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MATHEMATICAL TRIPOS Part IA
Thursday 1st June, 2006 9 to 12
PAPER 1
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in separate bundles, marked A, B, C, D, E and F according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter.
Attach a gold cover sheet to each bundle; write the code letter in the box marked ‘EXAMINER LETTER’ on the cover sheet.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Gold cover sheet None Green master cover sheet
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1B Algebra and Geometry Consider the cone K in R^3 defined by
x^23 = x^21 + x^22 , x 3 > 0.
Find a unit normal n = (n 1 , n 2 , n 3 ) to K at the point x = (x 1 , x 2 , x 3 ) such that n 3 > 0. Show that if p = (p 1 , p 2 , p 3 ) satisfies
p^23 > p^21 + p^22
and p 3 > 0 then p · n > 0.
2A Algebra and Geometry
Express the unit vector er of spherical polar coordinates in terms of the orthonormal Cartesian basis vectors i, j , k.
Express the equation for the paraboloid z = x^2 + y^2 in (i) cylindrical polar coordinates (ρ, φ, z) and (ii) spherical polar coordinates (r, θ, φ).
In spherical polar coordinates, a surface is defined by r^2 cos 2θ = a , where a is a real non-zero constant. Find the corresponding equation for this surface in Cartesian coordinates and sketch the surfaces in the two cases a > 0 and a < 0.
3F Analysis Let an ∈ R for n > 1. What does it mean to say that the infinite series
n an converges to some value A? Let sn = a 1 + · · · + an for all n > 1. Show that if
n an converges to some value A, then the sequence whose n-th term is
(s 1 + · · · + sn) /n
converges to some value A˜ as n → ∞. Is it always true that A = A˜? Give an example where (s 1 + · · · + sn) /n converges but
n an^ does not.
4D Analysis
Let
n=0 anz
n (^) and ∑∞ n=0 bnz
n (^) be power series in the complex plane with radii of
convergence R and S respectively. Show that if R 6 = S then
n=0(an^ +^ bn)z
n (^) has radius
of convergence min(R, S). [Any results on absolute convergence that you use should be clearly stated.]
Paper 1
7B Algebra and Geometry
(i) Let u, v be unit vectors in R^3. Write the transformation on vectors x ∈ R^3
x 7 →
u · x
u + v × x
in matrix form as x 7 → Ax for a matrix A. Find the eigenvalues in the two cases (a) when u · v = 0, and (b) when u, v are parallel.
(ii) Let M be the set of 2 × 2 complex hermitian matrices with trace zero. Show that if A ∈ M there is a unique vector x ∈ R^3 such that
A = R(x) =
x 3 x 1 − ix 2 x 1 + ix 2 −x 3
Show that if U is a 2 × 2 unitary matrix, the transformation
A 7 → U −^1 AU
maps M to M, and that if U −^1 R(x)U = R(y), then ‖x‖ = ‖y‖ where ‖ · ‖ means ordinary Euclidean length. [Hint: Consider determinants.]
Paper 1
8A Algebra and Geometry
(i) State de Moivre’s theorem. Use it to express cos 5θ as a polynomial in cos θ.
(ii) Find the two fixed points of the M¨obius transformation
z 7 −→ ω =
3 z + 1 z + 3
that is, find the two values of z for which ω = z.
Given that c 6 = 0 and (a−d)^2 +4bc 6 = 0, show that a general M¨obius transformation
z 7 −→ ω =
az + b cz + d
, ad − bc 6 = 0 ,
has two fixed points α, β given by
α =
a − d + m 2 c
, β =
a − d − m 2 c
where ±m are the square roots of (a − d)^2 + 4bc.
Show that such a transformation can be expressed in the form
ω − α ω − β
= k
z − α z − β
where k is a constant that you should determine.
9E Analysis State and prove the Intermediate Value Theorem.
Suppose that the function f is differentiable everywhere in some open interval containing [a, b], and that f ′(a) < k < f ′(b). By considering the functions g and h defined by
g(x) =
f (x) − f (a) x − a
(a < x 6 b) , g(a) = f ′(a)
and
h(x) =
f (b) − f (x) b − x
(a 6 x < b) , h(b) = f ′(b),
or otherwise, show that there is a subinterval [a′, b′] ⊆ [a, b] such that
f (b′) − f (a′) b′^ − a′^
= k.
Deduce that there exists c ∈ (a, b) with f ′(c) = k. [You may assume the Mean Value Theorem.]
Paper 1 [TURN OVER
