Turks & Caicos Islands Community CollegeP O Box 236,Grand Turks,

Turks & Caicos IslandsBritish West Indies

Department of Mathematics

MTH 2111: College Algebra

Tutorial Questions

Compiled By

Dr. J. O. Mubenwafor

©2007 Dr. J O Mubenwafor

1.1 – Exponential Functions

1. Solve the given equations (if possible) using algebraic method:

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

(p) (q) (r)

(s) (t) (u)

(v) (w) (x)

(y) (z)

[Answers: (a) 6 (b) 3 (c) (d) 4 (e) No solution (f) (g) 3 (h) (i) (j) 8

(k) (l) (m) 6 (n) (o) 10 (p) 3 (q) (r) -5 (s) -1 (t) (u) -½ (v) -23

(w) 3 (x) 5/3 (y) -3/2 (z) -2/3]

2. Solve the following equations:

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

[Answers: (a) -4 (b) 0.5 (c) x = 1, y = -1 (d) 31 (e) 32 (f) -4/3 (g) 7 (h) 4 (i) 0]

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3. Solve the following:(a) (b) (c)

(d) (e) (f)

(g) (h)

[Answers: (a) 3 (b) (c) (d) 3 (e) (f) 4 (g) -0.5 (h) -3]

4. (a) If , compute . (b) If , compute .(c) If , compute . (d) If , what is the value of ?(e) Given that , what is the value of ?(f) Given that , compute .(g) Given that , what is the value of ?

[Answers: (a) 64 (b) 125 (c) (d) (e) (f) (g) ]

5. World population is approximately described by the exponential function:

, where P(t) is the population in billions and t

is time in years, measured from 1998.(i) Predict world population for the year 2020, in billions, rounded to two

decimal places;(ii) What was the population in 1970?[Answers: (i) 8.70 billion (ii) 3.66 billion]

6. World industrial production is growing exponentially according to the equation:

,

Production is measured on a scale, with the 1963 production equal to 100. Will production surpass 1000 by the year 2000?

[Answer: Yes, 1299.6]

7. The number of bacteria present in a certain culture after t minutes is given by:

, where N is the number of bacteria.

(a) How many bacteria are present initially?(b) Approximately how many bacteria are present after 3 minutes?

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[Answer: (a) 300 (b) 711]

8. The world population P, at time t, is currently observed to be doubling and can be described by the function:

, for suitable values of k and c.

World population in August 1998 was approximately 5.94 billion and is observed to be doubling every 40 years. Predict the world population for the:(i) year 2050; (ii) next 80 years.[Answer: (i) 14.63 billion (ii) 23.76 billion]

9. If a single pane of glass obliterates 3% of the light passing through it, then the percent p of the light that passes through n successive panes is given approximately by the function:

.What percent of light will pass through:(a) 10 panes? (b) 25 panes?[Answer: (a) 74% (b) 47%]

10. The function:can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug has been administered. How many milligrams will be present after:(i) 1 hour? (ii) 6 hours?[Answer: (i) 3.35 mg (ii) 0.45 mg]

11. The projected population, P, of a city is given by:

where ‘t’ is the number of years after 1990.Predict the population for the year 2010. [Answer: 271828]

12. A radioactive element decays such that after t days the number of milligrams present is given by:

where N0 represents the amount of elements present at time t = 0.

Given that N0 = 100, how many milligrams are present after 10 days?[Answer: 53.8]

13. Pierre de Fermat (1601 – 1665) conjectured that the function:, for x = 1, 2, 3, …, would always have a value

equal to a prime number. But Leonhard Euler (1707 – 1783) showed that this formula fails for x = 5. Determine the prime numbers produced by f for x = 1, 2, 3, 4, then show that f(5) = 641 x 6,700,417, which is not prime.

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[Answer: 5, 17, 257, 65537, 4294967297]

14. Sales for a new magazine are expected to grow according to the equation:,

where t is given in weeks.Calculate the numbers of magazines sold after:(a) one week; (b) 20, 45 and 52 weeks.[Answer: (a) 9754, (b) 126, 179, 185]

15. A consumption function is modeled by the equation:, where ‘t’ is income level.

Calculate the consumption value when income level is:(a) 12 (b) 20. [Answer: (a) 486, (b) 499]

16. The total value of sales, S (in $000), is given by the equation:where ‘t’ is given in weeks.

(a) Find the total sales at the:(i) beginning of week one,(ii) end of week one, (iii) end of week 5;

(b) Calculate the time, in weeks, taken for sales to reach $400,000.(c) What is the maximum sale attainable?[Answer: (a) (i) 0, (ii) 73.93, (iii) 275.34; (b) 10.1 wks (c) $500,000 (never reached]

17. A virus is spreading through a chicken farm according to the function:

, where N is the number of infected chickens after t days.

(a) How many chickens are infected:(i) initially?(ii) after 20, 60, and 100 days?

(b) Will all the chickens become infected eventually? Give reasons for your answer.

[Answer: (a) (i) 1 (ii) 7, 268, 772 (b) As t increases, in the denominator will approach zero, therefore N = 800 infected chickens.]

18. Scientists estimate that the maximum carrying capacity of a lake is 6000 and that the number of fish increase according to the function:

,

where “t” is given in years and “P(t)” is the fish population at time “t”.

(a) Find the fish population in the lake:(i) initially; (ii) after 4, and 10 years.

(b) Calculate the time taken for the fish population to reach 1000, 3000, 4000.

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[Answer: (a) (i) 200 (ii) 875, 3919 (b) 4.39 yrs, 8.42 yrs, 10.15 yrs]

19. The table below describes an exponential function:

.

X 0 12 24f(x) 8 16 32

Find the function and give its initial value and its doubling time.

[Answer: , 8, 12]

20. In a study by molecular biologists in experiments involving genetic manipulation of the important bacterium E. coli, it was found that the bacterium has a doubling time of 17 minutes at 370C in a laboratory with an initial count of 1000 bacteria.(a) Compute the number of cells of the bacterium after 1 hour 42 minutes;(b) Determine how long it takes the population to grow to 20,000[Answer: (a) 64,000, (b) 73.47 minutes]

21. The bacterium Bacillus subtilis can be found in soil and in the content of intestines. Its doubling time is 26 minutes.(a) Give the function that describes the growth of the bacterium for an initial

count of 200 bacteria;(b) Hence, determine when the population will reach 3200.

[Answer: (a) , (b) 104 minutes]

22. The population of the United States in 1997 was 268.5 million. The population is growing exponentially, with a doubling time of about 65 years. If the current trend continues, when is the population predicted to reach 300 million?[Answer: 2007]

23. The cities of Penton and Trent have populations of 150 thousand and 200 thousand, respectively. They are growing with doubling times of 20 and 30 years, respectively.(a) After how many years (to the nearest year) will the population of Penton

equal that of Trent, if current trends continue?(b) What will the populations then be, to the nearest thousand?[Answer: (a) 25, (b) 356 thousand]

24. The worldwide consumption of crude oil has been growing exponentially with a doubling time of 19.8 years. The amount of oil consumed in 1995 was 7,841,777 thousand metric tons.(a) Construct the exponential function that describes the consumption of oil;(b) Predict the consumption of oil in the year 2005.

[Answer: (a) , (b) 11,128,838.48 thousand metric tons]

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25. (a) If and , find the value of ‘a’.(b) Use the laws of indices to evaluate: 200 x 36-½ x 64⅔ x 81-¾ x (⅓)-2

[Answer: (a) 1, (b) ]

1.2: Logarithmic Functions

1. Compute the value of each of the following logarithms:

(a) (b) (c)

(d) (e) (f)

[Answer: (a) 2 (b) 3 (c) -3 (d) -4 (e) -5 (f) -1/3]

2. Using the properties of logarithms, compute the following, to four decimal places:(a) (b)(c) (d)(e)[Answer: (a) 3.2362 (b) 0.4307 (c) -1.0969 (d) 0.3562 (e) 6.1530]

3. Solve the following equations, correct to two decimal places:(a)(b)(c)(d)[Answer: (a) 5.46 (b) 2.87 (c) 2.90 (d) -3.211]

4. Use the properties of logarithms to solve the following equations for x:(a) (b)(c) (d)(e) (f)(g) (h)(i) (j)[Answer: (a) 2 (b) 2/7 (c) 20/7 (d) 5 (e) 3.5 (f) 16/3 (g) 4 (h) 1 (i) 12 (j) 3]

5. Solve the following equations for x, correct to four decimal places:(a)(b)(c)[Answer: (a) 50.1377 (b) 218.7266 (c) 745.9895]

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6. Solve the following equations (if possible) to four decimal places:(a) (b) (c)(d) (e) (f) \(g) (h) (i)(j) (k)[Answer: (a) 0.5365 (b) -0.8959 (c) 0.3715 (d) -4.3167 (e) 2.0000(f) 0.5570 ((g) 0.9186 (h) -0.6788 (i) 2.3900 (j) 0.2619 (k) -1.8293]

7. The population growth of the Turks & Caicos Islands is described by the function:

,

where P(t) is the population in thousands at time t in years, measured from 1980.If the current growth continues in this manner, when will the population reach 160 thousands?[Answer: t = 30, in 2010]

8. The population growth of a certain country is described by:

,

where P(t) is the population in units of one million and t is the time measured from 1985. In which year is the population predicted to reach 40 million?

[Answer: t = 18, in 2003]

9. The population of Century City is described by the function:

,

where P(t) is the population in thousands at time t in years, measured from 1990.If the current growth continues, determine in which year the population will reach 50 thousand. [Answer: t = 12.86, in the 12th year, 2002]

10. The number of watts w provided by a space satellite’s power supply after a period of d days is given by the function:

.(a) How much power will be available after:

(i) 30 days?(ii) 1 year?

(b) How long will it take for the available power to drop to:(i) 30 watts?(ii) only 5 watts?

[Answer: (a) (i) 44.3 watts (ii) 11.6 watts (b) (i) 128 days (ii) 576 days]

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11. A model for the number of people N in a college community who have heard a certain rumor is:

, where P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a community of 1000 students:

(a) how many students will have heard the rumor after 3 days?(b) how many days will elapse before 450 students have heard the rumor?[Answer: (a) ≈ 362 students (b) 4 days]

12. The growth of a certain bacterium is described by:

.

How long (to the nearest minute) will it take the bacterium to reach 5000?[Answer: 58 minutes]

13. The growth of the bacterium Pseudomonas natriegens is described by the function:

.

How long (to the nearest minute) will it take an initial count of 500 to grow to 2500?[Answer: 23 minutes]

14. The inverse of the function of world population growth given in problem 5 of section 1.1 above, measured from 1998 is:

.

