Math Answers

7. An aeroplane heads due north at 500 km/h. It experiences a 80 km/h crosswind flowing in

the direction N60oE.

(a) Find the true velocity of the aeroplane. (7)

(b) Determine the speed of the aeroplane. (Leave your answer in terms of square root

6. Four forces act on an object such that the object is at rest. Three of the forces are given by

F1 = 2i −2j, F2 = i −4j, F4 = −3i −5j. Determine F3 and its magnitude

5. Water is flowing downhill at 15.0m/s through a pipe that is at an angle of 75o with the horizontal. What are the components of its velocity?

3. Use De Moivre’s Theorem to determine the cube root of Z and leave your answer in polar

form with the angle in radians

(a) Z = 1+i√3

3. Let Z = i

(i) Write Z in a polar form (2)

(ii) Use De Moivre’s Theorem to determine Z4

5. Decompose

(i)

x2 +x +1

(x +3)(x2 −x +1)

(ii)

x4 −x3 −2x2 +4x +1

x (x −1)2

3. Find a polynomial, P(x), of degree 3 with zeros of 4,1 and −1, if P(0) = 8.

2. Let P (x) = 2x4 +15x3 +31x2 +20x +4

(a) Determine whether (x −1) is a factor of P (x). (2)

(b) Find all the possible rational zeros of P (x) by using the Rational Zeros Theorem. (2)

(c) Solve P (x) = 0

It has been reported that 70% of university students do volunteer work during their summer vacation. Four students are randomly selected to do volunteer work.

a. The probability that at least 1 student will do volunteer work this summer (correct to 3 decimal places) is 

b. The probability that exactly 3 graduates will not do any volunteer work this summer (correct to 4 decimal places) is 

c. The expected number of students (correct to 1 decimal place) who will not do volunteer work this summer is

Based on past experience, it is assumed that the number of flaws of per metre in rolls of wrapping paper follows a Poisson distribution with a mean of 2 flaws per 4 metres of paper. The probability (correct to 2 decimal places) that more than 2 flaws will be observed in 5 metres of wrapping paper produced is: