Sketch on a single Argand diagram the loci given by
| z - 1 + 2 i | = 3,
[2]
| z + 1 | = | z - 2 |.
[2]
Indicate, by shading, the region of the Argand diagram for which | z - 1 + 2 i | ⩽ 3 and
| z + 1 | ⩽ | z - 2 |.
[2]
You are given that f(x) = tan-1(1 + x).
Find the value of f (0).
[1]
Determine the value of f ′ (0).
[2]
Hence find the Maclaurin series for f(x) up to and including the term in x2.
[2]
A function f(z) is defined on all complex numbers z by f(z) = z3 - 3z2 + kz - 5 where k is a real constant. The roots of the equation f(z) = 0 are α, β and γ. You are given that α2 + β2 + γ2 =-5.
Explain why f (z) = 0 has only one real root.
[3]
Points A, B and C have coordinates (4, 2, 0), (1, 5, 3) and (1, 4, - 2) respectively. The line l passes through A and B.
Find a cartesian equation for l.
[3]
M is the point on l that is closest to C.
Find the coordinates of M.
[4]
Find the exact area of the triangle ABC.
[4]
Use de Moivre’s theorem to find the constants A, B and C in the identity
sin5θ ≡ A sin θ + B sin 3θ + C sin 5θ.
[4]
O is the origin of a coordinate system whose units are cm. The points A, B, C and D have coordinates (1, 0), (1, 4), (6, 9) and (0, 9) respectively. The arc BC is part of the curve with equation x2 + (y - 10)2 = 37. The closed shape OABCD is formed, in turn, from the line segments OA and AB, the arc BC and the line segments CD and DO (see diagram). A funnel can be modelled by rotating OABCD by 2 π radians about the y-axis.
Find the volume of the funnel according to the model.
[3]
The diagram below shows the curve with polar equation r = sin 3θ for 0 ⩽ θ ⩽ 
.
Find the polar coordinates of the point on the curve where r takes its maximum value.
[2]
In this question you must show detailed reasoning.
Find the exact area enclosed by the curve.
[4]
Given that sin 3θ = 3 sin θ - 4 sin3θ, find a cartesian equation for the curve.
[2]
You are given that f (x) = 4 sinh x + 3 cosh x.
Show that the curve y = f (x) has no turning points.
[3]
Determine the exact solution of the equation f (x) = 5.
[5]
Determine whether the line with equation y = kx in part (a) is a line of invariant points
under T.
[1]
[8]
The displacement of a door from its equilibrium (closed) position is measured by the angle, θ radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that θ can take positive and negative values. The door is released from rest from an open position at time t = 0.
A proposed differential equation to model the motion of the door for t ⩾ 0 is
According to the model, for what value of λ will the motion of the door be simple
harmonic?
[1]
Explain briefly why modelling the motion of the door as simple harmonic is unlikely to
be realistic.
[1]
Find the range of values of λ for which the model predicts that the door will never pass through
the equilibrium position.
[2]
Sketch a possible graph of θ against t when λ lies outside the range found in part (b) but the
motion is not simple harmonic.
[1]
The paper as printed
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