Math Problem Solver

Question 1 [20 marks]

1. Express the set A consisting of all even numbers less than 14 in:

(a) Statement form

(b) Roster form

(c) Set builder form

Question 1(a): Set A is the set of all even numbers less than 14.

Question 1(b): 2, 4, 6, 8, 10, 12

Question 1(c): x \ x is an even number and x < 14 (2 marks)

(2 marks)

(2 marks)

2. Evaluate the Summation (4 marks)

6

∑(5 + 3i)

i=1

3. A firm’s total revenue function is: (10 marks)

R (x) = x2 - 7x + 10

Required:

1) Factorise the function

2) Find the roots

3) Interpret the result economically

Question 2 [30 marks]

1. Express with Positive Indices (4 marks)

1. y-4

2. a-3b2

2. Evaluate (6 marks)

i. 32 x 34

ii. 161/2

iii. 641/3

3. Simplify (14 marks)

(a) 45 ÷ 42

(b) (52)3)

(c) (3x2)2

(d) 𝑥6

𝑥3

(e) (ab3)2

(f) 𝑚5𝑛3

𝑚2𝑛

4. Write in Index Form (6 marks)

1. √𝑦

2. 3√𝑥2

3. 1

𝑍5

Question 3 [10 marks]

1. A committee of 8 members must elect:

• President

• Treasurer

• Secretary

How many different ways can this be done? (5 marks)

2. A company has 12 employees. It wants to select a 4-member project team. How many different teams can be formed? (5 marks)

Question 4 [14 marks]

1. Given: (6 marks)

i. TC = FC + VC

ii. Y = C + I + G

iii. C = 40 + 0.75y

Identify the definitional equation, identity and behavioural equation.

2. Given (8 marks)

C = 40 + 0.75y

a) Identify the endogenous variable

b) Identify the exogenous variable

c) If income increases by 20 units, what happens to consumption?

d) Explain the meaning of 0.75 in the equation

Question 5 [16 marks]

1. If

Find 𝑑𝑦

𝑑𝑥

y = x2(4 - x)

(8 marks)

2. If:

2

y = 5𝑥

3−2𝑥

Find the derivative of y. (8 marks)

Question 6 [40 marks]

1. Profit function (20 marks)

𝜋(x) = x3 - 6x2 - 15x + 4 Find:

a) First derivative

b) Critical points

c) Nature of turning points

d) Maximum or minimum profit

2. Cost function (20 marks)

C(x) = x3 - 12x2 + 36x

Determine:

a) Stationary points

b) Points of inflection

c) Economic interpretation

Question 7 [10 marks]

Marginal Revenue:

MR(x) = 40x1/2

Find the total revenue function if (R = 0) when (x = 0).