Question
As a Business Mathematics Professor and senior Lecturerer experinced in setting exams at Diploma Level at National Institute of Public Administration in Zambia, what are the most important and key topics you must tell your students concetrate on as you want them to get Distinctions in Business Mathematics upon finishing the course. Advise and set 7 exam questions under each topic with well explained questions and answers to prepare your students as an experienced seasoned Professor or lecture in the course by using the course content given below.
1. UNIT ONE – SOLVING ARITHMETIC
2. UNIT TWO – INDICES
3. LOGARITHMS
4. NATURAL LOGARITHMS
5. COMMON LOGARITHMS
6. UNIT THREE – SERIES
7. GEOMETIC PROGRESSION (GP)
8. UNIT FOUR – FINANCIAL MATHEMATICS
9. DEPRECIATION
10. UNIT FIVE – STATISTICS
11. UNIT SIX – MATRICES
12. UNIT SEVEN – DIFFERENTIATION
1. UNIT ONE – SOLVING ARITHMETIC
2. UNIT TWO – INDICES
3. LOGARITHMS
4. NATURAL LOGARITHMS
5. COMMON LOGARITHMS
6. UNIT THREE – SERIES
7. GEOMETIC PROGRESSION (GP)
8. UNIT FOUR – FINANCIAL MATHEMATICS
9. DEPRECIATION
10. UNIT FIVE – STATISTICS
11. UNIT SIX – MATRICES
12. UNIT SEVEN – DIFFERENTIATION
Answers
GPT-4o mini
To help your students excel in Business Mathematics and achieve distinction, it's crucial to focus on foundational concepts while also preparing them for practical applications. Here are the key topics from your syllabus along with suggested exam questions and comprehensive answers.
### **UNIT ONE – SOLVING ARITHMETIC**
**Key Topics:**
- Basic operations (addition, subtraction, multiplication, division)
- Order of operations
- Application of arithmetic in business scenarios
**Exam Questions:**
1. **Question:** Calculate 15% of $250 and determine the total amount after including the calculated percentage.
- **Answer:** \( 15\% \times 250 = \frac{15}{100} \times 250 = 37.5 \). Therefore, the total amount is \( 250 + 37.5 = 287.5 \).
2. **Question:** If a product costs $40 and is marked up by 20%, what is the new selling price?
- **Answer:** \( 20\% \text{ of } 40 = \frac{20}{100} \times 40 = 8 \). Selling price = \( 40 + 8 = 48 \).
3. **Question:** A company sold 150 units at $20 each. What was the total revenue?
- **Answer:** Revenue = \( 150 \times 20 = 3000 \).
4. **Question:** If you buy 30 shares at $50 each and sell them at $55 each, what is your profit?
- **Answer:** Cost = \( 30 \times 50 = 1500 \) and the Selling Price = \( 30 \times 55 = 1650 \). Profit = \( 1650 - 1500 = 150 \).
5. **Question:** Convert 250 hours into minutes.
- **Answer:** \( 250 \times 60 = 15000 \) minutes.
6. **Question:** If 5 products cost $250, what is the cost of 12 products?
- **Answer:** Cost per product = \( \frac{250}{5} = 50 \). Cost for 12 products = \( 12 \times 50 = 600 \).
7. **Question:** A person has $600 and wants to divide it among 4 friends evenly. How much will each friend receive?
- **Answer:** Amount per friend = \( \frac{600}{4} = 150 \).
### **UNIT TWO – INDICES**
**Key Topics:**
- Definition and properties of indices
- Simplifying expressions involving indices
- Applications in business calculations
**Exam Questions:**
1. **Question:** Simplify \( 3^4 \times 3^2 \).
- **Answer:** \( 3^{4+2} = 3^6 = 729 \).
2. **Question:** Calculate \( (2^3)^4 \).
- **Answer:** \( 2^{3 \times 4} = 2^{12} = 4096 \).
3. **Question:** Express \( 81 \) as a power of \( 3 \).
- **Answer:** \( 81 = 3^4 \).
4. **Question:** Simplify \( \frac{4^5}{4^2} \).
- **Answer:** \( 4^{5-2} = 4^3 = 64 \).
5. **Question:** If \( x^2 = 16 \), find \( x \).
- **Answer:** \( x = \pm 4 \).
6. **Question:** Simplify \( 2^5 \times 2^{-3} \).
- **Answer:** \( 2^{5-3} = 2^2 = 4 \).
7. **Question:** Solve for \( y \) in \( 5^y = 125 \).
- **Answer:** \( 125 = 5^3 \) thus \( y = 3 \).
### **LOGARITHMS**
**Key Topics:**
- Definition and properties of logarithms
- Change of base formula
- Solving logarithmic equations
**Exam Questions:**
1. **Question:** Simplify \( \log_2(8) \).
- **Answer:** \( 8 = 2^3, \text{ so } \log_2(8) = 3 \).
