Edexcel FP2 (Further Pure Mathematics 2)

Find the set of values for which $$|x - 1| > 6x - 1.$$ [5]

1. Find the general solution of the differential equation $$t \frac{dv}{dt} - v = t, \quad t > 0$$ and hence show that the solution can be written in the form \(v = t(\ln t + c)\), where \(c\) is an arbitrary constant. [6]
2. This differential equation is used to model the motion of a particle which has speed \(v\) m s\(^{-1}\) at time \(t\) s. When \(t = 2\) the speed of the particle is \(3\) m s\(^{-1}\). Find, to \(3\) significant figures, the speed of the particle when \(t = 4\). [4]

1. Show that \(y = \frac{1}{2}x^2e^x\) is a solution of the differential equation $$\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + y = e^x.$$ [4]
2. Solve the differential equation $$\frac{d^2 y}{dx^2} - 2\frac{dy}{dx} + y = e^x,$$ given that at \(x = 0\), \(y = 1\) and \(\frac{dy}{dx} = 2\). [9]

The curve \(C\) has polar equation \(r = 3a \cos \theta\), \(-\frac{\pi}{2} \leq \frac{\pi}{2}\). The curve \(D\) has polar equation \(r = a(1 + \cos \theta)\), \(-\pi \leq \theta < \pi\). Given that \(a\) is a positive constant,

1. sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line. [4] The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
2. Find the polar coordinates of \(P\) and \(Q\). [3]
3. Use integration to find the exact value of the area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac{\pi}{3}\). [7] The region \(R\) contains all points which lie outside \(D\) and inside \(C\). Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac{\pi}{3}\) is $$\frac{3a^2}{16}(2\pi - 3\sqrt{3}),$$
4. show that the area of \(R\) is \(\pi a^2\). [4]

Using algebra, find the set of values of \(x\) for which $$2x - 5 > \frac{3}{x}.$$ [7]

1. Find the general solution of the differential equation $$\cos x \frac{dy}{dx} + (\sin x)y = \cos^3 x.$$ [6]
2. Show that, for \(0 \leq x \leq 2\pi\), there are two points on the \(x\)-axis through which all the solution curves for this differential equation pass. [2]
3. Sketch the graph, for \(0 \leq x \leq 2\pi\), of the particular solution for which \(y = 0\) at \(x = 0\). [3]

1. Find the general solution of the differential equation $$2\frac{d^2 y}{dt^2} + 7\frac{dy}{dt} + 3y = 3t^2 + 11t.$$ [8]
2. Find the particular solution of this differential equation for which \(y = 1\) and \(\frac{dy}{dt} = 1\) when \(t = 0\). [5]
3. For this particular solution, calculate the value of \(y\) when \(t = 1\). [1]

\includegraphics{figure_1} The curve \(C\) shown in Fig. 1 has polar equation $$r = a(3 + \sqrt{5} \cos \theta), \quad -\pi \leq \theta < \pi$$

1. Find the polar coordinates of the points \(P\) and \(Q\) where the tangents to \(C\) are parallel to the initial line. [6] The curve \(C\) represents the perimeter of the surface of a swimming pool. The direct distance from \(P\) to \(Q\) is \(20\) m.
2. Calculate the value of \(a\). [3]
3. Find the area of the surface of the pool. [6]

1. The point \(P\) represents a complex number \(z\) in an Argand diagram. Given that $$|z - 2i| = 2|z + i|,$$
   1. find a cartesian equation for the locus of \(P\), simplifying your answer. [2]
   2. sketch the locus of \(P\). [3]
2. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is a translation \(-7 + 11i\) followed by an enlargement with centre the origin and scale factor \(3\). Write down the transformation \(T\) in the form $$w = az + b, \quad a, b \in \mathbb{C}.$$ [2]

$$y\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + y = 0.$$

1. Find an expression for \(\frac{d^3 y}{dx^3}\). [5] 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 an including the term in \(x^3\). [5]
3. Comment on whether it would be sensible to use your series solution to give estimates for \(y\) at \(x = 0.2\) and at \(x = 50\). [2]

$$z = 4\left(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right) \text{ and } w = 3\left(\cos \frac{2\pi}{3} + i\sin \frac{2\pi}{3}\right).$$ Express \(zw\) in the form \(r(\cos \theta + i \sin \theta)\), \(r > 0\), \(-\pi < \theta < \pi\). [3]

