AQA FP1 (Further Pure Mathematics 1) 2012 January

2 Show that only one of the following improper integrals has a finite value, and find that value:

1. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 2 } { 3 } } \mathrm {~d} x\);
2. \(\quad \int _ { 8 } ^ { \infty } x ^ { - \frac { 4 } { 3 } } \mathrm {~d} x\).

3

1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
   1. \(x ^ { 2 } + 9 = 0\);
   2. \(( x + 2 ) ^ { 2 } + 9 = 0\).
2. 1. Expand \(( 1 + x ) ^ { 3 }\).
   2. Express \(( 1 + 2 \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
   3. Given that \(z = 1 + 2 \mathrm { i }\), find the value of $$z ^ { * } - z ^ { 3 }$$

4

1. Use the formulae for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( 4 r - 3 ) = k n ( n + 1 ) \left( 2 n ^ { 2 } - 1 \right)$$ where \(k\) is a constant.
2. Hence evaluate $$\sum _ { r = 20 } ^ { 40 } r ^ { 2 } ( 4 r - 3 )$$ (2 marks)

5 The diagram below (not to scale) shows a part of a curve \(y = \mathrm { f } ( x )\) which passes through the points \(A ( 2 , - 10 )\) and \(B ( 5,22 )\).

1. 1. On the diagram, draw a line which illustrates the method of linear interpolation for solving the equation \(\mathrm { f } ( x ) = 0\). The point of intersection of this line with the \(x\)-axis should be labelled \(P\).
   2. Calculate the \(x\)-coordinate of \(P\). Give your answer to one decimal place.
2. 1. On the same diagram, draw a line which illustrates the Newton-Raphson method for solving the equation \(\mathrm { f } ( x ) = 0\), with initial value \(x _ { 1 } = 2\). The point of intersection of this line with the \(x\)-axis should be labelled \(Q\).
   2. Given that the gradient of the curve at \(A\) is 8 , calculate the \(x\)-coordinate of \(Q\). Give your answer as an exact decimal.
      \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-3_876_1063_1779_523}

6 Find the general solution of each of the following equations:

1. \(\quad \tan \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { \sqrt { 3 } }\);
2. \(\quad \tan ^ { 2 } \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { 3 }\).

7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$

1. Find the equations of the asymptotes of \(H\).
2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3
   0 \end{array} \right]\).
   1. Write down the equation of the translated curve.
   2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\).
   \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}

8 The diagram below shows a rectangle \(R _ { 1 }\) which has vertices \(( 0,0 ) , ( 3,0 ) , ( 3,2 )\) and \(( 0,2 )\).

1. On the diagram, draw:
   1. the image \(R _ { 2 }\) of \(R _ { 1 }\) under a rotation through \(90 ^ { \circ }\) clockwise about the origin;
   2. the image \(R _ { 3 }\) of \(R _ { 2 }\) under the transformation which has matrix $$\left[ \begin{array} { l l } 4 & 0
      0 & 2 \end{array} \right]$$
2. Find the matrix of:
   1. the rotation which maps \(R _ { 1 }\) onto \(R _ { 2 }\);
   2. the combined transformation which maps \(R _ { 1 }\) onto \(R _ { 3 }\).
      \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-5_913_910_1228_598}

9 A curve has equation $$y = \frac { x } { x - 1 }$$

1. Find the equations of the asymptotes of this curve.
2. Given that the line \(y = - 4 x + c\) intersects the curve, show that the \(x\)-coordinates of the points of intersection must satisfy the equation $$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
3. It is given that the line \(y = - 4 x + c\) is a tangent to the curve.
   1. Find the two possible values of \(c\).
      (No credit will be given for methods involving differentiation.)
   2. For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.