Using this inverse function, predict when the world population will reach 10 billion.[Answer: t(10) = 30.06, year 2028]

15. Wood from the foundation of a fortification wall in a mound at Alisher Huyuh, Turkey, was found to have radioactivity R = 10.26 units. If after t years of decay the amount of radioactivity R(t) given off by 1 gram of a radioactive substance is:

,

determine the approximate age of the wall.[Answer: 3,210 years]

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16. The population growth of a village, measured from 1980, is given by the function:,

where P is the number of persons in the population at time t.(a) Estimate the population in:

(i) 1990(ii) 2000

(b) In what year will the population reach 1750 persons?[Answer: (a) (i) 1016 (ii) 1372 (b) 2008]

17. The population of a village at the beginning of the year 1800 was 240. The population increased so that, after a period of n years, the new population was 240(1.06)n. Find:(i) the population at the beginning of 1820;(ii) the year in which the population first reached 2500.[Answer: (i) 770 (ii) t = 40, 1840]

18. A liquid cools from its original temperature of 90oC to a temperature ToC in x minutes. Given that T = 90(0.98)x, find the value of:(i) T when x = 10;(ii) x when T = 27[Answer: (i) 73.5oC (ii) 60]

19. (a). Calculate the value of x, to four decimal places, given that:

(b) Solve the following equation for x, correct to 3 decimal places:

[Answer: (a) -1.4741 (b) -2.486]

20. (a) Solve the following for x:

(b) Given that: , express y in terms of x.

[Answer: (a) or (b) ]

21. If and , find the value of

,

in terms of a and c.[Answer: 7a + 4c]

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22. If and , show that:

23. Given that: :

(i) find the value of N;(ii) find also the value of when a = ⅔[Answer: (i) (ii) 3]

24. Given that: , and , express

in terms of p and q.

[Answer: ]

25. Given that log x = a and log y = b, express in

terms of a and b.[Answer: ½(3 + 3a – b)]

26. If , and c = a4, prove that: q = abc.

2.0 - ALGEBRAIC EQUATIOS

2.1 – Linear Equations

1. Solve the following equations:(a) (b)(c) (d)(e) (f) 3(2x – 5) – 2(x – 3) = 3(g)[Answer: (a) 3 (b) 1 (c) 3 (d) 6/5 (e) 3 (f) 3 (g) 3/14]

2. Find the value of x in each of the following:

(a)

(b)

[Answer: (a) 19.5 (b) 2]

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3. Solve the following equations:

(a) (b)

(c) (d)

(e) (f)

[Answer: (a) (b) (c) (d) (e) (f) ]

4. Solve the following equations for x:

(a) (b)

(c) (d)

(e) (f)

(g) (h)

[Answer: (a) -7 (b) 4 (c) 19.41 (d) 5.91 (e) 0.41 (f) 9 (g) 0.0682 (h) – 0.9405]

5. Paula and Sam want to buy a house, so they have decided to save one-fourth of each of their salaries. Paula earns $22.00 per hour and receives an extra $8.00 a week because she declined company benefits, and Sam earns 26.00 per hour plus benefits. They want to save at least $350.00 each week. How many hours must each work each week?[Answer: 29 hours]

6. A day care center’s total monthly revenue from the care of x toddlers is given by:r = 650x,

and its total monthly costs are given by:c = 400x + 2000.

How many toddlers need to be needs to be enrolled each month to break even?[Answer: 8]

7. $380 is divided between two students, A and B, so that A gets $144 less than B. Calculate how much each of them got.[Answer: A =$118, B = $262]

8 A farmer shared 496 plums among her three workers, Paula, Greta and Gertrude. Greta received 16 more than Paula. Gertrude received three times as many as Paula. Calculate the number of plums Paula received.[Answer: Paula = 96 plums]

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9. The Anderson Company produces a product for which the variable cost per unit is $6 and fixed cost is $80,000. Each unit has a selling price of $10. Determine the number of units that must be sold for the company to earn a profit of $60,000.[Answer: 35,000units]

10. Sportcraft manufactures women’s sportswear and is planning to sell its new line of slacks to retail outlets. The cost to the retailer will be $33 per pair of slacks. As a convenience to the retailer, Sportcraft will attach a price tag to each pair. What amount should be marked on the price tag so that the retailer may reduce this price by 20% during a sale and still make a profit of $15% on the cost?[Answer: $47.44]

11. A total of $100.00 was invested in two business ventures, A and B. At the end of the first year, A and B yielded returns of 6% and 5¾%, respectively, on the original investments. How was the original amount allocated if the total amount earned was $588.75?[Answer: $5500 at 6%, $4500 at 5¾%]

12. The geometric Products Company produces a product at a variable cost per unit of $2.20. If fixed costs are $95,000 and each unit sells for $3., how many units must be sold for the company to have a profit of $50,000?[Answer: 181,250]

13. The Clark Company management would like to know the total sales units that are required for the company to earn a profit of $100,000. The following data are available: Unit selling price of $20; variable cost per unit of $15; total fixed cost of $600,000. From these data determine the required sales units.[Answer: 140,000]

14. A person wishes to invest $20,000 in two enterprises so that the total income per year will be $1440. One enterprise pays 6% annually; the other has more risk and pays 7½% annually; How much must be invested in each?[Answer: 4,000 at 6%, 16,000 at 7½%]

15. A person invested $20,000, part an interest rate of 6% annually and the reminder at 7% annually. The total interest at the end of 1 year was equivalent to an annual 6¾% rate on the entire $20,000. How much was invested at each rate?[Answer: 5,000 at 6%, 15,000 at 6¾%]

16. Candy has $70,000 to invest and requires an overall rate of return of 9%. She can invest in a safe, government-insured certificate of deposit, but it only pays 8%. To obtain 9%, she agrees to invest some of her money in noninsured corporate bonds paying 12%. How much should be placed in each investment to achieve her goal?[Answer: $17,000 in bonds, $52,500 in certificates]

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17. A total of $18,000 is invested, some in stocks and some in bonds. If the amount invested in bonds is half that invested in stocks, how much is invested in each category?[Answer: $12,000 in stocks, $6,000 in bonds]

18. Shannon grossed $435 one week by working 52 hours. Her employer pays time-and-a-half foe all hours worked in excess of 40 hours. With this information, can you determine Shannon’s regular hourly wage?[Answer: $7.50]

19. A total of $20.000 is to be invested, some in bonds and some in certificates of deposit (CDs). If the amount invested in bonds is to exceed that in CDs by $3000, how much will be invested in each type of investment?[Answer: $11,500 in bonds, $8,500 in CDs]

20. An inheritance of $900,000 is to be divided among Scott, Alice and Tricia in the following manner: Alice is to receive ¾ of what Scott gets, while Tricia gets ½ of what Scott gets. How much does each receive?[Answer: Scott: $400,000; Alice: $300,000; Tricia: $200,000]

21. Tickets for a concert cost either 50¢ each or 70¢ each. If the number of cheaper tickets sold was twice the number of dearer tickets and the total receipts were $340, find the number of cheaper tickets sold.[Answer: 400]

22. A sum of money consists of 40 coins some of which are 20¢ pieces and the rest of which are 10¢ pieces. If the total value of the coin is $5, find the number of 20¢ pieces.[Answer: 10]

23. A women had $200. She went to a meatshop, a bookstore and a drugstore. She spent four times as much money at the meatshop as she did at the drugstore. She spent $15 less at the bookstore than at the drugstore. She had $5 left. Obtain an equation for the total amount of money spent and hence calculate the amount she spent at the drugstore.[Answer: 6x – 15 = 195, $35.00]

2.2 – Simultaneous Linear Equations: Two unknowns

1. Solve the following simultaneous equations:(a) 6x – 5y = -7 (b) 4x + 2y + 8 = 0

3x + 4y = 16 6x = 2y – 27

(c) (d)

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(e) 4x – 3y = 5 (f) 4x + 2y + 11 = 06x + y =2 3x + 4y + 5 = 0

(g) 3x + 7y = 13 (h) 3x + 4y = 142x = 4 + 7y -2x + y= 9

[Answer: (a) x = 4/3, y = 3 (b) x = - 3.5, y = 3, (c) x = 40, y = 35, (e) x = ½, y = -1 (g) x = 3.4, y = 0.4 (h) x =-2, y=5]

2. Solve the following simultaneous equations:

(a) (b)

(c) (d)

(e)

[Ans.: (a) x = 12, y = - 12, (b) x = 3, y= 4, (c) x = - 6, y =8, (d) x = 3/2, y = - ⅔ (e) x = -3/5, y = -4/5]

3. Solve the following simultaneous equation:(a) x + 4y = 3, (b) 3x – 4y = 13,

3x – 2y = -5 2x + 3y = 3.(c) 5v + 2w = 36 (d) x – 2y = 8,

8v – 3w = -54 5x + 3y = 1[Answer: (a) x = -1, y = 1, (b) x = 3, y= -1, (c) v = 0, w = 18, (d) x = 2, y = - 3]

4. Solve, for x and y, the equations:

(a) (b)

[Answer: (a) x = 4, y = - 2 (b) x = 6.5, y = 2.5]

5. Solve for the unknowns in the following sets of equations:

(a) (b)

(c) (d)

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(e) (f)

(g) (h)

(i) (j)

[Answer: (l) 6.5, 1.5]

6. Solve the simultaneous equations: y = 3x + 5; 2y + 3x = 28 for x and y.[Answer: x = 2, y = 11]

7. Use the method of elimination to find the value of x and y that satisfy the following simultaneous equations:

(a) x + y = 4 (b) 3x + 2y = 19 (c) 3x + 2y =73x – y = 8 3x – 3y = -6 x + y =3

(d) x + 3y = 7 (e) 7x – 4y = 372x – 2y = 6 6x + 3y =51

[Answer: (a) x = 3, y = 1, (b) x = 3, y = 5, (c) x = 1, y = 2, (d) x = 4, y = 1,(e) x = 7, y = 3]

8. Solve the following simultaneous equation:

(a) (b) (c)

[Ans.: (a) x = 19/14, y = -19 (b) x = ½, y = 1 (c) x = ¼, y = 1]

9. A bill for $74 was paid with $5 and $1 notes, a total of 50 notes being used. How many $5 notes were there?[Answer: 6]

10. A company manufactures two products X and Y by means of two processes A and B. The maximum capacity of process A is 1750 hours and of process B 4000 hours. Each unit of product X requires 3 hours in A and 2 hours in B, while each unit of product Y requires 1 hour in A and 4 in B. Use the algebraic method to calculate how many units of products X and Y are produced if the maximum capacity available is utilized. [Answer: 300 units of x and 85 units of y]

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11. Two numbers are such that one third of the smaller number plus half the larger number is 19. If the difference between the two numbers is 13, find the two numbers. [Answer: x = 28, y = 15]

12. 500 tickets were sold for a concert, some at 40¢ each and the remainder at 25¢ each. The money received for the dearer tickets was $70.00 more than for the cheaper tickets. Find the number of dearer tickets sold.[Answer: 300]

13. 300 tickets were sold for a concert, some at $8 each and the remainder at $5 each. The money received for the dearer tickets was $60 more than for the cheaper tickets. Find the number of dearer tickets which were sold.[Answer: 120]

