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MATHEMATICAL TRIPOS Part IA
Thursday 2 June 2005 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 REQUIRMENTS 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
1C Algebra and Geometry
Convert the following expressions from suffix notation (assuming the summation convention in three dimensions) into standard notation using vectors and/or matrices, where possible, identifying the one expression that is incorrectly formed:
(i) δij ,
(ii) δii δij ,
(iii) δll ai bj Cij dk − Cik di, (iv) ijk ak bj ,
(v) ijk aj ak.
Write the vector triple product a×(b×c) in suffix notation and derive an equivalent expression that utilises scalar products. Express the result both in suffix notation and in standard vector notation. Hence or otherwise determine a × (b × c) when a and b are orthogonal and c = a + b + a × b.
2B Algebra and Geometry
Let n ∈ R^3 be a unit vector. Consider the operation
x 7 → n × x.
Write this in matrix form, i.e., find a 3 × 3 matrix A such that Ax = n × x for all x, and compute the eigenvalues of A. In the case when n = (0, 0 , 1), compute A^2 and its eigenvalues and eigenvectors.
3F Analysis Define the supremum or least upper bound of a non-empty set of real numbers.
Let A denote a non-empty set of real numbers which has a supremum but no maximum. Show that for every > 0 there are infinitely many elements of A contained in the open interval (sup A − , sup A).
Give an example of a non-empty set of real numbers which has a supremum and maximum and for which the above conclusion does not hold.
Paper 1
SECTION II
5C Algebra and Geometry
Give the real and imaginary parts of each of the following functions of z = x + iy, with x, y real,
(i) ez^ ,
(ii) cosz,
(iii) logz,
(iv)
z
z¯
(v) z^3 + 3z^2 z¯ + 3z z¯^2 + ¯z^3 − ¯z,
where ¯z is the complex conjugate of z.
An ant lives in the complex region R given by |z − 1 | ≤ 1. Food is found at z such that
(logz)^2 = −
π^2 16
Drink is found at z such that
z + 12 ¯z ( z − 12 z¯
) 2 = 3,^ z^6 = 0.
Identify the places within R where the ant will find the food or drink.
6B Algebra and Geometry
Let A be a real 3 × 3 matrix. Define the rank of A. Describe the space of solutions of the equation Ax = b , (†)
organizing your discussion with reference to the rank of A.
Write down the equation of the tangent plane at (0, 1 , 1) on the sphere x^21 + x^22 + x^23 = 2 and the equation of a general line in R^3 passing through the origin (0, 0 , 0).
Express the problem of finding points on the intersection of the tangent plane and the line in the form (†). Find, and give geometrical interpretations of, the solutions.
Paper 1
7A Algebra and Geometry
Consider two vectors a and b in Rn. Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to b and the other perpendicular to b. By setting the former equal to cos θ|a|ˆb, where bˆ is a unit vector along b, show that
cos θ =
a · b |a||b|
Explain why this is a sensible definition of the angle θ between a and b.
Consider the 2n^ vertices of a cube of side 2 in Rn, centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the 2 n−^1 diagonals of the cube. Show that no two diagonals can be perpendicular if n is odd.
For n = 4, what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.
8A Algebra and Geometry Given a non-zero vector vi, any 3 × 3 symmetric matrix Tij can be expressed as
Tij = Aδij + Bvivj + (Civj + Cj vi) + Dij
for some numbers A and B, some vector Ci and a symmetric matrix Dij , where
Civi = 0, Dii = 0, Dij vj = 0 ,
and the summation convention is implicit.
Show that the above statement is true by finding A, B, Ci and Dij explicitly in terms of Tij and vj , or otherwise. Explain why A, B, Ci and Dij together provide a space of the correct dimension to parameterise an arbitrary symmetric 3 × 3 matrix Tij.
Paper 1 [TURN OVER
11E Analysis
Prove that if f is a continuous function on the interval [a, b] with f (a) < 0 < f (b) then f (c) = 0 for some c ∈ (a, b).
Let g be a continuous function on [0, 1] satisfying g(0) = g(1). By considering the function f (x) = g(x + 12 ) − g(x) on [0, 12 ], show that g(c + 12 ) = g(c) for some c ∈ [0, 12 ]. Show, more generally, that for any positive integer n there exists a point cn ∈ [0, n− n 1 ] for which g(cn + (^) n^1 ) = g(cn).
12E Analysis
State and prove Rolle’s Theorem. Prove that if the real polynomial p of degree n has all its roots real (though not necessarily distinct), then so does its derivative p′. Give an example of a cubic polynomial p for which the converse fails.
END OF PAPER
Paper 1
