Questions — Edexcel FP2 (360 questions)

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Edexcel FP2 Q37
7 marks Standard +0.8
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]
Edexcel FP2 Q38
10 marks Standard +0.3
  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]
Edexcel FP2 Q39
12 marks Challenging +1.3
  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]
Edexcel FP2 Q40
13 marks Standard +0.8
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]
Edexcel FP2 Q41
5 marks Standard +0.3
  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]
Edexcel FP2 Q42
7 marks Standard +0.3
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]
Edexcel FP2 Q43
12 marks Standard +0.3
  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]
Edexcel FP2 Q44
14 marks Standard +0.3
  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]
Edexcel FP2 Q45
13 marks Challenging +1.3
\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]
Edexcel FP2 Q46
11 marks Standard +0.3
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]