Exam 2016, questions PDF

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All questions may be attempted but only marks obtained on the best four solutions will count. The use of an electronic calculator is not permitted in this examination.

1. (a) Carefully define the scalar product a · b and the vector product a × b of the two vectors a and b. (b) State the scalar product a ·b and the vector product a ×b using index notation and define the symbols δij and εij k . (c) Find the value of εijk εijk δmn δmn . (d) Two unit vectors ˆ a and bˆ have an angle γ between them. Show that the vector ˆ ˆ b − (ˆ a · b)ˆ a is perpendicular to ˆ a and is of length | sin γ|.

2. Let A, B, C, D be the corners of a trapezoid with AB k CD and diagonals AC and BD. The letters S, T , U1 , U2 stand for the areas of the respective triangles indicated in the figure.

D

C S U1

U2 T

A

B

(a) Using vectors, show that the triangle ACB has the same area as the triangle ADB . Deduce that U1 = U2 . [No marks will be awarded for methods that do not use vectors.] (b) Using any appropriate method, show that ST = U1 U2 . (c) Let K be the area of the trapezoid, derive the formula √ √ √ K= S+ T.

MATH1401

PLEASE TURN OVER 1

3. (a) Show that sinh(3x) cosh(3x) = 2. − cosh(x) sinh(x) (b) Derive the relations sin(ix) = i sinh(x) ,

and

cos(ix) = cosh(x) ,

for x ∈ R.

(c) Let z be a complex number. A curve is defined by the equation αz z¯ + βz + β¯z¯ + γ = 0 , with α, γ ∈ R and β ∈ C. Describe the shape of the curve for α = 0 and α = 6 0 by finding its Cartesian form.

4. (a) Find the partial fractions decomposition of x4

1 . −1

(b) Find Z (c) Find

Z

sin(3x) cos(5x)dx . dx

p

sinh(x)(sech(x) + tanh(x))

.

5. Find the general solution for each differential equation: (a)

3y − 4x − 2 dy = , dx y+x+1

(b)

d2 y dy + 3y = 5 sin2 (x) + e−2x + e−x . +4 2 dx dx

If necessary, solutions can be left in implicit form in the transformed variables.

MATH1401

CONTINUED 2

6. (a) A hunter is shooting pheasants and misses with a probability of 75%. He continues hunting until his first hit. (i) Describe a suitable sample space. (ii) Find the probability pn that the hunter hits on the n-th shot. P∞ (iii) Verify that n=1 pn = 1.

(b) The probability density describing the location of a particle is given by (   −1 < x < 1 , c −x4 + 45 x2 + 51 , f (x) = 0, otherwise . (i) Determine the value of the normalisation constant c. (ii) Find the mean and the variance. (iii) Consider the interval I = (−1/2, 1/2). Is the particle more likely to be inside or outside this interval?

MATH1401

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