Edexcel FP1 (Further Pure Mathematics 1) 2013 June

1. $$\mathbf { M } = \left( \begin{array} { c c } x & x - 2
3 x - 6 & 4 x - 11 \end{array} \right)$$ Given that the matrix \(\mathbf { M }\) is singular, find the possible values of \(x\).

2. $$\mathrm { f } ( x ) = \cos \left( x ^ { 2 } \right) - x + 3 , \quad 0 < x < \pi$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 2.5,3 ]\).
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2. Use linear interpolation once on the interval [2.5,3] to find an approximation for \(\alpha\), giving your answer to 2 decimal places.

3. Given that \(x = \frac { 1 } { 2 }\) is a root of the equation $$2 x ^ { 3 } - 9 x ^ { 2 } + k x - 13 = 0 , \quad k \in \mathbb { R }$$ find

1. the value of \(k\),
2. the other 2 roots of the equation.

4. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 4\) The point \(P \left( 2 t , \frac { 2 } { t } \right)\) lies on \(H\), where \(t \neq 0\)

1. Show that an equation of the normal to \(H\) at the point \(P\) is $$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$ The normal to \(H\) at the point where \(t = - \frac { 1 } { 2 }\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the point \(Q\).

5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } ( r + 2 ) ( r + 3 ) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 9 n + 26 \right)$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 3 n } ( r + 2 ) ( r + 3 ) = \frac { 2 } { 3 } n \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be found.

6. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x , \quad a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(Q \left( a q ^ { 2 } , 2 a q \right)\) lie on \(C\), where \(p \neq 0 , q \neq 0 , p \neq q\).

1. Show that an equation of the tangent to the parabola at \(P\) is $$p y - x = a p ^ { 2 }$$
2. Write down the equation of the tangent at \(Q\). The tangent at \(P\) meets the tangent at \(Q\) at the point \(R\).
3. Find, in terms of \(p\) and \(q\), the coordinates of \(R\), giving your answers in their simplest form. Given that \(R\) lies on the directrix of \(C\),
4. find the value of \(p q\).

7. $$z _ { 1 } = 2 + 3 \mathrm { i } , \quad z _ { 2 } = 3 + 2 \mathrm { i } , \quad z _ { 3 } = a + b \mathrm { i } , \quad a , b \in \mathbb { R }$$

1. Find the exact value of \(\left| z _ { 1 } + z _ { 2 } \right|\). Given that \(w = \frac { z _ { 1 } z _ { 3 } } { z _ { 2 } }\),
2. find \(w\) in terms of \(a\) and \(b\), giving your answer in the form \(x + \mathrm { i } y , \quad x , y \in \mathbb { R }\) Given also that \(w = \frac { 17 } { 13 } - \frac { 7 } { 13 } \mathrm { i }\),
3. find the value of \(a\) and the value of \(b\),
4. find \(\arg w\), giving your answer in radians to 3 decimal places.

8. $$\mathbf { A } = \left( \begin{array} { c c } 6 & - 2
- 4 & 1 \end{array} \right)$$ and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

1. Prove that $$\mathbf { A } ^ { 2 } = 7 \mathbf { A } + 2 \mathbf { I }$$
2. Hence show that $$\mathbf { A } ^ { - 1 } = \frac { 1 } { 2 } ( \mathbf { A } - 7 \mathbf { I } )$$ The transformation represented by \(\mathbf { A }\) maps the point \(P\) onto the point \(Q\).
   Given that \(Q\) has coordinates \(( 2 k + 8 , - 2 k - 5 )\), where \(k\) is a constant,
3. find, in terms of \(k\), the coordinates of \(P\).

9. (a) A sequence of numbers is defined by $$\begin{aligned} & u _ { 1 } = 8
& u _ { n + 1 } = 4 u _ { n } - 9 n , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 4 ^ { n } + 3 n + 1$$ (b) Prove by induction that, for \(m \in \mathbb { Z } ^ { + }\), $$\left( \begin{array} { l l } 3 & - 4
1 & - 1 \end{array} \right) ^ { m } = \left( \begin{array} { c c } 2 m + 1 & - 4 m
m & 1 - 2 m \end{array} \right)$$