Edexcel FP2 (Further Pure Mathematics 2)

1. Express \(\frac{1}{t(t+2)}\) in partial fractions. [1]
2. Hence show that \(\sum_{n=1}^{\infty} \frac{4}{n(n+2)} = \frac{n(3n+5)}{(n+1)(n+2)}\) [5]

1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
2. Using your answer to part (a) and the method of differences, show that $$\sum_{r=1}^n \frac{3}{(3r-1)(3r+2)} = \frac{3n}{2(3n+2)}$$ [3]
3. Evaluate \(\sum_{r=1}^{30} \frac{3}{(3r-1)(3r+2)}\), giving your answer to 3 significant figures. [2]

Find the set of values of \(x\) for which $$\frac{3}{x+3} > \frac{x-4}{x}.$$ [7]

Find the set of values of \(x\) for which $$|x^2 - 4| > 3x.$$ [5]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + 2i}{iz}$$ The transformation maps points on the real axis in the \(z\)-plane onto a line in the \(w\)-plane. Find an equation of this line. [4]

1. Express \(\frac{2}{(2r + 1)(2r + 3)}\) in partial fractions. [2]
2. Using your answer to (a), find, in terms of \(n\), $$\sum_{r=1}^n \frac{2}{(2r + 1)(2r + 3)}$$ [3]

Give your answer as a single fraction in its simplest form.

Solve the equation $$z^2 = 4\sqrt{2} - 4\sqrt{2}i,$$ giving your answers in the form \(r(\cos \theta + i \sin \theta)\), where \(-\pi < \theta \leq \pi\). [6]

The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation $$\frac{d^2 x}{dt^2} + x + \cos x = 0.$$ When \(t = 0\), \(x = 0\) and \(\frac{dx}{dt} = \frac{1}{2}\). Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t^4\). [5]

$$\frac{d^2 y}{dx^2} = e^x \left(2x \frac{dy}{dx} + y^2 + 1\right).$$

1. Show that $$\frac{d^4 y}{dx^4} = e^x \left[2x \frac{d^3 y}{dx^3} + 4 \frac{d^2 y}{dx^2} + 6y \frac{dy}{dx} + y^2 + 1\right],$$ where \(k\) is a constant to be found. [3]

Given that, at \(x = 0\), \(y = 1\) and \(\frac{dy}{dx} = 2\),

1. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x^4\). [4]

The curve \(C\) has polar equation $$r = 1 + 2 \cos \theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$ At the point \(P\) on \(C\), the tangent to \(C\) is parallel to the initial line. Given that \(O\) is the pole, find the exact length of the line \(OP\). [7]

Use algebra to find the set of values of \(x\) for which $$\frac{6x}{3 - x} > \frac{x + 1}{1}$$ [7]

\(z = 5\sqrt{3} - 5i\) Find

1. \(|z|\), [1]
2. \(\arg(z)\), in terms of \(\pi\). [2]

$$w = 2\left[\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right]$$ Find

1. \(\left|\frac{w}{z}\right|\), [1]
2. \(\arg\left(\frac{w}{z}\right)\), in terms of \(\pi\). [2]

Find the general solution of the differential equation $$\sin x \frac{dy}{dx} - y \cos x = \sin 2x \sin x$$ giving your answer in the form \(y = f(x)\). [8]

1. Find the set of values of \(x\) for which $$x + 4 > \frac{2}{x+3}$$ [6]
2. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \left|\frac{2}{x+3}\right|$$ [1]

Find the general solution of the differential equation $$x \frac{dy}{dx} + 5y = \frac{\ln x}{x}, \quad x > 0,$$ giving your answer in the form \(y = f(x)\). [8]

1. Express the complex number \(-2 + (2\sqrt{3})i\) in the form \(r(\cos \theta + i \sin \theta)\), \(-\pi < \theta \leq \pi\) [3]
2. Solve the equation $$z^3 = -2 + (2\sqrt{3})i$$ giving the roots in the form \(r(\cos \theta + i \sin \theta)\), \(-\pi < \theta \leq \pi\). [5]