2. **Question:** If \( \log_10(100) = x \), find \( x \).
- **Answer:** Since \( 100 = 10^2, x = 2 \).
3. **Question:** Evaluate \( \log_5(25) \).
- **Answer:** \( 25 = 5^2, \text{ thus } \log_5(25) = 2 \).
4. **Question:** Solve for \( x \) in the equation \( \log_3(x) = 4 \).
- **Answer:** \( x = 3^4 = 81 \).
5. **Question:** Convert \( \log_2(16) \) to base 10.
- **Answer:** Using base change, \( \log_2(16) = \frac{\log_{10}(16)}{\log_{10}(2)} = \frac{4 \cdot \log_{10}(2)}{\log_{10}(2)} = 4 \).
6. **Question:** Find the value of \( x \) if \( \log_x(100) = 2 \).
- **Answer:** \( x^2 = 100 \) implies \( x = 10 \).
7. **Question:** Solve \( 2 \log_5(x) = 4 \).
- **Answer:** \( \log_5(x) = 2 \Rightarrow x = 5^2 = 25 \).
### **NATURAL LOGARITHMS**
**Key Topics:**
- Properties of natural logarithms
- Relation to exponential functions
- Applications in real-world scenarios
**Exam Questions:**
1. **Question:** Calculate \( \ln(e^3) \).
- **Answer:** \( \ln(e^3) = 3 \).
2. **Question:** Simplify \( \ln(1) \).
- **Answer:** \( \ln(1) = 0 \).
3. **Question:** If \( \ln(x) = 5 \), what is \( x \)?
- **Answer:** \( x = e^5 \).
4. **Question:** Evaluate \( \ln(2) + \ln(3) \).
- **Answer:** Using properties of logarithms, \( \ln(2) + \ln(3) = \ln(6) \).
5. **Question:** Find \( x \) if \( e^x = 20 \).
- **Answer:** \( x = \ln(20) \).
6. **Question:** Calculate \( \frac{d}{dx} [\ln(x)] \).
- **Answer:** The derivative is \( \frac{1}{x} \).
7. **Question:** Solve for \( y \) in \( \ln(y) = -2 \).
- **Answer:** \( y = e^{-2} \).
### **COMMON LOGARITHMS**
**Key Topics:**
- Definition and properties
- Relation of common logarithms to the decimal system
- Applications in finance and statistics
**Exam Questions:**
1. **Question:** Simplify \( \log_{10}(1000) \).
- **Answer:** \( 1000 = 10^3, \text{ thus } \log_{10}(1000)=3 \).
2. **Question:** If \( \log_{10}(x) = 3 \), find \( x \).
- **Answer:** \( x = 10^3 = 1000 \).
3. **Question:** Evaluate \( \log_{10}(50) + \log_{10}(2) \).
- **Answer:** \( \log_{10}(100) = 2 \).
4. **Question:** Convert \( \log_{10}(5) \) to natural logarithm.
- **Answer:** \( \log_{10}(5) = \frac{\ln(5)}{\ln(10)} \).
5. **Question:** Solve for \( x \) in \( 10^x = 100 \).
- **Answer:** \( x = \log_{10}(100) = 2 \).
6. **Question:** What is \( \log_{10}(100000) \)?
- **Answer:** \( \log_{10}(10^5) = 5 \).
7. **Question:** Use the property of logarithms to express \( \log_{10}(30) \).
- **Answer:** \( \log_{10}(30) = \log_{10}(3) + \log_{10}(10) = \log_{10}(3) + 1 \).
### **UNIT THREE – SERIES**
**Key Topics:**
- Arithmetic series
- Geometric series
- Applications in financial calculations
**Exam Questions:**
1. **Question:** Calculate the sum of the first 10 natural numbers.
- **Answer:** The sum \( S_n = \frac{n(n+1)}{2} = \frac{10 \times 11}{2} = 55 \).
2. **Question:** Find the 20th term of an arithmetic sequence where the first term is 4 and the common difference is 3.
- **Answer:** \( a_n = a + (n-1)d = 4 + (20-1)3 = 4 + 57 = 61 \).
3. **Question:** If the sum of the first \( n \) terms of a geometric series is \( \frac{a(1 - r^n)}{1 - r} \), find the sum for \( a = 5, r = 2, n = 4 \).
- **Answer:** \( S_n = \frac{5(1 - 2^4)}{1 - 2} = \frac{5(1 - 16)}{-1} = \frac{5 \times -15}{-1} = 75 \).
4. **Question:** Sum the series \( 2 + 4 + 6 + ... + 20 \).
- **Answer:** The number of terms \( n = 10 \): \( S_n = n/2 \cdot (first term + last term) = 10/2 \cdot (2 + 20) = 5 \cdot 22 = 110 \).
5. **Question:** If the first term of a geometric series is 3 and the common ratio is 1.5, find the sixth term.
