CAIE FP1 (Further Pure Mathematics 1) 2014 June

1 The equation \(x ^ { 3 } + p x + q = 0\), where \(p\) and \(q\) are constants, with \(q \neq 0\), has one root which is the reciprocal of another root. Prove that \(p + q ^ { 2 } = 1\).

2 Expand and simplify \(( r + 1 ) ^ { 4 } - r ^ { 4 }\). Use the method of differences together with the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$

3 Prove by mathematical induction that, for all non-negative integers \(n\), $$11 ^ { 2 n } + 25 ^ { n } + 22$$ is divisible by 24 .

4 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 195 \sin 2 t$$

5 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\). Draw a sketch of \(C\). Find the exact value of the area of the region enclosed by \(C\) and the half-lines \(\theta = \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 2 } { 3 } \pi\).

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & - 1 & 1 & 3
2 & 0 & 0 & 5
6 & - 2 & 2 & 11
10 & - 3 & 3 & 19 \end{array} \right)$$

1. Find the rank of \(\mathbf { M }\) and state a basis for the range space of T .
2. Obtain a basis for the null space of T .

7 Use de Moivre's theorem to show that $$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$ where \(t = \tan \theta\). Deduce that the roots of the equation \(t ^ { 4 } - 10 t ^ { 2 } + 5 = 0\) are \(\pm \tan \frac { 1 } { 5 } \pi\) and \(\pm \tan \frac { 2 } { 5 } \pi\). Hence show that \(\tan \frac { 1 } { 5 } \pi \tan \frac { 2 } { 5 } \pi = \sqrt { } 5\).

8 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find

1. the arc length of \(C\),
2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

9 The matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 2 & 2
2 & 1 & 2
- 3 & - 6 & - 7 \end{array} \right)$$ has an eigenvector \(\left( \begin{array} { r } 0
1
- 1 \end{array} \right)\). Find the corresponding eigenvalue. It is given that if the eigenvalues of a general \(3 \times 3\) matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l l } a & b & c
d & e & f
g & h & i \end{array} \right)$$ are \(\lambda _ { 1 } , \lambda _ { 2 }\) and \(\lambda _ { 3 }\) then $$\lambda _ { 1 } + \lambda _ { 2 } + \lambda _ { 3 } = a + e + i$$ and the determinant of \(\mathbf { A }\) has the value \(\lambda _ { 1 } \lambda _ { 2 } \lambda _ { 3 }\). Use these results to find the other two eigenvalues of the matrix \(\mathbf { M }\), and find corresponding eigenvectors.

10 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 2 n } x } { \cos x } \mathrm {~d} x\), where \(n \geqslant 0\). Show that $$I _ { n } - I _ { n + 1 } = \frac { 2 ^ { - \left( n + \frac { 1 } { 2 } \right) } } { 2 n + 1 }$$ Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 6 } x } { \cos x } \mathrm {~d} x = \ln ( 1 + \sqrt { } 2 ) - \frac { 73 } { 120 } \sqrt { } 2\).

11 The line \(l _ { 1 }\) passes through the points \(A ( 2,3 , - 5 )\) and \(B ( 8,7 , - 13 )\). The line \(l _ { 2 }\) passes through the points \(C ( - 2,1,8 )\) and \(D ( 3 , - 1,4 )\). Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\). The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(D\). The plane \(\Pi _ { 2 }\) passes though the points \(A , C\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.

The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = ( 2 - t ) ^ { \frac { 1 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$ Find

1. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\),
2. the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 4\),
3. the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the \(x\)-axis and the \(y\)-axis.

The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$ where \(a , b , c\) and \(d\) are constants. The curve cuts the \(y\)-axis at \(( 0 , - 2 )\) and has asymptotes \(x = 2\) and \(y = x + 1\).

1. Write down the value of \(d\).
2. Determine the values of \(a , b\) and \(c\).
3. Show that, at all points on \(C\), either \(y \leqslant 3 - 2 \sqrt { 6 }\) or \(y \geqslant 3 + 2 \sqrt { 6 }\).