1. Express \(\frac{2}{(r + 1)(r + 3)}\) in partial fractions. [2]
2. Hence show that $$\sum_{r=1}^{\infty} \frac{2}{(r + 1)(r + 3)} = \frac{n(5n + 13)}{6(n + 2)(n + 3)}$$ [4]
3. Evaluate \(\sum_{r=1}^{30} \frac{2}{(r + 1)(r + 3)}\), giving your answer to 3 significant figures. [2]

$$\frac{d^2 y}{dx^2} + 4y - \sin x = 0$$ Given that \(y = \frac{1}{2}\) and \(\frac{dy}{dx} = \frac{1}{8}\) at \(x = 0\), find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x^5\). [5]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3\cos \theta, \quad a > 0, \quad 0 \leq \theta < 2\pi.$$ The area enclosed by the curve is \(\frac{10\pi}{2}\). Find the value of \(a\). [8]

\(z = -8 + (8\sqrt{3})i\)

1. Find the modulus of \(z\) and the argument of \(z\). [3]

Using de Moivre's theorem,

1. find \(z^3\). [2]
2. find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [5]

Given that $$(2r + 1)^3 = Ar^3 + Br^2 + Cr + 1,$$

1. find the values of the constants \(A\), \(B\) and \(C\). [2]
2. Show that $$(2r + 1)^3 - (2r - 1)^3 = 24r^2 + 2.$$ [2]
3. Using the result in part (b) and the method of differences, show that $$\sum_{r=1}^n r^2 = \frac{1}{6}n(n + 1)(2n + 1).$$ [5]

Find the general solution of the differential equation $$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2 \cos t - \sin t.$$ [9]

Given that $$y \frac{d^3 y}{dx^3} + \left(\frac{dy}{dx}\right)^2 + 5y = 0$$

1. find \(\frac{d^3 y}{dx^3}\) in terms of \(\frac{d^2 y}{dx^2}\), \(\frac{dy}{dx}\) and \(y\). [4]

Given that \(y = 2\) and \(\frac{dy}{dx} = 2\) at \(x = 0\),

1. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x^3\). [5]

1. Given that $$z = r(\cos n\theta + i \sin n\theta), \quad r \in \mathbf{R}$$ prove, by induction, that \(z^n = r^n(\cos n\theta + i \sin n\theta)\), \(n \in \mathbf{Z}^+\). [5]
2. Find the exact value of \(w^2\), giving your answer in the form \(a + ib\), where \(a, b \in \mathbf{R}\). [2]

\(y = \sec^2 x\)

1. Show that \(\frac{d^2 y}{dx^2} = 6 \sec^4 x - 4 \sec^2 x\). [4]
2. Find a Taylor series expansion of \(\sec^2 x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\), up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [6]

\includegraphics{figure_1} Figure 1 Figure 1 shows the curves given by the polar equations $$r = 2, \quad 0 \leq \theta \leq \frac{\pi}{2},$$ and $$r = 1.5 + \sin 3\theta, \quad 0 \leq \theta \leq \frac{\pi}{2}.$$

1. Find the coordinates of the points where the curves intersect. [3]

The region \(S\), between the curves, for which \(r > 2\) and for which \(r < (1.5 + \sin 3\theta)\), is shown shaded in Figure 1.

1. Find, by integration, the area of the shaded region \(S\), giving your answer in the form \(a\pi + b\sqrt{3}\), where \(a\) and \(b\) are simplified fractions. [7]

The point \(P\) represents the complex number \(z\) on an Argand diagram, where $$|z - i| = 2.$$ The locus of \(P\) as \(z\) varies is the curve \(C\).

