CAIE FP1 (Further Pure Mathematics 1) 2016 June

1 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 7 = 0\) are \(\alpha , \beta\) and \(\gamma\). Using the substitution \(y = 1 + \frac { 1 } { x }\), or otherwise, find the cubic equation whose roots are \(1 + \frac { 1 } { \alpha } , 1 + \frac { 1 } { \beta }\) and \(1 + \frac { 1 } { \gamma }\), giving your answer in the form \(a y ^ { 3 } + b y ^ { 2 } + c y + d = 0\), where \(a , b , c\) and \(d\) are constants to be found.

2 Express \(\frac { 4 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\).

3 Prove by mathematical induction that, for all positive integers \(n , 10 ^ { n } + 3 \times 4 ^ { n + 2 } + 5\) is divisible by 9 .

4 A curve \(C\) has polar equation \(r ^ { 2 } = 8 \operatorname { cosec } 2 \theta\) for \(0 < \theta < \frac { 1 } { 2 } \pi\). Find a cartesian equation of \(C\). Sketch \(C\). Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\), the half-line \(\theta = \frac { 1 } { 6 } \pi\) and the half-line \(\theta = \frac { 1 } { 3 } \pi\).
[0pt] [It is given that \(\int \operatorname { cosec } x \mathrm {~d} x = \ln \left| \tan \frac { 1 } { 2 } x \right| + c\).]

5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \sin ^ { 2 } x \mathrm {~d} x\), for \(n \geqslant 0\). By differentiating \(\cos ^ { n - 1 } x \sin ^ { 3 } x\) with respect to \(x\), prove that $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 } \quad \text { for } n \geqslant 2$$ Hence find the exact value of \(I _ { 4 }\).

6 Use de Moivre's theorem to express \(\cot 7 \theta\) in terms of \(\cot \theta\). Use the equation \(\cot 7 \theta = 0\) to show that the roots of the equation $$x ^ { 6 } - 21 x ^ { 4 } + 35 x ^ { 2 } - 7 = 0$$ are \(\cot \left( \frac { 1 } { 14 } k \pi \right)\) for \(k = 1,3,5,9,11,13\), and deduce that $$\cot ^ { 2 } \left( \frac { 1 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 3 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 5 } { 14 } \pi \right) = 7$$

7 A curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(0 < y < 8\). Sketch \(C\), giving the coordinates of the turning points.

8 Find a cartesian equation of the plane \(\Pi _ { 1 }\) passing through the points with coordinates \(( 2 , - 1,3 )\), \(( 4,2 , - 5 )\) and \(( - 1,3 , - 2 )\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - y + 2 z = 5\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

9 Find the value of the constant \(k\) such that \(y = k x ^ { 2 } \mathrm { e } ^ { 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 \mathrm { e } ^ { 2 x }$$ Hence find the general solution of ( \(*\) ). Find the particular solution of ( \(*\) ) such that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2\) when \(x = 0\).

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 1 \\ 0 & - 1 & 2 \\ 0 & 0 & 1 \end{array} \right)$$ and find corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), and hence find the matrix \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
[0pt] [Question 11 is printed on the next page.]

A curve \(C\) has parametric equations $$x = \mathrm { e } ^ { 2 t } \cos 2 t , \quad y = \mathrm { e } ^ { 2 t } \sin 2 t , \quad \text { for } - \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi .$$ 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.

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 3 & - 4 \\ 2 & - 4 & 7 & - 9 \\ 4 & - 8 & 14 & - 18 \\ 5 & - 10 & 17 & - 22 \end{array} \right)$$ Find the rank of \(\mathbf { M }\). Obtain a basis for the null space \(K\) of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right)$$ and hence show that any solution of $$\mathbf { M x } = \left( \begin{array} { l } 15 \\ 33 \\ 66 \\ 81 \end{array} \right)$$

has the form \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }\), where \(\lambda\) and \(\mu\) are scalars and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). Hence obtain a solution \(\mathbf { x } ^ { \prime }\) of ( \(*\) ) such that the sum of the components \(\mathbf { x } ^ { \prime }\) is 6 and the sum of the squares of the components of \(\mathbf { x } ^ { \prime }\) is 26 . {www.cie.org.uk} after the live examination series. }