- **Answer:** \( a_n = ar^{n-1} = 3 \times (1.5)^{5} = 3 \times 7.59375 = 22.78125 \).
6. **Question:** Calculate the sum of the first 15 terms of the series, where the first term is 10 and the common difference is 5.
- **Answer:** \( S_{15} = \frac{15}{2} (2 \times 10 + (15 - 1)5) = \frac{15}{2} (20 + 70) = \frac{15}{2} \times 90 = 675 \).
7. **Question:** Given a geometric series with first term 5 and a total of five terms where the sum is 31.25, find the common ratio.
- **Answer:** \( S_n = \frac{a(1 - r^n)}{1 - r} \Rightarrow 31.25 = \frac{5(1 - r^5)}{1 - r} \). Rearranging gives \( r^5 = 1 - \frac{31.25(1 - r)}{5} \). (Use numeric methods for specific value).
### **GEOMETRIC PROGRESSION (GP)**
**Key Topics:**
- Identifying geometric sequences
- Sum of finite and infinite geometric series
- Applications in investment and finance
**Exam Questions:**
1. **Question:** Determine the first five terms of a geometric sequence where \( a = 2 \) and \( r = 3 \).
- **Answer:** Terms are \( 2, 6, 18, 54, 162 \).
2. **Question:** Find the sum of the first 6 terms of the GP where \( a = 1 \) and \( r = 2 \).
- **Answer:** \( S_n = \frac{a(1 - r^n)}{1 - r} = \frac{1(1 - 2^6)}{1 - 2} = \frac{1(1 - 64)}{-1} = 63 \).
3. **Question:** Calculate the sum of an infinite GP with \( a = 4 \) and \( |r| < 1 \) where \( r = 0.5 \).
- **Answer:** \( S_{\infty} = \frac{a}{1 - r} = \frac{4}{1 - 0.5} = 8 \).
4. **Question:** If the first term of a geometric series is \( 5 \) and the common ratio is \( 2 \), find the 7th term.
- **Answer:** \( a_n = 5 \times 2^{6} = 5 \times 64 = 320 \).
5. **Question:** Determine the common ratio if the third term of a GP is \( 12 \) and the first term is \( 3 \).
- **Answer:** The third term is given by \( ar^2 = 12 \Rightarrow 3r^2 = 12 \Rightarrow r^2 = 4 \Rightarrow r = 2 \) or \( r = -2 \).
6. **Question:** Find the sum of the first n terms of a GP series with first term \( 7 \) and \( r = 4 \).
- **Answer:** \( S_n = \frac{7(1 - 4^n)}{1 - 4} = \frac{7(1 - 4^n)}{-3} = \frac{7(4^n - 1)}{3} \).
7. **Question:** If the sum of the first four terms of a geometric series is 45 and the first term is \( 5 \), what is the common ratio?
- **Answer:** \( 5(1 + r + r^2 + r^3) = 45 \Rightarrow 1 + r + r^2 + r^3 = 9 \).
### **UNIT FOUR – FINANCIAL MATHEMATICS**
**Key Topics:**
- Simple and compound interest
- Present and future value calculations
- Amortization and loan calculations
**Exam Questions:**
1. **Question:** Compute the simple interest on a loan of $1000 at a rate of 5% per annum for 3 years.
- **Answer:** \( I = P \times r \times t = 1000 \times 0.05 \times 3 = 150 \).
2. **Question:** If you invest $8000 at an interest rate of 6% compounded annually, how much will it be worth in 5 years?
- **Answer:** \( A = P(1 + r)^t = 8000(1 + 0.06)^5 \approx 8000 \times 1.338226 = 10705.81 \).
3. **Question:** Calculate the present value of $5000 due in 3 years at an interest rate of 8%.
- **Answer:** \( PV = \frac{FV}{(1 + r)^t} = \frac{5000}{(1 + 0.08)^3} \approx \frac{5000}{1.259712} \approx 3961.62 \).
4. **Question:** A loan of $10,000 is repaid with equal monthly payments over 3 years at an annual interest rate of 5%. What is the monthly payment?
- **Answer:** Using the formula \( M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \), where \( r = \frac{0.05}{12} \) and \( n = 36 \): Plugging into the formula returns approximately \( 299.71 \).
5. **Question:** How long will it take for a $1000 investment at an interest rate of 10% compounded annually to double in value?
- **Answer:** The Rule of 72 gives a rough estimate \( \frac{72}{10} = 7.2 \) years.
6. **Question:** Find the total amount after investing $4000 at a compound interest rate of 4% for 10 years.
- **Answer:** \( A = 4000(1 + 0.04)^{10} \approx 4000 \times 1.48024 = 5920.96 \).
7. **Question:** If an amount of $3000 is to be paid in 5 years with a 7% interest rate, what is the present value?
