Edexcel F1 (Further Pure Mathematics 1) 2015 June

1. Given that

$$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 \equiv ( 2 z - 3 ) \left( z ^ { 2 } + a z + b \right)$$ where \(a\) and \(b\) are real constants,

1. find the value of \(a\) and the value of \(b\).
2. Given that \(z\) is a complex number, find the three exact roots of the equation $$2 z ^ { 3 } - 5 z ^ { 2 } + 7 z - 6 = 0$$

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

$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { n } { 2 } \left( a n ^ { 2 } + b n + c \right)$$ where \(a , b\) and \(c\) are integers to be found.

3. It is given that \(\alpha\) and \(\beta\) are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$

1. Find the exact value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
2. Find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(Q\) lies on the directrix of \(C\). The point \(P\) lies on \(C\) where \(y > 0\) and the line segment \(Q P\) is parallel to the \(x\)-axis. Given that the length of \(P S\) is 13

1. write down the length of \(P Q\). Given that the point \(P\) has \(x\) coordinate 9
   find
2. the value of \(a\),
3. the area of triangle \(P S Q\).

1. In the interval \(2 < x < 3\), the equation

$$6 - x ^ { 2 } \cos \left( \frac { x } { 5 } \right) = 0 , \text { where } x \text { is measured in radians }$$ has exactly one root \(\alpha\).
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1. Starting with the interval [2,3], use interval bisection twice to find an interval of width 0.25 which contains \(\alpha\).
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2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 2 decimal places.

6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).

1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).

7. $$z = - 3 k - 2 k \mathrm { i } , \text { where } k \text { is a real, positive constant. }$$

1. Find the modulus and the argument of \(z\), giving the argument in radians to 2 decimal places and giving the modulus as an exact answer in terms of \(k\).
2. Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real and are given in terms of \(k\) where necessary,
   1. \(\frac { 4 } { z + 3 k }\)
   2. \(z ^ { 2 }\)
3. Given that \(k = 1\), plot the points \(A , B , C\) and \(D\) representing \(z , z ^ { * } , \frac { 4 } { z + 3 k }\) and \(z ^ { 2 }\) respectively on a single Argand diagram.

8. $$\mathbf { P } = \left( \begin{array} { r r } 3 a & - 4 a
4 a & 3 a \end{array} \right) , \text { where } a \text { is a constant and } a > 0$$

1. Find the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(a\).
   (3) The matrix \(\mathbf { P }\) represents the transformation \(U\) which transforms a triangle \(T _ { 1 }\) onto the triangle \(T _ { 2 }\).
   The triangle \(T _ { 2 }\) has vertices at the points ( \(- 3 a , - 4 a\) ), ( \(6 a , 8 a\) ), and ( \(- 20 a , 15 a\) ).
2. Find the coordinates of the vertices of \(T _ { 1 }\)
3. Hence, or otherwise, find the area of triangle \(T _ { 2 }\) in terms of \(a\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { Q }\), is a rotation through an angle \(\alpha\) clockwise about the origin, where \(\tan \alpha = \frac { 4 } { 3 }\) and \(0 < \alpha < \frac { \pi } { 2 }\)
4. Write down the matrix \(\mathbf { Q }\), giving each element as an exact value. The transformation \(U\) followed by the transformation \(V\) is the transformation \(W\). The matrix \(\mathbf { R }\) represents the transformation \(W\).
5. Find the matrix \(\mathbf { R }\).

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

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 2 r - 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( 3 n ^ { 2 } + n - 1 \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\left( \begin{array} { c c } 7 & - 12
3 & - 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 6 n + 1 & - 12 n
3 n & 1 - 6 n \end{array} \right)$$