 5 } \pi\).

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 \(x \rightarrow \infty\).