14. 76 is divided into two parts, x and y respectively, so that ⅔ of the first part is equal to of the second part. Find the two parts x and y.[Answer: x = 36, y = 40]

15. A chemical manufacturer wishes to fill an order for 500 liters of a 25% acid solution. (Twenty-five percent by volume is acid.) If solutions of 30% and 18% are available in stock, how many liters of each must be mixed to fill the order?[Answer: 291⅔ liters of 30%, 208⅓ liters of 18%]

16. A box contains 44 coins, made up of dimes (10c) and nickels (5c coins). If the total value of the coins is $3.50, calculate the number of dimes in the box.[Answer: 26]

17. A man went to a post office to buy some stamps. If he bought x, 50 cents stamps and y, 25 cents stamps, the total cost would have been $3.50. If however he bought twice as many 50 cent stamps and half as many 25 cents stamps, then the total cost would have been $4.75. Find x and y.[Answer: x = 4, y = 6]

18. An invoice clerk receives a bill for £37.50 for ten blank ledger books and three special filing trays. On phoning the supplier (after discovering the order had been written out incorrectly) the clerk agrees to return three of the ledger books in return for the supplier sending an extra filing tray. Given that there will be an extra £2.50 to pay and that there is a 10% discount for orders of 10 ledger books or more:(a) derive linear equations in x and y to represent the components of the

original and revised invoices, where x is the price of a single ledger book and y is the price of a filing tray, and

(b) solve the two equation simultaneously for x and y.[Answer: (a) 10(0.9)x + 3y = 37.5, 7x + 4y = 40, (b) x = 2, y = 6.5]

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19. A furniture factory manufactures two type of coffee table, A and B. each table goes through two distinct costing stages, assembly and finishing. The maximum capacity for assembly is 195 hours and for finishing, 165 hours. Each A table requires 4 hours assembly and 3 hours finishing, while a B table requires 1 hour for assembly and 2 hours for finishing. Calculate the number of A and B tables to be produced to ensure that the maximum capacity available is utilized.[Answer: 45A and 15B tables]

20. A chemical manufacturer wishes to fill an order for 700 gallon of a 24% acid solution. Solutions of 20% and 30% are in stock. How many gallons of each solution must be mixed to fill the order?[Answer: 420 (20%), 280 (30%)]

21. A manufacturer of dining-room sets produces two styles: early American and contemporary. From past experience, management has determined that 20% more of the early American styles can be sold than the contemporary styles. A profit of $250 is made on each early American set sold, whereas a profit of $350 is made on each contemporary set. If, in the forthcoming year, management desires a total profit of $130.000, how many units of each style must be sold?[Answer: 240(EA), 200]

22 When petrol cost 10¢ a litre and oil 25¢ a litre, a motorist found that his cost in petrol and oil was $10.50 for each 1000km. When petrol was increased to 10½¢ a litre and oil to 26¢ a litre, the cost was $11.02. Find how many litres of petrol and oil he used to travel 1000km.[Answer: 100 litres of petrol; 2 litres of oil]

23. A sum of $2800 is invested, partly at 5% and partly at 4%. If the total income is $128 per annum, find the amount invested at 5%. [Answer: $1600]

24. A supervisor and 7 workers together earn $520 per week, whilst 2 supervisors and 17 workers together earn $1220 per week. Find the weekly wage of a supervisor and of a worker. [Answer: $100, $60]

25. In the year 300 B.C., the Mathematician Euclid gave the following problem to his student: ‘A mule and a donkey were going to a market laden with wheat. The mule said: “If you give me one measure, I should carry twice as much as you, but if I give you one measure we shall have equal burden.”’ Calculate the burden of the mule and that of the donkey.[Answer: 7; 5]

26. A bird laid a certain number of eggs in a nest, and a collector took away two-thirds of them. The bird laid some more, and another collector took two-thirds of them. If the bird laid 12 eggs altogether and collectors too 10 altogether, how many eggs were there to begin with? [Answer: 9]

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27. A sum of $3000 is invested, partly at 3½% and partly at 4%. If the annual income from the two investments $112, find how much is invested at 4%.[Answer: $1400]

28. The charge for electricity is 3¢ per unit for lighting and ½¢ per unit for heating. A man’s bill for a quarter should have been $8. By mistake, his lighting was charged at ½¢ per unit and his heating at 3¢ per unit. The amount of the bill was $13. How many units of electricity did he use altogether?[Answer: 600]

29. At a concert, tickets were 70¢ or 50¢ each. Three quarters of those who paid 70¢ bought programmes and half the 50¢ seat-holders bought programmes. The receipts from the tickets were $580 and from the programme $60. If the programmes cost 10¢ each, how many paid 70¢ for the concert?[Answer: 400]

30. The cost $C of a journey on a certain railway is given by the formula: C = a + bx, where ‘x’ is the number of kilometers traveled and ‘a’ and ‘b’ are constants. A journey of 50km costs $2, and a journey of 70km costs $2.50. Find the cost of a journey of 90km.[Answer: $3]

31. If I invest $2000 at x% and $2500 at y%, my annual income is $160. Had I invested $2500 at x% and $2000 at y%, my income would have been $155. Find x and y. [Answer: 3; 4]

32. I invest $x at 4% and $y at 5%. My annual income is $230. Had I invested $x at 5% and $y at 4%, my annual income would have been $220. Find x and y.[Answer: 2000; 3000]

33. At a meeting of a cricket club to elect a captain, 75 members were present, and the captain was elected by a majority of 13, all voting. How many voted for and against? [Answer: 44 for; 31 against]

34. The daily wages of 10 men and 7 boys amount to $40.32; if a man earns in two days as much as a boy earns in seven days, find what each earns per day.[Answer: $3.36; $0.96]

35. If Ada were to give Ellis twelve dollars, Ada would have half the sum which Ellis then has; but, if Ellis were to give Ada thirteen dollars, Ellis would have one-third of what Ada has. How much money has each originally?[Answer: $32; $28]

36. A number of two digits is such that the sum of its digits is 10. When the digits are reversed, the number is increased by 36. Find the number. [Answer: 37]

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37. A number of two digits is such that twice the ten digit is 6 greater than the unit digit. When the digits are reversed, the number is increased by 9. What is the number?[Answer: 78]

38. A number with three digits has the hundred digit three times the unit digit and the sum of the digits is 19. If the digits are written in reverse order, the value of the number is decreased by 594. Find the number.[Answer: 973]

39. Solve the simultaneous equations:

and

[Answer: x = 2, y = 1]

40. Given that: and ,

find the value of x and of y.[Answer: x = 2, y = 2.5]

41. If and ,calculate the value of x and of y.[Answer: , ]

2.3 – Quadratic Equations

1 . S o lve t he quad ra t i c equa t ions :( a ) (b )( c ) (d )( e ) ( f )(g ) (h )( i ) ( j )( k ) ( l )(m) (n )(o) (p)

(q ) x ≠ 0 (r)

[Ans w er : ( a ) 1 , 5 (b ) –3 , 0 . 5 ( c ) -⅔ , 2½ (d ) –4 , 2 ( e ) 3 /2 , - 1 ( f ) 2 . 22 , - 0 .22 (g ) 3 tw ice (h ) 5 , - 1 ( i ) 3 . 29 , 0 . 71 ( j ) - 2 , 5 / 2 (k ) 2 , 3 ( l ) ⅓ tw ic e (m) -4 , - 1 (n ) 6 / 5 (o ) 3 , 4 (p ) 2 , ½ (q ) -⅔ , ⅓ ( r ) 6 . 34 , 0 . 16 ]

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2. Determine the solutions of the following equations rounded to 4 decimal places:(a)(b)[Answer: (a) 2.7656, -1.2656 (b) 2.5734, -0.9067]

3. Solve the following equations to 2 decimal places:(a) (b)(c) (d)(e) (f)(g) (h)(i) (j)(k) (l)[Answer: (a) -5.24, -0.76 (b) -4.30, -0.70 (c) -3.24, 1.2 (d) -3.56, 0.56(e) -0.73, 2.73 (f) -0.30, 3.30 (g) -2.28, -0.22 (h) 0.28, 2.39 (i) 0.57, 1.77(j) -3.85, -0.65 (k) -0.85, 2.35 (l) -0.43, 0.77]

4. Solve the following equations:(a) (b)(c) (d)(e) (f) (g)(h) (i) (j)(k)[Answer: ]

5. By using suitable substitutions, or otherwise, find the values of x (correct to two decimal places where necessary) such that:(a) (b)(c) (d)(e) (f)(g) (h)

(i) (j)(k) (l)[Answer: (g) 2.58 (h) 0, 0.63 (i) 1, 0.36 (j) 0, -1 (k) -4, 1 (l) -2, 1]

6. Solve the following equations for x:(a) (e)(b) (f)(c) (g)

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(d) (h)[Answer: (h) 1]

7. Find two positive integers that have sum 10 and product 21. [Answer: 3, 7]

8. From each corner of a square piece of sheet metal, remove a square of side 9 cm. Turn up the edges to form an open box. If the box is to hold 144 cm3, what should be the dimensions of the piece of sheet metal? [Answer: 22 cm by 22cm]

9. When the square of a natural number is increased by 3 the result is 52. Determine the number. [Answer: 7]

10. Jonathan bought x shirts at $(x – 4) each. He spent $96 on shirts. Determine the number of shirts bought. [Answer: 12]

11. Henry bought 2x pens at $(x – 5) each. He sold half of them at $(x – 3) each and the other half at $4 each. If he made a profit of $24, determine the number of pens he bought. . [Answer: 16]

12. The area of a rectangular floor is 72 m2. If the width is half the length, determine the dimensions of the floor. [Answer: 12 m, 6 m]

13. The area of a rectangle is 96 cm2. If the length exceeds its width by 4 cm, determine the dimensions of the rectangle. [Answer: 8 cm, 12 cm]

14. The total area of the walls of a rectangular room is 624 m2. If the room is 10 m longer than its width and its height is 6 m less than its length, determine the dimensions of the room. [Answer: 18m by 8m by 12m]

15. Two squares have a total area of 149 cm2. If the side of one square is 3 cm shorter than the other, determine the dimensions of each square.[Answer: 10cm by 10cm, 7cm by 7cm]

16. The positive square root of a number is less than the number by 6. Determine the number. [Answer: 9]

17. If the product of two consecutive numbers is 42, find the numbers.[Answer: 6, 7]

18. If the product of two consecutive even numbers is 80, find the numbers.[Answer: 8, 10]

19. Two sisters are 9 and 5 years old. In how many years’ time will the product of their ages be 221? [Answer: 8]

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20. The sum of the squares of three consecutive numbers is 77. Form an equation for x and, by solving it, find the three numbers. [Answer: 4, 5, 6 (or – 6, - 5, - 4)]

