Edexcel F1 (Further Pure Mathematics 1) 2024 June

1. 1. The matrix \(\mathbf { A }\) is defined by

$$\mathbf { A } = \left( \begin{array} { c c } 3 k & 4 k - 1
2 & 6 \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
2. determine \(\mathbf { A } ^ { - 1 }\) in terms of \(k\), giving your answer in simplest form.
   (ii) The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \left( \begin{array} { l l } p & 0
   0 & q \end{array} \right)$$ where \(p\) and \(q\) are integers.
   State the value of \(p\) and the value of \(q\) when \(\mathbf { B }\) represents
3. an enlargement about the origin with scale factor - 2
4. a reflection in the \(y\)-axis.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$ Given that \(z = 3\) is a root of the equation \(f ( z ) = 0\)

1. show that \(p = - 87\)
2. Use algebra to determine the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\), giving your answers in simplest form. On an Argand diagram
   • the root \(z = 3\) is represented by the point \(P\)
   • the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are represented by the points \(Q\) and \(R\)
   • the number \(z = - 9\) is represented by the point \(S\)
   • Show on a single Argand diagram the positions of \(P , Q , R\) and \(S\)
   • Determine the perimeter of the quadrilateral \(P Q S R\), giving your answer as a simplified surd.

3. $$\mathrm { f } ( x ) = x ^ { 3 } - 5 \sqrt { x } - 4 x + 7 \quad x \geqslant 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 0.25,1 ]\)

1. Use linear interpolation once on the interval [ \(0.25,1\) ] to determine an approximation to \(\alpha\), giving your answer to 3 decimal places. The equation \(\mathrm { f } ( x ) = 0\) has another root \(\beta\) in the interval [1.5, 2.5]
2. Determine \(\mathrm { f } ^ { \prime } ( x )\)
3. Hence, using \(x _ { 0 } = 1.75\) as a first approximation to \(\beta\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to determine a second approximation to \(\beta\), giving your answer to 3 decimal places.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} The complex number \(z\) is defined by $$2 = - 3 + 4 i$$

1. Determine \(\left| z ^ { 2 } - 3 \right|\)
2. Express \(\frac { 50 } { z ^ { * } }\) in the form \(k z\), where \(k\) is a positive integer.
3. Hence find the value of \(\arg \left( \frac { 50 } { z ^ { * } } \right)\) Give your answer in radians to 3 significant figures.

1. The equation \(5 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\frac { 1 } { p }\) and \(\frac { 1 } { q }\)
   1. Without solving the equation,
      1. show that \(p q = \frac { 5 } { 2 }\)
      2. determine the value of \(p + q\)
   2. Hence, without finding the values of \(p\) and \(q\), determine a quadratic equation with roots
   $$\frac { p } { p ^ { 2 } + 1 } \text { and } \frac { q } { q ^ { 2 } + 1 }$$ giving your answer in the form \(a x ^ { 2 } + b x + c = 0\) where \(a , b\) and \(c\) are integers.

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

$$\left( \begin{array} { l l } 1 & r
0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & \left( 2 ^ { n } - 1 \right) r
0 & 2 ^ { n } \end{array} \right)$$ where \(r\) is a constant. $$\mathbf { M } = \left( \begin{array} { l l } 4 & 0
0 & 5 \end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r } 1 & - 2
0 & 2 \end{array} \right) ^ { 4 }$$ The transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\) is represented by the matrix \(\mathbf { B }\)
(b) (i) Determine \(\mathbf { N }\) in the form \(\left( \begin{array} { l l } a & b
c & d \end{array} \right)\) where \(a , b , c\) and \(d\) are integers.
(ii) Determine B Hexagon \(S\) is transformed onto hexagon \(S ^ { \prime }\) by matrix \(\mathbf { B }\)
(c) Given that the area of \(S ^ { \prime }\) is 720 square units, determine the area of \(S\)

1. In this question use the standard results for summations.
   1. Show that for all positive integers \(n\)
   $$\sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = A n ^ { 3 } + B n ^ { 2 }$$ where \(A\) and \(B\) are integers to be determined.
2. Hence determine the value of \(n\) for which $$\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = 270$$

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

$$f ( n ) = 7 ^ { n - 1 } + 8 ^ { 2 n + 1 }$$ is divisible by 57
(6)

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\)
   The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)