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

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


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.


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]


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


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


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.


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


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.]


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

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