Edexcel F1 (Further Pure Mathematics 1) 2018 June

1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { n } { a } ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are integers to be found.

1. The transformation represented by the \(2 \times 2\) matrix \(\mathbf { P }\) is an anticlockwise rotation about the origin through 45 degrees.
   1. Write down the matrix \(\mathbf { P }\), giving the exact numerical value of each element.
   $$\mathbf { Q } = \left( \begin{array} { c c } k \sqrt { 2 } & 0 \\ 0 & k \sqrt { 2 } \end{array} \right) \text {, where } k \text { is a constant and } k > 0$$
2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf { Q }\). The combined transformation represented by the matrix \(\mathbf { P Q }\) transforms the rhombus \(R _ { 1 }\) onto the rhombus \(R _ { 2 }\). The area of the rhombus \(R _ { 1 }\) is 6 and the area of the rhombus \(R _ { 2 }\) is 147
3. Find the value of the constant \(k\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the parabola \(C\) which has cartesian equation \(y ^ { 2 } = 6 x\). The point \(S\) is the focus of \(C\).

1. Find the coordinates of the point \(S\). The point \(P\) lies on the parabola \(C\), and the point \(Q\) lies on the directrix of \(C\). \(P Q\) is parallel to the \(x\)-axis with distance \(P Q = 14\)
2. State the distance \(S P\). Given that the point \(P\) is above the \(x\)-axis,
3. find the exact coordinates of \(P\).

4. $$\mathbf { A } = \left( \begin{array} { c c } 2 p & 3 q \\ 3 p & 5 q \end{array} \right)$$ where \(p\) and \(q\) are non-zero real constants.

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(p\) and \(q\). Given \(\mathbf { X A } = \mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { c c } p & q \\ 6 p & 11 q \\ 5 p & 8 q \end{array} \right)$$
2. find the matrix \(\mathbf { X }\), giving your answer in its simplest form.

5. Given that $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 \equiv \left( z ^ { 2 } + 9 \right) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real numbers,

1. find the value of \(a\) and the value of \(b\).
2. Hence find the exact roots of the equation $$z ^ { 4 } - 6 z ^ { 3 } + 34 z ^ { 2 } - 54 z + 225 = 0$$
3. Show your roots on a single Argand diagram.

6. $$f ( x ) = \frac { 2 \left( x ^ { 3 } + 3 \right) } { \sqrt { x } } - 9 , \quad x > 0$$ The equation \(\mathrm { f } ( x ) = 0\) has two real roots \(\alpha\) and \(\beta\), where \(0.4 < \alpha < 0.5\) and \(1.2 < \beta < 1.3\)

1. Taking 0.45 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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2. Use linear interpolation once on the interval [1.2, 1.3] to find an approximation to \(\beta\), giving your answer to 3 decimal places.

7. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(5 x ^ { 2 } - 4 x + 3 = 0\) Without solving the quadratic equation,

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

1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\)

$$\left( \begin{array} { l l } a & 0 \\ 1 & b \end{array} \right) ^ { n } = \left( \begin{array} { c c } a ^ { n } & 0 \\ \frac { a ^ { n } - b ^ { n } } { a - b } & b ^ { n } \end{array} \right)$$ where \(a\) and \(b\) are constants and \(a \neq b\).

9. Given that $$\frac { z - k \mathrm { i } } { z + 3 \mathrm { i } } = \mathrm { i } \text {, where } k \text { is a positive real constant }$$

1. show that \(z = - \frac { ( k + 3 ) } { 2 } + \frac { ( k - 3 ) } { 2 } \mathrm { i }\)
2. Using the printed answer in part (a),
   1. find an exact simplified value for the modulus of \(z\) when \(k = 4\)
   2. find the argument of \(z\) when \(k = 1\). Give your answer in radians to 3 decimal places, where \(- \pi < \arg z < \pi\)

10. The rectangular hyperbola \(H\) has equation \(x y = 144\). The point \(P\), on \(H\), has coordinates \(\left( 12 p , \frac { 12 } { p } \right)\), where \(p\) is a non-zero constant.

1. Show, by using calculus, that the normal to \(H\) at the point \(P\) has equation $$y = p ^ { 2 } x + \frac { 12 } { p } - 12 p ^ { 3 }$$ Given that the normal through \(P\) crosses the positive \(x\)-axis at the point \(Q\) and the negative \(y\)-axis at the point \(R\),
2. find the coordinates of \(Q\) and the coordinates of \(R\), giving your answers in terms of \(p\).
3. Given also that the area of triangle \(O Q R\) is 512 , find the possible values of \(p\).
   

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