AQA FP1 (Further Pure Mathematics 1) 2010 June

1 A curve passes through the point ( 1,3 ) and satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + x ^ { 3 }$$ Starting at the point ( 1,3 ), use a step-by-step method with a step length of 0.1 to estimate the \(y\)-coordinate of the point on the curve for which \(x = 1.3\). Give your answer to three decimal places.
(No credit will be given for methods involving integration.)

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

1. Find, in terms of \(x\) and \(y\), the real and imaginary parts of $$( 1 - 2 i ) z - z ^ { * }$$
2. Hence find the complex number \(z\) such that $$( 1 - 2 \mathrm { i } ) z - z ^ { * } = 10 ( 2 + \mathrm { i } )$$

   PART	REFERENCE
   \(\_\_\_\_\)	\(\_\_\_\_\)

3 Find the general solution, in degrees, of the equation $$\cos \left( 5 x - 20 ^ { \circ } \right) = \cos 40 ^ { \circ }$$

4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants.
The following approximate values of \(x\) and \(y\) have been found.

1. Complete the table below, showing values of \(X\), where \(X = x ^ { 2 }\).
2. On the diagram below, draw a linear graph relating \(X\) and \(y\).
3. Use your graph to find estimates, to two significant figures, for:
   1. the value of \(x\) when \(y = 15\);
   2. the values of \(a\) and \(b\).

      1. \(\boldsymbol { x }\)	2	4	6	8
         \(\boldsymbol { X }\)
         \(\boldsymbol { y }\)	6.0	10.5	18.0	28.2

      2. \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-05_771_1586_1772_274}

5 A curve has equation \(y = x ^ { 3 } - 12 x\).
The point \(A\) on the curve has coordinates ( \(2 , - 16\) ).
The point \(B\) on the curve has \(x\)-coordinate \(2 + h\).

1. Show that the gradient of the line \(A B\) is \(6 h + h ^ { 2 }\).
2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
   \includegraphics[max width=\textwidth, alt={}]{763d89e4-861a-4754-a93c-d0902987673f-06_1894_1709_813_153}

6 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } } \end{array} \right]$$ Describe fully the geometrical transformation represented by each of the following matrices:

1. A ;
2. B ;
3. \(\quad \mathbf { A } ^ { 2 }\);
4. \(\quad \mathbf { B } ^ { 2 }\);
5. AB.
   □

7

1. 1. Write down the equations of the two asymptotes of the curve \(y = \frac { 1 } { x - 3 }\).
   2. Sketch the curve \(y = \frac { 1 } { x - 3 }\), showing the coordinates of any points of intersection with the coordinate axes.
   3. On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line \(y = 2 x - 5\).
2. 1. Solve the equation $$\frac { 1 } { x - 3 } = 2 x - 5$$
   2. Find the solution of the inequality $$\frac { 1 } { x - 3 } < 2 x - 5$$ □ \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}

8 The quadratic equation $$x ^ { 2 } - 4 x + 10 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } = \frac { 2 } { 5 }\).
3. Find a quadratic equation, with integer coefficients, which has roots \(\alpha + \frac { 2 } { \beta }\) and \(\beta + \frac { 2 } { \alpha }\).

9 A parabola \(P\) has equation \(y ^ { 2 } = x - 2\).

1. 1. Sketch the parabola \(P\).
   2. On your sketch, draw the two tangents to \(P\) which pass through the point \(( - 2,0 )\).
2. 1. Show that, if the line \(y = m ( x + 2 )\) intersects \(P\), then the \(x\)-coordinates of the points of intersection must satisfy the equation $$m ^ { 2 } x ^ { 2 } + \left( 4 m ^ { 2 } - 1 \right) x + \left( 4 m ^ { 2 } + 2 \right) = 0$$
   2. Show that, if this equation has equal roots, then $$16 m ^ { 2 } = 1$$
   3. Hence find the coordinates of the points at which the tangents to \(P\) from the point \(( - 2,0 )\) touch the parabola \(P\).