21. Mrs. Jones is now five times as old as her son Tony. Four years ago, the product of their ages was 52. Find their ages now. [Answer: 30 and 6 years]

22. If a reduction of $2 a dozen in the price of eggs will give 6 more for $42, find the price per dozen. [Answer: $14]

23. A man bought x oxen for $120. Another bought 3 more for the same money. If the difference was $2 per ox, what were the numbers bought? [Answer: 12, 15]

24. A bill of $80 was shared equally between x persons. If two were excused and this made a difference of $2 to each, what was x? [Answer: 10]

25. A man spends $5 on bars of chocolate. Had they been 2½¢ cheaper, he could have bought 10 more. How many did he buy? [Answer: 40]

26. A candy company makes the popular rectangular-shaped Dandy Bar with 10 cm long, 5 cm wide, and 2 cm thick. Because of increasing costs, the company has decided to cut the volume of the bar by a drastic 28%. The thickness will be the same, but the length and width will be reduced by equal amounts. What will be the length and width of the new bar? [Answer: 9cm by 4cm]

27. Toni must spend $2.88 on vegetable oil per month. If the price of vegetable oil increases by 6¢ per bottle, she buys 4 bottles less, but she spends $2.88 monthly non-the-less. How many bottles does she consume at the original price?[Answer 16]

28. The cost of hiring a minibus for a party of children to have a day’s outing is $11.20. If two more children had been able to go, each child would have had to pay 10¢ less. How many children are there in the party and how much would each child pay? [Answer: 14, 80¢]

29. A man allowed $120 for his holiday. He afterwards calculated that if he had spent $1 less a day, he could have extended his holiday by 4 days on the same money. How long a holiday did he have? [Answer: 20 days]

30. When the price of sugar is raised by 10¢ per kg, the weight of sugar that can be bought for $30 is decreased by 10kg. Find the original price of the sugar per kg.[Answer: 50¢]

31. A farmer bought some pigs for $1800 at a cost of $x each. He also bought a herd of cows for $4500 at a cost of $(2x – 50) each. If he bought 30 animals altogether, find the cost of each kind of animal. [Answer: $150, $250]

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32. Oyo Transport Limited spends a total of $45 per month on petrol but since the time its price rose by 2.5¢ per litre, the company consumed 10 litres less per month for the same total distance. Find the present monthly consumption rate.[Answer: 139.3 litres, 32.3¢]

33. When the price of a product is $p each, a manufacturer will supply 3p2 - 4p units of the product to the market and consumers will demand 24 – p2 units. Find the value of p at market equilibrium.[Answer: $3]

34. For security reasons, a company will enclose a rectangular area of 11,200 ft2 in the rear of its plant. One side will be bounded by the building and the other three sides by fencing. If 300 ft of fencing will be used, what will be the dimensions of the rectangular area?[Answer: 80ft by 140ft]

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3.0 NUMBER SYSTEMS

1. Convert the following denary numbers to binary numbers:(a) 25 (b) 18 (c) 21 (d) 67[Answer: (a) 11001; (b) 10010; (c) 10101; (d) 1000011]

2. Convert 645ten into a number in base six.[Answer: 2553]

3. Convert: (i) 132four (ii) 100110two (iii) 11001two

(iv) 10010two (v) 10101two

to base ten.[Answer: (i) 30; (ii) 38; (iii) 25; (iv) 18; (v) 21]

4. Find positive values for x if:(a) 34x = 19ten (b) 45x = 29 ten

(c) 62x = 50ten, (d) 115x = 47ten

(e) 110x = 1020four (f) 126x = 86ten

(g) 142x = 47ten (h) 23x = 1111two

(i) 132five = Xsix (j) 244x = 1022four

[Answer: (a) 5; (b) 6; (c) 8; (d) 6; (e) 8; (f) 8; (g) 5; (h) 6; (i) 110; (j) 5]

5. Do the following addition if all numbers are in the binary system:(a) 110110 + 1011101 + 11111(b) 11101 + 10111 + 111001(c) 110101 + 1101 + 101111[Answer: (a) 10110010; (b) 1101101; (c) 1110001]

6. Evaluate:(a) 1011 + 1101 – 110, where all the numbers are in the binary system;(b) 10 + 13 + 17 + 21, where all the numbers are in the octal scale;

giving your answer in the same system.[Answer: (a) 10010; (b) 63]

7. Find the number bases x and y from the following simultaneous equations:

(a) (b)

(c)

[Answer: (a) x = 6, y = 6; (b) x = 6, y = 4; (c) x = 6, y = 4]

8. Add 25103six and 55504six together. [Answer: 125011six]

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9. Find the value of:(a) 1101two – 101111two + 110101two

(b) 1011101two + 11111two – 110110two

(c) [{(1101two – 101two) – (101two – 11two)} – 11two][Answer: (a) 10011two; (b) 1000110two; (c) 11two]

10. Calculate the value of x if:(a) 52x – 24x = 25x;(b) 61x – 43x = 16x

[Answer: (a) 7; (b) 8]

11. Solve the following equations, if all the numbers are in binary:(a) (b)

(c) (d)

[Answer: (a) x = 2 or 1; (b) x = 11 or 1; (c) x = 11 or 10; (d) x = 11]

12. Find the range of values of x for which:(a) , where all numbers are in base 4;(b) , where all numbers are in base 8[Answer: (a) ; (b) ]

13. State the value of m:

(a) If in base 2;

(b) If in base seven

[Answer: (a) m = 1; (b) m = 24seven]

14. Simplify: 212three + 110111two and express your answer in base eight.[Answer: 116eight]

15. (a) If all the numbers are in base three, solve the equation:121x + 11 = 1100

(b) Solve for x: 111x + 1011 = 101110 if all numbers are in binary scale. [Answer: (a) 2three; (b) 101two]

16. Multiply:(i) 101two by 11two (ii) 1011two by 111two

(iii) 256eight by 63eight (iv) 152six by 103six

(v) 43five by 32five (vi) 152six by 412six

(vii) 384nine by 615nine (viii) 246seven by 315seven

[Answer: (i) 1111two; (ii) 1001101 two; (iii) 21252eight; (iv) 20140six; (v) 3031five; (vi) 115504six; (vii) 262712nine; (viii) 115122seven]

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17. Divide:(i) 1010two by 101two (ii) 11011101two by 1101two

(iii) 4611seven by 24seven (iv) 22021four by 23four

(v) 313123five by 324five (vi) 33166nine by 384nine

[Answer: (i) 10 two; (ii) 10001 two; (iii) 162seven; (iv) 323four; (v) 432five; (vi) 76nine]

18. A man said that he had 15eight children. A computer which works on a binary system interpreted this information. What is the interpretation?

[Answer: 1101]

19. In Quinto land, where all numbers are in base five and all calculations are performed in that base, Dr. Cooley died leaving a sum of $131400 for his wife, 2 daughters and 13 sons. The wife’s share was $4300, each daughter received $3200 and the rest was shared equally among the sons. How much did each son receive?[Answer: $3400]

20. In Eleme land, where all numbers are in base five and all calculations are performed in that base, Dr. Nwafor-Ewee died leaving a sum of $210130 for his wife, 3 daughters and 14 sons. The wife’s share was $4430, each daughter received $3410 and the rest was shared equally among the sons. How much did each son receive?[Answer: $4130]

21. In a country where all numbers are in base six and all calculations are performed in that base, Dr. Allotments died leaving a sum of $40105 for his wife, 2 daughters and 12 sons. The wife’s share was $2355, each daughter received $1545 and the rest was shared equally among the sons. How much did each son receive?[Answer: $2111]

22. In a certain country, where the number system is base 5, the price of soap is increased by 40%. Later the new price is decreased by 40%. What percent of the original price is the final price?[Answer: 341five %]

23. The currency in Kalabari Kingdom is called Igbiki. The numeration system is in base eight. If 56 shirts cost 16 Igbiki each and they were sold for a total gain of 134 Igbiki, for how much was each shirt sold?[Answer: 20eight Igbiki]

24. The currency in a certain country is called Wados. The numeration in the country is in base six. If 35 shirts cost 11 Wados each and they were sold for a total gain of 114 Wados, for how much was each shirt sold?[Answer: 13six Wados]

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25. In Saraveca, South America, the quinary scale is used. The weights of four Saravecans are: 22, 24, 32, and x units respectively. If their average weight in the quinary scale is 32, calculate the value of x.

[Answer: 100]

26. Convert:(a) 17ten

(b) 2376eight to binary[Answer: (a) 10001, (b) 10011111110

27. Solve the simultaneous equations if all numbers are binary:

[Answer: x = 100, y = 11]

28. A number is written as 52x. Four times the number is written as 301x. What is x?[Answer: 7]

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4.0 SERIES & PROGRESSIONS

4.1 ARITHMETIC PROGRESSION

1. Find the 8th and 17th of the series 12, 15, 18, 21…….[Answer: 33, 60]

2. Which term of the series –40, -38, -36, … is equal to 4?[Answer: 23rd]

3. The 3rd term of an A.P. is –8 and 16th is 57. Find(a) the 10th term,(b) the sum of the first 14 terms.[Answer: (a) 27 (b) 203]

4. How many terms in the series 5.6, 4.1, 2.6, … must be taken so that their sum is –3.6?[Answer: 9]

5. In boring a well 300 m deep, the cost of boring the first meter is $200,000 whilst each subsequent metre costs $250. What is the cost of boring the entire well?[Answer: $274750]

6. Find the number of terms and the sum of the following finite arithmetic progressions:(a) 5.5, 6.25, …, 13.75 (c) 4/3, 2,…, (b) 14.2, 12.9, …, -14.4 (d) 17, 16⅜,…., -18(e) 11, 18, 25, …, 109[Answer: (a) 12, 115.5; (b) 23, -2.3; (c) 14, 29.6; (d) 57, -28.5; (e)15, 900]

7. Find the sum of the indicated number of terms of the following arithmetic progressions:(a) 5, 8, 11, … 20 terms (c) 11, 17, 23, … 17 terms(b) 15, 9, 3, … 10 terms (d) 18, 14, 10, … 8 terms(e) 6, 14, 22, … 15 terms[Answer: (a) 670; (b) –120; (c) 1003; (d) 32; (e) 930]

8. Show that the sum of the odd number from 1 to 125 inclusive is equal to the sum of the odd number from 169 to 209 inclusive.[Answer: 3969]

9. How many terms of the A.P. 1 + 3 + 5 … are needed to give a sum greater than 500?[Answer: 23]

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10. An arithmetic progression contains 20 terms. Given that the 8th term is 25, and that the sum of the last 8 terms is 404, calculate the sum of the first 8 terms.[Answer: 116]

11. The number of terms in an arithmetic progression is 40 and the last term is – 54. Given that the sum of the first 15 terms added to the sum of the first 30 terms is zero, calculate:(a) the first term and the common difference.(b) the sum of the progression.[Answer: (a) a=24, d=-2; (b) -600]

12. An arithmetic progression is such that the sum of the first 10 terms is 3 times the sum of the first 5 terms. Find the ratio of the 10th term to the 5th term. Given further that the 5th term is 0.14, calculate the sum of the first 200 terms.[Answer: or 12/7, 410]

13. In a new production process, the number of units produced during the first week is 450. If the efficiency gradually increases so that ten additional units can be produced each week, find: (a) how many will be produced in the tenth week;(b) the expected total production after ten weeks.[Answer: (a) 540 units, (b) 4950 units]

14. Justin Time Ltd has won a contract to supply 109,200 units of a product to a customer. The customer wants final delivery within 12 weeks. The production planning department of Justin Time Ltd has estimated that production in the first week will be 8,000 units, but that by recruiting and training some extra staff, output will rise by 200 units each week.(a) Will Justin Time Ltd be able to complete the contract by the time required

by the customer?(b) If the recruitment of staff does not go as well as expected so that after six

weeks, weekly production remains constant at the level achieved in the sixth week, how long would it take to complete the contract?

