Edexcel FP1 (Further Pure Mathematics 1) 2018 June

1. $$f ( z ) = 2 z ^ { 3 } - 4 z ^ { 2 } + 15 z - 13$$ Given that \(\mathrm { f } ( z ) \equiv ( z - 1 ) \left( 2 z ^ { 2 } + a z + b \right)\), where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Hence use algebra to find the three roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

2. $$f ( x ) = \frac { 3 } { 2 } x ^ { 2 } + \frac { 4 } { 3 x } + 2 x - 5 , \quad x < 0$$ The equation \(\mathrm { f } ( x ) = 0\) has a single root \(\alpha\).

1. Show that \(\alpha\) lies in the interval \([ - 3 , - 2.5 ]\)
2. Taking - 3 as a first approximation to \(\alpha\), apply the Newton-Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
3. Use linear interpolation once on the interval \([ - 3 , - 2.5 ]\) to find another approximation to \(\alpha\), giving your answer to 3 decimal places.

3. (i) Given that $$\mathbf { A } = \left( \begin{array} { r r } - 2 & 3
1 & 1 \end{array} \right) , \quad \mathbf { A } \mathbf { B } = \left( \begin{array} { r r r } - 1 & 5 & 12
3 & - 5 & - 1 \end{array} \right)$$

1. find \(\mathbf { A } ^ { - 1 }\)
2. Hence, or otherwise, find the matrix \(\mathbf { B }\), giving your answer in its simplest form.
   (ii) Given that $$\mathbf { C } = \left( \begin{array} { r r } 0 & 1
   - 1 & 0 \end{array} \right)$$
3. describe fully the single geometrical transformation represented by the matrix \(\mathbf { C }\).
4. Hence find the matrix \(\mathbf { C } ^ { 39 }\)

1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = \frac { 1 } { 3 } n ( n - a ) ( n + a )$$ where \(a\) is a positive integer to be determined.
(b) Hence, or otherwise, state the positive value of \(n\) that satisfies $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 8 \right) = 0$$ Given that $$\sum _ { r = 3 } ^ { 17 } \left( k r ^ { 3 } + r ^ { 2 } - r - 8 \right) = 6710 \quad \text { where } k \text { is a constant }$$ (c) find the exact value of \(k\).

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

Given that \(P \left( c t , \frac { c } { t } \right) , t \neq 0\), is a general point on \(H\),

1. use calculus to show that the equation of the tangent to \(H\) at \(P\) can be written as $$t ^ { 2 } y + x = 2 c t$$ The points \(A\) and \(B\) lie on \(H\).
   The tangent to \(H\) at \(A\) and the tangent to \(H\) at \(B\) meet at the point \(\left( - \frac { 8 c } { 5 } , \frac { 3 c } { 5 } \right)\).
   Given that the \(x\) coordinate of \(A\) is positive,
2. find, in terms of \(c\), the coordinates of \(A\) and the coordinates of \(B\).

6. $$\mathbf { M } = \left( \begin{array} { r r } 8 & - 1
- 4 & 2 \end{array} \right)$$

1. Find the value of \(\operatorname { det } \mathbf { M }\) The triangle \(T\) has vertices at the points \(( 4,1 ) , ( 6 , k )\) and \(( 12,1 )\), where \(k\) is a constant.
   The triangle \(T\) is transformed onto the triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of triangle \(T ^ { \prime }\) is 216 square units,
2. find the possible values of \(k\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\).

The straight line \(l\) passes through the point \(S\) and meets the directrix of \(C\) at the point \(D\).
Given that the \(y\) coordinate of \(D\) is \(\frac { 24 a } { 5 }\),

1. show that an equation of the line \(l\) is $$12 x + 5 y = 12 a$$ The point \(P \left( a k ^ { 2 } , 2 a k \right)\), where \(k\) is a positive constant, lies on the parabola \(C\).
   Given that the line segment \(S P\) is perpendicular to \(l\),
2. find, in terms of \(a\), the coordinates of the point \(P\).

1. Prove by induction that

$$f ( n ) = 2 ^ { n + 2 } + 3 ^ { 2 n + 1 }$$ is divisible by 7 for all positive integers \(n\).

1. 1. Given that

$$\frac { 3 w + 7 } { 5 } = \frac { p - 4 \mathrm { i } } { 3 - \mathrm { i } } \quad \text { where } p \text { is a real constant }$$

1. express \(w\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. Give your answer in its simplest form in terms of \(p\). Given that arg \(w = - \frac { \pi } { 2 }\)
2. find the value of \(p\).
   (ii) Given that $$( z + 1 - 2 i ) ^ { * } = 4 i z$$ find \(z\), giving your answer in the form \(z = x + i y\), where \(x\) and \(y\) are real constants.
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