MATH1055 Numbers and Vectors 201920
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MATH105501
Numbers and Vectors
Semester One 201920
SECTION A
A1. Use mathematical induction to prove the following inequality
(1 + x)N ≥ 1 + Nx when x > − 1
for all natural numbers N .
A2. Write the complex number
z = + e —iT/6
A3. Sketch the following set M in the complex plane
M = {z ∈ C : Im(z2 ) ≥ 0, |z| ≤ |z − i|}.
A4. Is the sequence (an ) deined by an = n + cos(nT) for n ≥ 1 bounded (bounded above, bounded below)? Is it convergent? Is it decreasing or increasing?
n + n3
n→∞ 2n + n4 n5 − n .
A6. When is an ininite series called absolutely convergent? What can you say about the convergence of an absolutely convergent ininite series?
A7. Find the intersection of the unit sphere (radius 1, centre at origin) and the plane passing through the points A = (1, 1, 1), B = (1, 1, 0), C = (1, 0, 0).
A8. Compute the vector product of a = (1, 2, 0) and b = (2, 2, − 1) and ind a unit vector
SECTION B
B1. (a) Deine the modulus |z| and argument arg(z) of the complex number z = x + iy . Write the modulus and argument of z2 w in terms of |z|, arg(z), |w| and arg(w).
(b) Find two complex numbers z and w such that
z2 w = − 1 + i^3, arg(z) = T/4, |w| = 1.
(c) Sketch the set M deined by
M = {z ∈ C : |Re(z)| < 1, 0 < arg(z2 ) < T/2}.
(d) State Euler’s formula and use it to derive the trigonometric addition formulae for the sine and cosine.
(e) Find all complex solutions of the equation
z9 − z6 + z3 = 1.
Hint: z = 1 is a solution.
B2. (a) Consider the ininite series
+ + + · · · + + ...
Show – using induction or otherwise – that for the partial sums sn =对k(n)=1 , we have
sn = 12 ( 1 − ) .
Finally, determine the sum of the series.
(b) Discuss the convergence of the series 对 qk and how this depends on q ∈ R. Determine its sum for the values of q such that the series converges.
(c) Geometric series can be used to obtain expressions for rational numbers, given in
decimal form, as the ratio of two coprime integers.
For r = 0.105510551055 · · · = 0.1055 we can also write
r = 1055/10000 + 1055/100000000 + 1055/1000000000000 + ...
Use this to express r as the ratio of two coprime integers.
(d) Consider the power series
n!nnxn .
Using the ratio test, determine the values of x for which the series converges. What can you say about the convergence when |x| is equal to the radius of convergence?
B3 . Consider the sphere S given by (x − 1)2 + (y + 2)2 + z2 = 24, the line L given by
(t + 2, 2t, − 1 − t) , t ∈ R passing through the centre of S , and the point P = (5, 0, 2)
(a) Find the coordinates of the intersection points A,B of S and L. (b) Find the coordinate equation of the plane T touching S at P .
(c) The line L intersects T in the point Q. Find Q and compute the distance between P and Q.
(d) Find the cosine of the angle between L and the line connecting P to Q. (e) Obtain the equation of the sphere with centre Q which touches the sphere S .
2022-08-09