Secondary 1 Mathematics Algebra Functions Quiz

Answer Key — Secondary 1 Mathematics Quiz: Algebra Functions

Total Marks: 40

Note: Answers include teaching explanations for students new to algebraic concepts.

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Section A: Basic Algebraic Concepts

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1. Simplify $3a + 5b - 2a + 7b$. [2 marks]

Answer: $a + 12b$

Working and Explanation:

• Key concept: "Like terms" have exactly the same variable part. We can only combine terms with the same letters.
• Step 1: Group the $a$ terms: $3a - 2a = 1a = a$ [1 mark]
• Step 2: Group the $b$ terms: $5b + 7b = 12b$ [1 mark]
• Final answer: $a + 12b$

Common mistake: Forgetting that $-2a$ means subtract, not that the coefficient is 2. Also, students sometimes try to combine $a$ and $b$ terms — these are unlike terms and cannot be combined.

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2. Evaluate $4x^2 - 3x + 7$ when $x = -2$. [2 marks]

Answer: 29

Working and Explanation:

• Key concept: Substitution means replacing the variable with the given value. Use brackets to keep track of negatives.
• Step 1: Substitute $x = -2$: $4(-2)^2 - 3(-2) + 7$ [0.5 mark for correct substitution]
• Step 2: Evaluate powers first: $(-2)^2 = 4$, so $4 \times 4 = 16$ [0.5 mark]
• Step 3: Evaluate $-3(-2) = +6$ [0.5 mark]
• Step 4: Add: $16 + 6 + 7 = 29$ [0.5 mark]

Common mistake: Writing $-2^2 = -4$ instead of $(-2)^2 = 4$. The brackets around $-2$ are essential — without them, only the 2 is squared.

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3. Expand and simplify: $2(3m + 5) - 3(2m - 4)$ [2 marks]

Answer: 22

Working and Explanation:

• Key concept: Expanding brackets uses the distributive law: multiply each term inside by the factor outside. Watch the negative signs carefully.
• Step 1: Expand first bracket: $2 \times 3m + 2 \times 5 = 6m + 10$ [0.5 mark]
• Step 2: Expand second bracket (note $-3 \times -4 = +12$): $-3 \times 2m + (-3) \times (-4) = -6m + 12$ [0.5 mark]
• Step 3: Combine: $6m + 10 - 6m + 12 = 22$ [1 mark]
• The $m$ terms cancel out, leaving just a constant.

Common mistake: Sign errors with $-3(2m - 4)$. Many students get $-6m - 12$ instead of $-6m + 12$. Remember: negative × negative = positive.

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4. Factorise completely: $6p^2q - 9pq^2$ [2 marks]

Answer: $3pq(2p - 3q)$

Working and Explanation:

• Key concept: Factorise means "write as a product." Find the HCF of all terms, then divide each term by this HCF.
• Step 1: Find HCF of coefficients: HCF of 6 and 9 is 3 [0.5 mark]
• Step 2: Find HCF of $p$ terms: $p^2$ and $p$, so HCF is $p$ [0.5 mark]
• Step 3: Find HCF of $q$ terms: $q$ and $q^2$, so HCF is $q$ [0.5 mark]
• Step 4: Overall HCF is $3pq$; divide each term by $3pq$: $\frac{6p^2q}{3pq} = 2p$ and $\frac{-9pq^2}{3pq} = -3q$
• Result: $3pq(2p - 3q)$ [0.5 mark]

Common mistake: Taking out only the numerical HCF (3) but missing the common variables. Always check: "What do both terms have in common?"

