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NST IA Mathematics II (B course) Lent Term 2006Examples Class I
lecturer: Professor Peter Haynes ([email protected])
February 15, 2006
1
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 2
1 Probability
961043
4B
(a) I drop a piece of bread and jam repeatedly. It lands either jam-side up orjam-side down and I know that the probability it will land jam-side down is p.
(i) What is the probability that it falls jam-side down for the first n drops?
(ii) What is the probability that it falls jam-side up for the first time on thenth drop?
(iii) What is the probability that it falls jam-side up for the second time on thenth drop?
(iv) I continue dropping it until it falls jam-side up for the first time. Writedown an expression for the expected number of drops. By considering thebinomial expansion for (1 − p)−2, or otherwise, show that the expectednumber is 1/(1− p).
(v) Give a rough sketch of the probability distribution function for the numberof times it falls jam-side down in N drops, where N is large.
(b) I am playing a game of cards in which 52 distinct cards are allocated randomlyto four players (one of whom is me), each player receiving 13 cards. Four of thecards are aces; one of these is called the ace of spades and another is called the aceof clubs.
(i) What is the probability that I receive the ace of spades?
(ii) Show that the probability that I receive both the ace of spades and the aceof clubs is 1/17.
(iii) What is the probability that I receive all four aces?
(iv) What is the probability that I receive neither the ace of spades nor the aceof clubs?
(v) What is the probability that I receive at least one ace?
5C
In this question r, θ and φ are the usual spherical polar coordinates.
(a) The mass density of a gas which fills the whole of space is given by
ρ(r) =( r
a
)2ρ0 exp
(−2r
a
),
where ρ0 and a are constants. Sketch the form of the function ρ(r) and find thetotal mass of the gas.
(b) Give a rough sketch the surface described by r = a cos θ where a is a constant,and 0 ! θ ! π/2 and 0 ≤ φ ≤ 2π. The mass density of a gas which fills the volumeenclosed by this surface is given by
ρ(r, θ) =ρ0r
a cos3 θ,
where ρ0 is a constant. Find the total mass of gas enclosed.
[TURN OVER
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 3
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 4
97212
7
12F
A biased coin has probability p of coming down heads and probability q = 1 − pof coming down tails.
(a) Find the probability that the first head is obtained on the nth toss.
(b) Write down an expression for the probability of obtaining k heads in n tosses.
(c) Calculate, in terms of p, the expectation value for the number of tosses neededto obtain the first head.
(d) I play a coin-tossing game, which lasts at most N tosses, and start with a stakeof £1. Each time the coin comes down tails my money is doubled. The first timeit lands on heads my money is reduced to the original £1 stake, and if it lands onheads a second time I lose everything. The game ends after N tosses or after thesecond head.
Find the expectation value of my total money at the end of the game.
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 5
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 6
01211
8
11F
A bag contains 2 red and 5 green counters.
(a) In a trial, counters are repeatedly drawn from the bag and replaced each time.Find the probability that a red counter is drawn on the n-th draw for the firsttime.
[6]
(b) In another trial counters are now drawn without being replaced. Let E1 be theevent that the first drawn is red, and E2 the event that the second drawn is red.If P (E) denotes the probability of event E, find the following probabilities:
(i) P (E1);
(ii) P (E2);
(iii) P (E1 ∩ E2).
Hence or otherwise find
(iv) P (E1 ∪ E2);
(v) P (E1|E2);
where E1|E2 denotes the event “E1 given E2”.[14]
12F*
(a) In each of the following cases state whether the function has a finite limit as xtends to zero, and if so find its value:
(i)1x
sin 2x ;
(ii) x cos1x
;
(iii)x
1− exp(−x).
[7]
(b) Explain what is meant by the statement that a series∑
un is
(i) convergent;
(ii) absolutely convergent.[6]
(c) Show whether or not the series∑
un is convergent when un =n4
2n.
[7]
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 7
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 8
02204
3
3B
Express the Cartesian coordinates x, y, z in terms of spherical polar coordinates r,θ, φ. Write down the standard volume element in spherical polar coordinates.
[4]
(a) Fluid is contained within a sphere of radius a and centre the origin. The densityof the fluid is ρ = µ (2 + (z/r)) where µ is constant. Calculate the total massof fluid.
[6]
(b) A distribution of electric charge has charge density (i.e., charge per unit volume)ρ = λxy with λ a constant. It occupies the region of space with r ! a andx, y, z " 0. Calculate the total charge.
[10]
4B
Consider n independent events, each with two possible outcomes, one called‘success’, which occurs with probability p, and the other called ‘failure’, whichoccurs with probability q = 1− p.
Write down the probability pr that exactly r of the n events are successes and showthat the sum of these probabilities for 0 ! r ! n is equal to one.
[6]
Under certain conditions, with n large, the discrete distribution above can beapproximated by a normal distribution having the same mean and variance. Theapproximation is
pr ≈ P (r − 12 ! x ! r + 1
2 )
where
P (α ! x ! β) = (2πσ2)−12
β∫α
exp [−(x− µ)2/ 2σ2] dx .
Write down expressions for µ and σ in terms of n, p and q.[3]
A student sits a multiple choice exam and guesses the answer to each questionrandomly from a selection of 4 possible answers. If the total number of questionsis 60, what is the expected number of correct answers? Show, using the normalapproximation above, that there is a probability greater than 1
2 that the numberof correct answers will lie in the range 13 to 17 inclusive.
