Edexcel FP1 (Further Pure Mathematics 1) 2015 June

1. $$f ( x ) = 9 x ^ { 3 } - 33 x ^ { 2 } - 55 x - 25$$ Given that \(x = 5\) is a solution of the equation \(\mathrm { f } ( x ) = 0\), use an algebraic method to solve \(\mathrm { f } ( x ) = 0\) completely.
(5)

2. In the interval \(13 < x < 14\), the equation $$3 + x \sin \left( \frac { x } { 4 } \right) = 0 , \text { where } x \text { is measured in radians, }$$ has exactly one root, \(\alpha\).
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1. Starting with the interval [13,14], 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 [13,14] to find an approximate value for \(\alpha\). Give your answer to 3 decimal places.

3. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 4 ) = \frac { n } { 3 } ( n + 4 ) ( n + 5 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) ( r + 4 ) = \frac { n } { 3 } ( n + 1 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be found.

4. $$z _ { 1 } = 3 \mathrm { i } \text { and } z _ { 2 } = \frac { 6 } { 1 + \mathrm { i } \sqrt { 3 } }$$

1. Express \(z _ { 2 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real numbers.
2. Find the modulus and the argument of \(z _ { 2 }\), giving the argument in radians in terms of \(\pi\).
3. Show the three points representing \(z _ { 1 } , z _ { 2 }\) and \(\left( z _ { 1 } + z _ { 2 } \right)\) respectively, on a single Argand diagram.

5. The rectangular hyperbola \(H\) has equation \(x y = 9\) The point \(A\) on \(H\) has coordinates \(\left( 6 , \frac { 3 } { 2 } \right)\).

1. Show that the normal to \(H\) at the point \(A\) has equation $$2 y - 8 x + 45 = 0$$ The normal at \(A\) meets \(H\) again at the point \(B\).
2. Find the coordinates of \(B\).

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

$$\left( \begin{array} { r r } 1 & 0
- 1 & 5 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 0
- \frac { 1 } { 4 } \left( 5 ^ { n } - 1 \right) & 5 ^ { n } \end{array} \right)$$ (ii) Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\), $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

$$\mathbf { A } = \left( \begin{array} { r r } 5 k & 3 k - 1
- 3 & k + 1 \end{array} \right) , \text { where } k \text { is a real constant. }$$ Given that \(\mathbf { A }\) is a singular matrix, find the possible values of \(k\).
(ii) $$\mathbf { B } = \left( \begin{array} { l l } 10 & 5
- 3 & 3 \end{array} \right)$$ A triangle \(T\) is transformed onto a triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { B }\). The vertices of triangle \(T ^ { \prime }\) have coordinates \(( 0,0 ) , ( - 20,6 )\) and \(( 10 c , 6 c )\), where \(c\) is a positive constant. The area of triangle \(T ^ { \prime }\) is 135 square units.

1. Find the matrix \(\mathbf { B } ^ { - 1 }\)
2. Find the coordinates of the vertices of the triangle \(T\), in terms of \(c\) where necessary.
3. Find the value of \(c\).

1. The point \(P \left( 3 p ^ { 2 } , 6 p \right)\) lies on the parabola with equation \(y ^ { 2 } = 12 x\) and the point \(S\) is the focus of this parabola.
   1. Prove that \(S P = 3 \left( 1 + p ^ { 2 } \right)\)
   The point \(Q \left( 3 q ^ { 2 } , 6 q \right) , p \neq q\), also lies on this parabola.
   The tangent to the parabola at the point \(P\) and the tangent to the parabola at the point \(Q\) meet at the point \(R\).
2. Find the equations of these two tangents and hence find the coordinates of the point \(R\), giving the coordinates in their simplest form.
3. Prove that \(S R ^ { 2 } = S P \cdot S Q\)