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5. Solve for $x$: $\frac{x}{3} + 5 = 8$ [2 marks]

Answer: $x = 9$

Working and Explanation:

• Key concept: Solve means find the value of $x$ that makes the equation true. Do the same operation to both sides to isolate $x$.
• Step 1: Subtract 5 from both sides: $\frac{x}{3} = 8 - 5 = 3$ [1 mark]
• Step 2: Multiply both sides by 3: $x = 3 \times 3 = 9$ [1 mark]

Checking: $\frac{9}{3} + 5 = 3 + 5 = 8$ ✓

Common mistake: Multiplying by 3 first — this would incorrectly give $x + 5 = 24$. Always deal with additions/subtractions before multiplications/divisions when isolating the variable, or multiply every term by 3.

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6. Write an algebraic expression for: "The product of $n$ and the sum of $n$ and 7." [1 mark]

Answer: $n(n + 7)$ or $n^2 + 7n$

Working and Explanation:

• Key concept: Translate words to math carefully. "Sum of $n$ and 7" = $(n + 7)$. "Product of $n$ and [that sum]" = $n \times (n + 7)$.
• "Sum of $n$ and 7" means $n + 7$ [must have brackets to show it's the whole sum being multiplied]
• "Product of $n$ and" this sum: $n(n + 7)$

Common mistake: Writing $n + n + 7 = 2n + 7$ (confusing "product" with "sum") or $n \times n + 7 = n^2 + 7$ (missing brackets, so only first term is multiplied).

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7. If $a = 3$ and $b = -1$, find the value of $2a^2b - ab^2$. [2 marks]

Answer: $-21$

Working and Explanation:

• Step 1: Substitute carefully with brackets: $2(3)^2(-1) - (3)(-1)^2$ [0.5 mark]
• Step 2: Evaluate powers: $(3)^2 = 9$ and $(-1)^2 = 1$ [0.5 mark]
• Step 3: Calculate each term: $2 \times 9 \times (-1) = -18$ and $3 \times 1 = 3$ [0.5 mark]
• Step 4: Combine: $-18 - 3 = -21$ [0.5 mark]

Common mistake: Sign errors! $(-1)^2 = 1$ (positive), but $2a^2b = 2 \times 9 \times (-1) = -18$ (negative because of the single $b = -1$).

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8. Simplify $\frac{15x^3y}{5xy^2}$. [1 mark]

Answer: $\frac{3x^2}{y}$ or $3x^2y^{-1}$

Working and Explanation:

• Key concept: When dividing powers with the same base, subtract exponents: $\frac{x^m}{x^n} = x^{m-n}$.
• Step 1: Divide coefficients: $\frac{15}{5} = 3$ [0.3 mark]
• Step 2: For $x$: $\frac{x^3}{x^1} = x^{3-1} = x^2$ [0.3 mark]
• Step 3: For $y$: $\frac{y^1}{y^2} = y^{1-2} = y^{-1} = \frac{1}{y}$ [0.4 mark]

Common mistake: Subtracting exponents in wrong order, getting $y^{2-1} = y$ instead of $y^{-1}$. Remember: top exponent minus bottom exponent.

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Section B: Equations and Formulae

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9. Solve: $5(2x - 3) = 2(3x + 7)$ [2 marks]

Answer: $x = \frac{29}{4} = 7.25$

Working and Explanation:

• Step 1: Expand both sides: $10x - 15 = 6x + 14$ [1 mark for correct expansion]
• Step 2: Rearrange: $10x - 6x = 14 + 15$ [bring $x$ to left, numbers to right; add 15 to both sides, subtract 6x from both sides]
• Step 3: $4x = 29$, so $x = \frac{29}{4} = 7.25$ [1 mark]

Checking: LHS = $5(2 \times 7.25 - 3) = 5(14.5 - 3) = 5 \times 11.5 = 57.5$; RHS = $2(3 \times 7.25 + 7) = 2(21.75 + 7) = 2 \times 28.75 = 57.5$ ✓

Common mistake: Sign error expanding to $10x - 15 = 6x + 14$ — wait, that's correct. Actually, common error is $10x - 15 = 6x + 7$ (forgetting to multiply 2 × 7 = 14).

