Edexcel F1 (Further Pure Mathematics 1) 2020 June

1. $$f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1.4,1.5]
2. Determine \(\mathrm { f } ^ { \prime } ( x )\) .
3. Using \(x _ { 0 } = 1.4\) as a first approximation to \(\alpha\) ,apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to calculate a second approximation to \(\alpha\) ,giving your answer to 3 decimal places.
   \(f ( x ) = x ^ { 3 } - \frac { 10 \sqrt { x } - 4 x } { x ^ { 2 } } \quad x > 0\)
4. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval[1.4,1.5]
5. Determine \(\mathrm { f } ^ { \prime } ( x )\) .

2
2. The quadratic equation $$5 x ^ { 2 } - 2 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,

1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
2. determine, giving each answer as a simplified fraction, the value of
   1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
3. determine a quadratic equation that has roots $$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.

3. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are integers.
The complex numbers \(3 + \mathrm { i }\) and \(- 1 - 2 \mathrm { i }\) are roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down the other roots of this equation.
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Determine the values of \(a , b , c\) and \(d\).

   VILU SIHI NI JIIIM ION OC

   VIUV SIHI NI III M M I ON OO

   VIAV SIHI NI JIIIM I ION OC

4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$ for all positive integers \(n\).
(b) Hence find the exact value of the sum of the squares of the odd numbers between 200 and 500 \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-13_2255_50_314_34}

1. The rectangular hyperbola \(H\) has equation \(x y = 64\)

The point \(P \left( 8 p , \frac { 8 } { p } \right)\), where \(p \neq 0\), lies on \(H\).

1. Use calculus to show that the normal to \(H\) at \(P\) has equation $$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$ The normal to \(H\) at \(P\) meets \(H\) again at the point \(Q\).
2. Determine, in terms of \(p\), the coordinates of \(Q\), giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-17_2255_50_314_34}

6. (i) $$\mathbf { A } = \left( \begin{array} { l l } 1 & 0
0 & 3 \end{array} \right)$$

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(45 ^ { \circ }\) clockwise about the origin.
2. Write down the matrix \(\mathbf { B }\), giving each element of the matrix in exact form. The transformation represented by matrix \(\mathbf { A }\) followed by the transformation represented by matrix \(\mathbf { B }\) is represented by the matrix \(\mathbf { C }\).
3. Determine \(\mathbf { C }\).
   (ii) The trapezium \(T\) has vertices at the points \(( - 2,0 ) , ( - 2 , k ) , ( 5,8 )\) and \(( 5,0 )\), where \(k\) is a positive constant. Trapezium \(T\) is transformed onto the trapezium \(T ^ { \prime }\) by the matrix $$\left( \begin{array} { r r } 5 & 1
   - 2 & 3 \end{array} \right)$$ Given that the area of trapezium \(T ^ { \prime }\) is 510 square units, calculate the exact value of \(k\).

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The line \(l\) with equation \(3 x - 4 y + 48 = 0\) is a tangent to \(C\) at the point \(P\).

1. Show that \(a = 9\)
2. Hence determine the coordinates of \(P\). Given that the point \(S\) is the focus of \(C\) and that the line \(l\) crosses the directrix of \(C\) at the point \(A\),
3. determine the exact area of triangle \(P S A\).
   \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-25_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

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

$$\sum _ { r = 1 } ^ { n } \frac { 2 r ^ { 2 } - 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } } = \frac { n ^ { 2 } } { ( n + 1 ) ^ { 2 } }$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 12 ^ { n } + 2 \times 5 ^ { n - 1 }$$ is divisible by 7 \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-29_2255_50_314_34}
\includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-31_2255_50_314_34}