- **Answer:** \( PV = \frac{FV}{(1+r)^t} = \frac{3000}{(1.07)^5} \approx \frac{3000}{1.403} \approx 2135.60 \).
### **UNIT FIVE – STATISTICS**
**Key Topics:**
- Descriptive statistics (mean, median, mode)
- Probability concepts
- Simple linear regression
**Exam Questions:**
1. **Question:** Calculate the mean of the data set: {10, 15, 20, 25, 30}.
- **Answer:** \( \text{Mean} = \frac{10 + 15 + 20 + 25 + 30}{5} = \frac{100}{5} = 20 \).
2. **Question:** Determine the median of the data set: {3, 5, 7, 9, 11, 13}.
- **Answer:** Arrange the numbers: {3, 5, 7, 9, 11, 13}. The median is the average of 7 and 9, which is 8.
3. **Question:** What is the mode of the data set: {1, 2, 2, 3, 4, 4, 5}?
- **Answer:** The mode is 2 and 4 (bimodal).
4. **Question:** If the probability of an event occurring is \( 0.2 \), what is the probability of the event not occurring?
- **Answer:** \( P(\text{not } A) = 1 - P(A) = 1 - 0.2 = 0.8 \).
5. **Question:** Given the dataset: {2, 3, 3, 6, 7}, calculate the standard deviation.
- **Answer:** Mean = \( \frac{2+3+3+6+7}{5} = 4.2 \). \( \sigma = \sqrt{\frac{(2-4.2)^2 + (3-4.2)^2 + (3-4.2)^2 + (6-4.2)^2 + (7-4.2)^2}{5}} \approx 1.67 \).
6. **Question:** In a simple linear regression \( y = mx + b \), if \( m = 3 \) and \( b = 1 \), what is \( y \) when \( x = 4 \)?
- **Answer:** \( y = 3(4) + 1 = 12 + 1 = 13 \).
7. **Question:** Calculate the variance for the following data: {4, 8, 6, 5}.
- **Answer:** Mean \( = 5.75 \). Variance \( = \frac{(4-5.75)^2 + (8-5.75)^2 + (6-5.75)^2 + (5-5.75)^2}{4} \approx 2.6875 \).
### **UNIT SIX – MATRICES**
**Key Topics:**
- Basic matrix operations (addition, subtraction, multiplication)
- Determinants and inverse of matrices
- Applications in business problems
**Exam Questions:**
1. **Question:** If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \), calculate \( A + B \).
- **Answer:** \( A + B = \begin{pmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} \).
2. **Question:** Find the product of matrices \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \).
- **Answer:** \( AB = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix} \).
3. **Question:** Calculate the determinant of \( A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \).
- **Answer:** \( det(A) = (1)(4) - (3)(2) = 4 - 6 = -2 \).
4. **Question:** Find the inverse of the matrix \( A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix} \).
- **Answer:** \( A^{-1} = \frac{1}{-10} \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -0.6 & 0.7 \\ 0.2 & -0.4 \end{pmatrix} \).
5. **Question:** Solve the system of equations using matrices:
1. \( x + 2y = 3 \)
2. \( 3x + 4y = 10 \)
- **Answer:** Represent as \( AX = B \) where \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, X = \begin{pmatrix} x \\ y \end{pmatrix}, B = \begin{pmatrix} 3 \\ 10 \end{pmatrix} \). Calculate \( A^{-1} \) and find \( X = A^{-1}B \).
6. **Question:** If \( C = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \) and \( D = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \), what is \( C \times D \)?
- **Answer:** \( C \times D = D = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \).
7. **Question:** Verify whether the two matrices \( A \) and \( B \) are invertible.
- **Answer:** Both matrices must have non-zero determinants. If \( det(A) \neq 0 \) and \( det(B) \neq 0 \), the matrices are invertible.
### **UNIT SEVEN – DIFFERENTIATION**
**Key Topics:**
- Basic rules of differentiation
- Application in business problems (e.g., cost, revenue, profit functions)
- Marginal analysis
**Exam Questions:**
1. **Question:** If \( f(x) = x^2 + 3x + 5 \), find \( f'(x) \).
- **Answer:** \( f'(x) = 2x + 3 \).
2. **Question:** Calculate the derivative of \( g(x) = 5x^3 - 2x + 1 \).
- **Answer:** \( g'(x) = 15x^2 - 2 \).
3. **Question:** If \( R(x) = 20x - x^2 \), find the marginal revenue \( R'(x) \) at \( x = 10 \).
- **Answer:** \( R'(x) = 20 - 2x; R'(10) = 20 - 20 = 0 \).
4. **Question:** Determine \( \frac{d}{dx} [\ln(x^2 + 1)] \).
- **Answer:** Using the chain rule, \( \frac{d}{dx} [\ln(u)] = \frac{1}{u} \frac{du}{dx} = \frac{1}{x^2 + 1}(2x) = \frac{2x}{x^2 + 1} \).
