Edexcel GCE Further Pure Mathematics FP2 June 2011 exam answers review

Find the set of values of x for which

(7)

1. Show that

   d³y/dx³= e^x[2yd²y/dx² + 2(dy/dx)² + kydy/dx + y² + 1]

   where k is a constant to be found.

   (3)

Given that, at x=0, y=1 and dy/dx = 2

2. find a series solution for y in ascending powers of x, up to and including the term in x²(4)

Find the general solution of the differential equation

giving your answer in the form y = f(x).

(8)

Given that

1. find the values of the constants A, B and C.

2. Show that

   (2r + 1)³ – (2r – 1)³ = 24r² + 2

   (2)

3. Using the result in part (b) and the method of differences, show that
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The point P represents the complex number z on an Argand diagram, where

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 = z + i/3 + iz, z ≠ 3i

The point Q is mapped by T onto the point R. Given that R lies on the real axis,

3. show that Q lies on C.(5)

The curve C shown in Figure 1 has polar equation

At the point A on C, the value of r is 5/2

The point N lies on the initial line and AN is perpendicular to the initial line.

The finite region R, shown shaded in Figure 1, is bounded by the curve C, the initial line and the line AN.

Find the exact area of the shaded region R.

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1. Use de Moivre’s theorem to show that

   sin5θ = 16sin⁵θ – 20sin³θ + 5sinθ

   (5)

Hence, given also that sin3θ = 3sinθ – 4sin³θ

2. find all the solutions of

   sin5θ = 5sin3θ

   in the interval 0≤θ≤2π Give your answers to 3 decimal places.

   (6)

The differential equation

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=½ and dx/dt = 0

   (5)

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

3. Find approximate values for the coordinates of A.(2)