(a) Let S = {1 , 2 , 3 , 4 , 5 , 6}, A = {1 , 3 , 5}, B = {4 , 6} and C = {1 , 4}. Compute the…
(a) Let S= {1,2,3,4,5,6}, A= {1,3,5}, B= {4,6} and C= {1,4}. Compute the following set operations
(i) An B (1 mark)
(ii) B? C (1 mark)
(iii) An (B n C) (2 marks)
(iv) (A? B)C. (2 marks)
(b) Solve for xin the following inequality and give the integral values;
2(x– 1) >3(2x+ 3)
(2 marks)
(c) Solve the following equations simultaneously;
Y+ 2X= 3
X2+ Y2= 2
(3 marks)
(d) Compute the following limits
(2 marks)
(3 marks)
(e) The second, the fourth and the seventh terms of an Arithmetic Progression (AP) are the first three consecutive terms of a Geometric Progression (GP). The common difference of the AP is 2. Find
(i) The common ratio (3 marks)
(ii) The sum of the first eight terms of the GP. (3 marks)
(f) A customer deposited Ksh.48,000 at the beginning of a year in a bank account. The rate of interest is 8% p.a. compounded semi-annually for the first five years and 10 % p.a. compounded annually for the subsequent 5 years. Calculate the total amount of
|
money earned after the 10 years. (g) Solve for xin the following equations. |
(4 marks) |
. (2 marks)
(2 marks)QUESTION TWO (20 Marks)
(a) A business person buys a car at a cost of Ksh.960,000 using installment plan. The plan consists of 30 monthly payments of Ksh.40,000 plus 5% interest on the unpaid balance for that month. If payments commences at the end of the first month, calculate
(i) The amount of the first and second installments. (4 marks)
(ii) The amount paid on the 30th month. (3 marks)
(iii) The total amount paid for the car. (3 marks)
(b) Find the simple interest that would accrue when a welfare group lends to a member
Ksh.15,000 at a rate of 12% p.a. in 2 years. (3 marks)
(c) An employee’s salary is 300,000 p.a the salary increases by 10% annually. Calculate the total amount the employee will have earned in 15 years. (4 marks)
(d) Calculate the amount of money one needs to deposit in the account today at 9% annual interest compounded monthly to have 12,000 in the account after 6 years. (3 marks)
QUESTION THREE (20 Marks)
(a) (i) Define the break even point as used in Business. (1 mark)
(ii) The revenue Rin terms of the number of items produced is given by R(x) = 12xand the cost Cby C(x) = 7x+ 85. Find the break-even point and the break even price per unit item. (3 marks)
(b) The market supply function of sugar is q= 160 + 8p, where qdenotes the quantity of sugar supplied andpdenotes its market price. The unit cost of production is Ksh.4. It is felt that the total profit should be Ksh.5,000. What market price has to be fixed in order to achieve this profit? (4 marks)
(c) Using the quadratic formula, solve the following quadratic equation; 6x2– 11x+ 4 = 0.
(4 marks)
(d) Consider the arithmetic sequence given by {1,4, 7,10,13}
(i) Compute the sum of the first 10 terms.
(ii) How many terms of the arithmetic series must be taken so that their sum of 590 ?
(4 marks)
(e) Consider the sequence given by {1,3,9,27,81,···}
(i) Compute the sum of the first 15 terms
(ii) How many terms of the G.P must be taken so that their sum is 3280 ?
(4 marks)QUESTION FOUR (20 Marks)
(a) In a class of 82 students, 10 students were found to perform well in all the three subjects offered; Mathematics, English and Science. 32 students excelled in both Mathematics and English and 18 excelled in both Science and English. The number of students who excelled in Mathematics only exceeded those who excelled in Science only by 2. The number of students who excelled in English only was twice the number of students who excelled in Science.
(i) Represent the information in a well labeled venn diagram. (2 marks)
(ii) Determine the number of students who excelled in Mathematics only. (3 marks)
(ii) Determine the number of students who excelled in Science only. (3 marks)
(ii) Determine the number of students who excelled in English and at least one other subject. (4 marks)
(b) Factorize the polynomial x3– 17x2+ 54x– 8, given (x– 1) is a factor. (3 marks)(c) Solve for xin the following equations;
log6(x+ 4) + log6(x– 2) = log6(4x)
(3 marks)d) Simplify the following expression; . (2 marks)
QUESTION FIVE (20 Marks)
(a) (i) Define the term annuity. (2 marks)
(ii) Find the future value of an ordinary annuity with Ksh.1500 monthly payments and 6% annual interest for 12 years. (4 marks)
(b) The model A= P(1 + r)2describes the amount of money Ain an account after 2 years when Pshillings are invested at interest rate rcompounded annually. At what interest rate will 10,000 grow to 12,100 in 2 years? (3 marks)
(c) (i) Define the term payback period as used in project appraisal. (1 mark)
(ii) Explain any two problems of using net present value to appraise an investment project. (2 marks)
(iii) BRITZ insurance has identified that it could male operating cost savings in production by partnering with advertising agencies. There are two suitable such agencies on the market, MagEvents and OneWorld. The relevant data relating to each of these are as follows:
|
MagEvents |
OneWorld | |
|
Cost(Payable immediately) Annual Savings: |
20,000 |
25,000 |
|
Year 1 |
4,000 |
8,000 |
|
2 |
6,000 |
6,000 |
|
3 |
6,000 |
5,000 |
|
4 |
7,000 |
6,000 |
|
5 |
6,000 |
8,000 |
The annual savings are, in effect, opportunity cash inflows in that they represent savings rom the cash outflows that would occur if the investment were not undertaken. Which, if either, of these agencies should be partnered with if the financing cost is a constant 12% per annum. (8 marks)
