Grade 10 — Term 2

LCM of Algebraic Expressions

What you must know

• HCF vs LCM confusion: Highest Common Factor (HCF) takes common factors with LOWEST powers; LCM takes ALL factors with HIGHEST powers.
• Factorise binomials first: Before finding LCM of (x² - 4) and (x + 2)², factorise them into (x - 2)(x + 2) and (x + 2)²; LCM is (x - 2)(x + 2)².
• Leave LCM in factored form: In Paper 1 Part A, it is standard practice to leave the algebraic LCM in factorised form; do not waste time expanding brackets.

Try one question

Find the Least Common Multiple (LCM) of the algebraic expressions: 4x²y, 6xy³, and 8x³y².

1. A. 12x³y³
2. B. 24x²y
3. C. 48x²y²
4. D. 24x³y³

A model question written by Idasara on the exam pattern. It is not from a past paper.

Algebraic Fractions

What you must know

• The illegal cancellation blunder: In (x + 2) / (x + 5), NEVER cancel the x's; terms separated by + or - cannot be cancelled individually.
• Sign change trap in subtraction: In (a/b) - ((c - d)/e), the negative sign applies to the ENTIRE numerator: - (c - d) = -c + d.
• Restricted denominator values: Algebraic fractions are undefined when the denominator equals zero (e.g., 3 / (x - 2) is undefined when x = 2).

Try one question

Simplify the sum of algebraic fractions: 3/(2x) + 5/(3x).

1. A. 19/(6x)
2. B. 15/(6x²)
3. C. 19/(5x)
4. D. 8/(5x)

A model question written by Idasara on the exam pattern. It is not from a past paper.

Percentages

What you must know

• Cost Price vs Selling Price baseline trap: Profit and loss percentages MUST be calculated against Cost Price, NEVER against Selling Price (unless explicitly requested).
• Quarterly rates division: Remember that municipal rates are paid in 4 equal quarterly instalments; divide the annual rate by 4 to find the quarterly payment.
• Time in years for simple interest: If time is given in months, divide by 12 (e.g., 9 months = 9/12 = 3/4 years).

Try one question

A trader buys a television for Rs. 50,000 and sells it at a profit of 15%. What is the selling price of the television?

1. A. Rs. 55,000
2. B. Rs. 57,500
3. C. Rs. 60,000
4. D. Rs. 58,000

A model question written by Idasara on the exam pattern. It is not from a past paper.

Equations

What you must know

• Sign-checking in elimination: If matching coefficients have IDENTICAL signs, SUBTRACT the equations; if they have OPPOSITE signs, ADD the equations.
• Verification step: Always substitute your calculated values of x and y back into BOTH original equations to confirm they satisfy both.
• Do not stop at finding x: A frequent error in Paper 1 Part B is finding x and forgetting to substitute back to calculate y and state contextual quantities.

Parallelograms I

What you must know

• Sufficient conditions to prove a quadrilateral is a parallelogram: Show ONE pair of opposite sides is BOTH equal and parallel, OR both pairs of opposite sides are equal, OR diagonals bisect each other.
• Formal geometric citations: When using parallelogram properties in riders, cite 'opp. sides of //gram =', 'opp. angles of //gram =', or 'diagonals of //gram bisect'.
• Consecutive interior angles: Remember that consecutive angles add up to 180° because they are allied (co-interior) angles between parallel lines.

Try one question

In a parallelogram ABCD, angle A = 70°. Find the magnitude of angle B.

1. A. 90°
2. B. 140°
3. C. 70°
4. D. 110°

A model question written by Idasara on the exam pattern. It is not from a past paper.

Parallelograms II

What you must know

• Rhombus diagonal properties: The diagonals of a rhombus meet at 90°, creating four congruent right-angled triangles; this enables applying Pythagoras' theorem to find side length from diagonals.
• Rectangle diagonals vs Rhombus diagonals: Rectangle diagonals are EQUAL but not perpendicular; rhombus diagonals are PERPENDICULAR but not equal in length.
• Proving a figure is a square: First prove it is a rectangle, then prove adjacent sides are equal (or diagonals are perpendicular).

Try one question

Which of the following conditions is sufficient to prove that a quadrilateral is a parallelogram?

1. A. The diagonals are perpendicular to each other.
2. B. Two adjacent sides are equal in length.
3. C. One pair of opposite sides is both equal and parallel.
4. D. One pair of opposite angles is equal.

A model question written by Idasara on the exam pattern. It is not from a past paper.

Sets

What you must know

• The double-counting trap: In word problems, 'students who like tea' includes those who like BOTH tea and coffee; students who like ONLY tea = n(A) - n(A ∩ B).
• Venn diagram region labelling: Fill the intersection region n(A ∩ B) first before populating the 'only' crescent regions.
• Outside the circles region: Never forget elements belonging to neither set: n((A ∪ B)') = n(ε) - n(A ∪ B).

Try one question

In a class of 40 students, 25 study Science, 20 study Commerce, and 5 study neither subject. How many students study both subjects?

1. A. 15
2. B. 10
3. C. 5
4. D. 12

A model question written by Idasara on the exam pattern. It is not from a past paper.

Logarithms I

What you must know

• The pseudo-distributive logarithm error: log(x + y) NEVER equals log x + log y; similarly log(x - y) NEVER equals log x - log y.
• Quotient law vs fraction of logs: log(x / y) = log x - log y, which is completely different from (log x) / (log y).
• Converting between forms: Master switching rapidly between index form (2⁵ = 32) and log form (log₂ 32 = 5) for Paper 1 Part A questions.

Try one question

Convert the logarithmic equation log₃(81) = 4 into its equivalent index (exponential) form.

1. A. 3⁴ = 81
2. B. 4³ = 81
3. C. 81³ = 4
4. D. 3 × 4 = 81

A model question written by Idasara on the exam pattern. It is not from a past paper.

Logarithms II

What you must know

• Bar notation representation: A negative characteristic is written with a bar over it (e.g., 2̄.4532 = -2 + 0.4532 = -1.5468); never write -2.4532 because mantissa MUST remain positive.
• Subtracting logs with bar numbers: When subtracting a bar number, remember that subtracting a negative becomes an addition: 1.2 - 2̄.5 = 1.2 - (-2 + 0.5) = 1.2 + 2 - 0.5 = 2.7.
• Four-figure table mean differences: Add mean difference values to the main table reading; never subtract mean differences.

Try one question

Given that log₁₀(2) = 0.3010 and log₁₀(3) = 0.4771, calculate the value of log₁₀(18).

1. A. 1.5562
2. B. 0.7781
3. C. 1.2552
4. D. 1.0791

A model question written by Idasara on the exam pattern. It is not from a past paper.

Linear Relationships

What you must know

• Gradient subtraction order consistency: In m = (y₂ - y₁) / (x₂ - x₁), maintain the same point order in numerator and denominator; mixing them flips the gradient sign.
• Positive vs negative gradient: A line sloping upwards from left to right has m > 0; a line sloping downwards has m < 0; horizontal line has m = 0; vertical line has undefined gradient.
• Rearranging to standard form: Before reading off m and c, always rearrange the equation into subject y form: from 2y - 4x = 6 to y = 2x + 3 (m = 2, c = 3).

Try one question

What is the gradient (m) and y-intercept (c) of the straight line represented by 2y - 6x = 8?

1. A. m = 3, c = 4
2. B. m = -3, c = 4
3. C. m = 3, c = 8
4. D. m = 6, c = 8

A model question written by Idasara on the exam pattern. It is not from a past paper.