[Answer: (a) 12 weeks, (b) 12.47 weeks]

15. Pete intends to increase the distance he runs each week by two miles. If he now runs 15 miles per week, in how many weeks will he be running 63 miles per weeks?[Answer: 25 weeks]

16. Mary starts a diet that will cause her to lose 1½ lb per week. If she now weighs 168 lb, how many weeks will it take her to meet her goal of 135 lb?[Answer: 23 weeks]

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17. Joanna and Phil are hired by the Ray Taco Diner. Joanna is hired with the understanding she will get $220 per week with an increase of $5 per week every week. Phil’s salary will be $220 per week with a raise of $20 per week at the end of every four weeks. After 12 weeks who has earned more money? How much more? [Answer: Joanna, $90]

18. Ms. Bond and Mr. Stox are hired by a brokerage firm. Both start at $300 per week. Ms. Bond is to receive a weekly raise of $10. Mr. Stox’ weekly salary will be increased by $40 at the end of every four weeks. After 52 weeks, whose total income is the larger? How much larger?[Answer: Ms. Bond, $10,140]

19. Willy is stacking blocks. How many blocks are there in his stack if there are 18 blocks in the bottom row, 17 in the second row, 16 in the third row, and so on, until there is 1 block in the top row?[Answer: 171 blocks]

20. At a club meeting with 300 people present, everyone shook hands with every other person exactly once. How many handshakes were there? Hint: Person A shook hands with how many people, person B shook hands with how many people not already counted, and so on.[Answer: 44,850]

21. Mary learned 20 new French words on January 1, 24 new French words on January 2, 28 new French words on January 3, and so on, through January 31. By how much did she increase her French vocabulary during January?[Answer: 2480]

22. A pile of logs has 70 logs in the bottom layer, 69 logs in the second layer, and so on to the top layer with 10 logs. How many logs are in the pile?[Answer: 2440]

23. If Ronnie is paid $10 on January 1, $20 on January 2, $30 on January 3 and so on, how much does he earn during January?[Answer: $4960]

24. A contractor employs 150 men on a certain Monday and added 12 men to his staff every successive working day. He continued to do so for 8 weeks, Saturday counting as a full working day and no work being done on Sundays.(a) How many employees had he at the end of the 8th week?(b) If the wages of each man were $3 per day, what was the total wage bill for

the 8 weeks?[Answer: (a) 714; (b) $62,208]

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25. A contractor undertakes to bore a well at $1 for the first 5 m, $1.25 for the next 5m, $1.50 for the next 5 m, and so on.(i) What is the cost of sinking a well 200m deep?(ii) What is the depth of a well which cost $100 to bore?[Answer: (i) $235; (ii) 125m]

26. On the ground are placed 20 stones in line at intervals of 5m. How far will a person travel who, starting from the position of the first stone, brings them one by one to a basket placed at the first stone?[Answer: 1900 m]

27. An arithmetic progression is such that the sum of the first 8 terms is 156, and the sum of the first 10 terms is 215. Find the fifth term.[Answer: 20.5]

28. A 62 m length of rope is cut into pieces whose lengths are in arithmetic progression with a common difference of d m. Given that the lengths of the shortest and longest pieces are 0.5 m and 3.5 m respectively, find the number of pieces and the value of d.[Answer: 31, 0.1]

29. A man saves $60 of his salary in a certain year and in each succeeding year saves $20 more than the previous year. How much does he save in 20 years?[Answer: $5000]

30. A lorry delivers gravel from a base on a road to dumps on the road which are 20 yards apart. The first dump is 40 yards from the base and the second 60 yards from base. The lorry starts from the base, delivers the load to the first dump, returns to base, delivers the second load to the second dump and so on. Find how far the lorry has traveled when it has just returned to base after delivering the 20th load.[Answer: 9200 yards]

31. A salesgirl is able to save $50 of her salary in a certain year and afterwards saves in any one year $20 more than she saved in the preceding year. How long does it take her to save $4370?[Answer: 19years]

32. Find the:(a) 10th term of: 3 + 11 + …(b) 23rd term of the A. P.: -7, -3, +1, …(c) the number of terms in the following A.P.:

2 + 4 + 6 + …… + 46(d) value of ‘n’ given that 77 is the nth terms of the A.P.:

3½, 7, 10½, …[Answer: (a) 75; (b) 81; (c) 23; (d) 22]

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33. Find the sum of the 12 terms of:(a) 4 + 10 + 16 + …(b) 7 + 11 + 15 + …[Answer: (a) 444; (b) 348]

34. (a) Find the sum of: 1 + 3 + 5 + ….. + 101(b) How many terms of the series: 15 + 13 + 11 + … make 55?[(Answer: (a) 2601; (b) 5 or 11]

35. In an A.P. the sum of the first 8th terms is 164 and the sum of the next 6 terms is 333. Find the first term, the common difference and hence the A.P.[Answer: 3; 5; A.P = 3, 8, 13, 18, 23, …]

36. A clerk starting with a salary of $100, has a salary of $105 in his second year, $110 in his third year, $115 in his fourth year, and so on. Find his salary in his twenty-first year of service.[Answer: $200]

37. A workman is to be paid $1.20 for his 1st day’s work, $1.30 for his 2nd day, $1.40 for his 3rd day, and so on. Find how much more he earns in the 2nd week than in the 1st, if he works 6 days in the week.[Answer: $3.60]

38. Prove that log x, log (xy), log (xy2), …, is an arithmetic progression and show, without using tables, that when x = 160 and y = ½, the sum of the first nine terms of the progression is 9.

39. The 5th term of an A.P. is -5, and the 11th term is -23, find the 30th term and the sum of 30 terms.[Answer: -80, -1095]

40. (a) The 4th term of an A.P. is 18 and the 7th term is 30. Final the 10th term and the sum of 20 terms.

(b) Find the value of n given that -49 is the nth term of the A.P: 11, 8, 5 …[Answer: (a) 42, 880; (b) 21]

41. (a) If -5, p, q, 16 are in A.P., find p and q(b) Find the sum of 18 terms of the series: 2½ + 3¾ + 5 + …[Answer: (a) 2, 9; (b) 236¼]

42. The sum of the first eight terms of an arithmetic progression is 60 and the sum of the next six terms (from the ninth to the fourteenth) is 108. Find the first termand the common difference. [Answer: 2¼. 1½]

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43. A man receives $60 the 1st week and $3 more each week than the preceding. What does he get in 20 weeks? [Answer: $1770]

44. The 2nd term of an A.P. is 15 and the 5th term is 21. Find the common difference, the first term and the sum of the first ten terms.[Answer: 2; 13; 220]

45. A straight path is made by laying paving slabs end to end. The slabs are laid in order of size and their lengths are in A.P. Given that the shortest slabs is 0.79 meters long, that the longest slab is 1.24 meters long and that the difference in the lengths of adjoining slabs is 0.025 meters, calculate:(i) the number of slabs used;(ii) the length of the path.

The costs of the slabs are also in A. P. and vary from $2.16 for the smallest to $3.24 for the largest. Find the total cost of the slabs in the path.[Answer: (i) 19; (ii) 19.284 meters; total cost = $51.30]

46. A long–distance runner runs the first kilometer of a race in 3 minutes 45 seconds but finds that his speed drops steadily so that each kilometer takes him 12 seconds more than the preceding one. Find:(i) his time for the 10th kilometer;(ii) the time taken to cover the first 12 kilometers.[Answer: (i) 5.55 minutes or 5 minutes 33 seconds; (ii) 58.2 minutes]

47. Find the acute angle θ for which, sinθ, 2cosθ and 2sinθ respectively, are the first three terms of an A.P.;

Find the sum of the first 20 terms of this progression.[Answer: 53.10; 92]

48. An A.P. is such that the fifth term is 22 and the sum of the first eight terms is 166. Find the first term.[Answer: 12]

49. The number of terms in an arithmetic progression is 40 and the last term is – 45. Given that the sum of the first 15 terms added to the sum of the first 30 terms is zero, calculate the first term and the common difference.[Answer: a = 20, d = – 5/3]

50. An arithmetic progression is such that the 10th term is 40 and the sum of the first ten terms is 265. Find the sum of the first 20 terms.[Answer: 830]

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51. The sixth term of an Arithmetic Progression is twice the third term. If the first term is 3, find the common difference and the tenth term.[Answer: 3, 30]

52. Find the sum of the even numbers, divisible by 3, lying between 400 and 500.[Answer: 7650]

4.2 GEOMETRIC PROGRESSION

1. Find the seventh term of the series 2, 6, 18,…[Answer: 1458]

2. Write the first four terms of the geometric progression, and find the indicated term:(i) a = 60, r = .1. Find 8th term (iii) a = 30, r = ⅓. Find 7th term.(ii) a = 1, r = 3. Find 6th term (iv) a = 4, r = ½. Find 7th term.[Answer: (i) 60, 6, 0.6, 0.06; 0.000006 (ii) 1, 3, 9, 27; 243

(iii) 30, 10, 10/3, 10/9; 10/243 (iv) 4, 2, 1, ½; 1/16]

3. Find the number of terms and the sum of the following finite geometric progressions.(a) 1/6, ⅓,… 42⅔ (c) 6, 12,……..,1536(b) 80, 20, …, 5/64 (d) 1/9, ⅓, …, 243[Answer: (a) 9 terms, 511/6 (b) 6 terms, 6825/64

(c) 9, 3066 (d) 8, 3280/9]

4. Find the sum of the indicated number of terms of the following geometric progressions:(a) 10, 20, 40, ... 11 terms (b) 2, -6, 18, ... 6 terms[Answer: (a) 20470 (b) – 364]

5. Find the sum(i) of the even numbers from 50 to 100 inclusive,(ii) of the first ten terms of the geometric series which has a first term of 8 and

a common ratio of 1-5,(iii) to infinity of the geometric series whose second term is ½ and third term is

