Download University Math Exam: Vector Calc, Fourier Series, Linear Algebra, & Probability Theory and more Exams Mathematics in PDF only on Docsity!
NATURAL SCIENCES TRIPOS Part IA
Monday 11 June 2001 9 to 12
MATHEMATICS (1)
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 will be 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).
Answers must be tied up 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, a blue cover sheet must be completed and attached to each bundle, with the appropriate letter written in the section box.
A separate yellow master cover sheet listing all the questions attempted must also be completed.
Every cover sheet must bear your examination number and desk number.
1A The points A, B and C have position vectors (relative to the origin)
a = 2ˆı − ˆ − 2 ˆk b = −ˆı − 2ˆ + ˆk c = −ˆı + ˆ.
(a) Find the equation of a straight line that passes through points A and B; (^) [2]
(b) determine a unit vector perpendicular to the plane containing the vectors a and b; (^) [4]
(c) evaluate the area of the triangle with A, B and C at its vertices; (^) [6]
(d) find an equation for the plane containing the triangle ABC and calculate the perpendicular distance of this plane from the origin. (^) [8]
2A Sketch the even function
f (x) =
1 + (x/π) for − π 6 x 6 0 1 − (x/π) for 0 6 x 6 π. (^) [3]
Find the cosine Fourier series of f (x) in the range −π 6 x 6 π and hence show that
π^2 8
∑^ ∞
n=
(2n − 1)^2
[17]
3B Consider the matrix
A =
cos θ sin θ sin θ cos θ
(a) Determine the eigenvalues of A. (^) [7]
(b) Determine the corresponding eigenvectors. Normalise these eigenvectors and show they are orthogonal. (^) [7]
(c) Determine the eigenvalues and corresponding eigenvectors of the matrix B = A^2. (^) [6]
Paper 1
6C Evaluate
S F^. dS^ in the following cases. (a) F = (xy^2 , yz^2 , zx^2 ) and S is the boundary of the region |x| 6 a, |y| 6 b, |z| 6 c. (^) [8]
(b) F = r = (x, y, z) and S is the surface defined by
r = (a cos φ sin θ, b sin φ sin θ, c cos θ)
with 0 6 θ 6 π and 0 6 φ 6 2 π. (^) [12]
[Use dS = ∂ ∂θr × (^) ∂φ∂r dθdφ.]
7D
(a) Evaluate:
(i)
(ii)
(iii) ( 4 8 1 )
(iv)
[8]
(b) Find the inverse of
Hence, or otherwise, solve the following linear equations when λ = 3:
x + y + z = 5 x + 2y + λz = 13 x + 4y + λ^2 z = 35.
Find the values of λ for which these equations have no solution. (^) [12]
Paper 1
8D* Let r = (x, y, z) be the Cartesian position vector, p a fixed vector, and E = |r|−^3 (3|r|−^2 (p. r)r − p). Use the divergence theorem to show that the surface integral
∫
A
E. ds [10]
vanishes for any closed surface A not enclosing the origin. Verify directly that the integral vanishes when p = (0, 0 , 1) and A is the surface of the infinite cylinder x^2 + y^2 = a^2. (^) [10]
9E Write down the Taylor expansion of f (x) about the point x = x 0. (^) [2]
Find, by any method, the first three non-zero terms in the Taylor expansion about x = 0 of the following functions, where a is a real constant:
(a) ln(1 + x) (^) [4]
(b) √x 2 x+a (^2) [4]
(c) exp{−(x − a)^2 } (^) [5]
(d) ln (^) 1+2^1 −xx 2. (^) [5]
Paper 1 [TURN OVER
12F x is a continuous real-valued random variable. The probability of x having a value in the range (x, x + dx) is P (x)dx.
(a) Define the mean and variance of x. (^) [3]
(b) Find the mean and variance of x in the case
P (x) =
1 − |x| − 1 6 x 6 1 0 otherwise
[7]
(c) Suppose that x is now normally distributed with mean zero and variance 1, so that x has distribution function
P (x) =
2 π
exp(−x^2 /2). [10]
Using integration by parts or otherwise
(i) show that x^3 has mean value zero; (ii) derive the mean value of x^4.
Paper 1
