CAIE FP1 (Further Pure Mathematics 1) 2011 November

1 Verify that \(\frac { 1 } { n ^ { 2 } } - \frac { 1 } { ( n + 1 ) ^ { 2 } } = \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } }\). Let \(S _ { N } = \sum _ { r = 1 } ^ { N } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\). Express \(S _ { N }\) in terms of \(N\). Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\). Find the least value of \(N\) such that \(S - S _ { N } < 10 ^ { - 16 }\).

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

3 The equation $$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$ has roots \(\alpha , \beta , \gamma\). Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\). Hence show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta
\alpha & 1 & \gamma
\beta & \gamma & 1 \end{array} \right)\) is singular.

4 A curve has parametric equations $$x = 2 \sin 2 t , \quad y = 3 \cos 2 t$$ for \(0 < t < \frac { 1 } { 2 } \pi\). For the point on the curve where \(t = \frac { 1 } { 3 } \pi\), find the value of

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

5 Use de Moivre's theorem to express \(\cos ^ { 4 } \theta\) in the form $$a \cos 4 \theta + b \cos 2 \theta + c$$ where \(a , b , c\) are constants to be found. Hence evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta d \theta$$ leaving your answer in terms of \(\pi\).

6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = \sin 2 t$$ Describe the behaviour of \(x\) as \(t \rightarrow \infty\), justifying your answer.

7 Show that \(\frac { \mathrm { d } } { \mathrm { d } t } \left( t \left( 1 + t ^ { 3 } \right) ^ { n } \right) = ( 3 n + 1 ) \left( 1 + t ^ { 3 } \right) ^ { n } - 3 n \left( 1 + t ^ { 3 } \right) ^ { n - 1 }\). Let \(I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + t ^ { 3 } \right) ^ { n } \mathrm {~d} t\). Using the above result, or otherwise, show that $$( 3 n + 1 ) I _ { n } = 2 ^ { n } + 3 n I _ { n - 1 }$$ Hence evaluate \(I _ { 3 }\).

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.

9 Find a cartesian equation of the plane \(\Pi\) containing the lines $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - 7 \mathbf { j } + 10 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )$$ The line \(l\) passes through the point \(P\) with position vector \(6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\). Find

1. the position vector of the point where \(l\) meets \(\Pi\),
2. the perpendicular distance from \(P\) to \(\Pi\),
3. the acute angle between \(l\) and \(\Pi\).

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.]

The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { \frac { 1 } { 2 } } ( 3 - x )\), for \(0 \leqslant x \leqslant 3\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 3\). Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( x ^ { - \frac { 1 } { 2 } } + x ^ { \frac { 1 } { 2 } } \right)$$ where \(s\) denotes arc length, and find the arc length of \(C\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

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.