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FIRST PUBLIC EXAMINATION Trinity Term Preliminary Examination in Physics Paper CP3: MATHEMATICAL METHODS 1 also Preliminary Examination in Physics and Philosophy Wednesday 6 June 2012, 2.30 pm – 5.00 pm Time allowed: 2 21 hours
Answer all of Section A and three questions from Section B. Start the answer to each question on a fresh page. The use of calculators is not permitted. A list of physical constants, mathematical formulae and conversion factors will not be provided for this paper. The numbers in the margin indicate the weight that the Moderators expect to assign to each part of the question.
Do NOT turn over until told that you may do so.
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Section A 1. Write the following in the form a + i b where a and b are real. e
iπ/2
,
−2i
i
,
1 + 3i 1 − 3i
,
√
3+i
3
. [4]
2. Solve 3 cos(ix) − i sin(ix) − 3 = 0.
[4]
3. Solve z 3 = 8 and show your solutions on an Argand diagram.
[4]
4. For b ⊥ c and a = b × c, prove that a × b = |b|2 c.
[4]
5. A plane is defined as 4x + 4y + 3z = 3. What is the closest distance of the plane to the origin?
[4]
6. Derive a condition x and y must satisfy to ensure that the matrix A below is non-singular. 1 x −1 A = 3 0 2 y 2 0 Invert the matrix A using a choice of x and y for which it has real eigenvalues.
[8]
7. Solve the differential equation: dx + xy − x = 0. dy
[6]
8. Solve the differential equation: dx = x−n − 2 x dt for x(t) over the interval 0 < x < ∞ for n > 0 and x(1) = 1. [You might try multiplying both sides by xn and changing variables.]
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Section B 9. Points A, B, C, and D have position vectors a = 2 i + 2 k, b = i + 3 j, c = i + 2 j + 2k, d = 4 j + 3 k, respectively. i, j, k are an orthonormal set of basis vectors. (a) Find the area of triangle ABC.
b0
[5]
(b) Find the perpendicular distance of D to the plane containing A, B and C.
[5]
(c) Find the volume of the tetrahedron ABCD.
[5]
(d) The points A0 , B0 , C0 and D0 are defined = Rb, c0 = Rc and d0 = Rd, respectively, where √ √1/2 − 3/2 R = 3/2 1/2 0 0
by the position vectors a0 = Ra, 0 0 . 1
What are the positions of the shifted points? What is the volume of the tetrahedron A0 B0 C0 D0 ?
[5]
10. (a) The projection matrix P has the property that P2 = P. Prove that this matrix only has eigenvalues 1 or 0.
[4]
(b) The exponential of a matrix A is defined as: ∞ X Am . exp (A) = m! m=0
[4]
Carry out the sum above when A = P. (c) Let A be a Hermitian n × n matrix. Show that the matrix exp (A) is also Hermitian. Let A be a diagonal matrix. Find the matrix elements of exp (A) in terms of the matrix elements Aij .
[4]
(d) Let B and C be complex n × n matrices. Show that when BC=CB, exp (B + C) = exp (B) exp (C) . [8] 4277
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x 11. The operator L acting on a vector has the property y
dx 2 dy + + 2y dt x dt L = y dy x− −y dt
.
(a) Find the complementary function (CF) that satisfies: x 0 L = . y 0 [8] (b) Find the particular integral (PI) for which x 3t L = , y 2
[8]
and find the full solution when 0 x(t = 0) . = 0 y(t = 0) [4]
12. An operator L acting on a function f (t) has the property Lf (t) = α2
d2 f (t) df (t) + α1 + α0 f (t) = h(t), 2 dt dt
where h(t) is a function and the αi are real numbers. (a) Under what conditions do solutions to Lf = 0 have oscillatory components? (b) Solve the particular integral (PI) for h(t) = 1 + t + α0 = 2. (c) Solve the PI for h(t) =
2et ,
t3 ,
α2 = 1, α1 = 3 and [5]
α2 = 1, α1 = −2 and α0 = 1.
[5]
(d) What is the limiting behaviour of f (t) as t → ∞ for h(t) = A cos(2 t), α2 = 1, α1 = 1 and α0 = 1?
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