1. Find a cartesian equation of \(C\). [2]
2. Sketch the curve \(C\). [2]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z + i}{3 + iz}, \quad z \neq 3i.$$ The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,

1. show that \(Q\) lies on \(C\). [5]

$$x \frac{dy}{dx} = 3x + y^2.$$

1. Show that $$\frac{d^2 y}{dx^2} + (1 - 2y) \frac{dy}{dx} = 3.$$ [2]

Given that \(y = 1\) at \(x = 1\),

1. find a series solution for \(y\) in ascending powers of \((x - 1)\), up to and including the term in \((x - 1)^3\). [8]

1. Find, in the form \(y = f(x)\), the general solution of the equation $$\frac{dy}{dx} = 2y \tan x + \sin 2x, \quad 0 < x < \frac{\pi}{2}$$ [6]

Given that \(y = 2\) at \(x = \frac{\pi}{6}\),

1. find the value of \(y\) at \(x = \frac{\pi}{4}\), giving your answer in the form \(a + k \ln b\), where \(a\) and \(b\) are integers and \(k\) is rational. [4]

1. Find the general solution of the differential equation $$x \frac{dy}{dx} + 2y = 4x^2$$ [5]
2. Find the particular solution for which \(y = 5\) at \(x = 1\), giving your answer in the form \(y = f(x)\). [2]
3. Find the exact values of the coordinates of the turning points of the curve with equation \(y = f(x)\), making your method clear. [???]
4. Sketch the curve with equation \(y = f(x)\), showing the coordinates of the turning points. [5]

A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{z}{z-i}, \quad z \neq i.$$ The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).

1. Show that \(C\) is a circle and find its centre and radius. [8]

The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.

1. Shade the region \(R\) on an Argand diagram. [2]

A complex number \(z\) is represented by the point \(P\) in the Argand diagram.

1. Given that \(|z - 6| = |z|\), sketch the locus of \(P\). [2]
2. Find the complex numbers \(z\) which satisfy both \(|z - 6| = |z|\) and \(|z - 3 - 4i| = 5\). [3]

The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac{30}{z}\).

1. Show that \(T\) maps \(|z - 6| = |z|\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle. [5]

\includegraphics{figure_1} The differential equation $$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 9x = \cos 3t, \quad t \geq 0,$$ describes the motion of a particle along the \(x\)-axis.

1. Find the general solution of this differential equation. [8]
2. Find the particular solution of this differential equation for which, at \(t = 0\), \(x = \frac{1}{2}\) and \(\frac{dx}{dt} = 0\). [5]

On the graph of the particular solution defined in part (b), the first turning point for \(T > 30\) is the point \(A\).

1. Find approximate values for the coordinates of \(A\). [2]

1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
2. Hence prove, by the method of differences, that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [6]

The point \(Q\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6| = 2|z - 3|.$$

1. Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6]

where \(a\) and \(b\) are constants to be found.

1. Hence show that $$\sum_{r=1}^{2n} \frac{1}{r(r + 2)} = \frac{n(4n + 5)}{4(n + 1)(n + 2)},$$ [3]
2. Find the complex number for which both \(|z - 6| = 2|z - 3|\) and \(\arg(z - 6) = -\frac{3\pi}{4}\). [4]

The complex number \(z = e^{i\theta}\), where \(\theta\) is real.

1. Use de Moivre's theorem to show that $$z^n + \frac{1}{z^n} = 2\cos n\theta$$ where \(n\) is a positive integer. [2]
2. Show that $$\cos^n \theta = \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta)$$ [5]
3. Hence find all the solutions of $$\cos 5\theta + 5\cos 3\theta + 12\cos \theta = 0$$ in the interval \(0 \leq \theta < 2\pi\). [4]

1. Use algebra to find the exact solutions of the equation $$|2x^2 + 6x - 5| = 5 - 2x$$ [6]
2. On the same diagram, sketch the curve with equation \(y = |2x^2 + 6x - 5|\) and the line with equation \(y = 5 - 2x\), showing the \(x\)-coordinates of the points where the line crosses the curve. [3]
3. Find the set of values of \(x\) for which $$|2x^2 + 6x - 5| > 5 - 2x$$ [3]