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10. Make $t$ the subject of the formula: $v = u + at$ [2 marks]

Answer: $t = \frac{v - u}{a}$ (where $a \neq 0$)

Working and Explanation:

• Key concept: "Make $t$ the subject" means rearrange so $t =$ [expression without $t$]. This is a physics formula relating velocity ($v$), initial velocity ($u$), acceleration ($a$), and time ($t$).
• Step 1: Subtract $u$ from both sides: $v - u = at$ [1 mark]
• Step 2: Divide both sides by $a$: $t = \frac{v - u}{a}$ [1 mark]

Common mistake: Dividing by $v$ or $u$ instead of isolating the $at$ term first. Always isolate the term containing the target variable before dividing.

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11. Solve the inequality $3(2y - 1) < 21$ and illustrate on the number line. [2 marks]

Answer: $y < 4$

Working and Explanation:

• Key concept: Solving inequalities is like solving equations, EXCEPT when you multiply or divide by a negative number, you must reverse the inequality sign. Here we don't need to, so the method matches equation solving.
• Step 1: Expand: $6y - 3 < 21$ [0.5 mark]
• Step 2: Add 3: $6y < 24$ [0.5 mark]
• Step 3: Divide by 6: $y < 4$ [0.5 mark]
• Number line: Open circle at 4, arrow pointing left (towards smaller numbers) [0.5 mark]

Image pending generation: number_line for Q11.

Common mistake: Using a closed circle (filled dot) at 4. Since the inequality is strict ($<$ not $\leq$), $y = 4$ is NOT a solution. The circle must be open.

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12. The perimeter of a rectangle is 48 cm. If the length is $(3x + 2)$ cm and the breadth is $(2x - 1)$ cm, find $x$. [3 marks]

Answer: $x = 3$

Working and Explanation:

• Key concept: Perimeter of rectangle = $2 \times$ length + $2 \times$ breadth = $2(\text{length} + \text{breadth})$.
• Step 1: Set up equation: $2[(3x + 2) + (2x - 1)] = 48$ [1 mark for correct set-up]
• Step 2: Simplify inside brackets: $(3x + 2) + (2x - 1) = 5x + 1$ [0.5 mark]
• Step 3: So $2(5x + 1) = 48$, giving $10x + 2 = 48$ [0.5 mark]
• Step 4: Solve: $10x = 46$... wait, let me recheck. $2(5x+1) = 10x + 2 = 48$, so $10x = 46$, $x = 4.6$?

Let me recheck: Perimeter = $2L + 2B = 2(3x+2) + 2(2x-1) = 6x+4 + 4x-2 = 10x + 2 = 48$. So $10x = 46$, $x = 4.6$.

Hmm, this is getting messy. Let me recalculate with cleaner numbers, or proceed with $x = 4.6$ if that's correct, or adjust.

Actually: $2(3x+2 + 2x-1) = 2(5x+1) = 10x+2 = 48$. So $10x = 46$, $x = 4.6$. This is valid, just not a "nice" number. Let me verify: Length = 3(4.6)+2 = 15.8, Breadth = 2(4.6)-1 = 8.2. Perimeter = 2(15.8) + 2(8.2) = 31.6 + 16.4 = 48. ✓

I could adjust, but $x = 4.6$ is fine. Let's continue.

• Actually, let me use a cleaner version. I'll adjust the numbers mentally for the answer key to work backwards. If I want $x = 3$: length = 11, breadth = 5, perimeter = 32. Not 48.

Let me recalculate with $x = 5$: length = 17, breadth = 9, perimeter = 52. Still not 48.

With original: $x = 4.6$. I'll present this clearly.

Correction to question setup for cleaner answer: Let's use the values as given and solve exactly.

• Step 1: $2(3x+2) + 2(2x-1) = 48$ or $2[(3x+2)+(2x-1)] = 48$ [1 mark]
• Step 2: Simplify: $2(5x + 1) = 48$ or $10x + 2 = 48$ [1 mark]
• Step 3: $10x = 46$, so $x = 4.6 = \frac{23}{5}$ [1 mark]

Actually, let me verify my expansion once more: $2(2x-1) = 4x - 2$. And $2(3x+2) = 6x+4$. Sum: $10x + 2$. Yes.

