CAIE FP1 (Further Pure Mathematics 1) 2017 June

1 It is given that \(\sum _ { r = 1 } ^ { n } u _ { r } = n ^ { 2 } ( 2 n + 3 )\), where \(n\) is a positive integer.

1. Find \(\sum _ { r = n + 1 } ^ { 2 n } u _ { r }\).
2. Find \(u _ { r }\).

2 Prove, by mathematical induction, that \(5 ^ { n } + 3\) is divisible by 4 for all non-negative integers \(n\).

3 A curve \(C\) has equation \(\tan y = x\), for \(x > 0\).

1. Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
2. Hence find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( 1 , \frac { 1 } { 4 } \pi \right)\) on \(C\).

4

1. Find the value of \(k\) for which the set of linear equations $$\begin{aligned} x + 3 y + k z & = 4 \\ 4 x - 2 y - 10 z & = - 5 \\ x + y + 2 z & = 1 \end{aligned}$$ has no unique solution.
2. For this value of \(k\), find the set of possible solutions, giving your answer in the form $$\left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \mathbf { a } + t \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors and \(t\) is a scalar.

5 The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & - 2 \\ 6 & 4 & - 6 \\ 6 & 5 & - 7 \end{array} \right)$$ has eigenvalues \(1 , - 1\) and - 2 .

1. Find a set of corresponding eigenvectors.
2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \mathbf { A } - 2 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Write down the eigenvalues of \(\mathbf { B }\), and state a set of corresponding eigenvectors.

6 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\).

1. Prove that, for \(n \geqslant 2\), $$I _ { n } + n ( n - 1 ) I _ { n - 2 } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } .$$
2. Calculate the exact value of \(I _ { 1 }\) and deduce the exact value of \(I _ { 3 }\).

7 By finding a cubic equation whose roots are \(\alpha , \beta\) and \(\gamma\), solve the set of simultaneous equations $$\begin{aligned} \alpha + \beta + \gamma & = - 1 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 29 , \\ \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - 1 . \end{aligned}$$

8

1. Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\) and hence express \(16 \sin ^ { 5 } \theta\) in the form \(\sin 5 \theta + p \sin 3 \theta + q \sin \theta\), where \(p\) and \(q\) are integers to be determined.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta\).

9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 6 } { 1 - x }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the turning points of \(C\).
3. Find the coordinates of any intersections with the coordinate axes.
4. Sketch \(C\).

10 It is given that \(x = t ^ { \frac { 1 } { 2 } }\), where \(x > 0\) and \(t > 0\), and \(y\) is a function of \(x\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 t ^ { \frac { 1 } { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 t \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Hence show that the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 8 x + \frac { 1 } { x } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 12 x ^ { 2 } y = 4 x ^ { 2 } \mathrm { e } ^ { - x ^ { 2 } }$$ reduces to the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = \mathrm { e } ^ { - t }$$
3. Find the general solution of ( \(*\) ), giving \(y\) in terms of \(x\).

11 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\) for \(- \pi < \theta \leqslant \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\).
3. Show that the length of the arc of \(C\) from the pole to the point furthest from the pole is given by $$s = ( \sqrt { } 2 ) a \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \sqrt { } ( 1 + \sin \theta ) \mathrm { d } \theta$$
4. Show that the substitution \(u = 1 + \sin \theta\) reduces this integral for \(s\) to \(( \sqrt { } 2 ) a \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { } ( 2 - u ) } \mathrm { d } u\). Hence evaluate \(s\).

The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant 4\).

1. The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = 4\). Find, in terms of e, the coordinates of the centroid of the region \(R\).
2. Show that \(\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\), where \(s\) denotes the arc length of \(C\), and find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

The position vectors of the points \(A , B , C , D\) are $$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad 5 \mathbf { i } - 5 \mathbf { j } + \alpha \mathbf { k } ,$$ respectively, where \(\alpha\) is a positive integer. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is equal to \(2 \sqrt { } 2\).

1. Show that the possible values of \(\alpha\) are 3 and 5 .
2. Using \(\alpha = 3\), find the shortest distance of the point \(D\) from the line \(A C\), giving your answer correct to 3 significant figures.
3. Using \(\alpha = 3\), find the acute angle between the planes \(A B C\) and \(A B D\), giving your answer in degrees.
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