Edexcel F1 (Further Pure Mathematics 1) 2024 January

1. $$\mathbf { M } = \left( \begin{array} { c c } 2 k + 1 & k
k + 7 & k + 4 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Show that \(\mathbf { M }\) is non-singular for all real values of \(k\).
2. Determine \(\mathbf { M } ^ { - 1 }\) in terms of \(k\).

2. $$f ( z ) = 2 z ^ { 3 } + p z ^ { 2 } + q z - 41$$ where \(p\) and \(q\) are integers.
The complex number \(5 - 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. Write down another complex root of this equation.
2. Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
3. Determine the value of \(p\) and the value of \(q\). When plotted on an Argand diagram, the points representing the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) form the vertices of a triangle.
4. Determine the area of this triangle.

1. The 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 \(H\).
The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).

1. Determine, in terms of \(c\) and \(t\),
   1. the coordinates of \(A\),
   2. the coordinates of \(B\). Given that the area of triangle \(A O B\), where \(O\) is the origin, is 90 square units,
2. determine the value of \(c\), giving your answer as a simplified surd.

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

1. Describe the single geometrical transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a rotation of \(210 ^ { \circ }\) anticlockwise about centre \(( 0,0 )\).
2. Write down the matrix \(\mathbf { B }\), giving each element 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. Find \(\mathbf { C }\). The hexagon \(H\) is transformed onto the hexagon \(H ^ { \prime }\) by the matrix \(\mathbf { C }\).
4. Given that the area of hexagon \(H\) is 5 square units, determine the area of hexagon \(H ^ { \prime }\)

1. The quadratic equation

$$2 x ^ { 2 } - 3 x + 7 = 0$$ 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 the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
3. find a quadratic equation which has roots $$\left( \alpha - \frac { 1 } { \beta ^ { 2 } } \right) \text { and } \left( \beta - \frac { 1 } { \alpha ^ { 2 } } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.

$$f ( x ) = x - 4 - \cos ( 5 \sqrt { x } ) \quad x > 0$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [2.5, 3.5]
   [0pt]
2. Use linear interpolation once on the interval [2.5, 3.5] to find an approximation to \(\alpha\), giving your answer to 2 decimal places.
   (ii) $$\operatorname { g } ( x ) = \frac { 1 } { 10 } x ^ { 2 } - \frac { 1 } { 2 x ^ { 2 } } + x - 11 \quad x > 0$$
3. Determine \(\mathrm { g } ^ { \prime } ( x )\). The equation \(\mathrm { g } ( x ) = 0\) has a root \(\beta\) in the interval [6,7]
4. Using \(x _ { 0 } = 6\) as a first approximation to \(\beta\), apply the Newton-Raphson procedure once to \(\mathrm { g } ( x )\) to find a second approximation to \(\beta\), giving your answer to 3 decimal places.

1. The parabola \(C\) has equation \(y ^ { 2 } = \frac { 4 } { 3 } x\)

The point \(P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)\), where \(t \neq 0\), lies on \(C\).

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$3 t x + 3 y = t ^ { 3 } + 2 t$$ The normal to \(C\) at the point where \(t = 9\) meets \(C\) again at the point \(Q\).
2. Determine the exact coordinates of \(Q\).

1. (a) Use the standard results for summations to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n + 1 ) ^ { 2 } ( n - 2 )$$ (b) Hence show that, for all positive integers \(n\), $$\sum _ { r = n } ^ { 2 n } r \left( 2 r ^ { 2 } - 3 r - 1 \right) = \frac { 1 } { 2 } n ( n - 1 ) ( a n + b ) ( c n + d )$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be determined.

1. Given that

$$\frac { 3 z - 1 } { 2 } = \frac { \lambda + 5 i } { \lambda - 4 i }$$ where \(\lambda\) is a real constant,

1. determine \(z\), giving your answer in the form \(x + y i\), where \(x\) and \(y\) are real and in terms of \(\lambda\). Given also that \(\arg z = \frac { \pi } { 4 }\)
2. find the possible values of \(\lambda\).

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

$$\left( \begin{array} { r r } 5 & - 1
4 & 1 \end{array} \right) ^ { n } = 3 ^ { n - 1 } \left( \begin{array} { c c } 2 n + 3 & - n
4 n & 3 - 2 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 8 ^ { 2 n + 1 } + 6 ^ { 2 n - 1 }$$ is divisible by 7