Verification: Length = 3(4.6)+2 = 15.8 cm, Breadth = 2(4.6)-1 = 8.2 cm. Perimeter = 2(15.8+8.2) = 2(24) = 48. ✓

Common mistake: Forgetting to multiply both length and breadth by 2, writing $(3x+2)+(2x-1)=48$ directly. Also, sign errors with $-2 \times -1 = -2$ instead of $+2$... wait, $2 \times (2x-1) = 4x-2$, the $-2$ comes from $2 \times (-1) = -2$.

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13. Solve: $\frac{2x - 1}{3} = \frac{x + 5}{2}$ [2 marks]

Answer: $x = 17$

Working and Explanation:

• Key concept: Eliminate fractions by cross-multiplying, or multiply both sides by the LCM of denominators (which is 6).
• Method — Cross-multiplication: $2(2x-1) = 3(x+5)$ [1 mark for correct cross-multiplication]
• Expand: $4x - 2 = 3x + 15$
• Rearrange: $4x - 3x = 15 + 2$
• Solve: $x = 17$ [1 mark]

Checking: LHS = $\frac{2(17)-1}{3} = \frac{33}{3} = 11$; RHS = $\frac{17+5}{2} = \frac{22}{2} = 11$ ✓

Common mistake: Incorrect cross-multiplication like $2(2x-1) = x+5$ (multiplying only one side by 2) or $(2x-1) \times (x+5) = 3 \times 2$ (multiplying numerators together and denominators together — this is wrong!).

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14. Make $a$ the subject of $A = \frac{1}{2}(a + b)h$. [3 marks]

Answer: $a = \frac{2A}{h} - b$ or $a = \frac{2A - bh}{h}$

Working and Explanation:

• Key concept: This is the trapezium area formula. We need to "unravel" the operations step by step, working from the outside in.
• Step 1: Multiply both sides by 2: $2A = (a + b)h$ [1 mark]
• Step 2: Divide both sides by $h$: $\frac{2A}{h} = a + b$ [1 mark]
• Step 3: Subtract $b$: $a = \frac{2A}{h} - b$ [1 mark]

Alternative form: $a = \frac{2A - bh}{h}$ (single fraction)

Common mistake: Dividing by $h$ before multiplying by 2, leading to $\frac{A}{h} = \frac{1}{2}(a+b)$ then struggling with the fraction. Also, forgetting that $(a+b)$ is a group — you must divide the entire $2A$ by $h$, not just part of it.

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Section C: Problem Solving and Applications

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15. A number is multiplied by 4, then 7 is subtracted. The result is 5 more than twice the original number. Find the original number. [2 marks]

Answer: 6

Working and Explanation:

• Key concept: Translate the word problem into an equation by reading carefully and defining the unknown.
• Step 1: Let the number be $n$ [0.5 mark]
• Step 2: "Multiplied by 4, then 7 subtracted": $4n - 7$ [0.5 mark]
• Step 3: "5 more than twice the original number": $2n + 5$ [0.5 mark]
• Step 4: Set equal: $4n - 7 = 2n + 5$
• Step 5: Solve: $2n = 12$, so $n = 6$ [0.5 mark]

Checking: $4(6) - 7 = 24 - 7 = 17$ and $2(6) + 5 = 12 + 5 = 17$ ✓

Common mistake: Reversing the operations ("then 7 is subtracted" means $-7$, not $7-$). Also writing "5 more than twice" as $2(n+5)$ instead of $2n+5$.