5. **Question:** If the cost function is \( C(x) = 3x^3 + 5x^2 + 2 \), find the marginal cost.
- **Answer:** \( C'(x) = 9x^2 + 10x \).
6. **Question:** Compute the derivative of \( P(x) = 4e^x + x^3 \).
- **Answer:** \( P'(x) = 4e^x + 3x^2 \).
7. **Question:** A revenue function is given by \( R(x) = 7x^2 - 4x + 10 \). Find the maximum revenue.
- **Answer:** Set \( R'(x) = 0 \) and solve to find critical points, then test to confirm the maximum.
The outlined structure of topics and corresponding questions prepares students not only to tackle examinations but also to apply these mathematical principles in real-world business scenarios. Focus on these topics, practice solving similar problems, and encourage collaboration and ability to understand concepts for mastering Business Mathematics.
### **UNIT ONE – SOLVING ARITHMETIC**
**Key Topics:**
- Basic operations (addition, subtraction, multiplication, division)
- Order of operations
- Application of arithmetic in business scenarios
**Exam Questions:**
1. **Question:** Calculate 15% of $250 and determine the total amount after including the calculated percentage.
- **Answer:** \( 15\% \times 250 = \frac{15}{100} \times 250 = 37.5 \). Therefore, the total amount is \( 250 + 37.5 = 287.5 \).
2. **Question:** If a product costs $40 and is marked up by 20%, what is the new selling price?
- **Answer:** \( 20\% \text{ of } 40 = \frac{20}{100} \times 40 = 8 \). Selling price = \( 40 + 8 = 48 \).
3. **Question:** A company sold 150 units at $20 each. What was the total revenue?
- **Answer:** Revenue = \( 150 \times 20 = 3000 \).
4. **Question:** If you buy 30 shares at $50 each and sell them at $55 each, what is your profit?
- **Answer:** Cost = \( 30 \times 50 = 1500 \) and the Selling Price = \( 30 \times 55 = 1650 \). Profit = \( 1650 - 1500 = 150 \).
5. **Question:** Convert 250 hours into minutes.
- **Answer:** \( 250 \times 60 = 15000 \) minutes.
6. **Question:** If 5 products cost $250, what is the cost of 12 products?
- **Answer:** Cost per product = \( \frac{250}{5} = 50 \). Cost for 12 products = \( 12 \times 50 = 600 \).
7. **Question:** A person has $600 and wants to divide it among 4 friends evenly. How much will each friend receive?
- **Answer:** Amount per friend = \( \frac{600}{4} = 150 \).
### **UNIT TWO – INDICES**
**Key Topics:**
- Definition and properties of indices
- Simplifying expressions involving indices
- Applications in business calculations
**Exam Questions:**
1. **Question:** Simplify \( 3^4 \times 3^2 \).
- **Answer:** \( 3^{4+2} = 3^6 = 729 \).
2. **Question:** Calculate \( (2^3)^4 \).
- **Answer:** \( 2^{3 \times 4} = 2^{12} = 4096 \).
3. **Question:** Express \( 81 \) as a power of \( 3 \).
- **Answer:** \( 81 = 3^4 \).
4. **Question:** Simplify \( \frac{4^5}{4^2} \).
- **Answer:** \( 4^{5-2} = 4^3 = 64 \).
5. **Question:** If \( x^2 = 16 \), find \( x \).
- **Answer:** \( x = \pm 4 \).
6. **Question:** Simplify \( 2^5 \times 2^{-3} \).
- **Answer:** \( 2^{5-3} = 2^2 = 4 \).
7. **Question:** Solve for \( y \) in \( 5^y = 125 \).
- **Answer:** \( 125 = 5^3 \) thus \( y = 3 \).
### **LOGARITHMS**
**Key Topics:**
- Definition and properties of logarithms
- Change of base formula
- Solving logarithmic equations
**Exam Questions:**
1. **Question:** Simplify \( \log_2(8) \).
- **Answer:** \( 8 = 2^3, \text{ so } \log_2(8) = 3 \).
2. **Question:** If \( \log_10(100) = x \), find \( x \).
- **Answer:** Since \( 100 = 10^2, x = 2 \).
3. **Question:** Evaluate \( \log_5(25) \).
- **Answer:** \( 25 = 5^2, \text{ thus } \log_5(25) = 2 \).
4. **Question:** Solve for \( x \) in the equation \( \log_3(x) = 4 \).
- **Answer:** \( x = 3^4 = 81 \).
5. **Question:** Convert \( \log_2(16) \) to base 10.
- **Answer:** Using base change, \( \log_2(16) = \frac{\log_{10}(16)}{\log_{10}(2)} = \frac{4 \cdot \log_{10}(2)}{\log_{10}(2)} = 4 \).
6. **Question:** Find the value of \( x \) if \( \log_x(100) = 2 \).
