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MAS140
SCHOOL OF MATHEMATICS AND STATISTICS
MAS140
Spring Semester
2012–2013
Mathematics (Chemical)
3 hours
Attempt ALL questions.
Each question in Section A carries 3 marks,
each question in Section B carries 8 marks.
Section A
(8 − ax)1/3 − 2
A1 Use the binomial theorem to find the value of a for which lim
= 1.
x→0
x
A2 If f (x) =
3x
, x ≥ 0, what is the range of f (x)? Find f −1 (y).
x+2
A3 Find the first three nonzero terms in the Maclaurin expansion of
f (x) = e−2x cos 3x.
x cos 3x
.
x→π/2 2x − π
A4 Evaluate lim
A5 Find all the complex numbers z for which Re(z) + Im(z) = 0 and |z − 1 + i| =
√
2.
A6 If a = (3, 1, −1) and b = (1, −2, 1), show that a is perpendicular to b and find
a × b.
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1
Turn Over
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A7 Use integration by parts to find the indefinite integral
Z
xn ln(x)dx
where n 6= −1.
A8 Compute the definite integral
Z 1
sin2
−1
πx
dx.
2
A9 Find the determinant of A−1 B, where A and B are given by
"
A=
1 −1
2 −4
#
"
,
B=
4 1
1 1
#
.
A10 Consider the homogeneous linear system of equations
3x + y = 0,
ax − 4y = 0,
which has the trivial solution x = y = 0. Find the value of a for which the system
has additional solutions.
A11 Find the general solution of the following differential equation:
d2 y
dy
+ 5 + 6y = 0.
2
dx
dx
A12 Show that the general solution of the differential equation
dy
y
=
,
dx
1 − x2
s
is given by y = A
MAS140
1+x
, where A is a constant.
1−x
2
Continued
MAS140
Section B
B1 Find the quadratic Q(x) for which the polynomial P (x) = x3 − x2 − x − 15 can
be written in the form
P (x) = (x − 3) Q(x).
Find the roots of Q(x) and plot them on an Argand diagram. Find the constants
A and B such that
Q(x) = (x − A)2 + B.
Sketch the curve y = Q(x).
B2 Determine the modulus and principal argument of the complex numbers
z1 = −3 + 4i and z2 = −4 − 3i.
By expressing z1 and z2 in exponential form, or otherwise, show that
1
z13
= z¯1 ,
4
z2
25
where z¯1 is the complex conjugate of z1 .
B3 Find the values of x at which the function
f (x) = x3 + 2x2 − 4x + 1
has stationary points. Determine the nature of each of the stationary points and
sketch the curve y = f (x).
B4 A function f (x, y) of two variables is given by
f (x, y) = 2x5 −
1
cos 5y.
x5
Calculate the first-order partial derivatives fx and fy . Calculate the second-order
partial derivatives fxx and fyy and demonstrate that
x2 fxx + xfx + fyy = 0.
B5 Let
dx
+ 2x + a
where a is a constant. Find the indefinite integral I in the following cases,
(i) a = 1,
I=
Z
x2
(ii) a = 0,
(iii) a = 2.
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3
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B6 Let


1 0 2


A =  0 3 1 .
1 −4 1
(a) Find |A|, the determinant of A.
(b) Find A−1 , the inverse of A.
(c) Use A−1 to find x, y and z which are solutions to the linear equations
x + 2z = 5,
3y + z = 11,
x − 4y + z = −9.
B7 Let
"
A=
1 2
2 1
#
.
(a) Find the eigenvalues and normalized eigenvectors of A.
(b) Show that the eigenvectors are orthogonal.
(c) Four points P QRS on a unit square are described by the column vectors
"
0
0
#
"
,
1
0
#
"
,
0
1
#
"
,
1
1
#
,
respectively. The transformed points P 0 Q0 R0 S 0 are obtained by multiplying
the column vectors by matrix A. Draw a sketch of the xy plane showing the
original points P QRS, the transformed points P 0 Q0 R0 S 0 , and the direction of
the eigenvectors of A.
B8 Consider the following differential equation:
y 00 + 2y 0 + 2y = f (x)
(a) Find the general solution in the case f (x) = 0.
(b) Find a particular solution for f (x) = cos x.
(c) Find the solution for f (x) = cos x with initial conditions y(0) = y 0 (0) = 0.
End of Question Paper
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MAS140-151-152 Formula Sheet
These results may be quoted without proof unless proofs are asked for in the questions.
Trigonometry
Hyperbolic Functions
sin2 x + cos2 x = 1
sinh x = 21 (ex − e−x )
1 + tan2 x = sec2 x
cosh x = 21 (ex + e−x )
sinh x
cosh x
cosh x
coth x =
sinh x
1
sechx =
cosh x
1 + cot2 x = cosec2 x
tanh x =
2 sin2 x = 1 − cos 2x
2 cos2 x = 1 + cos 2x
2 sin x cos x = sin 2x
cos(x ± y) = cos x cos y ∓ sin x sin y
cosh2 x − sinh2 x = 1
sin(x ± y) = sin x cos y ± cos x sin y
2 cosh2 x = 1 + cosh 2x
tan(x ± y) =
tan x ± tan y
1 ∓ tan x tan y
2 sinh2 x = cosh 2x − 1
2 sinh x cosh x = sinh 2x
a cos x + b sin x = R cos (x − α)
√
where R = a2 + b2 ,
cos α =
a
R
and
sin α =
sech2 x = 1 − tanh2 x
√
sinh−1 x = ln(x + x2 + 1) , all x
√
cosh−1 x = ln(x + x2 − 1) , x ≥ 1
(
)
1
1+x
−1
tanh x = ln
, |x| < 1
2
1−x
b
R
2 cos x cos y = cos(x + y) + cos(x − y)
2 sin x sin y = cos(x − y) − cos(x + y)
2 sin x cos y = sin(x + y) + sin(x − y)
1 − tan2 (x/2)
cos x =
1 + tan2 (x/2)
sin x =
2 tan(x/2)
1 + tan2 (x/2)
tan x =
2 tan(x/2)
1 − tan2 (x/2)
1
Series
Sum of an arithmetic series:
first term + last term
× (number of terms)
2
1 − xn
Sum of a geometric series: 1 + x + x2 + . . . + xn−1 =
1 − x( )
n(n
−
1)
n r
2
Binomial theorem: (1 + x)n = 1 + nx +
x + ... +
x + ...
2!
r
( )
n(n − 1)(n − 2) . . . (n − r + 1)
n
=
where
r
r!
If n is a positive integer then the series terminates and the result is true for all x, otherwise, the series is infinite and only converges for |x| < 1.

