MAS156 Mathematics (Electrical and Aerospace) Spring Semester 2014-2015 PDF

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MAS156

SCHOOL OF MATHEMATICS AND STATISTICS MAS156 Mathematics (Electrical and Aerospace)

Spring Semester 2014–2015 2 hours 30 minutes

Attempt ALL questions. Each question in Section A carries 2 or 3 marks and each question in Section B carries 8 marks.

Section A

A1

Give an example of a function f which has a minimum at x = 0 but which has f ′′ (0) = 0. Give an example of a function g which has g ′ (2) = 0 but which does not have a minimum or a maximum at x = 2. (2 marks)

A2

Calculate

A3

Calculate the angle between the vectors (1, 2, 3) and (4, 5, 6).

A4

Find all complex numbers z that satisfy the equation |z − 1| = |z − j|. (3 marks)

A5

∂ 2f where f (x, y) = y sin(xy). ∂x∂y

Evaluate the indefinite integral

MAS156

Z

x2 dx. 1+x 1

(3 marks)

(3 marks)

(3 marks)

Turn Over

MAS156 A6

Find the general solution of the differential equation t

dy dt

+ 2y =

1 − 2s . s2 + 6s + 13

A7

Find the inverse Laplace transform of the function

A8

Find lim

A9

 1 −2 1 Evaluate the determinant of the matrix M =  3 0 2 . −1 −2 2

A10

√

x sin 2x . x→π/2 π − 2x

t. (3 marks)

(3 marks)

(3 marks)



Find the eigenvalues of the matrix M =



3 2 −1 1



.

(3 marks)

(3 marks)

Section B

B1

Find the solutions of z 4 = j and plot them on the Argand plane. Find the solutions of (w − 1)4 = j .

B2

State, without proof or justification, the limits sin(x) and x→0 x lim

cos(x) − 1 . x→0 x lim

Give the definition of the derivative of a function f at the point x = a. Use this to prove from first principles that if f (x) = sin(x) then f ′ (a) = cos(x).

B3

State the domain and range of each of the following functions: f (x) = cosh(x) and g(x) = arccosh(x). Suppose that y = arcsinh(x) and z = ey . Show that z 2 − 2xz − 1 = 0. Find z and hence deduce a formula for arcsinh(x) in terms of x.

MAS156

2

Continued

MAS156 B4

Let I(a) =

Z

x2

dx , where a is a constant. + 4x + a

Find the indefinite integral I(a) when (i) a = 0,

B5

(ii)

a = 4,

(iii)

a = 5.

Find the general solution of the differential equation d2 y dy + 6 + 13y = sin 2t. 2 dt dt Show that as t → ∞, y is approximately equal to α sin(2t − φ), and determine the values of α and φ.

B6

(i)

Sketch the function f (x) = e−x cos 10x,

x ≥ 0.

Show from your sketch that the equation e−x cos 10x = 2x, has a single solution x∗ ∈ (0, π/20). (ii)

x≥0

Devise the Newton-Raphson iterative scheme to solve the equation e−x cos 10x − 2x = 0,

x ≥ 0.

Using x0 = 0.1 as a first approximation, use this scheme to find the solution correct to four decimal places.

B7

(i)

Find the relationship between α and β if the system of equations x − 2y + z = 0

2x + y + z = 0 4x + αy + βz = 0

has a non-trivial solution. Find the general solution when this relationship holds. (ii)

Find the values of α and β for which the equations x − 2y + z = 0 2x + y + z = 5

4x + αy + βz = 0 have infinitely many solutions.

End of Question Paper MAS156

3

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Formula Sheet for MAS156 Trigonometry cos2 θ = (1 + cos 2θ)/2 sin2 θ = (1 − cos 2θ)/2 sin 2θ = 2 sin θ cos θ Hyperbolic Functions cosh2 θ = (1 + cosh 2θ)/2 sinh2 θ = −(1 − cosh 2θ)/2 sinh 2θ = 2 sinh θ cosh θ Binomial theorem   n(n − 1) 2 n r (1 + x) = 1 + nx + x + ... + x + ... 2! r n

  n(n − 1)(n − 2) . . . (n − r + 1) n . = where r! r If n is a positive integer then the series terminates and the result is true for all x, otherwise, the series is infinite and only converges for |x| < 1. Function sin x cos x tan x cosec x sec x cot x sinh x cosh x tanh x x  sin−1 a  x cos−1 a x tan−1 a x  sinh−1 a x cosh−1 a x tanh−1 a

Derivative cos x − sin x sec2 x −cosec x cot x sec x tan x −cosec2 x cosh x sinh x sech2 x 1 √ 2 a − x2 √

−1 a2 − x2

a a2 + x2 √

x2

√

x2

1 + a2

1 − a2

a a2 − x2

4

Integration-by-Parts Z Z ′ uv dx = uv − u′ v dx Substitution for a Rational Function of sin x and cos x  x 2t dx 2 1 − t2 then sin x = and = If t = tan , cos x = . 2 2 2 1+t dt 1 + t2 1+t Taylor expansion of f (x) about x = a f (a) + (x − a)f (1) (a) +

(x − a)n−1 (n−1) (x − a)2 (2) f (a) + . . . + f (a) + . . . 2! (n − 1)!

Newton-Raphson formula for the root of f (x) = 0 xn+1 = xn −

f (xn ) f ′ (xn )

Table of Laplace transforms f (t)

F (s) = L(f (t))

tn

n! (n = 0, 1, 2, . . .) sn+1

eat

1 s−a

sin ωt cos ωt sinh ωt cosh ωt

s2

ω + ω2

s2

s + ω2

ω s2 − ω 2 s2

s − ω2

eat f (t)

F (s − a) (shift theorem)

f ′ (t)

sF (s) − f (0)

f ′′ (t)

s2 F (s) − sf (0) − f ′ (0)

MAS156

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