1. Express \(\frac{2}{(r + 1)(r + 3)}\) in partial fractions. [2]
2. Hence prove that \(\sum_{r=1}^{n} \frac{2}{(r + 1)(r + 3)} = \frac{n(5n + 13)}{6(n + 2)(n + 3)}\). [5]

1. Sketch, on the same axes, the graphs with equation \(y = |2x - 3|\), and the line with equation \(y = 5x - 1\). [2]
2. Solve the inequality \(|2x - 3| < 5x - 1\). [3]

1. Use the substitution \(y = vx\) to transform the equation $$\frac{dy}{dx} = \frac{(4x + y)(x + y)}{x^2}, \quad x > 0 \quad \text{(I)}$$ into the equation $$x\frac{dv}{dx} = (2 + v)^2. \quad \text{(II)}$$ [4]
2. Solve the differential equation II to find \(v\) as a function of \(x\). [5]
3. Hence show that $$y = -2x - \frac{x}{\ln x + c}, \text{ where } c \text{ is an arbitrary constant,}$$ is a general solution of the differential equation I. [1]

1. Find the value of \(\lambda\) for which \(\lambda x \cos 3x\) is a particular integral of the differential equation $$\frac{d^2 y}{dx^2} + 9y = -12 \sin 3x.$$ [4]
2. Hence find the general solution of this differential equation. [4] The particular solution of the differential equation for which \(y = 1\) and \(\frac{dy}{dx} = 2\) at \(x = 0\), is \(y = g(x)\).
3. Find \(g(x)\). [4]
4. Sketch the graph of \(y = g(x)\), \(0 \leq x \leq \pi\). [2]

\includegraphics{figure_1} Figure 1 shows a sketch of the cardioid \(C\) with equation \(r = a(1 + \cos \theta)\), \(-\pi < \theta \leq \pi\). Also shown are the tangents to \(C\) that are parallel and perpendicular to the initial line. These tangents form a rectangle \(WXYZ\).

1. Find the area of the finite region, shaded in Fig. 1, bounded by the curve \(C\). [6]
2. Find the polar coordinates of the points \(A\) and \(B\) where \(WZ\) touches the curve \(C\). [5]
3. Hence find the length of \(WX\). [2] Given that the length of \(WZ\) is \(\frac{3\sqrt{3}a}{2}\),
4. find the area of the rectangle \(WXYZ\). [1] A heart-shape is modelled by the cardioid \(C\), where \(a = 10\) cm. The heart shape is cut from the rectangular card \(WXYZ\), shown in Fig. 1.
5. Find a numerical value for the area of card wasted in making this heart shape. [2]

1. Express as a simplified fraction \(\frac{1}{(r-1)^2} - \frac{1}{r^2}\). [2]
2. Prove, by the method of differences, that $$\sum_{r=2}^{n} \frac{2r-1}{r^2(r-1)^2} = 1 - \frac{1}{n^2}.$$ [3]

Solve the inequality \(\frac{1}{2x + 1} > \frac{x}{3x - 2}\). [6]

1. Using the substitution \(t = x^2\), or otherwise, find $$\int x^3 e^{-x^2} \, dx.$$ [6]
2. Find the general solution of the differential equation $$x\frac{dy}{dx} + 3y = xe^{-x^2}, \quad x > 0.$$ [4]

\includegraphics{figure_1} A logo is designed which consists of two overlapping closed curves. The polar equations of these curves are $$r = a(3 + 2\cos \theta) \quad \text{and}$$ $$r = a(5 - 2 \cos \theta), \quad 0 \leq \theta < 2\pi.$$ Figure 1 is a sketch (not to scale) of these two curves.

