AQA FP1 (Further Pure Mathematics 1) 2016 June

The quadratic equation \(x^2 - 6x + 14 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
2. Find a quadratic equation, with integer coefficients, which has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). [5 marks]

A curve \(C\) has equation \(y = (2 - x)(1 + x) + 3\).

1. A line passes through the point \((2, 3)\) and the point on \(C\) with \(x\)-coordinate \(2 + h\). Find the gradient of the line, giving your answer in its simplest form. [3 marks]
2. Show how your answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((2, 3)\). State the value of this gradient. [2 marks]

The variables \(y\) and \(x\) are related by an equation of the form $$y = a(b^x)$$ where \(a\) and \(b\) are positive constants. Let \(Y = \log_{10} y\).

1. Show that there is a linear relationship between \(Y\) and \(x\). [2 marks]
2. The graph of \(Y\) against \(x\), shown below, passes through the points \((0, 2.5)\) and \((5, 0.5)\). \includegraphics{figure_3}
   1. Find the gradient of the line. [1 mark]
   2. Find the value of \(a\) and the value of \(b\), giving each answer to three significant figures. [4 marks]

1. Given that \(\sin \frac{\pi}{3} = \cos \frac{\pi}{k}\), state the value of the integer \(k\). [1 mark]
2. Hence, or otherwise, find the general solution of the equation $$\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}$$ giving your answer, in its simplest form, in terms of \(\pi\). [4 marks]
3. Hence, given that \(\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}\), show that there is only one finite value for \(\tan x\) and state its exact value. [2 marks]

1. Use the formulae for \(\sum_{r=1}^n r^2\) and \(\sum_{r=1}^n r\) to show that \(\sum_{r=1}^n (6r - 3)^2 = 3n(4n^2 - 1)\). [5 marks]
2. Hence express \(\sum_{r=1}^{2n} r^3 - \sum_{r=1}^n (6r - 3)^2\) as a product of four linear factors in terms of \(n\). [4 marks]

A parabola with equation \(y^2 = 4ax\), where \(a\) is a constant, is translated by the vector \(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) to give the curve \(C\). The curve \(C\) passes through the point \((4, 7)\).

1. Show that \(a = 2\). [3 marks]
2. Find the values of \(k\) for which the line \(ky = x\) does not meet the curve \(C\). [6 marks]

1. Solve the equation \(x^2 + 4x + 20 = 0\), giving your answers in the form \(c + di\), where \(c\) and \(d\) are integers. [3 marks]
2. The roots of the quadratic equation $$z^2 + (4 + i + qi)z + 20 = 0$$ are \(w\) and \(w^*\).
   1. In the case where \(q\) is real, explain why \(q\) must be \(-1\). [2 marks]
   2. In the case where \(w = p + 2i\), where \(p\) is real, find the possible values of \(q\). [5 marks]

The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\).

1. 1. Find the matrix \(\mathbf{A}^2\). [1 mark]
   2. Describe fully the single geometrical transformation represented by the matrix \(\mathbf{A}^2\). [1 mark]
2. Given that the matrix \(\mathbf{B}\) represents a reflection in the line \(x + \sqrt{3}y = 0\), find the matrix \(\mathbf{B}\), giving the exact values of any trigonometric expressions. [2 marks]
3. Hence find the coordinates of the point \(P\) which is mapped onto \((0, -4)\) under the transformation represented by \(\mathbf{A}^2\) followed by a reflection in the line \(x + \sqrt{3}y = 0\). [6 marks]

A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).

1. Write down the equations of the asymptotes of \(C\). [2 marks]
2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]