CAIE FP1 (Further Pure Mathematics 1) 2006 November

1 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & - 2 \\ 0 & 2 & 1 \\ 0 & 0 & - 3 \end{array} \right)$$ Write down the eigenvalues of \(\mathbf { A }\) and find corresponding eigenvectors.

2 The integral \(I _ { n }\), where \(n\) is a non-negative integer, is defined by $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 3 } } \mathrm {~d} x$$ By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { n + 1 } \mathrm { e } ^ { - x ^ { 3 } } \right)\) or otherwise, show that $$3 I _ { n + 3 } = ( n + 1 ) I _ { n } - \mathrm { e } ^ { - 1 }$$ Hence find \(I _ { 6 }\) in terms of e and \(I _ { 0 }\).

3 Verify that if $$v _ { n } = n ( n + 1 ) ( n + 2 ) \ldots ( n + m )$$ then $$v _ { n + 1 } - v _ { n } = ( m + 1 ) ( n + 1 ) ( n + 2 ) \ldots ( n + m ) .$$ Given now that $$u _ { n } = ( n + 1 ) ( n + 2 ) \ldots ( n + m ) ,$$ find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(m\) and \(N\).

4 Prove by mathematical induction that, for all positive integers \(n , 10 ^ { 3 n } + 13 ^ { n + 1 }\) is divisible by 7 .

5 Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} & 2 x + 3 y + 4 z = - 5 \\ & 4 x + 5 y - z = 5 a + 15 \\ & 6 x + 8 y + a z = b - 2 a + 21 \end{aligned}$$ has a unique solution. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.

6 The roots of the equation $$x ^ { 3 } + x + 1 = 0$$ are \(\alpha , \beta , \gamma\). Show that the equation whose roots are $$\frac { 4 \alpha + 1 } { \alpha + 1 } , \quad \frac { 4 \beta + 1 } { \beta + 1 } , \quad \frac { 4 \gamma + 1 } { \gamma + 1 }$$ is of the form $$y ^ { 3 } + p y + q = 0$$ where the numbers \(p\) and \(q\) are to be determined. Hence find the value of $$\left( \frac { 4 \alpha + 1 } { \alpha + 1 } \right) ^ { n } + \left( \frac { 4 \beta + 1 } { \beta + 1 } \right) ^ { n } + \left( \frac { 4 \gamma + 1 } { \gamma + 1 } \right) ^ { n }$$ for \(n = 2\) and for \(n = 3\).

7 The curve \(C\) has equation $$r = 10 \ln ( 1 + \theta )$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Use the substitution \(w = \ln ( 1 + \theta )\) to show that the area of the sector bounded by the line \(\theta = \frac { 1 } { 2 } \pi\) and the arc of \(C\) joining the origin to the point where \(\theta = \frac { 1 } { 2 } \pi\) is $$50 \left( b ^ { 2 } - 2 b + 2 \right) \mathrm { e } ^ { b } - 100$$ where \(b = \ln \left( 1 + \frac { 1 } { 2 } \pi \right)\).

8 Given that $$2 y ^ { 3 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 12 y ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y ^ { 2 } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 17 y ^ { 4 } = 13 \mathrm { e } ^ { - 4 x }$$ and that \(v = y ^ { 4 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 34 v = 26 \mathrm { e } ^ { - 4 x }$$ Hence find the general solution for \(y\) in terms of \(x\).

9 With \(O\) as origin, the points \(A , B , C\) have position vectors $$\mathbf { i } , \quad \mathbf { i } + \mathbf { j } , \quad \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$ respectively. Find a vector equation of the common perpendicular of the lines \(A B\) and \(O C\). Show that the shortest distance between the lines \(A B\) and \(O C\) is \(\frac { 2 } { 5 } \sqrt { } 5\). Find, in the form \(a x + b y + c z = d\), an equation for the plane containing \(A B\) and the common perpendicular of the lines \(A B\) and \(O C\).

10 The curve \(C\) has equation $$y = x ^ { 2 } + \lambda \sin ( x + y ) ,$$ where \(\lambda\) is a constant, and passes through the point \(A \left( \frac { 1 } { 4 } \pi , \frac { 1 } { 4 } \pi \right)\). Show that \(C\) has no tangent which is parallel to the \(y\)-axis. Show that, at \(A\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 - \frac { 1 } { 64 } \pi ( 4 - \pi ) ( \pi + 2 ) ^ { 2 }$$

11 Prove de Moivre's theorem for a positive integral exponent: $$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$ Use de Moivre's theorem to show that $$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ Hence obtain the roots of the equation $$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$ in the form \(\cos q \pi\), where \(q\) is a rational number.

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where \(q\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
4. It is given that, for all values of the constant \(\lambda\), the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).

The curve \(C\) has equation $$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$ where \(\lambda > 0\) and \(0 \leqslant x \leqslant 3\). The length of \(C\) is denoted by \(s\). Prove that \(s = 2 \sqrt { } 3\). The area of the surface generated when \(C\) is rotated through one revolution about the \(x\)-axis is denoted by \(S\). Find \(S\) in terms of \(\lambda\). The \(y\)-coordinate of the centroid of the region bounded by \(C\), the axes and the line \(x = 3\) is denoted by h. Given that \(\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }\), show that $$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$