1. Write down the polar coordinates of the points \(A\) and \(B\) where the curves meet the initial line. [2]
2. Find the polar coordinates of the points \(C\) and \(D\) where the two curves meet. [4]
3. Show that the area of the overlapping region, which is shaded in the figure, is $$\frac{a^2}{3}(49\pi - 48\sqrt{3}).$$ [8]

$$\frac{d^2 y}{dt^2} - 6\frac{dy}{dt} + 9y = 4e^{3t}, \quad t \geq 0.$$

1. Show that \(Kte^{3t}\) is a particular integral of the differential equation, where \(K\) is a constant to be found. [4]
2. Find the general solution of the differential equation. [3] Given that a particular solution satisfies \(y = 3\) and \(\frac{dy}{dt} = 1\) when \(t = 0\),
3. find this solution. [4] Another particular solution which satisfies \(y = 1\) and \(\frac{dy}{dt} = 0\) when \(t = 0\), has equation $$y = (1 - 3t + 2t^2)e^{3t}.$$
4. For this particular solution draw a sketch graph of \(y\) against \(t\), showing where the graph crosses the \(t\)-axis. Determine also the coordinates of the minimum of the point on the sketch graph. [5]

1. 1. On the same Argand diagram sketch the loci given by the following equations. $$|z - 1| = 1,$$ $$\arg(z + 1) = \frac{\pi}{12},$$ $$\arg(z + 1) = \frac{\pi}{2}.$$ [4]
   2. Shade on your diagram the region for which $$|z - 1| \leq 1 \quad \text{and} \quad \frac{\pi}{12} \leq \arg(z + 1) \leq \frac{\pi}{2}.$$ [1]
2. 1. Show that the transformation $$w = \frac{z - 1}{z}, \quad z \neq 0,$$ maps \(|z - 1| = 1\) in the \(z\)-plane onto \(|w| = |w - 1|\) in the \(w\)-plane. [3] The region \(|z - 1| \leq 1\) in the \(z\)-plane is mapped onto the region \(T\) in the \(w\)-plane.
   2. Shade the region \(T\) on an Argand diagram. [2]

1. Use de Moivre's theorem to show that $$\cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.$$ [6]
2. Hence find \(3\) distinct solutions of the equation \(16x^5 - 20x^3 + 5x + 1 = 0\), giving your answers to \(3\) decimal places where appropriate. [4]

Prove by the method of differences that \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n + 1)(2n + 1)\), \(n > 1\). [6]

$$\frac{dy}{dx} + y\left(1 + \frac{3}{x}\right) = \frac{1}{x^2}, \quad x > 0.$$

1. Verify that \(x^3e^x\) is an integrating factor for the differential equation. [3]
2. Find the general solution of the differential equation. [4]
3. Given that \(y = 1\) at \(x = 1\), find \(y\) at \(x = 2\). [3]

1. Sketch, on the same axes, the graph of \(y = |(x - 2)(x - 4)|\), and the line with equation \(y = 6 - 2x\). [4]
2. Find the exact values of \(x\) for which \(|(x - 2)(x - 4)| = 6 - 2x\). [5]
3. Hence solve the inequality \(|(x - 2)(x - 4)| < 6 - 2x\). [2]

$$\frac{d^2 y}{dx^2} + 4\frac{dy}{dx} + 5y = 65 \sin 2x, \quad x > 0.$$

1. Find the general solution of the differential equation. [9]
2. Show that for large values of \(x\) this general solution may be approximated by a sine function and find this sine function. [2]

1. Sketch the curve with polar equation $$r = 3 \cos 2\theta, \quad -\frac{\pi}{4} \leq \theta < \frac{\pi}{4}.$$ [2]
2. Find the area of the smaller finite region enclosed between the curve and the half-line \(\theta = \frac{\pi}{6}\). [6]
3. Find the exact distance between the two tangents which are parallel to the initial line. [8]

