All questions may be attempted but only marks obtained on the best four solutions will
count.
The use of an electronic calculator is not permitted in this examination.
1. (a) Carefully define the scalar product a · b and the vector product a × b of the
two vectors a and b.
(b) State the scalar product a·b and the vector product a×b using index notation
and define the symbols δij and εijk .
(c) Find the value of εijk εijk δmn δmn .
(d) Two unit vectors â and b̂ have an angle γ between them. Show that the vector
b̂ − (â · b̂)â is perpendicular to â and is of length | sin γ|.
2. Let A, B, C, D be the corners of a trapezoid with AB k CD and diagonals AC and
BD. The letters S, T, U1 , U2 stand for the areas of the respective triangles indicated
in the figure.
D
C
S
U1
U2
T
A
B
(a) Using vectors, show that the triangle ACB has the same area as the triangle
ADB. Deduce that U1 = U2 . [No marks will be awarded for methods that do
not use vectors.]
(b) Using any appropriate method, show that ST = U1 U2 .
(c) Let K be the area of the trapezoid, derive the formula
√
√
√
K= S+ T.
MATH1401
PLEASE TURN OVER
1
3. (a) Show that
sinh(3x) cosh(3x)
−
= 2.
sinh(x)
cosh(x)
(b) Derive the relations
sin(ix) = i sinh(x) ,
and
cos(ix) = cosh(x) ,
for x ∈ R.
(c) Let z be a complex number. A curve is defined by the equation
αz z̄ + βz + β̄ z̄ + γ = 0 ,
with α, γ ∈ R and β ∈ C. Describe the shape of the curve for α = 0 and α 6= 0
by finding its Cartesian form.
4. (a) Find the partial fractions decomposition of
x4
1
.
−1
(b) Find
Z
sin(3x) cos(5x)dx .
(c) Find
Z
dx
p
.
sinh(x)(sech(x) + tanh(x))
5. Find the general solution for each differential equation:
(a)
dy
3y − 4x − 2
=
,
dx
y+x+1
(b)
dy
d2 y
+ 4 + 3y = 5 sin2 (x) + e−2x + e−x .
2
dx
dx
If necessary, solutions can be left in implicit form in the transformed variables.
MATH1401
CONTINUED
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6. (a) A hunter is shooting pheasants and misses with a probability of 75%. He
continues hunting until his first hit.
(i) Describe a suitable sample space.
(ii) Find the probability pn that the hunter hits on the n-th shot.
P
(iii) Verify that ∞
n=1 pn = 1.
(b) The probability density describing the location of a particle is given by
(
−1 < x < 1 ,
c −x4 + 54 x2 + 15 ,
f (x) =
0,
otherwise .
(i) Determine the value of the normalisation constant c.
(ii) Find the mean and the variance.
(iii) Consider the interval I = (−1/2, 1/2). Is the particle more likely to be
inside or outside this interval?
MATH1401
END OF PAPER
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