New FP2
1.
Paper 3
(a) Sketch, on the same axes, the graphs with equation y = 2x – 3, and the line with equation
y = 5x – 1.
(2)
(b) Solve the inequality 2x – 3 < 5x – 1.
2.
(3)
(a) Use the substitution y = vx to transform the equation
d y (4 x  y)( x  y)
=
,x>0
dx
x2
(I)
into the equation
x
dv
= (2 + v)2.
dx
(II)
(b) Solve the differential equation II to find v as a function of x.
(4)
(5)
(c) Hence show that
y = 2x 
x
, where c is an arbitrary constant,
ln x  c
is a general solution of the differential equation I.
3.
(1)
(a) Find the value of  for which x cos 3x is a particular integral of the differential equation
d2 y
+ 9y = 12 sin 3x.
dx 2
(4)
(b) Hence find the general solution of this differential equation.
The particular solution of the differential equation for which y = 1 and
(4)
dy
= 2 at x = 0, is y = g(x).
dx
(c) Find g(x).
(4)
(d) Sketch the graph of y = g(x), 0  x  .
(2)
4.
Figure 1
Figure 1 shows a sketch of the cardioid C with equation r = a(1 + cos  ),  <    . Also shown are
the tangents to C that are parallel and perpendicular to the initial line. These tangents form a
rectangle WXYZ.
(a) Find the area of the finite region, shaded in Fig. 1, bounded by the curve C.
(6)
(b) Find the polar coordinates of the points A and B where WZ touches the curve C.
(5)
(c) Hence find the length of WX.
(2)
Given that the length of WZ is
3 3a
,
2
(d) find the area of the rectangle WXYZ.
(1)
A heart-shape is modelled by the cardioid C, where a = 10 cm. The heart shape is cut from the
rectangular card WXYZ, shown in Fig. 1.
(e) Find a numerical value for the area of card wasted in making this heart shape.
(2)
n
5.
Prove by the method of differences that
r
r 1
6.
1n
dn
x
2
(
e
cos
x
)

2
Given that
ex cos (x +
dx n
1
4
2
=
1
6
n(n + 1)(2n + 1), n > 1.
(6)
n), n  1, find the Maclaurin series expansion of
ex cos x, in ascending powers of x, up to and including the term in x4.
(3)
7.
The transformation T from the complex z-plane to the complex w-plane is given by
w=
z 1
,
zi
z  i.
(a) Show that T maps points on the half-line arg(z) =

in the z-plane into points on the circle
4
w = 1 in the w-plane.
(4)
(b) Find the image under T in the w-plane of the circle z = 1 in the z-plane.
(6)
(c) Sketch on separate diagrams the circle z = 1 in the z-plane and its image under T in the
w-plane.
(2)
(d) Mark on your sketches the point P, where z = i, and its image Q under T in the w-plane.
8.
(2)
Solve the equation
z5 = i,
giving your answers in the form cos  + i sin .
(5)
The End