Edexcel F1 (Further Pure Mathematics 1) 2017 January

\(\mathrm { f } ( x ) = 2 ^ { x } - 10 \sin x - 2\), where \(x\) is measured in radians

1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between 2 and 3
   [0pt]
2. Use linear interpolation once on the interval [2,3] to find an approximation to \(\alpha\). Give your answer to 3 decimal places.

The quadratic equation $$2 x ^ { 2 } - x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,

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

3. $$f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 26 x ^ { 2 } + 32 x + 160$$ Given that \(x = - 1 + 3 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of \(\mathrm { f } ( x ) = 0\)
(Solutions based entirely on graphical or numerical methods are not acceptable.)

4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ( 2 r + 1 ) ( 3 r + 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
(b) Hence find the value of $$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$

1. The complex number \(z\) is given by

$$z = - 7 + 3 i$$ Find

1. \(| z |\)
2. \(\arg z\), giving your answer in radians to 2 decimal places. Given that \(\frac { z } { 1 + \mathrm { i } } + w = 3 - 6 \mathrm { i }\)
3. find the complex number \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers. You must show all your working.
4. Show the points representing \(z\) and \(w\) on a single Argand diagram.

6. $$f ( x ) = x ^ { 3 } - \frac { 1 } { 2 x } + x ^ { \frac { 3 } { 2 } } , \quad x > 0$$ The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.6, 0.7].

1. Taking 0.6 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. Show that your answer to part (a) is correct to 3 decimal places.

7. (i) $$\mathbf { A } = \left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)$$

1. Describe fully the single transformation represented by the matrix \(\mathbf { A }\). The matrix \(\mathbf { B }\) represents a stretch, scale factor 3 , parallel to the \(x\)-axis.
2. Find the matrix \(\mathbf { B }\).
   (ii) $$\mathbf { M } = \left( \begin{array} { r r } - 4 & 3
   - 3 & - 4 \end{array} \right)$$ The matrix \(\mathbf { M }\) represents an enlargement with scale factor \(k\) and centre ( 0,0 ), where \(k > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about ( 0,0 ).
3. Find the value of \(k\).
4. Find the value of \(\theta\), giving your answer in radians to 2 decimal places.
5. Find \(\mathbf { M } ^ { - 1 }\)

8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a t ^ { 2 } , 2 a t \right)\) lies on \(C\).

1. Using calculus, show that the normal to \(C\) at \(P\) has equation $$y + t x = a t ^ { 3 } + 2 a t$$ The point \(S\) is the focus of the parabola \(C\).
   The point \(B\) lies on the positive \(x\)-axis and \(O B = 5 O S\), where \(O\) is the origin.
2. Write down, in terms of \(a\), the coordinates of the point \(B\). A circle has centre \(B\) and touches the parabola \(C\) at two distinct points \(Q\) and \(R\). Given that \(t \neq 0\),
3. find the coordinates of the points \(Q\) and \(R\).
4. Hence find, in terms of \(a\), the area of triangle \(B Q R\).

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

$$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } - 3 r ^ { 2 } + r \right) = n ^ { 3 } ( n + 1 )$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 5 ^ { 2 n } + 3 n - 1$$ is divisible by 9