AQA FP1 (Further Pure Mathematics 1) 2014 June

A curve passes through the point \((9, 6)\) and satisfies the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$ Use a step-by-step method with a step length of \(0.25\) to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places. [5 marks]

The quadratic equation $$2x^2 + 8x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2 marks]
2. 1. Find the value of \(\alpha^2 + \beta^2\). [2 marks]
   2. Hence, or otherwise, show that \(\alpha^4 + \beta^4 = \frac{449}{2}\). [2 marks]
3. Find a quadratic equation, with integer coefficients, which has roots $$2\alpha^4 + \frac{1}{\beta^2} \text{ and } 2\beta^4 + \frac{1}{\alpha^2}$$ [5 marks]

Use the formulae for \(\sum_{r=1}^{n} r^3\) and \(\sum_{r=1}^{n} r^2\) to find the value of $$\sum_{r=3}^{60} r^2(r - 6)$$ [4 marks]

Find the complex number \(z\) such that $$5iz + 3z^* + 16 = 8i$$ Give your answer in the form \(a + bi\), where \(a\) and \(b\) are real. [6 marks]

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

1. Find the gradient of the line passing through the point \((-5, 10)\) and the point on \(C\) with \(x\)-coordinate \(-5 + h\). Give your answer in its simplest form. [3 marks]
2. Show how the answer to part (a) can be used to find the gradient of the curve \(C\) at the point \((-5, 10)\). State the value of this gradient. [2 marks]

A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).

1. Write down the equations of all the asymptotes of \(C\). [2 marks]
2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
   1. Find the \(y\)-coordinate of the stationary point. [1 mark]
   2. Sketch the curve \(C\). [2 marks]
3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]

1. Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
   1. a reflection in the line \(y = -x\); [1 mark]
   2. a stretch parallel to the \(y\)-axis of scale factor \(7\). [1 mark]
2. Hence find the matrix corresponding to the combined transformation of a reflection in the line \(y = -x\) followed by a stretch parallel to the \(y\)-axis of scale factor \(7\). [2 marks]
3. The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} -3 & -\sqrt{3} \\ -\sqrt{3} & 3 \end{bmatrix}\).
   1. Show that \(\mathbf{A}^2 = k\mathbf{I}\), where \(k\) is a constant and \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [1 mark]
   2. Show that the matrix \(\mathbf{A}\) corresponds to a combination of an enlargement and a reflection. State the scale factor of the enlargement and state the equation of the line of reflection in the form \(y = (\tan \theta)x\). [5 marks]

1. Find the general solution of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ giving your answer for \(x\) in terms of \(\pi\). [5 marks]
2. Use your general solution to find the sum of all the solutions of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ that lie in the interval \(0 \leqslant x \leqslant 20\pi\). Give your answer in the form \(k\pi\), stating the exact value of \(k\). [4 marks]

An ellipse \(E\) has equation $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$

1. Sketch the ellipse \(E\), showing the values of the intercepts on the coordinate axes. [2 marks]
2. Given that the line with equation \(y = x + k\) intersects the ellipse \(E\) at two distinct points, show that \(-5 < k < 5\). [5 marks]
3. The ellipse \(E\) is translated by the vector \(\begin{bmatrix} a \\ b \end{bmatrix}\) to form another ellipse whose equation is \(9x^2 + 16y^2 + 18x - 64y = c\). Find the values of the constants \(a\), \(b\) and \(c\). [5 marks]
4. Hence find an equation for each of the two tangents to the ellipse \(9x^2 + 16y^2 + 18x - 64y = c\) that are parallel to the line \(y = x\). [3 marks]