Edexcel F1 (Further Pure Mathematics 1) 2018 Specimen

1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r \left( r ^ { 2 } - 3 \right) = \frac { n } { 4 } ( n + a ) ( n + b ) ( n + c )$$ where \(a\), \(b\) and \(c\) are integers to be found.

1. A parabola \(P\) has cartesian equation \(y ^ { 2 } = 28 x\). The point \(S\) is the focus of the parabola \(P\).
   1. Write down the coordinates of the point \(S\).
   Points \(A\) and \(B\) lie on the parabola \(P\). The line \(A B\) is parallel to the directrix of \(P\) and cuts the \(x\)-axis at the midpoint of \(O S\), where \(O\) is the origin.
2. Find the exact area of triangle \(A B S\).

3. $$\mathrm { f } ( x ) = x ^ { 2 } + \frac { 3 } { x } - 1 , \quad x < 0$$ The only real root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval \([ - 2 , - 1 ]\).

1. Taking - 1.5 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 2 decimal places.
2. Show that your answer to part (a) gives \(\alpha\) correct to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-06_2250_51_317_1980}

1. Given that

$$\mathbf { A } = \left( \begin{array} { c c } k & 3 \\ - 1 & k + 2 \end{array} \right) \text {, where } k \text { is a constant }$$

1. show that \(\operatorname { det } ( \mathbf { A } ) > 0\) for all real values of \(k\),
2. find \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
   

5. $$2 z + z ^ { * } = \frac { 3 + 4 \mathrm { i } } { 7 + \mathrm { i } }$$ Find \(z\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show all your working.

1. The rectangular hyperbola \(H\) has equation \(x y = 25\)
   1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\).
   The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
3. Find the coordinates of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-13_2261_50_312_36}

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

7. $$\mathbf { P } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 } \\ \frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)$$

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 line with equation \(y = x\)
2. Write down the matrix \(\mathbf { Q }\). Given that the transformation \(V\) followed by the transformation \(U\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\),
3. find the matrix \(\mathbf { R }\).
4. Show that there is a value of \(k\) for which the transformation \(T\) maps each point on the straight line \(y = k x\) onto itself, and state the value of \(k\).
   
   \includegraphics[max width=\textwidth, alt={}, center]{38217fcb-8f26-49ac-9bb1-61c2f304006e-17_2261_54_312_34}

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

8. $$\mathrm { f } ( z ) = z ^ { 4 } + 6 z ^ { 3 } + 76 z ^ { 2 } + a z + b$$ where \(a\) and \(b\) are real constants.
Given that \(- 3 + 8 \mathrm { i }\) is a complex root of the equation \(\mathrm { f } ( z ) = 0\)

1. write down another complex root of this equation.
2. Hence, or otherwise, find the other roots of the equation \(\mathrm { f } ( z ) = 0\)
3. Show on a single Argand diagram all four roots of the equation \(\mathrm { f } ( z ) = 0\)

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

1. The quadratic equation

$$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation,

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

   VIAV SIHI NI BIIIM ION OC

   VGHV SIHI NI GHIYM ION OC

   VJ4V SIHI NI JIIYM ION OC

1. (i) A sequence of positive numbers is defined by

$$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = 3 u _ { n } + 2 , \quad n \geqslant 1 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$u _ { n } = 2 \times ( 3 ) ^ { n } - 1$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } \frac { 4 r } { 3 ^ { r } } = 3 - \frac { ( 3 + 2 n ) } { 3 ^ { n } }$$