⅓[Answer: (i) 1950 (ii) 907 (iii) 2.25]

6. An arithmetic progression has a first term of 8 and a common difference of d. Given that the 1st, 5th, and 21st terms of this progression are three successive terms of a geometric progression, find the value of d.[Answer: 6]

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7. The sum of an infinite geometric progression is 500. Given that the common ratio is 0.8, calculate(i) the first term(ii) the twentieth term.[Answer: (i) 100 (ii) 100(0.8)19 or 1.44]

8. The fourth term of a geometric progression is 6 and the seventh term is –48. Calculate (i) the common ratio(ii) the first term(iii) the sum of the first eleven terms.[Answer: (i) – 2 (ii) – ¾, (iii) – 512.25]

9. The first term of a geometric progression is a, and the common ratio is r. Given that a = 12r and that the sum to infinity is 4 calculate the third term.[Answer: 3/16]

10. The second and fourth terms of a geometric progression are 20 and 12.8 respectively. Given that all terms are positive find(i) the sum, to the nearest whole number, of the first 8 terms,(ii) the sum to infinity.[Answer: (i) 104 (ii) 125]

11. What is the sum to infinity of a geometric progression whose first term is 36, and in which the value of each successive term is:(a) one tenth of the previous term?(b) one quarter of the previous term?[Answer: (a) 40 (b) 48]

12. A man deposits £150 annually to accumulate at 5 per cent compound interest. How much money will he have to his credit immediately after making the tenth deposit?[Answer: £1886.68]

13. A man’s salary in 1980 was $500 per annum and it increased by 10% each year. Find how much he earned in the year 1980 to 1989 inclusive.[Answer: $7968.71]

14. The first and fourth terms of a geometric progression are 500 and 32 respectively. Find:(i) the values of the second and third terms,(ii) the sum to infinity of the progression.[Answer: (i) 200, 80 (ii) 833⅓]

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15. In the first week of production 1000 articles are made. If the rise in weekly production is 5% per week, how many weeks are necessary to produce a total of 15000 articles?[Answer: 11.47 weeks]

16. A man pays a premium of £30 per annum for 18 years. How much should he receive as a lump sum just after paying the 18th premium reckoning compound interest at 5 percent per annum?[Answer: £843.97]

17. Find the 8th term of the series: 1, 1.1, 1.21, …[Answer: 1.949]

18. Find the 8th term of the series which is a G.P. having a 2nd term of – 3 and a 5th term of 81.[Answer: -2187]

19. Find the sum of the series 60, 30, 15,… (a) to 6 terms (b) to infinity.[Answer: (a) 118.1 (b) 120]

20. The sum of the first 6 terms of a G.P. series is 189 and the common ratio is 2. Find the series.[Answer: 3, 6, 12, 24, 48, 96]

21. A firm starts work with 110 employees for the 1st week. The number of employees rises by 6% per week.(a) How many weeks will it take for 250 persons to be employed?(b) How many persons will be employed in the 20th week if the present rate of

expansion continues?[Answer: (a) 15.1 (b) 333]

22. Car production in the first week of a new model was 150. If it continues with a fall per week of 2%,(a) find the total output in 52 weeks from the commencement of production in

the new model.(b) find how many cars will be produced in the next 52 weeks.[Answer: (a) 4877 (b) 1706]

23. A contractor hires out machinery. In the first year of hiring out one piece of equipment the profit is £6000, but this diminishes by 5% on successive years. Show that the annual profits form a G.P. and find the total of all profits for:(a) the first 6 years,(b) all possible future years.[Answer: (a) £31,789 (b) £120,000]

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24. A man saves £100 each year, and invests it at the end of the year at 4 per cent compound interest. How much will the combined savings and interest amount to at the end of 15 years?[Answer: £1997]

25. Sam Slick’s brother Joe triples his bets to cover losses. How much will Joe need to bet to cover eight straight losses if his initial bet was $10?[Answer: $21,870]

26. A certain casino sets its slot machines so the average gambler loses 20% of his investment per hour. If a gambler starts out with $100, how much would he have left after five hours? [Answer: $32.77]

27. A sales manager has reported that sales in the next year will amount to 5,000 units of product. He also estimates that sales will then increase in volume by 40% a year until year 6. What sales volume would be expected in year 6?[Answer: 26,891 units]

28. A certain culture of bacteria doubles every week. If there are 100 bacteria now, how many will there be after 10 full weeks?[Answer: 102,400]

29. A water lily grows so rapidly that each day it covers twice the area it covers the day before. At the end of 20 days, it completely covers a pond. If we start with two lilies, how long will it take to cover the same pond?[Answer: 2,097,150]

30. Johnny is paid $1 on January 1, $2 on January 2, $4 on January 3, and so on. Approximately how much will he earn during January?[Answer: Over $2 billion]

31. If you were offered 1¢ today, 2¢ tomorrow, 4¢ the third day, and so on for 20 days or a lump sum of $10,000, which one would you choose? Show why?[Answer: 1¢ today, 2¢ tomorrow because for 20 days it will yield $10,485.7 which is more than $10,000]

32. Suppose Karen puts $100 in the bank today and $100 at the beginning of each of the following 9 years. If this money earns interest at 8 percent compounded annually, what will it be worth at the end of 10 years.[Answer: $1564.55]

33. Jose makes 40 deposits of $25 each in a bank at intervals of three months, making the first deposit today. If money earns interest at 8 percent compounded quarterly, what will it all be worth at the end of 10 years?[Answer: $1510.05]

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34. From a sufficiently long piece of string, ten portions are cut off successively, their lengths forming a geometrical progression. The first and second portions are respectively 10 m and 9½ m in length. What is the total length cut off to the nearest metre? If the process is continued as long as the portion cut is not less than ⅓ m how many portions are obtained?[Answer: 80m, 67 portions]

35. £500 is invested at 5 per cent. Compound interest. After how many years will the amount first exceed £750?[Answer: 9 years]

36. If 1, 2 and 21 are added to the first, second and third terms respectively, of an arithmetic progression with a difference d = 5, a geometric progression is obtained. Find the arithmetic progression and the common ratio of the geometric progression.[Answer: AP = 1, 6, 11, 16; GP = 2, 8, 32, r = 4]

37. A man put $10 in the bank for his son on each of his birthdays from the first to the Twentieth inclusive. If the money accumulates at 3% compound interest, what is the total value on the son’s twenty-first birthday?[Answer: $276]

38. A firm’s labour forces are growing at the rate of 2% per annum. The firm now employs 500 people. How many may it be expected to employ in five year’s time?[(Answer: 552]

39. The first term of a GP is a, and the common ratio, r, is positive. Given that the

sum of the second and third terms is , calculate the value of r.

Given also that the sum of the first four terms is 65, calculate the value of a and hence, calculate the sum to infinity of the GP.[Answer: ⅔, 27, 81]

40. The lengths of the sides of a triangle are in GP and the longest side has a length of 36cm. Given that the perimeter is 76cm, find the length of the shortest side.[Answer: 16]

41. The first three terms, all positive, of a certain GP are (x – 4), x, (5x – 12). Find:(i) the value of x;(ii) the value of the fourth term.[Answer: (i) 6 (ii) 54]

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42. If the production of a factory was 5000 tons of fertilizer in 1990 and it has increased by 10% per annum each year since then, find the total output for the years 1990 to 1999 inclusive.[Answer: 79687.12]

43. A precious metal is extracted from the mineral ore obtained from a particular mine. During the first year of operation, the ore obtained yields 8000kg of metal. With the increasing difficulty of mining, the production of metal in each subsequent year shows a decrease of 10% on the previous year’s production. Assuming that mining continues in the same way for an indefinite period of time, calculate M, the maximum amount of metal which could possibly be extracted.

For economic reasons, mining is abandoned once the annual output of metal falls below 1000kg. Calculate:(i) the number of years the mine is in operation;(ii) the total production of metal during this time;(iii) the percentage of M which was not extracted.[Answer: 80,000kg (i) 20 years (ii) 70,274kg (iii) 12.2%]

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5.0 SETS THEORY

1. Given that: , and , calculate: .[Answer: 5]

2. The Universal set

(i) Draw a Venn Diagram to represent the above information.(ii) List, using set notation, the members of the set .[Answer: (ii) {6, 8}]

3. If

find: (i) (ii) (iii) AC (iv) BC, where is the universal set, A and B are subsets of , and AC and BC are the complements of sets A and B respectively.

Hence, or otherwise, show that: .[Answer: (i) {2, 3, 4, 5, 6, 8} (ii) {2} (iii) {1, 3, 5, 7} (iv) {1, 4, 6, 7, 8}]

4. Prove the identity:

5. Prove that if A, B, C are sets in a universal set , then:

6. Let A and B be any two sets in a universal set . Explain in words the meaning of each of the following:(i) (ii) (iii)(iv) (v) (vi)

7. Prove that, if A, B and C are sets, then:(i)(ii)(iii)

8. A survey of 150 householders, all of whom kept a cat or a dog or both, showed that 85 kept cats and 70 kept dogs. How many kept both? [Answer: 5]

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9. In a school of 150 students, 85 take English and 115 take Mathematics. If each student takes at least one of these subjects, how many take both?[Answer: 50]

10. A newsagent sells three papers: Free Press, TC Weekly News and TC Sun, to 135 customers. Of these,

70 buy Free Press,60 buy TC Weekly News,50 buy TC Sun’17 buy Free Press and TC Weekly News,15 buy TC Weekly News and TC Sun, and16 buy TC Sun and Free Press.

If each customer buys at least one of the papers:(i) Find the number of customers who buy all three papers;(ii) Express these facts in a Venn diagram, showing clearly the number of

customers in each separate region.[Answer: 3]

11. Of 60 martial arts experts, 26 are karatekas, 23 are judokas and 16 are neither karatekas nor judokas. Estimate the number of martial arts experts who are:(i) both karatekas and judokas(ii) only karatekas(iii) only judokas. [Answer: (i) 5 (ii) 21 (iii) 18]

12. In a survey of 100 college students, 35 were registered in College Algebra, 52 were registered in Computer Science I, and 18 were registered in both courses.(i) How many students were registered in College Algebra or Computer

Science I?(ii) How many were registered in neither course?[Answer: (i) 69 (ii) 31]

13. In a certain country, there are three TV channels: TBN, BET and HBO. Fifty people were asked which channels they watched the previous evening. The results were as shown below:

24 watched TBN,11 watched BET,27 watched HBO, 7 watched both TBN and BET, 3 watched both BET and HBO,10 watched both HBO and TBN, 2 watched all three.

(a) Show the information in a Venn diagram;(b) How many watched:

(i) no TV? (ii) only BET?[Answer: (b) (i) 6 (ii) 3]

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14. 50 members of a certain club were asked which of the games: table tennis, draughts and chess they had played one evening.

29 had played table tennis,26 had played draughts, 9 had played chess,Everyone who played chess had also played draughts,No-one had played both table tennis and chess, 7 had played none of the three games.