Find the complete set of values of \(x\) for which $$|x^2 - 2| > 2x.$$ [7]

1. Find the general solution of the differential equation $$\frac{dy}{dx} + 2y = x.$$ [5] Given that \(y = 1\) at \(x = 0\),
2. find the exact values of the coordinates of the minimum point of the particular solution curve, [4]
3. draw a sketch of this particular solution curve. [2]

1. Find the general solution of the differential equation $$\frac{d^2 y}{dt^2} + 2\frac{dy}{dt} + 2y = 2e^{-t}.$$ [6]
2. Find the particular solution that satisfies \(y = 1\) and \(\frac{dy}{dt} = 1\) at \(t = 0\). [6]

\includegraphics{figure_1} Figure 1 is a sketch of the two curves \(C_1\) and \(C_2\) with polar equations $$C_1 : r = 3a(1 - \cos \theta), \quad -\pi \leq \theta < \pi$$ and $$C_2 : r = a(1 + \cos \theta), \quad -\pi \leq \theta < \pi.$$ The curves meet at the pole \(O\), and at the points \(A\) and \(B\).

1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\). [4]
2. Show that the length of the line \(AB\) is \(\frac{3\sqrt{3}}{2}a\). [2] The region inside \(C_2\) and outside \(C_1\) is shown shaded in Fig. 1.
3. Find, in terms of \(a\), the area of this region. [7] A badge is designed which has the shape of the shaded region. Given that the length of the line \(AB\) is \(4.5\) cm,
4. calculate the area of this badge, giving your answer to three significant figures. [3]

Given that \(y = \tan x\),

1. find \(\frac{dy}{dx}\), \(\frac{d^2 y}{dx^2}\) and \(\frac{d^3 y}{dx^3}\). [3]
2. Find the Taylor series expansion of \(\tan x\) in ascending powers of \(\left(x - \frac{\pi}{4}\right)\) up to and including the term in \(\left(x - \frac{\pi}{4}\right)^3\). [3]
3. Hence show that \(\tan \frac{3\pi}{10} \approx 1 + \frac{\pi}{10} + \frac{\pi^2}{200} + \frac{\pi^3}{3000}\). [2]

1. Prove by induction that $$\frac{d^n}{dx^n}(e^x \cos x) = 2^{\frac{1}{2}n} e^x \cos\left(x + \frac{1}{4}n\pi\right), \quad n \geq 1.$$ [8]
2. Find the Maclaurin series expansion of \(e^x \cos x\), in ascending powers of \(x\), up to and including the term in \(x^4\). [3]

The transformation \(T\) from the complex \(z\)-plane to the complex \(w\)-plane is given by $$w = \frac{z + 1}{z + i}, \quad z \neq -i.$$

1. Show that \(T\) maps points on the half-line \(\arg(z) = \frac{\pi}{4}\) in the \(z\)-plane into points on the circle \(|w| = 1\) in the \(w\)-plane. [4]
2. Find the image under \(T\) in the \(w\)-plane of the circle \(|z| = 1\) in the \(z\)-plane. [6]
3. Sketch on separate diagrams the circle \(|z| = 1\) in the \(z\)-plane and its image under \(T\) in the \(w\)-plane. [2]
4. Mark on your sketches the point \(P\), where \(z = i\), and its image \(Q\) under \(T\) in the \(w\)-plane. [2]

1. Sketch the graph of \(y = |x - 2a|\), given that \(a > 0\). [2]
2. Solve \(|x - 2a| > 2x + a\), where \(a > 0\). [3]

Find the general solution of the differential equation $$\frac{dy}{dx} + 2y \cot 2x = \sin x, \quad 0 < x < \frac{\pi}{2},$$ giving your answer in the form \(y = f(x)\). [7]