1. Sketch the graph of \(y = |x^2 - a^2|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes. [2]
2. Solve \(|x^2 - a^2| = a^2 - x\), \(a > 1\). [6]
3. Find the set of values of \(x\) for which \(|x^2 - a^2| > a^2 - x\), \(a > 1\). [4]

1. Show that the transformation \(z = y^{\frac{1}{2}}\) transforms the differential equation $$\frac{dy}{dx} - 4y \tan x = 2y^{\frac{1}{2}}$$ [I] into the differential equation $$\frac{dz}{dx} - 2z \tan x = 1$$ [II]
2. Solve the differential equation (II) to find \(z\) as a function of \(x\). [6]
3. Hence obtain the general solution of the differential equation (I). [1]

1. Use de Moivre's theorem to show that $$\sin 5\theta = 16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta$$ [5]

Hence, given also that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\),

1. find all the solutions of $$\sin 5\theta = 5 \sin 3\theta$$ in the interval \(0 \leq \theta < 2\pi\). Give your answers to 3 decimal places. [6]

1. Show that the substitution \(y = vx\) transforms the differential equation $$3xy^2 \frac{dv}{dx} = v^4 + y^3$$ [I] into the differential equation $$3x v^2 \frac{dv}{dx} = 1 - 2v^3$$ [II] [3]
2. By solving differential equation (II), find a general solution of differential equation (I) in the form \(y = f(x)\). [6]

Given that \(y = 2\) at \(x = 1\),

1. find the value of \(\frac{dy}{dx}\) at \(x = 1\). [2]

1. Find the value of \(z\) for which \(z^{2e^x}\) is a particular integral of the differential equation $$\frac{d^2 y}{dt^2} - 6 \frac{dy}{dt} + 9y = 6e^{3t}, \quad t \geq 0$$ [5]
2. Hence find the general solution of this differential equation. [3]

Given that when \(t = 0\), \(y = 5\) and \(\frac{dy}{dt} = 4\)

1. find the particular solution of this differential equation, giving your solution in the form \(y = f(t)\). [5]

1. Show that the transformation \(y = xv\) transforms the equation $$4x^2 \frac{d^2 y}{dx^2} - 8x \frac{dy}{dx} + (8 + 4x^2)y = x^4$$ [I] into the equation $$x^2 \frac{d^2 v}{dx^2} + 4v = x$$ [II] [6]
2. Solve the differential equation (II) to find \(v\) as a function of \(x\). [6]
3. Hence state the general solution of the differential equation (I). [1]

$$\frac{d^2 x}{dt^2} + 6 \frac{dx}{dt} + 6x = 2e^{-t}.$$ Given that \(x = 0\) and \(\frac{dx}{dt} = 2\) at \(t = 0\),

1. find \(x\) in terms of \(t\). [8]

The solution to part (a) is used to represent the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds, where \(t > 0\), \(P\) is \(x\) metres from the origin \(O\).

1. Show that the maximum distance between \(O\) and \(P\) is \(\frac{2\sqrt{3}}{9}\) m and justify that this distance is a maximum. [7]

1. Find the value of \(z\) for which \(y = zx \sin 5x\) is a particular integral of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [4]
2. Using your answer to part (a), find the general solution of the differential equation $$\frac{d^2 y}{dx^2} + 25y = 3 \cos 5x.$$ [3]

Given that at \(x = 0\), \(y = 0\) and \(\frac{dy}{dx} = 5\),

1. find the particular solution of this differential equation, giving your solution in the form \(y = f(x)\). [5]
2. Sketch the curve with equation \(y = f(x)\) for \(0 \leq x \leq \pi\). [2]

\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3(\cos 2\theta)^{\frac{1}{2}}, \quad \text{where } -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}, \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}.$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.

1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4]
2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9]