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16. Simplify $\frac{2x^2 - 8}{x + 2}$ for $x \neq -2$. [2 marks]

Answer: $2(x - 2)$ or $2x - 4$

Working and Explanation:

• Key concept: Factorise the numerator first, then cancel common factors. The condition $x \neq -2$ is needed because the original expression is undefined at $x = -2$ (division by zero).
• Step 1: Factor out 2 from numerator: $\frac{2(x^2 - 4)}{x + 2}$ [0.5 mark]
• Step 2: Recognise $x^2 - 4$ as difference of squares: $x^2 - 4 = (x + 2)(x - 2)$ [0.5 mark]
• Step 3: So expression becomes: $\frac{2(x + 2)(x - 2)}{x + 2}$ [0.5 mark]
• Step 4: Cancel $(x + 2)$: $2(x - 2) = 2x - 4$ [0.5 mark]

Why difference of squares works: $a^2 - b^2 = (a+b)(a-b)$. Here $x^2 - 4 = x^2 - 2^2 = (x+2)(x-2)$.

Common mistake: Trying to cancel terms instead of factors, e.g., crossing out the $x$ in $2x^2$ with $x$ in $x+2$. You can only cancel factors (entire bracketed expressions multiplying everything else), not individual terms that are added/subtracted.

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17. The sum of three consecutive even numbers is 78. If the smallest is $n$, find the three numbers. [3 marks]

Answer: 24, 26, 28

Working and Explanation:

• Key concept: Consecutive even numbers differ by 2. If the smallest is $n$, the next is $n+2$, then $n+4$.
• Step 1: Set up equation: $n + (n+2) + (n+4) = 78$ [1 mark for correct setup]
• Step 2: Simplify: $3n + 6 = 78$ [0.5 mark]
• Step 3: Solve: $3n = 72$, so $n = 24$ [0.5 mark]
• Step 4: The numbers are: $n = 24$, $n+2 = 26$, $n+4 = 28$ [1 mark for all three correct]

Checking: $24 + 26 + 28 = 78$ ✓

Consecutive even vs odd: If we wanted consecutive odd numbers with first term $n$, they'd still be $n, n+2, n+4$. The difference is that $n$ itself would be odd. The "even" or "odd" property is determined by where you start, not by the gap (which is always 2 for either type of consecutive same-parity numbers).

Common mistake: Using $n, n+1, n+2$ for consecutive even numbers. That gives consecutive integers, not consecutive even integers. The gap must be 2.

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18. Given $T = 2\pi\sqrt{\frac{L}{g}}$, find $L$ when $T = 4$, $\pi = 3$, $g = 10$. [2 marks]

Answer: $L = \frac{40}{9} \approx 4.44$ (or exact: $\frac{40}{9}$)

Working and Explanation:

• Key concept: This is a formula from physics (pendulum period). We need to rearrange to make $L$ the subject first, or substitute immediately and solve. Substitution first is often easier.
• Method — Substitute first:
• Step 1: $4 = 2(3)\sqrt{\frac{L}{10}} = 6\sqrt{\frac{L}{10}}$ [0.5 mark]
• Step 2: Divide by 6: $\frac{4}{6} = \frac{2}{3} = \sqrt{\frac{L}{10}}$ [0.5 mark]
• Step 3: Square both sides: $\frac{4}{9} = \frac{L}{10}$ [0.5 mark]
• Step 4: Solve: $L = \frac{40}{9} = 4\frac{4}{9}$ [0.5 mark]

Alternative — Make $L$ subject first (harder but good practice):

• $T = 2\pi\sqrt{\frac{L}{g}}$
• $\frac{T}{2\pi} = \sqrt{\frac{L}{g}}$
• $\frac{T^2}{4\pi^2} = \frac{L}{g}$
• $L = \frac{gT^2}{4\pi^2} = \frac{10 \times 16}{4 \times 9} = \frac{160}{36} = \frac{40}{9}$

Common mistake: Forgetting to square when eliminating the square root, or squaring only one side. If $\frac{2}{3} = \sqrt{\frac{L}{10}}$, then squaring gives $\frac{4}{9} = \frac{L}{10}$, not $\frac{2}{3} = \frac{L}{10}$ or $\frac{4}{9} = \frac{L^2}{100}$.

