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NATURAL SCIENCES TRIPOS Part IA
Wednesday 14th June 2006 9 to 12
MATHEMATICS (2)
Before you begin read these instructions carefully:
You may submit answers to no more than six questions. All questions carry the same number of marks.
The approximate number of marks allocated to a part of a question is indicated in the right-hand margin.
Write on one side of the paper only and begin each answer on a separate sheet.
Questions marked with an asterisk (*) require a knowledge of B course material.
At the end of the examination:
Each question has a number and a letter (for example, 3B).
Tie up the answers in separate bundles, marked A, B, C, D, E or F according to the letter affixed to each question. Do not join the bundles together.
For each bundle, complete and attach a blue cover sheet, with the appropriate letter written in the section box.
Complete a separate yellow master cover sheet listing all the questions attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS 6 blue cover sheets and treasury tags None Yellow master cover sheet
Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1A
(a) If y = sin−^1
( (^) x √ 1 + x^2
find
dy dx
as a function of x. [4] (b) Find the first non-zero term in the Taylor series about x = 0 of
x sin(sin x) − sin^2 x x^4
[8]
(c) Evaluate (without using a calculator)
∫ (^) π/ 6
0
dx (sin x)
(^12) (cos x)
[8]
[Hint: substitute tan x = u.]
Paper 2
3B
(a) Which of the following vector fields, given in Cartesian coordinates, is conservative?
(i) F 1 = y i + [z cos(yz) + x] j + y cos(yz) k ,
(ii) F 2 = exp(xy) i + exp(x + y) j ,
(iii) F 3 = (2xyz + sin x) i + x^2 z j + x^2 y k.
In each case, if F is a conservative vector field, find a scalar potential φ such that F = ∇φ. (^) [10]
(b) Calculate directly the line integral ∫
C
F 3 · dr ,
where the integration path C is (i) a straight line from (0, 0 , 0) to (π, π, π), (^) [5]
(ii) the curve defined by a series of straight lines from (0, 0 , 0) to (0, 0 , π) then to (0, π, π) and finally to (π, π, π). (^) [5]
Paper 2
4B
The element of vector area for a surface S, given in Cartesian coordinates by the equation f (x, y, z) = 0, can be expressed as
dS =
n cos α
dx dy ,
where α < π/2 is the angle between the unit vector k in the z-direction and the unit normal n of S.
(a) Show how to construct from f (x, y, z) a vector field F(x, y, z) such that
dS = F dx dy
and F · k = 1. [6]
(b) Evaluate the element of vector area dS for the surface S given by
x^2 (1 + y) + y^2 z = 1
and bounded by 0 < x < 1, 0 < y < 2. (^) [6]
(c) The flux of a vector field G through the surface S, as specified in part (b), is defined by I =
S
G · dS.
Calculate the magnitude |I 1 − I 2 | of the difference between the fluxes of G 1 and G 2 through S, where G 1 and G 2 are:
G 1 = y^2 i + x k ,
G 2 = y^2 i + y^3 j + x k. (^) [8]
Paper 2 [TURN OVER
7D
Solve the following differential equations, by determining which are exact and using an integrating factor for those that are not.
(a) (2xy^2 + 4) dx + 2(x^2 y − 3) dy = 0 , (^) [6]
(b) (y^2 − x) dx + 2y dy = 0 , (^) [8]
(c) (cos x − x sin x + y^2 ) dx + 2xy dy = 0. (^) [6]
8D
(a) A species of bird always lays a nest of four eggs. Each egg may be white (with probability p) or brown (with probability 1 − p).
(i) Using the notation WnB (^4) −n, list all possible nest contents together with their probabilities of occurrence. (^) [6]
(ii) Taking p = 3/4, find the most common nest content. (^) [2]
(b) The discrete variable X assumes values xi = i (i = 1,... , 6) with probabilities pi = 1/6. Calculate: (i) the expectation value of X , (^) [2]
(ii) the expectation value of X^2 , (^) [2]
(iii) the variance of X. (^) [2]
(c) The continuous variable X in the interval [1, 6] has the probability distribution function
f (x) =
{ (^) α , 1 6 x 6 3 , 0 , 3 < x < 4 , α , 4 6 x 6 6. Calculate:
(i) the value of α , (^) [2]
(ii) the variance of X. (^) [4]
Paper 2 [TURN OVER
9E
(a) Use the method of separation of variables to show that the general solution of the differential equation dy dx
where a and b are constants and b 6 = a, is
y =
a ea(x+c)^ − b eb(x+c) ea(x+c)^ − eb(x+c)^
where c is an arbitrary constant. (^) [6]
(b) The function z(x) is related to y(x), as found in part (a), by
dz dx
= yz. (∗∗)
Find the general solution z(x). (^) [4]
(c) Find y(x) and z(x) in the special case when b = a. (^) [6]
(d) By eliminating y between equations (∗) and (∗∗), find the second-order linear differential equation with constant coefficients satisfied by z(x). (^) [4]
Paper 2
11F
Let f (x, y) be a function of two variables. x and y can be rewritten in terms of two new variables u = u(x, y) and v = v(x, y).
(a) Use the chain rule to find
∂f ∂x
y
in terms of
∂f ∂u
v
∂f ∂v
u
∂u ∂x
y
and ( ∂v ∂x
y
[3]
(b) Find expressions for
∂^2 f ∂x^2
and
∂^2 f ∂y^2
in terms of derivatives of f with respect to u and v. [6]
(c) Suppose that x = u cos v , y = u sin v.
Evaluate
∂^2 f ∂x^2
and
∂^2 f ∂y^2
in terms of derivatives of f with respect to u and v. [6] (d) A solution to ∂^2 f ∂x^2
∂^2 f ∂y^2
is x^2 − y^2. Use your results to show that u^2 cos(2v) is a solution to
∂^2 f ∂u^2
u
∂f ∂u
u^2
∂^2 f ∂v^2
[5]
12F
A function of two variables f (x, y) is given by
f (x, y) =
x + y x^2 + y^2 + 1
and represents the height of the point (x, y) above the (x, y)-plane.
(a) Find the extrema of this function. (^) [6]
(b) Determine, by examining the second derivatives of f , whether each extremum is a maximum, a minimum or a saddle point. (^) [8]
(c) Hence sketch a contour plot of f. (^) [6]
END OF PAPER
Paper 2
