Reference no: EM132614392

Question 1:

a. An oil company bores a hole 80m deep. Estimate the cost of boring if the cost is £30 for drilling the first meter with an Increase in cost of £2 per meter for each Succeeding meter.

b. A plant grows 1.67 cm in its first week. Each week it grows by 2% more than it did the week before. By how much does it grow in twenty weeks, including the first week?

I. As a farmer, can you come out with a general term that can be used to forecast the height of the plant at any particular time? Explain mathematically.

c. Consider a rectangle with perimeter 28 (units). Let the width of the rectangle be w (units) and let the Area of the region enclossed by the rectangle be A ( square units).

Express A as a function of w and state the domain and range of the function.

Question 2: a. Using the identities for sin (A + B) and cos (A+B) express
i. sin (2A) and cos (2A) interns of sin A and cos A.
ii. Also show that cos 3A = 4 cos³ A-3 cos A
iii. Generate a general rule in terms of n for Cos (nA)

b. The population P of a certain small-town t years from now is predicted to be p(t) = 35000 + 10000/(t+2)³. In your own method estimate the population of the town in the long run.

c. This Simplification of the expression below, where u, v, and w denote suitable positive realy numbers.

log. uv/w + log uw/v + log vw/u - log(uvw) is equal to zero (0). Explain what property makes the situation so.

Question 3.

a. A drilling machine is to have 8 speeds ranging from 100 rev/min to 1000 rev/min. If the speeds form a geometric progression determine their values, each correct to the nearest whole number and explain your results.

b. Prove the following

i.√((1 - sin x)/(1 + sin x)) secx - tan x. ii. 1/(1-cos²θ) = cosec θ

c. An arithmetic series has first term In 2 and common difference in 4. Show that the sum Sn of the first n terms is n²ln 2.

Question 4. a. The sum to infinity of a GP is twice the sum of the first two terms. Find possible values of the common ratio.

b. Express the function below as a sum or difference of sine and cosine.

sin 4x cos 3x

c. Let f(x) = 1/(1 - (2/(1+(1/(1-x)))))

i. Find f (1/2)
ii. Find the domain of f