Using a Venn diagram, find how many had played:(a) only table tennis,(b) only draughts. [Answer: (a) 17 (b) 5]

15. In a survey of 100 investors in the stock market,50 owned shares in IBM40 owned shares in AT&T45 owned shares in GE20 owned shares in both IBM and GE15 owned shares in both AT&T and GE20 owned shares in both IBM and AT&T5 owned shares in all three.

(i) How many owned just IBM shares?(ii) How many owned just GE shares?(iii) How many of the investors surveyed did not have shares in any of the

three companies?(iv) How many owned neither IBM nor GE?(v) How many owned either IBM or AT&T but no GE?[Answer: (i) 15 (ii) 15 (iii) 15 (iv) 25 (v) 40]

16. Out of three courses: Medicine, Engineering or Accountancy, 120 people were invited to state the one in which they were interested.

60 were interested in Medicine,86 were interested in Engineering,64 were interested in Studying Accountancy;32 were interested in Medicine and Engineering;38 were interested in Medicine and Accountancy; while34 were interested in Engineering and Accountancy.

How many were interested in all three courses?[Answer: 14]

17. Show, by using Venn Diagrams, that for finite sets A, B and C,

18. Every man in a certain club owns either a Mercedes or an Opel.

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23 own Mercedes,14 own Opels, and 5 own both.

How many men are there in the club?[Answer: 32]

19. Of 150 people questioned, 66 ate Rice, 57 ate Eba, and 58 ate Yam. If 13 ate Rice and Eba, and 10 ate Rice and Yam, and 11 ate Eba and Yam, how many ate all three?[Answer: 3]

20. Of pupils surveyed at the end of a term, 118 had applied for Florida Metropolitan University (FMU), 98 for University of Miami (UM), and 94 for Florida International University(FIU). 42 had applied for FMU and UM, 24 for FIU and UM and 34 for FIU and FMU, while 8 had applied for all three. How many pupils took part in the survey?[Answer: 218]

21. In a survey at an airport, 55 travelers said that last year they had been to Spain, 53 to France and 79 to Germany. 18 had been to Spain and France, 17 to Spain and Germany, and 25 to France and Germany, while 10 had been to all three countries. How many travelers took part in the survey?[Answer: 137]

22. Out of 136 companies in Providenciales, 67 manufacture Electrical appliances, 40 produce Domestic plants, and 56 produce Beverages. 11 manufacture both Electrical appliances and Domestic plants, and 12 produce Domestic plants and Beverages, while 9 manufacture Beverages and Electrical appliances. How many manufacture all the three products?[Answer: 5]

23. In a certain examination, 72 candidates offered Mathematics, 64 English and 62 French. 18 offered both Mathematics and English, 24 Mathematics and French, and 20 English and French. 8 candidates offered all three subjects. How many candidates were there for the examination?[Answer: 144]

24. In a certain school, there are 87 boys of a particular year. Of these, 43 play hockey, 42 play football and 47 play tennis; 15 play tennis and hockey, 17 tennis and football, and 21 hockey and football. If each boy plays at least one game, find the number of boys who play all three and express these facts in a Venn diagram, showing clearly the number of boys in each separate region.[Answer: 8]

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25. Of 60 girls’ readers of three magazines, 27 read WOMAN, 35 read JOY, and 30 read BEAUTY. 8 girls read all the three magazines. It was found that equal numbers read WOMAN and JOY, and WOMAN and BEAUTY. If four more girls than those who read WOMAN and JOY read JOY and BEAUTY, how many read WOMAN only? [Answer: 11]

26. In a random sample of 1000 men, the following facts emerged:271 of the men were bald;248 of the men wore spectacles;251 of the men had false teeth; 64 men were both bald and wore spectacles; 97 men were both bald and had false teeth; 59 wore spectacles and had false teeth;434 men were not bald, wore no spectacles and had all their own teeth.

How many were bald, wore spectacles and had false teeth? [Answer: 16]

27. A survey of 100 students gave the following data:41 students taking Spanish,29 students taking French,26 students taking Russian,15 students taking Spanish and French, 8 students taking French and Russian,19 students taking Spanish and Russian, and 5 students taking all the three languages.

Find the number of students among the 100 that are:(i) not taking any of the three languages,(ii) taking just one of the languages. [Answer: (i) 41 (ii) 27]

28. A survey of 300 investors showed the following information concerning the desirability of buying shares in one or more three Banks – ACB, PAB and NNB:

28 advised buying ACB and PAB, 98 advised buying ACB or PAB but not NNB, 42 advised buying PAB but not ACB or NNB,122 advised buying PAB or NNB but not ACB, 64 advised buying NNB but not ACB or PAB, 14 advised buying ACB and NNB but not PAB,100 advised buying none.

How many investors advised:(a) buying shares in all three Banks?(b) Buying ACB and PAB but not NNB?(c) Buying shares in only one bank?(d) Buying shares in ACB irrespective of PAB or NNB?(e) Buying shares in PAB irrespective of ACB or NNB?(f) Buying shares in NNB irrespective of ACB or PAB?[Answer: (a) 8 (b) 20 (c) 142 (d) 78 (e) 86 (f) 102]

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29. A sample poll of 1000 voters revealed the following information concerning three candidates – X, Y and Z of a certain political party vying for three different offices:

140 in favour of X and Y,610 in favour of Y or Z, but not X,460 in favour of X or Y, but not Z,320 in favour of Z but not X or Y,210 in favour of Y but not X or Z, 70 in favour of X and Z, but not Y.

How many of the voters were in favour of:(a) all the three candidates?(b) X and Y but not Z?(c) only one of the candidates?(d) X irrespective of Y or Z?(e) Y irrespective of X or Z?(f) Z irrespective of X or Y?[Answer: (a) 70 (b) 70 (c) 710 (d) 390 (e) 430 (f) 540]

30. A survey of a small town of 20,000 house-holds provided the following data:

60% of the households have a refrigerator,35% of the households have a telephone,15% of the households have a Colour TV set,18% of the households have both a refrigerator and a telephone,12% of the households have both a telephone and a Colour TV set,10% of the households have both a refrigerator and a Colour TV set, 8% of the households have a refrigerator, a telephone, & a Colour TV set

How many of these households have:(a) only a telephone?(b) only a refrigerator?(c) Only a Colour TV set?(d) Neither a telephone, nor a refrigerator, nor a Colour TV set?[Answer: (a) 2600 (b) 8000 (c) 200 (d) 4400]

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2.4 – Simultaneous Linear Equations: Three unknowns

1. Solve the 3x3 system of equations:(a) 3x – y + z = 5 (b) 4x – 3y + 2z = -7

2x + 2y + 3z = 2 2x + 2y – 3z = 27x + 3y - z = 11 2x – 5y – 4z = 11

(c) 2x + y + z = 3, (d) 2x + y + 6z = 3,-x + 2y + 2z = 1, x – y + 4z = 1x - y – 3z = -6. 3x + y – 2z = 2

(e) 5x – 7y + 4z = 2, (f) x + y + z = 63x + 2y – 2z = 3, 2x + 3y + z = 112x – y + 3z = 4 3x + 2y + 2x = 13

[Answer: (a) x = 3, y = 2, z = - 2, (c) x = 1, y = - 2, z = 3, (d) x = ½, y = ½, z = ¼, (e) x = 1, y = 1, z = 1]

2. Solve the following systems of liner equations for x, y and z.

(a) 3x + 2y – z = -1 (b) 3x + y – z = 10 (c) y:z = 2:1x + y + z = 6 x + 2y + z = 7 10x + y = 03x + y + 2z = 15 x – z = 3 5x + y + 2z = 15

[Answer: (a) x = 2, y = - 1, z = 5, (b) x = 2, y = 3, z = - 1, (c) x= - 1, y = 10, z = 5]

3. A company makes three types of patio furniture: chairs, rockets, and chaise lounges. Each requires wood, plastic, and aluminum, in the amounts shown in the following table. The company has in stock 400 units of wood, 600 units of plastic and 1500 units of aluminum. For its end-of-the-season production run, the company wants to use up all the stock. To do this, how many chairs, rockets, and chaise lounges should it make?

WOOD PLASTIC ALUMINUMCHAIR 1 unit 1 unit 2 unitsROCKER 1 unit 1 unit 3 unitsCHAISE LOUNGE 1 unit 2 units 5 units

[Answer: 100C, 100R, 200CL]

4. A father left $2000 to be shared among his three sons. Alfred and Ben received between them $200 more than Charles: Alfred got $200 more than Ben.(a) Write down three different equations representing the statement above.

(b) Calculate how much each of the sons got.

[Answer: x = 650, y = 450, z = 900]

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5. A father left $800 to be shared amongst his three sons. Alfred and Ben received between them $543: and Ben and Charles got between them $498.(a) Write down three different equations representing the statement above.(b) Calculate how much each of the sons got.[Answer: x = 302, y = 241, z = 257]

6. A company manufactures three products X, Y and Z, each of which must got through three processes A, B and c for the following times:

Product Time spent in processA B C

X 3 3 1Y 3 2 3Z 2 0 1

The maximum capacities of processes A, B and C are 130, 85 and 60 respectively. Calculate the number of units to be products X, Y and Z to ensure the utilization of maximum capacity.[Answer: 25x, 5y and 20z products]

2.5 – Simultaneous Linear Equations: One Linear, One Quadratic

1. Solve the following simultaneous equations:

(a) (b)

(c) (d)

(e) (f)

(g) (h)

(i) (j)

[Answer: (d) 2.5, 3; 3.5, 1 (e) -3, 1; 3.5, -4/9 (g) 3, 2; -5, -6 (h) 3, 5; -2, -10]

2. The square of a certain number exceeds twice the square of another number by 16. Find the numbers if the sum of their squares is 208.

