Edexcel FP2 (Further Pure Mathematics 2) 2008 June

Solve the differential equation \(\frac{dy}{dx} - 3y = x\) to obtain \(y\) as a function of \(x\). (Total 5 marks)

1. Simplify the expression \(\frac{(x + 3)(x + 9)}{x - 1} - (3x - 5)\), giving your answer in the form \(\frac{a(x + b)(x + c)}{x - 1}\), where \(a\), \(b\) and \(c\) are integers. (4)
2. Hence, or otherwise, solve the inequality \(\frac{(x + 3)(x + 9)}{x - 1} > 3x - 5\) (4)(Total 8 marks)

1. Find the general solution of the differential equation \(3\frac{d^2y}{dx^2} - \frac{dy}{dx} - 2y = x^2\) (8)
2. Find the particular solution for which, at \(x = 0\), \(y = 2\) and \(\frac{dy}{dx} = 3\). (6)(Total 14 marks)

The diagram above shows the curve \(C_1\) which has polar equation \(r = a(3 + 2\cos\theta)\), \(0 \leq \theta < 2\pi\) and the circle \(C_2\) with equation \(r = 4a\), \(0 \leq \theta < 2\pi\), where \(a\) is a positive constant.

1. Find, in terms of \(a\), the polar coordinates of the points where the curve \(C_1\) meets the circle \(C_2\).(4)

The regions enclosed by the curves \(C_1\) and \(C_2\) overlap and this common region \(R\) is shaded in the figure.

1. Find, in terms of \(a\), an exact expression for the area of the region \(R\).(8)
2. In a single diagram, copy the two curves in the diagram above and also sketch the curve \(C_3\) with polar equation \(r = 2a\cos\theta\), \(0 \leq \theta < 2\pi\) Show clearly the coordinates of the points of intersection of \(C_1\), \(C_2\) and \(C_3\) with the initial line, \(\theta = 0\).(3)(Total 15 marks)

\includegraphics{figure_4}

1. Find, in terms of \(k\), the general solution of the differential equation $$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 3x = kt + 5, \text{ where } k \text{ is a constant and } t > 0.$$ (7) For large values of \(t\), this general solution may be approximated by a linear function.
2. Given that \(k = 6\), find the equation of this linear function.(2)(Total 9 marks)

1. Find, in the simplest surd form where appropriate, the exact values of \(x\) for which $$\frac{x}{2} + 3 = \left|\frac{4}{x}\right|.$$ (5)
2. Sketch, on the same axes, the line with equation \(y = \frac{x}{2} + 3\) and the graph of $$y = \left|\frac{4}{x}\right|, x \neq 0.$$ (3)
3. Find the set of values of \(x\) for which \(\frac{x}{2} + 3 > \left|\frac{4}{x}\right|\). (2)(Total 10 marks)

1. Show that the substitution \(y = vx\) transforms the differential equation $$\frac{dy}{dx} = \frac{x}{y} + \frac{3y}{x}, x > 0, y > 0$$ (I) into the differential equation \(x\frac{dv}{dx} = 2v + \frac{1}{v}\). (II) (3)
2. By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = f(x)\). (7)

Given that \(y = 3\) at \(x = 1\), (c)find the particular solution of differential equation (I).(2)

The curve \(C\) shown in the diagram above has polar equation $$r = 4(1 - \cos\theta), 0 \leq \theta \leq \frac{\pi}{2}.$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the line \(\theta = \frac{\pi}{2}\).

1. Show that \(P\) has polar coordinates \(\left(2, \frac{\pi}{3}\right)\).(5)

The curve \(C\) meets the line \(\theta = \frac{\pi}{2}\) at the point \(A\). The tangent to \(C\) at the initial line at the point \(N\). The finite region \(R\), shown shaded in the diagram above, is bounded by the initial line, the line \(\theta = \frac{\pi}{2}\), the arc \(AP\) of \(C\) and the line \(PN\).

1. Calculate the exact area of \(R\). (8)

\includegraphics{figure_8}

$$(x^2 + 1)\frac{d^2y}{dx^2} = 2y^2 + (1 - 2x)\frac{dy}{dx}$$ (I)

1. By differentiating equation (I) with respect to \(x\), show that

$$(x^2 + 1)\frac{d^2y}{dx^2} = 2y^2 + (1 - 2x)\frac{dy}{dx}$$ (I)

1. By differentiating equation (I) with respect to \(x\), show that $$(x^2 + 1)\frac{d^3y}{dx^3} = (1 - 4x)\frac{d^2y}{dx^2} + (4y - 2)\frac{dy}{dx}.$$ (3) Given that \(y = 1\) and \(\frac{dy}{dx} = 1\) at \(x = 0\),
2. find the series solution for \(y\), in ascending powers of \(x\), up to and including the term in \(x_3\).(4)
3. Use your series to estimate the value of \(y\) at \(x = -0.5\), giving your answer to two decimal places.(1)

The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 3| = 2|z|.$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, and give the coordinates of the centre and the radius of the circle.(5)

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z + 3| = |z - i\sqrt{3}|.$$

1. Sketch, on the same Argand diagram, the locus of \(P\) and the locus of \(Q\) as \(z\) varies.(5)
2. On your diagram shade the region which satisfies $$|z - 3| \geq 2|z| \text{ and } |z + 3| \geq |z - i\sqrt{3}|.$$ (2)

De Moivre's theorem states that \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\) for \(n \in \mathbb{R}\)

1. Use induction to prove de Moivre's theorem for \(n \in \mathbb{Z}^+\). (5)
2. Show that \(\cos 5\theta = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta\) (5)
3. Hence show that \(2\cos\frac{\pi}{10}\) is a root of the equation $$x^4 - 5x^2 + 5 = 0$$ (3)