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19. Factorise $x^2 - 5x + 6$ and hence solve $x^2 - 5x + 6 = 0$. [2 marks]

Answer: $(x-2)(x-3) = 0$; $x = 2$ or $x = 3$

Working and Explanation:

• Key concept: To factorise $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$. For $x^2 - 5x + 6$, we need numbers multiplying to 6 and adding to -5.
• Step 1: Find factor pair of 6: $-2$ and $-3$ (since $(-2) \times (-3) = 6$ and $(-2) + (-3) = -5$) [0.5 mark]
• Step 2: Factorise: $(x - 2)(x - 3)$ [0.5 mark]
• Step 3: Solve: $x - 2 = 0$ or $x - 3 = 0$, so $x = 2$ or $x = 3$ [1 mark]

The "hence" meaning: "Hence" means "use the factorisation you just found." If

<stage5_quiz_answers_md>
# Answer Key — Secondary 1 Mathematics Quiz: Algebra Functions

**Total Marks:** 40

**Note:** Answers include teaching explanations for students new to algebraic concepts.

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## Section A: Basic Algebraic Concepts

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**1. Simplify $3a + 5b - 2a + 7b$. [2 marks]**

**Answer:** $a + 12b$

**Working and Explanation:**
- **Key concept:** "Like terms" have exactly the same variable part. We can only combine terms with the same letters.
- Step 1: Group the $a$ terms: $3a - 2a = 1a = a$ [1 mark]
- Step 2: Group the $b$ terms: $5b + 7b = 12b$ [1 mark]
- Final answer: $a + 12b$

**Common mistake:** Forgetting that $-2a$ means subtract, not that the coefficient is 2. Also, students sometimes try to combine $a$ and $b$ terms — these are unlike terms and cannot be combined.

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**2. Evaluate $4x^2 - 3x + 7$ when $x = -2$. [2 marks]**

**Answer:** 29

**Working and Explanation:**
- **Key concept:** Substitution means replacing the variable with the given value. Use brackets to keep track of negatives.
- Step 1: Substitute $x = -2$: $4(-2)^2 - 3(-2) + 7$ [0.5 mark for correct substitution]
- Step 2: Evaluate powers first: $(-2)^2 = 4$, so $4 \times 4 = 16$ [0.5 mark]
- Step 3: Evaluate $-3(-2) = +6$ [0.5 mark]
- Step 4: Add: $16 + 6 + 7 = 29$ [0.5 mark]

**Common mistake:** Writing $-2^2 = -4$ instead of $(-2)^2 = 4$. The brackets around $-2$ are essential — without them, only the 2 is squared.

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**3. Expand and simplify: $2(3m + 5) - 3(2m - 4)$ [2 marks]**

**Answer:** 22

**Working and Explanation:**
- **Key concept:** Expanding brackets uses the distributive law: multiply each term inside by the factor outside. Watch the negative signs carefully.
- Step 1: Expand first bracket: $2 \times 3m + 2 \times 5 = 6m + 10$ [0.5 mark]
- Step 2: Expand second bracket (note $-3 \times -4 = +12$): $-3 \times 2m + (-3) \times (-4) = -6m + 12$ [0.5 mark]
- Step 3: Combine: $6m + 10 - 6m + 12 = 22$ [1 mark]
- The $m$ terms cancel out, leaving just a constant.

**Common mistake:** Sign errors with $-3(2m - 4)$. Many students get $-6m - 12$ instead of $-6m + 12$. Remember: negative × negative = positive.

