Edexcel FP1 (Further Pure Mathematics 1) 2012 June

1. $$f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } - 7 x - 4$$

1. Show that \(\mathrm { f } ( 4 ) = 0\)
2. Use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.

2. (a) Given that $$\mathbf { A } = \left( \begin{array} { l l l } 3 & 1 & 3
4 & 5 & 5 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & 1
1 & 2
0 & - 1 \end{array} \right)$$ find \(\mathbf { A B }\).
(b) Given that $$\mathbf { C } = \left( \begin{array} { l l } 3 & 2
8 & 6 \end{array} \right) , \quad \mathbf { D } = \left( \begin{array} { r r } 5 & 2 k
4 & k \end{array} \right) , \text { where } k \text { is a constant }$$ and $$\mathbf { E } = \mathbf { C } + \mathbf { D }$$ find the value of \(k\) for which \(\mathbf { E }\) has no inverse.

3. $$f ( x ) = x ^ { 2 } + \frac { 3 } { 4 \sqrt { } x } - 3 x - 7 , \quad x > 0$$ A root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ 3,5 ]\).
Taking 4 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 2 decimal places.

4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } + 6 r - 3 \right) = \frac { 1 } { 4 } n ^ { 2 } \left( n ^ { 2 } + 2 n + 13 \right)$$ for all positive integers \(n\).
(b) Hence find the exact value of $$\sum _ { r = 16 } ^ { 30 } \left( r ^ { 3 } + 6 r - 3 \right)$$

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 8 x\). The point \(P\) lies on \(C\), where \(y > 0\), and the point \(Q\) lies on \(C\), where \(y < 0\) The line segment \(P Q\) is parallel to the \(y\)-axis. Given that the distance \(P Q\) is 12 ,

1. write down the \(y\)-coordinate of \(P\),
2. find the \(x\)-coordinate of \(P\). Figure 1 shows the point \(S\) which is the focus of \(C\).
   The line \(l\) passes through the point \(P\) and the point \(S\).
3. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

6. $$f ( x ) = \tan \left( \frac { x } { 2 } \right) + 3 x - 6 , \quad - \pi < x < \pi$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 1,2 ]\).
2. Use linear interpolation once on the interval \([ 1,2 ]\) to find an approximation to \(\alpha\). Give your answer to 2 decimal places.

7. $$z = 2 - \mathrm { i } \sqrt { } 3$$

1. Calculate \(\arg z\), giving your answer in radians to 2 decimal places. Use algebra to express
2. \(z + z ^ { 2 }\) in the form \(a + b \mathrm { i } \sqrt { } 3\), where \(a\) and \(b\) are integers,
3. \(\frac { z + 7 } { z - 1 }\) in the form \(c + d \mathrm { i } \sqrt { } 3\), where \(c\) and \(d\) are integers. Given that $$w = \lambda - 3 \mathrm { i }$$ where \(\lambda\) is a real constant, and \(\arg ( 4 - 5 \mathrm { i } + 3 w ) = - \frac { \pi } { 2 }\),
4. find the value of \(\lambda\).

8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a positive constant. The point \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\).

1. Show that an equation for the tangent to \(H\) at \(P\) is $$x + t ^ { 2 } y = 2 c t$$ The tangent to \(H\) at the point \(P\) meets the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Given that the area of the triangle \(O A B\), where \(O\) is the origin, is 36 ,
2. find the exact value of \(c\), expressing your answer in the form \(k \sqrt { } 2\), where \(k\) is an integer.

9. $$\mathbf { M } = \left( \begin{array} { r r } 3 & 4
2 & - 5 \end{array} \right)$$

1. Find \(\operatorname { det } \mathbf { M }\). The transformation represented by \(\mathbf { M }\) maps the point \(S ( 2 a - 7 , a - 1 )\), where \(a\) is a constant, onto the point \(S ^ { \prime } ( 25 , - 14 )\).
2. Find the value of \(a\). The point \(R\) has coordinates \(( 6,0 )\). Given that \(O\) is the origin,
3. find the area of triangle \(O R S\). Triangle \(O R S\) is mapped onto triangle \(O R ^ { \prime } S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
4. Find the area of triangle \(O R ^ { \prime } S ^ { \prime }\). Given that $$\mathbf { A } = \left( \begin{array} { r r } 0 & - 1
   1 & 0 \end{array} \right)$$
5. describe fully the single geometrical transformation represented by \(\mathbf { A }\). The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by \(\mathbf { M }\).
6. Find B.

10. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$f ( n ) = 2 ^ { 2 n - 1 } + 3 ^ { 2 n - 1 } \text { is divisible by } 5 .$$