Edexcel F1 (Further Pure Mathematics 1) 2021 June

1．（i） $$f ( x ) = x ^ { 3 } + 4 x - 6$$ （a）Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval［1，1．5］
（b）Taking 1.5 as a first approximation，apply the Newton Raphson process twice to \(\mathrm { f } ( x )\) to obtain an approximate value of \(\alpha\) ．Give your answer to 3 decimal places． Show your working clearly．
（ii） $$g ( x ) = 4 x ^ { 2 } + x - \tan x$$ where \(x\) is measured in radians． The equation \(\mathrm { g } ( x ) = 0\) has a single root \(\beta\) in the interval［1．4，1．5］
Use linear interpolation on the values at the end points of this interval to obtain an approximation to \(\beta\) ．Give your answer to 3 decimal places．

2. The complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) are given by $$\mathrm { z } _ { 1 } = 2 - \mathrm { i } \quad \mathrm { z } _ { 2 } = p - \mathrm { i } \quad \mathrm { z } _ { 3 } = p + \mathrm { i }$$ where \(p\) is a real number.

1. Find \(\frac { z _ { 2 } z _ { 3 } } { z _ { 1 } }\) in the form \(a + b \mathrm { i }\) where \(a\) and \(b\) are real. Give your answer in its simplest form in terms of \(p\). Given that \(\left| \frac { z _ { 2 } z _ { 3 } } { z _ { 1 } } \right| = 2 \sqrt { 5 }\)
2. find the possible values of \(p\).

1. The triangle \(T\) has vertices \(A ( 2,1 ) , B ( 2,3 )\) and \(C ( 0,1 )\).

The triangle \(T ^ { \prime }\) is the image of \(T\) under the transformation represented by the matrix $$\mathbf { P } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right)$$

1. Find the coordinates of the vertices of \(T ^ { \prime }\)
2. Describe fully the transformation represented by \(\mathbf { P }\) The \(2 \times 2\) matrix \(\mathbf { Q }\) represents a reflection in the \(x\)-axis and the \(2 \times 2\) matrix \(\mathbf { R }\) represents a rotation through \(90 ^ { \circ }\) anticlockwise about the origin.
3. Write down the matrix \(\mathbf { Q }\) and the matrix \(\mathbf { R }\)
4. Find the matrix \(\mathbf { R Q }\)
5. Give a full geometrical description of the single transformation represented by the answer to part (d).

1. A rectangular hyperbola \(H\) has equation \(x y = 25\)

The point \(P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0\), is a general point on \(H\).

1. Show that the equation of the tangent to \(H\) at \(P\) is \(t ^ { 2 } y + x = 10 t\) The distinct points \(Q\) and \(R\) lie on \(H\). The tangent to \(H\) at the point \(Q\) and the tangent to \(H\) at the point \(R\) meet at the point \(( 15 , - 5 )\).
2. Find the coordinates of the points \(Q\) and \(R\).

5. $$f ( x ) = \left( 9 x ^ { 2 } + d \right) \left( x ^ { 2 } - 8 x + ( 10 d + 1 ) \right)$$ where \(d\) is a positive constant.

1. Find the four roots of \(\mathrm { f } ( x )\) giving your answers in terms of \(d\). Given \(d = 4\)
2. Express these four roots in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).
3. Show these four roots on a single Argand diagram. \includegraphics[max width=\textwidth, alt={}, center]{d7689f4a-a41e-45be-911b-4a74e81997eb-21_2647_1840_118_111}

6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\) The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).

1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = 2 p ^ { 3 } + 4 p$$
2. Write down an equation of the normal to \(C\) at \(Q\) The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
3. Show that \(N\) has coordinates $$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$ The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
4. Find the value of \(( p + q ) ^ { 2 } - 3 p q\)

7. (a) Prove by induction that for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n } { 6 } ( n + 1 ) ( 2 n + 1 )$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 2 \right) = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.
(c) Using your answers to part (b), find the value of $$\sum _ { r = 10 } ^ { 25 } \left( r ^ { 2 } + 2 \right)$$

8. Prove by induction that \(4 ^ { n + 2 } + 5 ^ { 2 n + 1 }\) is divisible by 21 for all positive integers \(n\).