- **Answer:** \( x^2 = 100 \) implies \( x = 10 \).
7. **Question:** Solve \( 2 \log_5(x) = 4 \).
- **Answer:** \( \log_5(x) = 2 \Rightarrow x = 5^2 = 25 \).
### **NATURAL LOGARITHMS**
**Key Topics:**
- Properties of natural logarithms
- Relation to exponential functions
- Applications in real-world scenarios
**Exam Questions:**
1. **Question:** Calculate \( \ln(e^3) \).
- **Answer:** \( \ln(e^3) = 3 \).
2. **Question:** Simplify \( \ln(1) \).
- **Answer:** \( \ln(1) = 0 \).
3. **Question:** If \( \ln(x) = 5 \), what is \( x \)?
- **Answer:** \( x = e^5 \).
4. **Question:** Evaluate \( \ln(2) + \ln(3) \).
- **Answer:** Using properties of logarithms, \( \ln(2) + \ln(3) = \ln(6) \).
5. **Question:** Find \( x \) if \( e^x = 20 \).
- **Answer:** \( x = \ln(20) \).
6. **Question:** Calculate \( \frac{d}{dx} [\ln(x)] \).
- **Answer:** The derivative is \( \frac{1}{x} \).
7. **Question:** Solve for \( y \) in \( \ln(y) = -2 \).
- **Answer:** \( y = e^{-2} \).
### **COMMON LOGARITHMS**
**Key Topics:**
- Definition and properties
- Relation of common logarithms to the decimal system
- Applications in finance and statistics
**Exam Questions:**
1. **Question:** Simplify \( \log_{10}(1000) \).
- **Answer:** \( 1000 = 10^3, \text{ thus } \log_{10}(1000)=3 \).
2. **Question:** If \( \log_{10}(x) = 3 \), find \( x \).
- **Answer:** \( x = 10^3 = 1000 \).
3. **Question:** Evaluate \( \log_{10}(50) + \log_{10}(2) \).
- **Answer:** \( \log_{10}(100) = 2 \).
4. **Question:** Convert \( \log_{10}(5) \) to natural logarithm.
- **Answer:** \( \log_{10}(5) = \frac{\ln(5)}{\ln(10)} \).
5. **Question:** Solve for \( x \) in \( 10^x = 100 \).
- **Answer:** \( x = \log_{10}(100) = 2 \).
6. **Question:** What is \( \log_{10}(100000) \)?
- **Answer:** \( \log_{10}(10^5) = 5 \).
7. **Question:** Use the property of logarithms to express \( \log_{10}(30) \).
- **Answer:** \( \log_{10}(30) = \log_{10}(3) + \log_{10}(10) = \log_{10}(3) + 1 \).
### **UNIT THREE – SERIES**
**Key Topics:**
- Arithmetic series
- Geometric series
- Applications in financial calculations
**Exam Questions:**
1. **Question:** Calculate the sum of the first 10 natural numbers.
- **Answer:** The sum \( S_n = \frac{n(n+1)}{2} = \frac{10 \times 11}{2} = 55 \).
2. **Question:** Find the 20th term of an arithmetic sequence where the first term is 4 and the common difference is 3.
- **Answer:** \( a_n = a + (n-1)d = 4 + (20-1)3 = 4 + 57 = 61 \).
3. **Question:** If the sum of the first \( n \) terms of a geometric series is \( \frac{a(1 - r^n)}{1 - r} \), find the sum for \( a = 5, r = 2, n = 4 \).
- **Answer:** \( S_n = \frac{5(1 - 2^4)}{1 - 2} = \frac{5(1 - 16)}{-1} = \frac{5 \times -15}{-1} = 75 \).
4. **Question:** Sum the series \( 2 + 4 + 6 + ... + 20 \).
- **Answer:** The number of terms \( n = 10 \): \( S_n = n/2 \cdot (first term + last term) = 10/2 \cdot (2 + 20) = 5 \cdot 22 = 110 \).
5. **Question:** If the first term of a geometric series is 3 and the common ratio is 1.5, find the sixth term.
- **Answer:** \( a_n = ar^{n-1} = 3 \times (1.5)^{5} = 3 \times 7.59375 = 22.78125 \).
6. **Question:** Calculate the sum of the first 15 terms of the series, where the first term is 10 and the common difference is 5.
- **Answer:** \( S_{15} = \frac{15}{2} (2 \times 10 + (15 - 1)5) = \frac{15}{2} (20 + 70) = \frac{15}{2} \times 90 = 675 \).
7. **Question:** Given a geometric series with first term 5 and a total of five terms where the sum is 31.25, find the common ratio.
- **Answer:** \( S_n = \frac{a(1 - r^n)}{1 - r} \Rightarrow 31.25 = \frac{5(1 - r^5)}{1 - r} \). Rearranging gives \( r^5 = 1 - \frac{31.25(1 - r)}{5} \). (Use numeric methods for specific value).
