What is the Use of the Number System in Computers: Computational Mathematics Home work, UOL, Singapore

Question 1

(a) What is the use of the number system in computers?

(b) Why is it sometimes advantageous to use binary numbers instead of decimal?

(c) Express in base π the circumference of a circle of radius 1.

(d) Which digits from (0,1,2,…,9) are not allowed in octal representation

(e) Evaluate √ 618 as a number in the decimal system

Question 2

(a) The first three terms of an arithmetic sequence are x, 1 x , 1 where x < 0.

i. Workout the value of x
ii. Find the common difference
iii. Find the sum of the first 20 terms.

(b) Consider the number sequence where u1 = 500, u2 = 519, u3 = 538 and u4 = 557 etc.

i. Is this sequence represents arithmetic or geometrical sequence?
ii. Find the value of u12 [2]
iii. Find the sum of the first 12 terms of the sequence P12 n=1 un.

(c) Another number sequence is defined where w1 = 4, w2 = 8, and w3 = 16
and w4 = 32 etc.
i. Find the value of w8
ii. Find the sum of the first 10 terms of the sequence

Question 3

A manufacturer, which is producing 100 items each week, plans to increase its production.

The number of items produced is to be increased by 10 items each week from 100 in week 1 to 110 in week 2, to 120 in week 3 and so on, until it is producing 300 in week N.

i. Find the value of N [2]

The manufacturer then plans to continue to make 300 items each week.

ii. Find the total number of items that will be made in the first (52 weeks) starting from and including week 1.

Question 4

For each of the following series, say whether it converges or diverges, explain your answer.
i. P∞ n=1n 3 n5+3 [1]

ii. P∞ n=1 3 n 4n+4 [1]
iii. Let an = n 4 6n , does the series P∞ n=1 an converge or diverge? [1]
iv. Determine if the sequence an = 2n converges or diverges. If the sequence converges, what does it converge to?

Question 5

(a) Use mathematical induction to show if the following statements are correct.

i. For all n ≥ 1, 1 + 3 + 5 + … + (2n − 1) = n 2

ii. For all n ≥ 1, 1 2 + 22 + 32 + … + (2n)2 = n(2n+1)(4n+1) 3

(b) Use the principle of mathematical induction to show if 6 n −1 is divisible by 5 for any positive integer n.

Question 6

(a) Determine which of the following numbers are congruent to each other,
explain your answer.
i. 31 ≡ 1 (mod 10) [2]
ii. 43 ≡ 22 (mod 7) [2]
iii. 8 ≡ −8 (mod 3) [2]
iv. 91 ≡ 18 (mod 6) [2]

(b) What is −17(mod 10)?. [3]

(c) Find the remainder when the difference between 60002 and 601 is divided by 6.

Question 7

(a) Are the following polygons similar to each other? If they are, give the scale factor, figure not drawn to scale

(b) Are the following polygons similar to each other? If they are, give the scale factor, figure not drawn to scale

(c) Solve the triangle for side b, figure not drawn to scale

(d) If b = 14, c = 20 represents the sides of a triangle with ∠B = 40o

Workout the values of angles A, C and the side a. [5]

Question 8

f is a function f : Z×Z + → Q, defined as f(p, q) = p q , where Z denote the set of integers, Z + denotes to the set of positive integers and Q the set of rational number.
(a) Is f an injective function? Explain your answer? [2]
(b) Is f a surjective function? Explain your answer? [2]
(c) Is f a bijective function? Explain your answer?. [2]

Question 9

(a) Given the function f(x) = x 3 + x 2 − 10x + 8, f(1) = 0

i. Determine the x- and y-intercepts of f(x) [3]
ii. Draw a rough sketch of the graph. [3]
(b) Sketch the curves with equations y = x(x − 3) and y = x 2 (1 − x) On the same diagram. Find the coordinates of their points of intersection. [3]

Question 10

A racing car accelerates uniformly from 18.5 m/s to 46.1 m/s in 2.47 seconds. Determine the acceleration of the car and the distance traveled.