A survey showed that small engineering companies use three

Question

Answer all questions-

1. (a) If E = (X²-1/X²+1) and x= (a+b/a-b), find a simplified expression for E in terms of a and b.

(b) Transpose V2 = 64.4DH/D+4fl to make D the subject.

2. Solve the equations-

(i) x(x - 5) +1 = x² + 9x - 6

(ii) x/12(x-4) = 3/2x

(iii) 12.57x² - 27.14x + 7.776 = 0  (Answers correct to 2dp)

(iv) 3^x+1 = 27^x      

3. (a) Solve the simultaneous equations

2x + 2y = 14        

5y - 3x = 29                                        

(b) Factorise: rp - yr + qy - pq    

(c) Simplify: 3p² - 48q²/4pq + p²               

4. (i) Write the expression x² + 10x - 5 in the form (x+a)² + b by completing the square.

(ii) Hence solve the equation x² + 10x - 5 = 0, giving answers correct to 3 decimal places.

(iii) Using your result from part (i), determine the minimum value of the function f(x) = x² + 10x - 5.

5. Using BOTH of the following methods, find the solutions of the equation

X³ - 2x² - 9x + 18 = 0

(i) Method 1: Plot the graphs of both y = x³ and 2x² + 9x - 18 on the same axes between x = -4 and x = 4.

(ii) Method 2: Show that (x-3) is a factor of x³ - 2x² - 9x + 18 and find the other factor(s) algebraically.

State your solutions for each method.

6. Combine the following and give your answer as a single fraction in its simplest form:

(a) x + (x/(x-2)) - (2x/3(x+2))                    

(b) (a²-b²/a²b²) ÷ ab+b²/2a³b       

7. Express a single logarithm in simplest form:

(i) log₅16 + log₅4 - 3log₅2

(ii) log₁₀x3 + log₁₀x²y - 3log₁₀(x/y)

(iii) ln(x² - y²) - ln(x + y) 

8. A survey showed that small engineering companies use three fuels for heating workshops, electricity, gas and kerosene. 70 use electricity, 60 use gas and 50 use kerosene. 17 use both electricity and gas, 15 use gas and kerosene and 16 use kerosene and electricity. One firm had no heating. 135 firms were surveyed and x of them use all three fuels.

(i) Draw a Venn diagram showing the relationship between the sets.

(ii) Find x.

9. The volume of benzene leaking from a hole in a pipe is currently 5cm³ per hour. However the hole is decreasing in size so that the volume leaking through it each hour is ¾ of the previous hour's loss.

(i) Using a geometric progression find how much benzene is lost in the 5^th hour.

(ii) How much benzene is lost altogether, assuming the leak is never repaired?

10. (i) Expand (1-x/3)⁷ fully, using the Binomial Theorem, giving your expansion in its simplest form.

(ii) Hence, by making a suitable substitution for x, evaluate (0.9)⁷ correct to 4 decimal places.