Edexcel FP1 (Further Pure Mathematics 1) 2017 June

2. $$\mathbf { A } = \left( \begin{array} { r r } 2 & - 1
4 & 3 \end{array} \right) , \quad \mathbf { P } = \left( \begin{array} { r r } 3 & 6
11 & - 8 \end{array} \right)$$

1. Find \(\mathbf { A } ^ { - 1 }\)
   (2) The transformation represented by the matrix \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\) is equivalent to the transformation represented by the matrix \(\mathbf { P }\).
2. Find \(\mathbf { B }\), giving your answer in its simplest form.

3. The rectangular hyperbola \(H\) has parametric equations $$x = 4 t , \quad y = \frac { 4 } { t } \quad t \neq 0$$ The points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 4 }\) and \(t = 2\) respectively.
The line \(l\) passes through the origin \(O\) and is perpendicular to the line \(P Q\).

1. Find an equation for \(l\).
2. Find a cartesian equation for \(H\).
3. Find the exact coordinates of the two points where \(l\) intersects \(H\). Give your answers in their simplest form.

4. (i) The complex number \(w\) is given by $$w = \frac { p - 4 \mathrm { i } } { 2 - 3 \mathrm { i } }$$ where \(p\) is a real constant.

1. Express \(w\) in the form \(a + b i\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that \(\arg w = \frac { \pi } { 4 }\)
2. find the value of \(p\).
   (ii) The complex number \(z\) is given by $$z = ( 1 - \lambda i ) ( 4 + 3 i )$$ where \(\lambda\) is a real constant. Given that $$| z | = 45$$ find the possible values of \(\lambda\).
   Give your answers as exact values in their simplest form.
   II

5. (i) $$\mathbf { A } = \left( \begin{array} { l l } p & 2
3 & p \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } - 5 & 4
6 & - 5 \end{array} \right)$$ where \(p\) is a constant.

1. Find, in terms of \(p\), the matrix \(\mathbf { A B }\) Given that $$\mathbf { A B } + 2 \mathbf { A } = k \mathbf { I }$$ where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
2. find the value of \(p\) and the value of \(k\).
   (ii) $$\mathbf { M } = \left( \begin{array} { r r } a & - 9
   1 & 2 \end{array} \right) , \text { where } a \text { is a real constant }$$ Triangle \(T\) has an area of 15 square units.
   Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M. Given that the area of triangle \(T ^ { \prime }\) is 270 square units, find the possible values of \(a\).

6. Given that 4 and \(2 \mathrm { i } - 3\) are roots of the equation $$x ^ { 3 } + a x ^ { 2 } + b x - 52 = 0$$ where \(a\) and \(b\) are real constants,

1. write down the third root of the equation,
2. find the value of \(a\) and the value of \(b\).

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant and \(a > 0\) The point \(Q \left( a q ^ { 2 } , 2 a q \right) , q > 0\), lies on the parabola \(C\).

1. Show that an equation of the tangent to \(C\) at \(Q\) is $$q y = x + a q ^ { 2 }$$ The tangent to \(C\) at the point \(Q\) meets the \(x\)-axis at the point \(X \left( - \frac { 1 } { 4 } a , 0 \right)\) and meets the directrix of \(C\) at the point \(D\).
2. Find, in terms of \(a\), the coordinates of \(D\). Given that the point \(F\) is the focus of the parabola \(C\),
3. find the area, in terms of \(a\), of the triangle \(F X D\), giving your answer in its simplest form.

8. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 8 r + 3 \right) = \frac { 1 } { 2 } n ( 2 n + 5 ) ( n + 3 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 3 r ^ { 2 } + 8 r + 3 + k \left( 2 ^ { r - 1 } \right) \right) = 3520$$ (b) find the exact value of the constant \(k\).

9. (i) A sequence of numbers is defined by $$\begin{gathered} u _ { 1 } = 6 , \quad u _ { 2 } = 27
u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad n \geqslant 1 \end{gathered}$$ Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$u _ { n } = 3 ^ { n } ( n + 1 )$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 3 ^ { 3 n - 2 } + 2 ^ { 3 n + 1 } \text { is divisible by } 19$$ \includegraphics[max width=\textwidth, alt={}, center]{536d7ec7-91b0-4fda-a485-2ac4a72c7d59-29_56_20_109_1950}