### **GEOMETRIC PROGRESSION (GP)**
**Key Topics:**
- Identifying geometric sequences
- Sum of finite and infinite geometric series
- Applications in investment and finance
**Exam Questions:**
1. **Question:** Determine the first five terms of a geometric sequence where \( a = 2 \) and \( r = 3 \).
- **Answer:** Terms are \( 2, 6, 18, 54, 162 \).
2. **Question:** Find the sum of the first 6 terms of the GP where \( a = 1 \) and \( r = 2 \).
- **Answer:** \( S_n = \frac{a(1 - r^n)}{1 - r} = \frac{1(1 - 2^6)}{1 - 2} = \frac{1(1 - 64)}{-1} = 63 \).
3. **Question:** Calculate the sum of an infinite GP with \( a = 4 \) and \( |r| < 1 \) where \( r = 0.5 \).
- **Answer:** \( S_{\infty} = \frac{a}{1 - r} = \frac{4}{1 - 0.5} = 8 \).
4. **Question:** If the first term of a geometric series is \( 5 \) and the common ratio is \( 2 \), find the 7th term.
- **Answer:** \( a_n = 5 \times 2^{6} = 5 \times 64 = 320 \).
5. **Question:** Determine the common ratio if the third term of a GP is \( 12 \) and the first term is \( 3 \).
- **Answer:** The third term is given by \( ar^2 = 12 \Rightarrow 3r^2 = 12 \Rightarrow r^2 = 4 \Rightarrow r = 2 \) or \( r = -2 \).
6. **Question:** Find the sum of the first n terms of a GP series with first term \( 7 \) and \( r = 4 \).
- **Answer:** \( S_n = \frac{7(1 - 4^n)}{1 - 4} = \frac{7(1 - 4^n)}{-3} = \frac{7(4^n - 1)}{3} \).
7. **Question:** If the sum of the first four terms of a geometric series is 45 and the first term is \( 5 \), what is the common ratio?
- **Answer:** \( 5(1 + r + r^2 + r^3) = 45 \Rightarrow 1 + r + r^2 + r^3 = 9 \).
### **UNIT FOUR – FINANCIAL MATHEMATICS**
**Key Topics:**
- Simple and compound interest
- Present and future value calculations
- Amortization and loan calculations
**Exam Questions:**
1. **Question:** Compute the simple interest on a loan of $1000 at a rate of 5% per annum for 3 years.
- **Answer:** \( I = P \times r \times t = 1000 \times 0.05 \times 3 = 150 \).
2. **Question:** If you invest $8000 at an interest rate of 6% compounded annually, how much will it be worth in 5 years?
- **Answer:** \( A = P(1 + r)^t = 8000(1 + 0.06)^5 \approx 8000 \times 1.338226 = 10705.81 \).
3. **Question:** Calculate the present value of $5000 due in 3 years at an interest rate of 8%.
- **Answer:** \( PV = \frac{FV}{(1 + r)^t} = \frac{5000}{(1 + 0.08)^3} \approx \frac{5000}{1.259712} \approx 3961.62 \).
4. **Question:** A loan of $10,000 is repaid with equal monthly payments over 3 years at an annual interest rate of 5%. What is the monthly payment?
- **Answer:** Using the formula \( M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \), where \( r = \frac{0.05}{12} \) and \( n = 36 \): Plugging into the formula returns approximately \( 299.71 \).
5. **Question:** How long will it take for a $1000 investment at an interest rate of 10% compounded annually to double in value?
- **Answer:** The Rule of 72 gives a rough estimate \( \frac{72}{10} = 7.2 \) years.
6. **Question:** Find the total amount after investing $4000 at a compound interest rate of 4% for 10 years.
- **Answer:** \( A = 4000(1 + 0.04)^{10} \approx 4000 \times 1.48024 = 5920.96 \).
7. **Question:** If an amount of $3000 is to be paid in 5 years with a 7% interest rate, what is the present value?
- **Answer:** \( PV = \frac{FV}{(1+r)^t} = \frac{3000}{(1.07)^5} \approx \frac{3000}{1.403} \approx 2135.60 \).
### **UNIT FIVE – STATISTICS**
**Key Topics:**
- Descriptive statistics (mean, median, mode)
- Probability concepts
- Simple linear regression
**Exam Questions:**
1. **Question:** Calculate the mean of the data set: {10, 15, 20, 25, 30}.
- **Answer:** \( \text{Mean} = \frac{10 + 15 + 20 + 25 + 30}{5} = \frac{100}{5} = 20 \).
2. **Question:** Determine the median of the data set: {3, 5, 7, 9, 11, 13}.
- **Answer:** Arrange the numbers: {3, 5, 7, 9, 11, 13}. The median is the average of 7 and 9, which is 8.
3. **Question:** What is the mode of the data set: {1, 2, 2, 3, 4, 4, 5}?
