NST IA Mathematics II (B course) Lent Term 2006 Examples Class...
Transcript of NST IA Mathematics II (B course) Lent Term 2006 Examples Class...
NST IA Mathematics II (B course) Lent Term 2006Examples Class I
lecturer: Professor Peter Haynes ([email protected])
February 15, 2006
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P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 2
1 Probability
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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.
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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.
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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]
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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 . ]
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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]
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P.H.Haynes NST IA Mathematics II (B course) Lent Term 2006 12
2 Differentials
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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.
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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]
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3 Lagrange Multipliers
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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.]
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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]
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