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[Answer: (12, 8); (-12, -8); (12, -8); (-12, 8)]

3. The diagonal of a rectangle is 85cm. If the short side is increased by 11cm and the long side decreased by 7cm, the length of the diagonal remains the same. Find the dimensions of the original rectangle. [Answer: 40cm by 75cm]

4. The hypotenuse of a right triangle is 41cm long and the area of the triangle is 180cm2. Find the lengths of the two legs.[Answer: 9cm, 40cm]

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2.5 Nature of Roots, and Relationships between Roots and Coefficients, of a quadratic Equation

1. Distinguish between the natures of the following quadratic equations:(a) (b)(c) (d)(e) (f)(g)[Answer: (a) equal roots (b) no real roots (c) real and distinct roots (d) equal roots (e) imaginary roots (f) real roots (g) real roots]

2. Find the values of k for which each of the following equations has equal roots:(a) 3x2 + kx + 12 = 0 (b) x2 – 5x + k = 0(c) (k + 3)x2 – 12x + 2k = 0 (d) (k + 1)x2 + 4kx + 9 = 0(e)[Answer: (a) ±12 (b) 25/4 (e) 9]

3. Write down the sums and products of the roots of the following equations:(i) x2 – 3x + 2 = 0 (ii) 4x2 + 7x – 3 = 0(iii) x(x – 3) = x + 4 (iv) x2 – kx + k2 = 0

(v) (vi) ax2 – x(a + 2) – a = 0

[Answer:]

4. Write down the equations, the sum and product of whose roots are:(a) 3, 4 (b) – 2, ½ (c) ⅓, – 2/5 (d) – ¼, 0(e) a, a2 (f) – (k + 1), k2 – 3[Answer]

5. For what value of k are the roots of 3x2 + (k – 1)x – 2 = 0 equal and opposite?[Answer]

6. For what values of k is 9x2 + kx + 16 a perfect square?[Answer]

7. Prove that kx2 + 2x – (k – 2) = 0 has real roots for any value of k.[Answer]

8. Show that the roots of ax2 + (a + b)x + b = 0 are real for all values of a, and b.[Answer]

9. Find a relationship between p and q if the roots of px2 + qx + 1 = 0 are equal.[Answer: q2 = 4p]

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10. If x is real and , prove that p2 – 3(p + 3) ≥ 0.

[Answer]

11. The quadratic equation: ℓx2 + 5mx + 9ℓ = 0 has equal roots. Given that ℓ and m are positive, find the ratio: ℓ : m and solve the equation.[Answer]

12. In the equation: (p2 + 1)x2 + 2pqx + q2 = 0, p and q are real numbers. Show that the equation has no real roots for any values of p and q.[Answer]

13. Show that the equation x2 + (2 – k)x + k = 3 has real roots for all real values of k.[Answer]

14. Show that the equation: (p + 1)x2 + (2p + 3)x + (p + 2) = 0, has real roots for all real values of p.[Answer]

15. Find the range of values of a, for which (2 – 3a)x2 + (4 – a)x + 2 = 0 has no real roots.[Answer]

16. Calculate the smallest positive integer p for which the equation: 3x2 + px + 7 = 0 has real roots. [Answer]

17. Calculate the value of c for which the equation: 5x2 + 6x + c = 0 has equal roots.[Answer]

18. Show that the roots of the equation: 9x2 – 30x + 25 = 0 are equal.[Answer]

19. Given that a, b and c are real, prove that the roots of the equation:(x + a)(x + b) = c2 are real.[Answer]

20. State the condition for the equation: 3ax2 + 3bx + 2c = 0, a ≠ 0, to have real roots. Prove that if this condition is satisfied, then the roots of the equation:a2x2 + (3b2 – 4ac)x + 4c2 = 0 are real.[Answer]

21. Find the value of p if one root of is the square of the other.[Answer:]

22. The roots of the equation are . Find the value of:

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[Answer:]

23. If are the roots of , form the equation whose roots are:

[Answer:]

24. Calculate the discriminant of the equation: and state whether it has two distinct real roots, two identical roots, or no real roots.[Answer]

25. If one root of is treble the other, prove that:

[Answer]

26. Prove that the quadratic equation: has real roots for any value of k.

6.0 POLYNOMIALS

6. (a). Find the remainder when is divided by

(b). Given the (x – 2) is a factor of: ,

calculate the value of k.

(c) Use the Factor Theorem to find one of the factors of the cubic:

Hence, factorize the cubic into its linear factors

(b). The polynomial: has a factor of, and leaves a remainder of 3 when divided by .

(i) Determine the value of each of the constants, m and n.(ii) Find the exact value of the three roots of the equation p(x) = 0.

1. (a) The polynomial: is divisible by (x – 1) and leaves a remainder 12 when divided by (x – 2). Find the values of a and b.

(b) If (x – 1) is a factor of the polynomial: , where k is a constant, find:

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(i) the value of k,(ii) the remainder when P(x) is divided by (x – 3).

(c) Given that: ,(i) calculate the remainder when f(x) is divided by (x + 2),(ii) solve the equation f(x) = 0.

(d) Given that: , calculate:(i) the value of p for which f(x) is divisible by (x + 1),(ii) the value of p for which f(x) has a remainder of 31 when divide by

(x – p).

3. (a) The remainder when x3 – 3x2 – 2x + m is divided by x + 2 is twice the remainder when it is divided by x – 3. Find the value of m.

(b) Factorize completely the expression 4x3 – 13x – 6 and hence solve the equation:

(c) Find the value of p and of q for which x2 – 2x – 3 is a factor of

2x3 + px2 -12x +q.

3. (a). Resolve into partial fractions

(b). Express: in partial fractions.

(c). Resolve: into partial fractions.

7.0 BINOMIAL EXPANSION

1. Evaluate each of the following:

(i) (ii) (iii) (iv)

[Answer: (i) 120 (ii) 336 (iii) 7920 (iv) 2n(2n + 1)]

2. Expand:(i) (x + y)6 (ii) (2x + y)3 (iii) (x – y)6

(iv) (y + 3)4 (v) (x2 – y3)4

[Answer: (i) x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6

(ii) 8x3 + 12x2y + 6xy2 + y3

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(iii) x6 – 6x5y + 15x4y2 – 20x3y3 + 15x2y4 – 6xy5 + y6

(iv) y4 + 12y3 + 54y2 + 108y + 81(v) x8 – 4x6y3 + 6x4y6 – 4x2y9 + y12]

3. Find in ascending powers of x, the first four terms in the expansion of:(i) (1 – 3x)5

(ii) (1 + 5x)7

Hence, find the coefficient of x2 in the expansion (1 – 3x)5(1 + 5x)7

[Answer: (i) 1 – 15x + 90x2 – 270x3 (ii) 1 + 35x + 525x2 + 4375x3, 90]

4. Obtain the first four terms in the expansion of (1 + p)6 in ascending powers of p. By substituting p = x – x2 obtain the expansion of (1 + x – x2)6 as far as the term in x2. Find a value for x which would enable you to calculate (1.0099)6.[Answer: 1 + 6p + 15p2 + 20p3, 1 + 6x + 9x2, 0.0099]

5. Expand in ascending powers of x up to and including the term in , simplifying each coefficient in the expansion.Use your expansion to find an approximation to , stating clearly the substitution which you have used for x.[Answer: 1 – 20x + 180x2 – 960x3, 0.81704, x = 0.01]

6. Expand in ascending powers of x. Taking x = 0.1, and using

the first three terms of the expansion, find the value of , correct to three places of decimal.

[Answer: , 1.104]

7. Write down the expansion of in ascending powers of x. Taking the first three terms of the expansion, put x = 0.002, and find the values of as accurately as you can. Examine the fourth term of the expansion to find to how many places of decimal your answer is correct.[Answer: 64 – 192x + 240x2 – 160x3 + 60x4 – 12x5 + x6, 63.61696, 5 places]

8. Write down and simplify the expansion of (1 – p)5. Use this result to find the expansion of (1 – x – x2)5 in ascending powers of x as far as the term in x3. Find the value of x which would enable you to estimate (0.9899)5 from this expansion.[Answer: (1 – p5) + 10p2(1 – p) + 5p(p3 – 1), 1 – 5x + 5x2 + 10x3 + …, 0.0101]

9. Write down, and simplify, the first three terms of the expansion, in ascending powers of x, of:

(i) (ii) (2 – x)5

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Hence, or otherwise, obtain the coefficient of x2 in the expansion of

[Answer: (i) (ii) 32 – 80x + 80x2, 200]

10. Expand by the binomial theorem and reduce the terms to their simplest

form. Given that the first three terms in the expansion of are

32 – 16x + bx2, find the value of a, and of b.

[Answer: , a = ¾, b = – 10]

11. Evaluate the coefficient of x3 in the binomial expansion of .

[Answer: – 1701]

12. Find the first three terms in the expansion, in descending powers of x, of

. Hence, find the coefficient of x4 in the expansion of .

[Answer: x6 – 12x4 + 60x2, 156]

13. Given the expansion of (1 – 3x)2(1 + ax)8 in ascending powers of x as:1 + 10x + bx2 + …, calculate the value of a, and of b.[Answer: a = 2, b = 25]

14. Find the coefficient of x2 in the expansion of (2 + 5x + x2)(1 + x)7.[Answer]

15. Write down and evaluate the middle term of the binomial expansion of

[Answer]

16. Find, in ascending powers of x, the first three terms in the expansion of (2 – 3x)8. Use the expansion to find the value of (1.997)8 correct to the nearest whole number.[Answer: 256 – 3072x + 16128x2, 253]

17. Find the first three terms in the expansion, in ascending powers of x, of:(i) (1 – 3x)5 (ii) (2 + x)4

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Hence, find the coefficient of x2 in the expansion of (1 – 3x)5(2 + x)4.[Answer]

18. Find the first three terms in the expansion of (1 – 2x)5 in ascending powers of x, simplifying the coefficient. Given that the first three terms in the expansion of (a + bx)(1 – 2x)5 are 2 + cx + 10x2, state the value of a, and hence, find the value of b, and of c.[Answer: 1 – 10x + 40x2, a = 2, b = 7, c = – 13]

19. Find the first three terms of the expansion, in ascending powers of x, of:(i) (1 + 2x)6 (ii) (1 – 3x)6

Hence, obtain the coefficient of x2 in the expansion of (1 – x – 6x2)6.[Answer]

20. Find the first three terms of the expansion, in ascending powers of x, of (1 – 5x)6. Hence, obtain the coefficient of x2 in the expansion of (1 + 3x – 2x2)(1 – 5x)6.[Answer]

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26. Express 231.32five in base ten.[Answer: 66.68]

27. Express the following decimals as bicimals:(a) 0.75 (b) 0.375 (c) 0.875(d) 0.3125 (e) 26.25 (f) 0.625[Answer: (a) 0.11; (b) 0.011; (c) 0.111; (d) 0.0101; (e) 11010.01; (f) 0.101]

28. Convert:(a) (i) 17ten

(ii) 2376eight

(iii) 17.356ten

to base 2(b) 156FDFsixteen to base 10.(c) (i) 11101110011110two

(ii) 2367eight

to base 16.[Answer: (a) (i) 10001, (ii) 10011111110, (iii) 10001.0101; (b) 1404895;

(c) (i) 3B9E, (ii) 4F7]

29. Convert:(i) 19ten

(ii) 1110001001100two

(iii) 111001100.11001two

to base 8[Answer: (i) 16114 (ii) 714.62]

30. A number is written as 52x. Four times the number is written as 301x. What is x?[Answer: 7]

31. (a) Convert 3090base 10 to hexadecimal(b) Convert F9BD to decimal notation.[Answer: (a) C12sixteen (b) 63933ten]

32. (a) Subtract 8sixteen from 34sixteen

(b) Subtract ABsixteen from 929sixteen

[Answer: (a) 2Csixteen (b) 87Esixteen]

33. Multiply ABCsixteen by 49sixteen

[Answer: 30F9Csixteen]

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