Edexcel GCE Further Pure Mathematics FP2 June 2012 exam answers review

Find the set of values of x for which

The curve C has polar equation

At the point P on C, the tangent to C is parallel to the initial line.

Given that O is the pole, find the exact length of the line OP.

(7)

1. Express the complex number -2 + (2√3)i in the form r(cosθ + isinθ), -π≤θ≤π (3)

2. Solve the equation

   z² = -2 + (2√3)i

   giving the roots in the form r(cosθ + isinθ), -π≤θ≤π

   (5)

Find the general solution of the differential equation

1. Show that

   xd²y/dx² + (1 – 2y)dy/dx = 3

Given that y = 1 at x = 1,

2. find a series solution for y in ascending powers of (x – 1), up to and including the term in (x – 1)³(8)

1. Express 1/r(r + 2) in partial fractions.(2)
2. Hence prove, by the method of differences, that
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   where a and b are constants to be found.

   (6)

3. Hence show that

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1. Show that the substitution y = vx transforms the differential equation

   3xy²dy/dx = x³ + y³

   (I)

   into the differential equation

   3v²xdv/dx = 1 – 2v³

   (II)

2. By solving differential equation (II), find a general solution of differential equation (I) in the form y = f (x).(6)

Given that y = 2 at x = 1,

3. find the value of dy/dx at x=1(2)

The point P represents a complex number z on an Argand diagram such that

1. Show that, as z varies, the locus of P is a circle, stating the radius and the coordinates of the centre of this circle.(6)

The point Q represents a complex number z on an Argand diagram such that

arg (z − 6) = −3π/4
2. Sketch, on the same Argand diagram, the locus of P and the locus of Q as z varies.(4)
3. Find the complex number for which both z − 6i = 2 z − 3 and arg (z − 6) = –3π/4(4)