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NATURAL SCIENCES TRIPOS Part IA
Monday 12th June 2006 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 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) A matrix A satisfies A = 5AT^ , where T^ denotes the transpose. Find A. (^) [4] (b) If B = −BT^ and det(I + B) 6 = 0, where I is the unit matrix, show that C = (I − B)(I + B)−^1 is an orthogonal matrix. Is there any non-zero choice of B such that (i) C is symmetric? (ii) C is antisymmetric? In each case, if so, give an example; if not, explain why. (^) [7] (c) If D is a square matrix and det(I − D) 6 = 0, show that
(I − D)−^1 =
∑^ ∞
n=
Dn^ ,
where D^0 = I. (Assume that the right-hand side of the equation exists.) (^) [4] Use your result to find (I − D)−^1 if
D =
[5]
Paper 1
3B
(a) Give expressions for the following complex numbers in the form a + ib, where a and b are real: (i) 2 + i (1 − i)^2 ,^ [3] (ii) (^) ( i + 1 i − 1
[3]
(b) Find all solutions of the following equations and give the results in the form z = a + ib, where a and b are real: (i) z^3 = −8i , (^) [3] (ii) z^2 = 2(1 +
3 i) , (^) [3] (iii) z = i(1−i)^. [3] (c) Show that the solutions of the equation 2 sin z + cos z = i sin z are given by z =
nπ − π 8
− 14 i ln 2 , where n is an arbitrary integer. (^) [5]
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4B*
(a) State Stokes’ theorem, explaining carefully the meaning of any symbols you use and indicating the orientation of any integrations. (^) [4] (b) A vector field F is given in Cartesian coordinates by F = (2xz^3 , yz^2 , x^2 z^2 ). Evaluate ∇ × F. (^) [4] (c) Can F be obtained from a scalar potential φ such that F = ∇φ? (^) [2] (d) Verify Stokes’ theorem for F on the triangle ABC defined by A = (1, 1 , 0), B = (0, 1 , 0) and C = (0, 1 , 1). (^) [10]
5C
A, B and C are three points with position vectors a = a i, b = b j and c = c k respectively, where a, b and c are positive scalars. P is the plane passing through A, B and C. (a) Find a normal to the plane P. (^) [3] (b) What is the vector equation of P? (^) [3] (c) Define D as the point on P closest to the origin. What is the distance of D from the origin? (^) [4] (d) A position vector r is given by r = λ a + μ b + ν c where λ, μ and ν are scalars. What condition must λ, μ and ν satisfy if r is to lie in P? [3] (e) Find the values of λ, μ and ν corresponding to the point D. (^) [4] (f) If d is the position vector of D , find the angle between the vectors a − d and b − c. [3]
Paper 1 [TURN OVER
8D*
(a) Use the method of Lagrange multipliers to find the locations and values of the maxima and minima of the function x^2 y + 2y subject to the constraint x^2 + y^2 = 1. (^) [8] (b) The planes 2 x + y − z = 1 and x − y + z = 2 intersect in the line L. Without finding L explicitly, use the method of Lagrange multipliers to find the point on L closest to the origin. (^) [12]
9E
(a) Give Taylor’s formula for the series expansion of a function f (x) about the point x = a. [3] (b) Find, by any method, the first four non-vanishing terms in the Taylor series of: (i) ln x about x = 1 , (^) [5] (ii) arctan x about x = 1 , (^) [5] (iii) (^) x ex^ − 1 about^ x^ = 0^.^ [7]
Paper 1 [TURN OVER
10E
Solve the following differential equations for y(x), subject to y(0) = y′(0) = 0: (a) d^2 y dx^2 + 4y^ = cos^ x ,^ [6] (b) d^2 y dx^2 + 4y^ = cos
(^2) x , [6] (c) d^2 y dx^2 −^2
dy dx +^ y^ = (1 +^ x) e
x (^). [8]
11F
The periodic function f (x) of period 2π is defined by f (x) = sin 2x for 0 6 x 6 π , f (x) = − sin 2x for − π 6 x 6 0. (a) Sketch f (x). (^) [4] (b) Find the Fourier expansion of f (x). (^) [10] (c) Hence show that (^) ∞ ∑ m=
(2m + 1)^2 − 4 =
3. [6]
12F*
(a) Describe the method of separation of variables for Laplace’s equation in two dimensions, ∂^2 f ∂x^2 +^
∂^2 f ∂y^2 = 0^.^ [5] (b) A function f (x, y) satisfies Laplace’s equation inside a square region 0 6 x 6 a, 0 6 y 6 a. On the sides x = 0, x = a and y = 0, f vanishes. On the side y = a, f takes the constant value F. Find f (x, y) everywhere inside the square region. (^) [15]
END OF PAPER
Paper 1
