Edexcel F1 (Further Pure Mathematics 1) 2014 June

1. Find the value of

$$\sum _ { r = 1 } ^ { 200 } ( r + 1 ) ( r - 1 )$$

2. Given that \(- 2 + 3 \mathrm { i }\) is a root of the equation $$z ^ { 2 } + p z + q = 0$$ where \(p\) and \(q\) are real constants,

1. write down the other root of the equation.
2. Find the value of \(p\) and the value of \(q\).

3. $$\mathbf { A } = \left( \begin{array} { l l } 4 & - 2
a & - 3 \end{array} \right)$$ where \(a\) is a real constant and \(a \neq 6\)

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\). Given that \(\mathbf { A } + 2 \mathbf { A } ^ { - 1 } = \mathbf { I }\), where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(a\).

4. $$\mathrm { f } ( x ) = x ^ { \frac { 3 } { 2 } } - 3 x ^ { \frac { 1 } { 2 } } - 3 , \quad x > 0$$ Given that \(\alpha\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\),

1. show that \(4 < \alpha < 5\)
2. Taking 4.5 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to find a second approximation to \(\alpha\), giving your answer to 3 decimal places.
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3. Use linear interpolation once on the interval [4,5] to find another approximation to \(\alpha\), giving your answer to 3 decimal places.

1. Given that \(z _ { 1 } = - 3 - 4 \mathrm { i }\) and \(z _ { 2 } = 4 - 3 \mathrm { i }\)
   1. show, on an Argand diagram, the point \(P\) representing \(z _ { 1 }\) and the point \(Q\) representing \(z _ { 2 }\)
   2. Given that \(O\) is the origin, show that \(O P\) is perpendicular to \(O Q\).
   3. Show the point \(R\) on your diagram, where \(R\) represents \(z _ { 1 } + z _ { 2 }\)
   4. Prove that \(O P R Q\) is a square.
      

6. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(3 x ^ { 2 } + 5 x - 1 = 0\)

1. Find the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
2. Find a quadratic equation which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers.

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 }
\frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$

1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { P }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a reflection in the \(x\)-axis.
2. Write down the matrix \(\mathbf { Q }\). Given that \(V\) followed by \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a real number \(k\) for which the transformation \(T\) maps the point \(( 1 , k )\) onto itself. Give the exact value of \(k\) in its simplest form.

8. The hyperbola \(H\) has cartesian equation \(x y = 16\) The parabola \(P\) has parametric equations \(x = 8 t ^ { 2 } , y = 16 t\).

1. Find, using algebra, the coordinates of the point \(A\) where \(H\) meets \(P\). Another point \(B ( 8,2 )\) lies on the hyperbola \(H\).
2. Find the equation of the normal to \(H\) at the point (8, 2), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
3. Find the coordinates of the points where this normal at \(B\) meets the parabola \(P\).

9. (i) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 ) = \frac { n ( n + 1 ) ( n + 2 ) ( n + 3 ) } { 4 }$$ (ii) Prove by induction that, $$4 ^ { n } + 6 n + 8 \text { is divisible by } 18$$ for all positive integers \(n\).
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