- **Answer:** The mode is 2 and 4 (bimodal).
4. **Question:** If the probability of an event occurring is \( 0.2 \), what is the probability of the event not occurring?
- **Answer:** \( P(\text{not } A) = 1 - P(A) = 1 - 0.2 = 0.8 \).
5. **Question:** Given the dataset: {2, 3, 3, 6, 7}, calculate the standard deviation.
- **Answer:** Mean = \( \frac{2+3+3+6+7}{5} = 4.2 \). \( \sigma = \sqrt{\frac{(2-4.2)^2 + (3-4.2)^2 + (3-4.2)^2 + (6-4.2)^2 + (7-4.2)^2}{5}} \approx 1.67 \).
6. **Question:** In a simple linear regression \( y = mx + b \), if \( m = 3 \) and \( b = 1 \), what is \( y \) when \( x = 4 \)?
- **Answer:** \( y = 3(4) + 1 = 12 + 1 = 13 \).
7. **Question:** Calculate the variance for the following data: {4, 8, 6, 5}.
- **Answer:** Mean \( = 5.75 \). Variance \( = \frac{(4-5.75)^2 + (8-5.75)^2 + (6-5.75)^2 + (5-5.75)^2}{4} \approx 2.6875 \).
### **UNIT SIX – MATRICES**
**Key Topics:**
- Basic matrix operations (addition, subtraction, multiplication)
- Determinants and inverse of matrices
- Applications in business problems
**Exam Questions:**
1. **Question:** If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \), calculate \( A + B \).
- **Answer:** \( A + B = \begin{pmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} \).
2. **Question:** Find the product of matrices \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \).
- **Answer:** \( AB = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix} \).
3. **Question:** Calculate the determinant of \( A = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \).
- **Answer:** \( det(A) = (1)(4) - (3)(2) = 4 - 6 = -2 \).
4. **Question:** Find the inverse of the matrix \( A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix} \).
- **Answer:** \( A^{-1} = \frac{1}{-10} \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix} = \begin{pmatrix} -0.6 & 0.7 \\ 0.2 & -0.4 \end{pmatrix} \).
5. **Question:** Solve the system of equations using matrices:
1. \( x + 2y = 3 \)
2. \( 3x + 4y = 10 \)
- **Answer:** Represent as \( AX = B \) where \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, X = \begin{pmatrix} x \\ y \end{pmatrix}, B = \begin{pmatrix} 3 \\ 10 \end{pmatrix} \). Calculate \( A^{-1} \) and find \( X = A^{-1}B \).
6. **Question:** If \( C = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \) and \( D = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \), what is \( C \times D \)?
- **Answer:** \( C \times D = D = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \).
7. **Question:** Verify whether the two matrices \( A \) and \( B \) are invertible.
- **Answer:** Both matrices must have non-zero determinants. If \( det(A) \neq 0 \) and \( det(B) \neq 0 \), the matrices are invertible.
### **UNIT SEVEN – DIFFERENTIATION**
**Key Topics:**
- Basic rules of differentiation
- Application in business problems (e.g., cost, revenue, profit functions)
- Marginal analysis
**Exam Questions:**
1. **Question:** If \( f(x) = x^2 + 3x + 5 \), find \( f'(x) \).
- **Answer:** \( f'(x) = 2x + 3 \).
2. **Question:** Calculate the derivative of \( g(x) = 5x^3 - 2x + 1 \).
- **Answer:** \( g'(x) = 15x^2 - 2 \).
3. **Question:** If \( R(x) = 20x - x^2 \), find the marginal revenue \( R'(x) \) at \( x = 10 \).
- **Answer:** \( R'(x) = 20 - 2x; R'(10) = 20 - 20 = 0 \).
4. **Question:** Determine \( \frac{d}{dx} [\ln(x^2 + 1)] \).
- **Answer:** Using the chain rule, \( \frac{d}{dx} [\ln(u)] = \frac{1}{u} \frac{du}{dx} = \frac{1}{x^2 + 1}(2x) = \frac{2x}{x^2 + 1} \).
5. **Question:** If the cost function is \( C(x) = 3x^3 + 5x^2 + 2 \), find the marginal cost.
- **Answer:** \( C'(x) = 9x^2 + 10x \).
6. **Question:** Compute the derivative of \( P(x) = 4e^x + x^3 \).
- **Answer:** \( P'(x) = 4e^x + 3x^2 \).
7. **Question:** A revenue function is given by \( R(x) = 7x^2 - 4x + 10 \). Find the maximum revenue.
- **Answer:** Set \( R'(x) = 0 \) and solve to find critical points, then test to confirm the maximum.
The outlined structure of topics and corresponding questions prepares students not only to tackle examinations but also to apply these mathematical principles in real-world business scenarios. Focus on these topics, practice solving similar problems, and encourage collaboration and ability to understand concepts for mastering Business Mathematics.