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1.
For cach of the following linear transformations, write down its matrix
and describe its type.
(i) g(x,y) = (22, dy)
(ii) A(z, y) = (x, 52 + y)
(iii) A(z, y) = (y, 2)
2.
Use the matrices that you found in part 1. to show that the linear
transformation f = kohog is represented by the matrix
10 4
A= ( 2 5)
3.
Show that f is invertible, and find the matrix that represents f7!.
4,
Find the equation of the image f(C) of the unit circle C, in the form
ax” + bry + cy” = d,
where a, 6, c and d are integers whose values you should find.
5.
Calculate the area enclosed by f(C).
 

6.

The affine transformation f maps the points (0,0), (1,0) and (0,1) to

the points (—2,0), (—3,0) and (—2, —1), respectively.

(i) Determine f in the form f(x) = Ax +a, where A is a 2 x 2 matrix
and a is a column vector with two components.

(ii) The affine transformation f is an isometry. Find the fixed points (if
any) of f, and state whether f is a translation, rotation, reflection
or glide-reflection.

7.

The affine transformation k is the anticlockwise rotation through 7/2

about the point (7,8).

By using the translation h that maps the point (7,8) to the origin, and

its inverse h~!, find the transformation k in the form k(x) = Bx +b,

where B is a 2 x 2 matrix and b is a column vector with two
components.

8.

Find the partial fraction expansion of the rational expression
2r° + 5x? — 452 — 140

x? — 25
9.
Find the integral
2x? + 5x? — 45a — 140 4
———_—_.—____————- dz.
x? — 25
 

10.

. . x? +22 —7

Consider the function f(x) = ————.

L+A4

(a) Find the domain and intercepts of f in exact form.

(b) Find f'(z).

(c) Find the coordinates of any stationary points of f. Construct a table of
signs for f’(x), determine the intervals on which f is increasing and
those on which f is decreasing, and determine the nature(s) of the
stationary point(s).

(d) Determine the equations of the asymptotes of f.

(e) Determine whether f is an even or odd function, or neither.

(f) Hence sketch the graph of f.

11.

Use the identity cosh? x — sinh? x = 1 to find the integral
J sian’ cdr.

12,

Use a half-angle identity and then a product-to-sum identity to find the

integral
/ cos’ x cos(52x) dz.

13.

Consider the differential equation
dy _ 32 2
dt =-t*J/l—-y* (-l<y< 1).

(a) State the form of the differential equation and hence state which of the
methods described in Unit 8 for finding solutions of differential
cquations you would use to solve this equation.

(b) Find the general solution of the differential equation in explicit form.

14,

Consider the differential equation
dy 8y 1
—+—=— (@>0).
dr x 2x ( )

(a) State the form of the differential equation and hence state which of the
methods described in Unit 8 for finding solutions of differential
equations you would use to solve this equation.

(b) Find the general solution of the differential equation in explicit form.

15.

Water drains from a tank in such a way that the volume of water V

(in gallons) at time t (in minutes) since it started draining satisfies the

differential equation
dV .
>= -kv/4 (0<t<3, V>0),

where k is a positive constant.

(a) Find the general solution of this differential equation in explicit form.

(b) The tank is initially full, and contains 81 gallons of water. Find the
particular solution that describes the volume of water that the tank
contains as a function of time.

(c) The tank contains 16 gallons of water after 1 minute. Find the valuc
of k.

(d) Use your particular solution to calculate the time at which the tank is
one-third full. Give your answer in minutes to three significant figures.

(ec) Find the solution (in implicit form) to the initial value problem

dV ,
a= —4v3/4 (V>0), where V(0) = 81.
Check that your answer agrees with your answers to parts (b) and (c)