1. Express \(\frac{1}{r(r + 2)}\) in partial fractions. [2]
2. Hence prove, by the method of differences, that $$\sum_{r=1}^{n} \frac{4}{r(r + 2)} = \frac{n(3n + 5)}{(n + 1)(n + 2)}.$$ [5]
3. Find the value of \(\sum_{r=50}^{100} \frac{4}{r(r + 2)}\), to 4 decimal places. [3]

1. Show that the transformation \(y = xv\) transforms the equation $$x^2\frac{d^2 y}{dx^2} - 2x\frac{dy}{dx} + (2 + 9x^2)y = x^5, \quad \text{I}$$ into the equation $$\frac{d^2 v}{dx^2} + 9v = x^2. \quad \text{II}$$ [5]
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]

The curve \(C\) has polar equation \(r = 6 \cos \theta\), \(-\frac{\pi}{2} \leq \theta < \frac{\pi}{2}\), and the line \(D\) has polar equation \(r = 3 \sec\left(\frac{\pi}{3} - \theta\right)\), \(-\frac{\pi}{6} \leq \theta \leq \frac{5\pi}{6}\).

1. Find a cartesian equation of \(C\) and a cartesian equation of \(D\). [5]
2. Sketch on the same diagram the graphs of \(C\) and \(D\), indicating where each cuts the initial line. [3] The graphs of \(C\) and \(D\) intersect at the points \(P\) and \(Q\).
3. Find the polar coordinates of \(P\) and \(Q\). [5]

1. By expressing \(\frac{2}{4r^2 - 1}\) in partial fractions, or otherwise, prove that $$\sum_{r=1}^{n} \frac{2}{4r^2 - 1} = 1 - \frac{1}{2n + 1}.$$ [3]
2. Hence find the exact value of \(\sum_{r=11}^{20} \frac{2}{4r^2 - 1}\). [2]

Find the general solution of the differential equation $$(x + 1)\frac{dy}{dx} + 2y = \frac{1}{x}, \quad x > 0.$$ giving your answer in the form \(y = f(x)\). [7]

1. On the same diagram, sketch the graphs of \(y = |x^2 - 4|\) and \(y = |2x - 1|\), showing the coordinates of the points where the graphs meet the axes. [4]
2. Solve \(|x^2 - 4| = |2x - 1|\), giving your answers in surd form where appropriate. [5]
3. Hence, or otherwise, find the set of values of \(x\) for which of \(|x^2 - 4| > |2x - 1|\). [3]

1. Find the general solution of the differential equation $$2\frac{d^2 x}{dt^2} + 5\frac{dx}{dt} + 2x = 2t + 9.$$ [6]
2. Find the particular solution of this differential equation for which \(x = 3\) and \(\frac{dx}{dt} = -1\) when \(t = 0\). [4] The particular solution in part (b) is used to model the motion of a particle \(P\) on the \(x\)-axis. At time \(t\) seconds (\(t \geq 0\)), \(P\) is \(x\) metres from the origin \(O\).
3. Show that the minimum distance between \(O\) and \(P\) is \(\frac{1}{2}(5 + \ln 2)\) m and justify that the distance is a minimum. [4]

\includegraphics{figure_1} The curve \(C\) which passes through \(O\) has polar equation $$r = 4a(1 + \cos \theta), \quad -\pi < \theta \leq \pi$$ The line \(l\) has polar equation $$r = 3a \sec \theta, \quad -\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$ The line \(l\) cuts \(C\) at the points \(P\) and \(Q\), as shown in Figure 1.

1. Prove that \(PQ = 6\sqrt{3}a\). [6] The region \(R\), shown shaded in Figure 1, is bounded by \(l\) and \(C\).
2. Use calculus to find the exact area of \(R\). [7]

A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that $$|z - 3i| = 3,$$

1. sketch the locus of \(P\). [2]
2. Find the complex number \(z\) which satisfies both \(|z - 3i| = 3\) and \(\arg (z - 3i) = \frac{3}{4}\pi\). [4] The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by $$w = \frac{2i}{w}.$$
3. Show that \(T\) maps \(|z - 3i| = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line. [5]