MTH101 supports the mathematical reasoning needed in scientific and computing programmes. NOUN lists elementary set theory, algebra and trigonometry among its subject areas. Successful revision depends less on memorising final answers and more on understanding why each algebraic step is allowed.
These examples are independently developed teaching exercises, not copied NOUN examination questions. The exact course edition may organise topics differently, so use your official course material for the required unit order and notation.
What you will be able to do
- Find unions, intersections and complements for finite sets
- Solve a quadratic equation by factorisation and check roots
- Interpret the difference between degrees and radians
- Use a disciplined approach to multi-step mathematical questions
Lessons with worked examples
1. Sets and Venn-diagram reasoning
A set is a well-defined collection of distinct objects. If A and B are sets, A ∪ B means all elements that appear in either set, while A ∩ B means elements appearing in both. A \ B means elements in A that are not in B; reversing that order changes the result.
For a finite universal set U, the complement Aᶜ contains the elements of U that are not in A. In a two-set counting problem, avoid counting the overlap twice. The correct relationship is |A ∪ B| = |A| + |B| − |A ∩ B|.
- Write set members only once, even if repeated in the input list.
- Pay attention to the direction of set difference.
- For counting problems, subtract the overlap once.
Worked example — union, intersection and difference
Let A = {1, 2, 4, 6} and B = {2, 3, 6}. Then A ∪ B = {1, 2, 3, 4, 6}; A ∩ B = {2, 6}; A \ B = {1, 4}; B \ A = {3}. Check: 4 + 3 − 2 = 5 unique elements in the union.
Takeaway: The same two sets can have different differences depending on subtraction order.
2. Quadratic equations: reason before calculating
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Factorisation is convenient when the expression splits into simple factors. If an expression does not factor neatly, the quadratic formula provides a general method: x = (−b ± √(b² − 4ac)) / (2a).
The discriminant, b² − 4ac, indicates the number and nature of real roots: positive means two distinct real roots; zero means one repeated real root; negative means no real roots. Always substitute your solutions back into the original equation.
- Put the equation in standard form before finding a, b and c.
- Factor only when the factorisation can be checked by expanding.
- Substitute proposed roots to catch sign mistakes.
Worked example — factorisation
Solve x² − 5x + 6 = 0. Find two numbers with product +6 and sum −5: −2 and −3. Thus (x − 2)(x − 3) = 0, giving x = 2 or x = 3. Check x = 2: 4 − 10 + 6 = 0; x = 3: 9 − 15 + 6 = 0.
Takeaway: Checking both values confirms that they satisfy the original equation.
3. Angles, radians and fundamental ratios
In trigonometry, 180° equals π radians, so convert degrees to radians by multiplying by π/180 and radians to degrees by multiplying by 180/π. A right triangle supplies sine = opposite/hypotenuse, cosine = adjacent/hypotenuse and tangent = opposite/adjacent for an acute angle.
For angles outside the first quadrant, the sign of a trigonometric function depends on the quadrant. A drawing or unit-circle sketch is often safer than applying a memorised sign rule without checking the angle.
- Label the sides of a right triangle relative to the chosen angle.
- Use degree or radian mode consistently in your calculator.
- Check whether your numeric result is reasonable before finalising.
Worked example — angular conversion
Convert 60° to radians: 60 × π/180 = π/3 radians. Convert π/6 radians to degrees: (π/6) × 180/π = 30°. If a calculator gives an unexpected result for sin 30°, confirm that it is in degree mode.
Takeaway: Correct unit conversion is part of the mathematics, not merely calculator setup.
4. A repeatable method for solving mathematics questions
Mathematics assessment becomes more reliable when each solution is presented as a sequence of justified steps. First identify what is given and required. Next write a definition or formula, substitute carefully and perform an independent check.
Keep an error log with categories such as reading error, wrong formula, arithmetic slip, sign mistake and calculator mode. A student repeatedly losing marks through sign mistakes needs a different intervention from one who does not understand the underlying concept.
- Write a labelled known/unknown list for word problems.
- Perform calculations in a clear order and retain intermediate steps.
- Explain why the answer makes sense in the problem context.
Worked example — diagnosing a wrong answer
Suppose a student writes A \ B = {3} for the sets in the earlier example. The arithmetic is not the problem; the student reversed the order. Correcting the concept of set difference prevents the same error in many questions.
Takeaway: The best correction addresses the cause of the mistake, not just the final number.
Apply what you learned — MTH101 Practice Clinic
Original independent learning exercise. Try it before opening the worked explanation; this is not an official NOUN examination question.
Use inclusion–exclusion to count students
Situation: In a study group, 32 students study mathematics, 25 study statistics and 12 study both. Another 8 students study neither subject.
Your task: Find how many study at least one subject, how many study exactly one and the total group size.
Reveal the step-by-step solution
- At least one = 32 + 25 − 12 = 45 students, because the overlap was counted twice.
- Mathematics only = 32 − 12 = 20.
- Statistics only = 25 − 12 = 13.
- Exactly one = 20 + 13 = 33; total group = 45 + 8 = 53.
Answer: At least one: 45; exactly one: 33; total group: 53.
Common mistake to avoid: Do not add the overlap a second time when counting the union.
Test yourself — original practice questions
These are independently created learning questions, not official NOUN TMA or examination questions. Try each one before revealing the explanation.
Question 1. If A = {1,3,5} and B = {3,4}, what is A ∩ B?
- {1,4}
- {3}
- {1,3,4,5}
- {5}
Show answer and explanation
Answer B: {3}
The only shared member is 3.
Question 2. Solve x² − 7x + 10 = 0.
- x = 1 or 10
- x = −2 or −5
- x = 2 or 5
- x = 0 or 7
Show answer and explanation
Answer C: x = 2 or 5
Factor as (x−2)(x−5) = 0.
Question 3. What is 90° in radians?
- π/2
- π
- 2π
- π/4
Show answer and explanation
Answer A: π/2
90 × π/180 = π/2.
Question 4. How many elements are in A ∪ B if |A|=7, |B|=6 and |A ∩ B|=2?
- 15
- 13
- 11
- 9
Show answer and explanation
Answer C: 11
Use 7+6−2=11.
Three things to practise this week
- Write three set-difference examples and reverse the subtraction order for each.
- Solve two quadratics, then verify every root by substitution.
- Convert 30°, 45°, 90° and 180° into radians without a calculator.
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Verify the course with NOUN
Check your current official course material for assessment rules, units and session-specific instructions. Institutional pages below establish course identity and broad subject context; they are not citations for our original worked exercises.
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