[11]
[ You may assume (2π)−12
√5/3∫
0
exp (− 12 y2) dy > 1
4 . ]
[TURN OVER
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 9
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 10
03205
3
4B*
(a) State carefully the divergence theorem and Stokes’ theorem.[4]
(b) In Cartesian coordinates and components, the vector field F is given by
F = (x2yz , xy2z , xyz2) .
Evaluate∫S
F · dS , where S is the surface of the cube
0 ! x ! 1 , 0 ! y ! 1 , 0 ! z ! 1 .
[8]
(c) In Cartesian coordinates and components, the vector field G is given by
G = (4y , 3x , 2z) .
Evaluate∫S
(∇×G) · dS , where S is the open hemispherical surface
x2 + y2 + z2 = r2 , z " 0 .
[8]
5C
(a) It is known that n people out of a population of N suffer from a certain disease,and that the other N−n people do not. The test for the disease has a probabilitya of producing a correct positive result when used on a sufferer and a probabilityb of producing a false positive result when used on a non-sufferer. The test ispositive when done on me. What is the probability that I am a sufferer ?
[9]
(b) A random variable X has density function f(t) given by
f(t) = Ae−kt , for t ≥ 0 ,
where A and k are constants. Find, in terms of k :
(i) the value of A ;[2]
(ii) the probability that X ≥ 3 given that X ≥ 1 ;[5]
(iii) the expectation value of X .[4]
[TURN OVER
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 11
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 12
2 Differentials
00201
2
1A
(a) Give a necessary condition for the differential
P(x, y
)dx + Q
(x, y
)dy
to be exact.
Show that
w =[
1− y exp{
y
x + y
}]dx +
[1 + x exp
{y
x + y
}]dy
is not exact.
(b) Letx + y = u
y = uv .
Express dx and dy in terms of du and dv.
Hence express w in terms of u, v, du, and dv.
Find an integrating factor, µ, in terms of u and v, such that µw is exact.
Hence solve w = 0 , expressing your answer in terms of x and y.
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 13
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 14
01210
7
10E
Give a necessary condition for the expression
P (x, y)dx + Q(x, y)dy
to be an exact differential.[2]
For the thermodynamics of a gas, the internal energy U can be regarded as afunction of the entropy S and the volume V . It is given that:
dU = TdS − pdV
where T is the temperature and p the pressure. By considering the function
A = U − TS
or by some other method, show that(∂S
∂V
)T
=(
∂p
∂T
)V
.
[4]
Now, considering U as a function of T and V show that(∂U
∂V
)T
= T
(∂S
∂V
)T
− p .
[4]
Givenp =
nRT
V − nbexp
{ −an
V RT
}where a, b, n, R are constants, find
(∂U
∂V
)T
.
[6]
If, instead
p =nRT
V
and(
∂U
∂T
)V
= CV where CV is constant, find an expression for U .
[4]
[TURN OVER
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 15
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 16
3 Lagrange Multipliers
97202
2
1A
Polar co-ordinates (r, θ) are related to Cartesian co-ordinates (x, y) by x = r cos θ,y = r sin θ. A function f(x, y) can alternatively be written as a function of (r, θ).Show that (
∂f
∂x
)y
= cos θ
(∂f
∂r
)θ
− sin θ
r
(∂f
∂θ
)r
.
Obtain similar expressions for(∂f
∂y
)x
,∂2f
∂x2,
∂2f
∂y2.
(The last of these may be given without detailed calculations.)
Hence show that
∂2f
∂x2+
∂2f
∂y2=
∂2f
∂r2+
1r
∂f
∂r+
1r2
∂2f
∂θ2.
A function F (x, y) satisfies
∂2F
∂x2+
∂2F
∂y2= 0
and has the formF (x, y) = R(r)
4xy(x2 − y2)(x2 + y2)2
.
Express F as a function of r and θ only, and hence find the differential equationsatisfied by R(r).
2A
Two horizontal corridors, 0 ! x ! a, y " 0 and x " 0, 0 ! y ! b meet at rightangles. A ladder, which may be regarded as a stick of length L, is to be carriedhorizontally around the corner. Use the method of Lagrange multipliers to showthat the maximum possible length of the ladder is (a2/3 + b2/3)3/2.
[It is suggested that you place the ends of the ladder at the points (a + ξ, 0) and(0, b + η) and impose the condition that the corner (a, b) be on the ladder.]
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 17
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 18
02104
3
4B*
Explain, without proof, a method for finding the stationary points of a functionf(x, y, z) subject to simultaneous constraints g(x, y, z) = h(x, y, z) = 0.
[4]
A point is constrained to lie on the plane x − y + z = 0 and also on the ellipsoid
x2 +14
y2 +14
z2 = 1. Find the minimum and maximum distances of this point
from the origin, by considering the function f(x, y, z) = x2 + y2 + z2.[16]
5C
(a) Evaluate the definite integrals
∞∫0
e−x2dx ,
∞∫0
x2e−x2dx ,
as well as the indefinite integrals∫x e−x2
dx ,
∫x3e−x2
dx .
[10]
(b) Sketch the region R in the positive quadrant of the xy plane which is enclosedby the lines y = 0, x = 2, y = x and by the curve xy = 1. Evaluate∫ ∫
R
x2e−x2dx dy .
[10]
[TURN OVER
P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 19