Algebra Answers

In an Argand diagram, the point P represents the complex number z, where z = x+iy. Given that z +2 = λi(z +8), where λ is a real parameter, find the Cartesian equation of the locus of P as λ varies. If also z = µ(4 + 3i), where µ is real, prove that there is only one possible position for P.

Herbert and Dandreb can thoroughly clean a house in 3 hours. If each work, one of them would do the job in 1 hour less time than the other. How long would it take each person to do the cleaning?

Prove that ((x+y+z)/3)^(x+y+z) ≤ x^x y^y z^z ≤ ((x^2+y^2+z^2)/(x+y+z))^(x+y+z) Where x, y, z element of N

The sum of the first five terms of an AP is 35. The sum of the next five terms of this is AP is 85. Find the first term
and the common difference

A suburban newsagent is asked to collect data about the number of four popular daily newspapers
that he sells in his shop on weekdays.
The number of each paper sold and the daily taking are recorded for Monday to Thursday of the first
week:
Newspaper Takings
($) A B C D
Weekday Mon 123 108 48 12 348.00
Tue 136 b 43 15 377.60
Wed 115 133 36 11 345.30
Thur 145 128 45 18 405.50

Unfortunately, the number of newspaper B sold on Tuesday, b, was lost.

(a) (i) Write the number of each paper sold for the four days as a 4 × 4 matrix, P.

(ii) Write the takings for the four days as a column matrix, Q.


(b) If newspaper A costs $1.20, newspaper B costs $1.00, newspaper C costs $1.30 and newspaper D
Costs $2.50 write a suitable matrix X, representing the cost of each of the newspapers.

(c) Using matrix equation PX=Q to find the missing figure b; the number of newspaper B sold on
Tuesday. Show all working.

The cost of each of the components in the production of each size of pizza, and the selling price, is
given in the table below:

Cost of Production(cents) Selling
Price($) Size Base Topping Cooking Labour
Small 32 110 8 200 9.50
Medium 50 175 10 240 11.50
Large 72 250 12 280 13.50
Family 98 345 15 320 17.50

a) Use matrix methods to find the total cost of production for each size of the pizzas. Give your
answers in cents.


b) If Profit = Selling price – Cost of Production, write a matrix equation that will enable you to find
the profit and find the profit, in dollars correct to the nearest cent, made on each size of pizza.

c) use matrix methods and your answer to Q2 Part B Question (C) to find the total profit, in dollars
correct to the nearest cent, made in this shop for the week. (Hint: Convert matrix PA to a 1 × 4
matrix to find the profit.)

Number of pizzas sold

Size Mon. Tues. Wed. Thu. Fri. Sat. Sun.
Small 28 35 28 36 37 34 45
Mediu
m

36 37 47 32 36 38 47
Large 41 42 40 35 51 56 61
Family 30 33 39 33 78 76 83

a) Write the information in the table above as a 4 × 7 matrix, P.

b) Write down a suitable matrix, A, that when P is multiplied by A the resulting matrix will give the
number of each size of pizza sold in the week.

c) Calculate the matrix PA.


d) Write down a suitable matrix, B, that when B is multiplied by P the resulting matrix will give the
number of pizzas sold on each day of the week.

e) Calculate the matrix BP.

size
Mon
Tues
Wed
Thu
Fri
Sat
Sun
Small
28
35
28
36
37
34
45
Medium
36
37
47
32
36
38
47
large
41
42
40
35
51
56
61
Family
30
33
37
33
78
76
83
a) Write the information in the table above as a 4 × 7 matrix, P.

b) Write down a suitable matrix, A, that when P is multiplied by A the resulting matrix will give the
number of each size of pizza sold in the week.

c) Calculate the matrix PA.



d) Write down a suitable matrix, B, that when B is multiplied by P the resulting matrix will give the
number of pizzas sold on each day of the week.

e) Calculate the matrix BP.

A Pizza shop makes pizzas in four sizes:
Small, Medium, Large and Family.
In one particular week they have recorded the number of pizzas of each size that they have sold
each day. The data is recorded in the table below:

size
Mon
Tues
Wed
Thu
Fri
Sat
Sun
Small
28
35
28
36
37
34
45
Medium
36
37
47
32
36
38
47
large
41
42
40
35
51
56
61
Family
30
33
37
33
78
76
83
a) Write the information in the table above as a 4 × 7 matrix, P

b) Write down a suitable matrix, A, that when P is multiplied by A the resulting matrix will give the number of each size of pizza sold in the week?

Calculate the matrix PA?

Assume you use a revolutionary green car, whose fuel consumption per hour is proportional to the square root the car’ speed. You have to travel 750 km from Windhoek to a village at a constant speed, in a dust storm with a strong headwind of 25 km per hour. Assume that the minimum speed allowed on that road is 20km/h and the maximum speed (due to the storm) is 80 km/h. At what speed should you travel to minimize your fuel consumption. [20]