AQA FP1 (Further Pure Mathematics 1) 2006 June

1 The quadratic equation $$3 x ^ { 2 } - 6 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the numerical values of \(\alpha + \beta\) and \(\alpha \beta\).
2. 1. Expand \(( \alpha + \beta ) ^ { 3 }\).
   2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 4\).
3. Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\), giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.

2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \log _ { 10 } x$$ Starting at the point \(( 2,3 )\) on the curve, use a step-by-step method with a step length of 0.2 to estimate the value of \(y\) at \(x = 2.4\). Give your answer to three decimal places.

3 Show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r \right) = k n ( n + 1 ) ( n - 1 )$$ where \(k\) is a rational number.

4 Find, in radians, the general solution of the equation $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in terms of \(\pi\).

5 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
- \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$

1. Find the matrix:
   1. \(\mathbf { M } ^ { 2 }\);
   2. \(\mathbf { M } ^ { 4 }\).
2. Describe fully the geometrical transformation represented by \(\mathbf { M }\).
3. Find the matrix \(\mathbf { M } ^ { 2006 }\).

6 It is given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.

1. Write down, in terms of \(x\) and \(y\), an expression for $$( z + \mathrm { i } ) ^ { * }$$ where \(( z + \mathrm { i } ) ^ { * }\) denotes the complex conjugate of \(( z + \mathrm { i } )\).
2. Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form \(a + b \mathrm { i }\).

7

1. Describe a geometrical transformation by which the hyperbola $$x ^ { 2 } - 4 y ^ { 2 } = 1$$ can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
2. The diagram shows the hyperbola \(H\) with equation $$x ^ { 2 } - y ^ { 2 } - 4 x + 3 = 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-4_951_1216_824_402}
   By completing the square, describe a geometrical transformation by which the hyperbola \(H\) can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).

8

1. The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
   1. Express \(\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
   2. Use your answer to part (a)(i) to find the value of \(f ^ { \prime } ( 1 )\).
2. The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
   The graphs intersect at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(\mathrm { f } ( x ) = 0\), where f is the function defined in part (a).
   2. Taking \(x _ { 1 } = 1\) as a first approximation to the root of the equation \(\mathrm { f } ( x ) = 0\), use the Newton-Raphson method to find a second approximation \(x _ { 2 }\) to the root.
      (3 marks)
3. The region enclosed by the curve \(y = \frac { 1 } { x ^ { 2 } }\), the line \(x = 1\) and the \(x\)-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
   (3 marks)

9 A curve \(C\) has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the coordinates of the points where \(C\) intersects the \(x\)-axis. (2 marks)
   2. Write down the equations of all the asymptotes of \(C\).
2. 1. Show that, if the line \(y = k\) intersects \(C\), then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
   2. Given that there is only one stationary point on \(C\), find the coordinates of this stationary point.
      (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).