Find the interest rate necessary for shs. 20,000 to accrue to shs. 50,000 in 12years. (3 marks)c) Find the value of the 10th term and the sum of the first 12 terms of the sequence;6, 11, 16, 21 (3 marks)d) Find the derivative of the function:y=6x3 -18x2 + 5x + 12 (3 marks)e) Solve the simultaneous equations using matrices (4 marks)5x + 9y = -306x – 2y = 28f) A company will have to spend shs. 300,000 on new plant in two years from now.Currently investment rates are at a nominal rate of 10 %.i) What single sum should now be invested, if compounding is six-monthly.(3 marks)ii) Calculate the APR (2 marks)P.O. Box 90830100 Eldoret – KenyaTelephone: 0728458276Fax: 254-20-891084Email:[email protected]

Sure, let's break down each question:

a) To find the interest rate necessary for shs. 20,000 to accrue to shs. 50,000 in 12 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where: A = the amount of money accumulated after n years, including interest. P = principal amount (the initial amount of money) r = annual interest rate (in decimal) n = number of times that interest is compounded per year t = time the money is invested for in years

We know that A = 50,000, P = 20,000, n = 1 (compounded annually), and t = 12. We need to solve for r.

50,000 = 20,000(1 + r/1)^(1*12)

Solving this equation for r will give us the interest rate.

b) The sequence 6, 11, 16, 21 is an arithmetic sequence with a common difference of 5. The nth term of an arithmetic sequence can be found using the formula:

a_n = a_1 + (n - 1)d

Where a_n is the nth term, a_1 is the first term, d is the common difference, and n is the term number.

To find the 10th term, substitute n = 10, a_1 = 6, and d = 5 into the formula.

The sum of the first n terms of an arithmetic sequence can be found using the formula:

S_n = n/2(a_1 + a_n)

To find the sum of the first 12 terms, substitute n = 12, a_1 = 6, and a_n = a_12 into the formula.

c) To find the derivative of the function y = 6x^3 - 18x^2 + 5x + 12, use the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).

d) To solve the simultaneous equations using matrices, write the system of equations in matrix form, then use matrix operations to solve for the variables.

e) To find the single sum that should be invested now, use the formula for the future value of a lump sum:

FV = PV(1 + r/n)^(nt)

Where FV is the future value, PV is the present value, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years.

We know that FV = 300,000, r = 0.10, n = 2 (compounded semi-annually), and t = 2. We need to solve for PV.

f) The APR, or annual percentage rate, is a measure of the cost of a loan, expressed as a yearly interest rate. It includes the interest rate and other charges or fees. In this case, since there are no other charges or fees mentioned, the APR is the same as the nominal interest rate, 10%.