Questions — Edexcel FP2 (360 questions)

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Edexcel FP2 Q6
12 marks Standard +0.8
  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]
Edexcel FP2 Q7
13 marks Challenging +1.3
  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]
Edexcel FP2 2008 June Q1
Moderate -0.3
Solve the differential equation \(\frac{dy}{dx} - 3y = x\) to obtain \(y\) as a function of \(x\). (Total 5 marks)
Edexcel FP2 2008 June Q2
Standard +0.3
  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)
Edexcel FP2 2008 June Q3
Challenging +1.2
  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)
Edexcel FP2 2008 June Q4
Challenging +1.2
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}
Edexcel FP2 2008 June Q5
Standard +0.8
  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)
Edexcel FP2 2008 June Q6
Standard +0.3
  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)
Edexcel FP2 2008 June Q7
Challenging +1.2
  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)
Edexcel FP2 2008 June Q8
Challenging +1.3
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}
Edexcel FP2 2008 June Q9
Challenging +1.8
$$(x^2 + 1)\frac{d^2y}{dx^2} = 2y^2 + (1 - 2x)\frac{dy}{dx}$$ (I)
  1. By differentiating equation (I) with respect to \(x\), show that
Edexcel FP2 2008 June Q9
Challenging +1.2
$$(x^2 + 1)\frac{d^2y}{dx^2} = 2y^2 + (1 - 2x)\frac{dy}{dx}$$ (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)
Edexcel FP2 2008 June Q10
Standard +0.3
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)
Edexcel FP2 2008 June Q11
Challenging +1.2
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)
Edexcel FP2 Q1
5 marks Moderate -0.3
Find the set of values for which $$|x - 1| > 6x - 1.$$ [5]
Edexcel FP2 Q2
10 marks Standard +0.3
  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]
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]