x3 x5 x7


sin x = x −
+
−
+ ...



3!
5!
7!





2
4
6

x
x
x



cos x = 1 −
+
−
+ ...


2!
4!
6!





3
5
7
x
x
x
valid for all x
sinh x = x +
+
+
+ ...

3!
5!
7!






2
4
6

x
x
x


cosh x = 1 +
+
+
+ ...


2!
4!
6!






2
3

x
x

x
exp x = e = 1 + x +
+
+ ... 
2!
3!
x2 x3 x4
ln(1 + x) = x −
+
− + . . . (−1 < x ≤ 1)
2
3
4
2
Differentiation
Function
Derivative
Function
Derivative
sin x
cos x
cos x
− sin x
tan x
sec2 x
cot x
−cosec2 x
sin−1 x
√
1
, |x| < 1
1 − x2
1
1 + x2
cos−1 x
−√
cosh x
cosh x
sinh x
coth x
−
tan−1 x
sinh x
tanh x
sinh−1 x
tanh−1 x
coth−1 x
cot−1 x
1
cosh2 x
1
√
2
x +1
1
, |x| < 1
1 − x2
1
, |x| > 1
1 − x2
cosh−1 x
3
1
, |x| < 1
1 − x2
1
−
1 + x2
1
sinh2 x
1
√
, |x| > 1
2
x −1
Integration
In the following table the constants of integration have been omitted.
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
∫
xn+1
x dx =
n+1
n
x
e dx = e
∫
(n ̸= −1)
∫
x
∫
sin x dx = − cos x
∫
sec2 x dx = tan x
∫
sinh x dx = cosh x
∫
sech2 x dx = tanh x
x
dx
√
= sin−1
a
a 2 − x2
∫
(|x| < a)
dx
= ln |x|
x
ax
a dx =
ln a
x
(a > 0, a ̸= 1)
cos x dx = sin x
cosec2 x dx = − cot x
cosh x dx = sinh x
cosech2 x dx = − coth x
dx
1
−1 x
=
tan
a 2 + x2
a
a
∫
dx
dx
x
−1 x
√
√
= sinh
= cosh−1
a
a
x 2 − a2
x2 + a 2
a + x
dx
1
(= tanh−1 x if |x| < a)
=
ln 2
2
a −x
2a
a − x
a
(x)
cosecx dx = ln tan
or ln (cosecx − cot x)
2
(x π )
sec x dx = ln tan
or ln (sec x + tan x)
+
2 4
(x)
cosechx dx = ln tanh
2
4
(|x| > a)
Integration by parts
∫
∫
′
f (x) g (x) dx = f (x) g(x) − f ′ (x) g(x) dx
Newton-Leibnitz formula
∫ b
b
′
If F (x) = f (x), then
f (x) dx = F (b) − F (a) = F (x)
a
a
Variable substitution in definite integral
If x = φ(t) is a monotonic function in the interval [α, β] and a = φ(α), b = φ(β), then
∫ b
∫ β
f (x) dx =
f (φ(t)) φ′ (t) dt
a
α
Variable substitution for a rational function of sin x and cos x
Let t = tan
(x)
2
then sin x =
2t
,
1+t2
cos x =
1−t2
1+t2
and
dx
dt
=
2
.
1+t2
Area of planar figure
The area of a planar figure bounded by the graph of a continuous positive function f (x),
the x-axis, and the ordinates x = a and x = b is
∫ b
S=
f (x) dx
a
Volume of solid of revolution
The volume of a solid of revolution, obtained by rotation of a planar figure bounded by
the graph of a continuous positive function f (x), the x-axis, and the ordinates x = a and
x = b through a complete revolution around the x-axis, is
∫ b
V =π
f 2 (x) dx
a
5
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