Question 22
The function f satises the condition f (x) > 0 for a
a suitable change of variable, prove that
b
a
x
b, and g is the inverse of f. By making
f(x) dx = b a
g(y ) dy,
(1)
where = f(a) and =
Solution to Question 21
g(t)
The rst integral in equation (1)
in the question is equal to the
area under the lower curve in the
gure and the dierence between
the two integrals is equal to the
area bet
Question 21
Show by means of a sketch that if 0
f (t)
g (t) for 0
x
0
t
x
f (t) dt
0
g (t) dt.
1
10
0
(1)
0
Starting from the inequality 0 cos t 1, prove that if 0
1
that 1 2 x2 cos x 1. Deduce that
1
Solution to Question 20
(i) Since we are dealing with integers, a < b is the same as a
b 1. Therefore, bc
as c > 0.
Similarly, d < c implies that ad a(c 1), as a > 0, i.e. ad a(c 1).
(a + 1)c,
Putting
Question 20
Let a, b, c, d, p and q be positive integers. Prove that
(i) If a < b and d < c, then bc ad a + c;
a
c
(ii) If
< p < , then (bc ad)p a + c;
b
d
p
c
a+c
a
< < , then p
and q
(iii) If
b
q
d
Solution to Question 19
(i) We need to show that (C1 )T CT = I, because multiplying this on the right by (CT )1 gives the
required formula. Start from CC1 = I and take the transpose, using the given f
Question 19
Let A and C be n n matrices, and denote the n n unit matrix by I and the n n zero matrix
by O.
(i) Show that if C is non-singular then (CT )1 = (C1 )T , where CT denotes the transpose of C
Solution to Question 18
The general solution can be written in the form
x = A cos kt + B sin kt ,
where A and B are arbitrary constants.
At time t after the rope becomes taut, let y be the displacemen
Question 18
Write down the general solution of the equation
x = k 2 x,
where k is a constant.
A truck is towing a trailer of mass m across level ground by means of an elastic tow-rope of natural
lengt
Solution to Question 17
(i) The inverse of f(x) = ax is f 1 (y ) = y/a, since then f 1 f(x) = x as required. Substituting into
Lagranges formula gives
f 1 (y ) = y +
1
1 dn1
[y ay ]n
n! dy n1
= y+
= y
Question 17
If y = f(x), the inverse of f is given by Lagranges identity:
f 1 (y ) = y +
1
1 dn1
n
y f(y ) ,
n1
n! dy
when this series converges.
(i) Verify Lagranges identity when f(x) = ax.
(ii) Sho
Solution to Question 16
To obtain equation (4), we just subtract equation (3) from equation (2) and factorise:
b c = (y 2 zx) (z 2 xy ) = (y 2 z 2 ) + (xy xz ) = (y z )(y + z ) + (y z )x = (y z )(x +
Question 16
Given that x, y and z satisfy
x2 yz =
y 2 zx =
z 2 xy
=
a,
b,
(1)
(2)
c,
(3)
where a, b and c are real, not all equal, and a + b + c > 0, show that
b c = (y z )(x + y + z ) .
(4)
By consid
Solution to Question 15
1
For 0
tan < 1 , we have tan < 1,
4
so that tank is close to zero (i.e. much
smaller than 1) when k is large. This is
illustrated in the gure which shows three
cases: for k =
Question 15
Give rough sketches of the function tank for 0
1
4
Show that for any positive integer n
/4
0
tan2n+1 d = (1)n
and deduce that
ln 2 =
Show similarly that
1
2
in the two cases k = 1 and k
Solution to Question 14
We can isolate z (i.e. eliminate x and y ) in one go because of the special form of the equations:
(1) a (2) gives
=
z (1 ab) = a2 a
a(a 1)
z=
1 ab
provided ab = 1.
Note the co
Question 14
Find the simultaneous solution of the three linear equations
a2 x + ay + z
ax + y + bz
=
=
a2
1
(1)
(2)
a2 bx + y + bz
=
b
(3)
for all possible real values of a and b.
Discussion
The simpl
Solution to Question 13
This problem can be solved using tree diagrams; see Question 8 for an example of this kind of
solution. A more sophisticated (but not necessarily better) method is to use Bayes
Question 13
My two friends, who shall remain nameless, but whom I shall refer to as P and Q, both told me
this afternoon that there is a body in my fridge. Im not sure what to make of this, because P
Solution to Question 12
To prove the trig. identity, we can use the formulae for cos(a + b), etc:
cos 3 = cos(2 + ) = cos 2 cos sin 2 sin
and then use double angle formulae:
= (cos2 sin2 ) cos (2 sin
Question 12
Prove that cos 3 = 4 cos3 3 cos .
Show how the cubic equation
24x3 72x2 + 66x 19 = 0
()
can be reduced to the form
4z 3 3z = k
by means of the substitutions y = x + a and z = by , for suit
Solution to Question 11
Making the suggested change of variable, we have
(x )( x)
sin2 + sin2 cos2 cos2
=
=
and
(cos2 1) + sin2 (1 sin2 ) cos2
=
( )2 sin2 cos2
dx
= 2 cos sin + 2 sin cos = 2( ) si
Question 11
Show by means of the substitution x = cos2 + sin2 , or otherwise, that
1
(x )( x)
dx =
when < . What is the value of the integral when < ?
Using the substitution t = x1 , show that
b
a
dt
Solution to Question 10
a
The point represented by
a + (z a)ei
a + (z a)ei
is a rotation by anticlockwise about
the point represented by a of the point
represented by z .
z
There are a number of ways
Question 10
Explain the geometrical relationship between the points in the Argand diagram represented by the
complex numbers z and a + (z a)ei .
(i) Write down the necessary and sucient conditions tha
Solution to Question 9
T
a
2
a
T
The above diagram shows a horizontal cross-section of a segment of the orange.
Resolving the tension T in the ribbon as shown gives a horizontal component of force due
Question 9
A chocolate orange consists of a sphere of delicious smooth uniform chocolate of mass M and radius
a, sliced into segments by planes through a xed axis. It stands on a horizontal table with
Solution to Question 8
The tree diagram for the rst part is shown below.
black silk 1/3
brown 1/3
polka-dot 1/3
regimental 1/3
black silk 1/3
Sunday 1/7
grey 1/3
regimental 1/3
cravat
1/3
black silk 1
Question 8
Each day, I choose at random between my brown trousers, my grey trousers and my expensive but
fashionable designer jeans. Also in my wardrobe, I have a black silk tie, a rather smart brown
Solution to Question 7
For either snowball, the rate of change of volume dv/dt at any time is related to the surface area a
by
dv
= ka,
()
dt
where k is a positive constant. For a sphere of radius r,