Edexcel F1 (Further Pure Mathematics 1) 2023 January

1. Given that

$$\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 3
- 2 & 3 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r } 1 & k
0 & - 3
2 k & 2 \end{array} \right)$$ where \(k\) is a non-zero constant,

1. determine the matrix \(\mathbf { A B }\)
2. determine the value of \(k\) for which \(\operatorname { det } ( \mathbf { A B } ) = 0\)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that for all positive integers \(n\) $$\sum _ { r = 1 } ^ { n } ( 7 r - 5 ) ^ { 2 } = \frac { n } { 6 } ( 7 n + 1 ) ( A n + B )$$ where \(A\) and \(B\) are integers to be determined.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathrm { f } ( z ) = 4 z ^ { 3 } + p z ^ { 2 } - 24 z + 108$$ where \(p\) is a constant.
Given that - 3 is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(p\)
2. using algebra, solve \(\mathrm { f } ( \mathrm { z } ) = 0\) completely, giving the roots in simplest form,
3. determine the modulus of the complex roots of \(\mathrm { f } ( \mathrm { z } ) = 0\)
4. show the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.

4. $$f ( x ) = 1 - \frac { 1 } { 8 x ^ { 4 } } + \frac { 2 } { 7 \sqrt { x ^ { 7 } } } \quad x > 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root, \(\alpha\), that lies in the interval \([ 0.15,0.25 ]\)

1. 1. Determine \(\mathrm { f } ^ { \prime } ( x )\)
   2. Explain why 0.25 cannot be used as an initial approximation for \(\alpha\) in the Newton-Raphson process.
   3. Taking 0.15 as a first approximation to \(\alpha\) apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\) Give your answer to 3 decimal places.
2. Use linear interpolation once on the interval \([ 0.15,0.25 ]\) to find another approximation to \(\alpha\) Give your answer to 3 decimal places.

1. The quadratic equation

$$4 x ^ { 2 } + 3 x + k = 0$$ where \(k\) is an integer, has roots \(\alpha\) and \(\beta\)

1. Write down, in terms of \(k\) where appropriate, the value of \(\alpha + \beta\) and the value of \(\alpha \beta\)
2. Determine, in simplest form in terms of \(k\), the value of \(\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }\)
3. Determine a quadratic equation which has roots $$\frac { \alpha } { \beta ^ { 2 } } \text { and } \frac { \beta } { \alpha ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integer values in terms of \(k\)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola \(H\) has equation \(x y = 20\)
The point \(P \left( 2 t \sqrt { a } , \frac { 2 \sqrt { a } } { t } \right) , t \neq 0\), where \(a\) is a constant, is a general point on \(H\)

1. State the value of \(a\)
2. Show that the normal to \(H\) at the point \(P\) has equation $$t y - t ^ { 3 } x - 2 \sqrt { 5 } \left( 1 - t ^ { 4 } \right) = 0$$ The points \(A\) and \(B\) lie on \(H\)
   The point \(A\) has parameter \(t = c\) and the point \(B\) has parameter \(t = - \frac { 1 } { 2 c }\), where \(c\) is a constant. The normal to \(H\) at \(A\) meets \(H\) again at \(B\)
3. Determine the possible values of \(C\)

$$\mathbf { P } = \left( \begin{array} { r r } 0 & - 1
- 1 & 0 \end{array} \right)$$ The matrix \(\mathbf { P }\) represents a geometrical transformation \(U\)

1. Describe \(U\) fully as a single geometrical transformation. The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through \(240 ^ { \circ }\) anticlockwise about the origin followed by an enlargement about ( 0,0 ) with scale factor 6
2. Determine the matrix \(\mathbf { Q }\), giving each entry in exact numerical form. Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { R }\)
3. determine the matrix \(\mathbf { R }\)
   (ii) The transformation \(W\) is represented by the matrix $$\left( \begin{array} { c c } - 2 & 2 \sqrt { 3 }
   2 \sqrt { 3 } & 2 \end{array} \right)$$ Show that there is a real number \(\lambda\) for which \(W\) maps the point \(( \lambda , 1 )\) onto the point ( \(4 \lambda , 4\) ), giving the exact value of \(\lambda\) \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
   VILU SIHI NI JLIYM ION OC
   VEYV SIHI NI ELIYM ION OC

1. A parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.

The point \(S\) is the focus of \(C\)
The line \(l _ { 1 }\) with equation \(y = k\) where \(k\) is a positive constant, intersects \(C\) at the point \(P\)

1. Show that $$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$ The line \(l _ { 2 }\) passes through \(P\) and intersects the directrix of \(C\) on the \(x\)-axis.
   The line \(l _ { 2 }\) intersects the \(y\)-axis at the point \(A\)
2. Show that the \(y\) coordinate of \(A\) is \(\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }\) The line \(l _ { 1 }\) intersects the directrix of \(C\) at the point \(B\)
   Given that the areas of triangles \(B P A\) and \(O S P\), where \(O\) is the origin, satisfy the ratio $$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$
3. determine the exact value of \(a\)

1. Prove by induction that for all positive integers \(n\)

$$\sum _ { r = 1 } ^ { n } \log ( 2 r - 1 ) = \log \left( \frac { ( 2 n ) ! } { 2 ^ { n } n ! } \right)$$