Questions — Edexcel (10514 questions)

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Edexcel FP2 Q3
13 marks Standard +0.3
  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]
Edexcel FP2 Q4
18 marks Challenging +1.2
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]
Edexcel FP2 Q5
7 marks Standard +0.3
Using algebra, find the set of values of \(x\) for which $$2x - 5 > \frac{3}{x}.$$ [7]
Edexcel FP2 Q6
11 marks Standard +0.8
  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]
Edexcel FP2 Q7
14 marks Standard +0.8
  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]
Edexcel FP2 Q8
15 marks Challenging +1.8
\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]
Edexcel FP2 Q9
7 marks Standard +0.3
  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]
Edexcel FP2 Q10
12 marks Challenging +1.8
$$y\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + y = 0.$$
  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]
Edexcel FP2 Q11
3 marks Moderate -0.8
$$z = 4\left(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4}\right) \text{ and } w = 3\left(\cos \frac{2\pi}{3} + i\sin \frac{2\pi}{3}\right).$$ Express \(zw\) in the form \(r(\cos \theta + i \sin \theta)\), \(r > 0\), \(-\pi < \theta < \pi\). [3]
Edexcel FP2 Q12
7 marks Standard +0.8
  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]
Edexcel FP2 Q13
5 marks Moderate -0.8
  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]
Edexcel FP2 Q14
10 marks Challenging +1.2
  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]
Edexcel FP2 Q15
14 marks Standard +0.8
  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]
Edexcel FP2 Q16
16 marks Challenging +1.2
\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]
Edexcel FP2 Q17
5 marks Standard +0.8
  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]
Edexcel FP2 Q18
6 marks Standard +0.8
Solve the inequality \(\frac{1}{2x + 1} > \frac{x}{3x - 2}\). [6]
Edexcel FP2 Q19
10 marks Standard +0.8
  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]
Edexcel FP2 Q20
14 marks Challenging +1.2
\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]
Edexcel FP2 Q21
16 marks Standard +0.3
$$\frac{d^2 y}{dt^2} - 6\frac{dy}{dt} + 9y = 4e^{3t}, \quad t \geq 0.$$
  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]
Edexcel FP2 Q22
10 marks Standard +0.8
    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]
    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]
Edexcel FP2 Q23
10 marks Challenging +1.2
  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]
Edexcel FP2 Q24
6 marks Challenging +1.2
Prove by the method of differences that \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n + 1)(2n + 1)\), \(n > 1\). [6]
Edexcel FP2 Q25
10 marks Standard +0.3
$$\frac{dy}{dx} + y\left(1 + \frac{3}{x}\right) = \frac{1}{x^2}, \quad x > 0.$$
  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]
Edexcel FP2 Q26
11 marks Standard +0.3
  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]
Edexcel FP2 Q27
11 marks Standard +0.8
$$\frac{d^2 y}{dx^2} + 4\frac{dy}{dx} + 5y = 65 \sin 2x, \quad x > 0.$$
  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]