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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?
2. For a two-dimensional flow, the velocity potential function given by x2 − y2 calculate the stream function
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
2. Given z1 = 2∠45o
, ; z2 = 3∠120o and z3 = 4∠180o
. Determine the following and leave your
answers in rectangular form:
(i)
(z1)2 +z2
z2 +z3
(5)
(ii)
z1
z2z3
(5)
5. Decompose
(i)
x2 +x +1
(x +3)(x2 −x +1)
(ii)
x4 −x3 −2x2 +4x +1
x (x −1)2
4. Use the Upper and Lower Bounds Theorem to show that the real zeros of
(i)
P (x) = 7x8 −2x5 +x2 −2 lie between −1 and 1.
(ii)
P (x) = 2x3 −7x2 +4x +4 lie between −4 and 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
#340153 on Dec 2023