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**4. Factorise completely: $6p^2q - 9pq^2$ [2 marks]**

**Answer:** $3pq(2p - 3q)$

**Working and Explanation:**
- **Key concept:** Factorise means "write as a product." Find the HCF of all terms, then divide each term by this HCF.
- Step 1: Find HCF of coefficients: HCF of 6 and 9 is 3 [0.5 mark]
- Step 2: Find HCF of $p$ terms: $p^2$ and $p$, so HCF is $p$ [0.5 mark]
- Step 3: Find HCF of $q$ terms: $q$ and $q^2$, so HCF is $q$ [0.5 mark]
- Step 4: Overall HCF is $3pq$; divide each term by $3pq$: $\frac{6p^2q}{3pq} = 2p$ and $\frac{-9pq^2}{3pq} = -3q$
- Result: $3pq(2p - 3q)$ [0.5 mark]

**Common mistake:** Taking out only the numerical HCF (3) but missing the common variables. Always check: "What do both terms have in common?"

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**5. Solve for $x$: $\frac{x}{3} + 5 = 8$ [2 marks]**

**Answer:** $x = 9$

**Working and Explanation:**
- **Key concept:** Solve means find the value of $x$ that makes the equation true. Do the same operation to both sides to isolate $x$.
- Step 1: Subtract 5 from both sides: $\frac{x}{3} = 8 - 5 = 3$ [1 mark]
- Step 2: Multiply both sides by 3: $x = 3 \times 3 = 9$ [1 mark]

**Checking:** $\frac{9}{3} + 5 = 3 + 5 = 8$ ✓

**Common mistake:** Multiplying by 3 first — this would incorrectly give $x + 5 = 24$. Always deal with additions/subtractions before multiplications/divisions when isolating the variable, or multiply every term by 3.

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**6. Write an algebraic expression for: "The product of $n$ and the sum of $n$ and 7." [1 mark]**

**Answer:** $n(n + 7)$ or $n^2 + 7n$

**Working and Explanation:**
- **Key concept:** Translate words to math carefully. "Sum of $n$ and 7" = $(n + 7)$. "Product of $n$ and [that sum]" = $n \times (n + 7)$.
- "Sum of $n$ and 7" means $n + 7$ [must have brackets to show it's the whole sum being multiplied]
- "Product of $n$ and" this sum: $n(n + 7)$

**Common mistake:** Writing $n + n + 7 = 2n + 7$ (confusing "product" with "sum") or $n \times n + 7 = n^2 + 7$ (missing brackets, so only first term is multiplied).

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**7. If $a = 3$ and $b = -1$, find the value of $2a^2b - ab^2$. [2 marks]**

**Answer:** $-21$

**Working and Explanation:**
- Step 1: Substitute carefully with brackets: $2(3)^2(-1) - (3)(-1)^2$ [0.5 mark]
- Step 2: Evaluate powers: $(3)^2 = 9$ and $(-1)^2 = 1$ [0.5 mark]
- Step 3: Calculate each term: $2 \times 9 \times (-1) = -18$ and $3 \times 1 = 3$ [0.5 mark]
- Step 4: Combine: $-18 - 3 = -21$ [0.5 mark]

**Common mistake:** Sign errors! $(-1)^2 = 1$ (positive), but $2a^2b = 2 \times 9 \times (-1) = -18$ (negative because of the single $b = -1$).

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**8. Simplify $\frac{15x^3y}{5xy^2}$. [1 mark]**

**Answer:** $\frac{3x^2}{y}$ or $3x^2y^{-1}$

**Working and Explanation:**
- **Key concept:** When dividing powers with the same base, subtract exponents: $\frac{x^m}{x^n} = x^{m-n}$.
- Step 1: Divide coefficients: $\frac{15}{5} = 3$ [0.3 mark]
- Step 2: For $x$: $\frac{x^3}{x^1} = x^{3-1} = x^2$ [0.3 mark]
- Step 3: For $y$: $\frac{y^1}{y^2} = y^{1-2} = y^{-1} = \frac{1}{y}$ [0.4 mark]

**Common mistake:** Subtracting exponents in wrong order, getting $y^{2-1} = y$ instead of $y^{-1}$. Remember: top exponent minus bottom exponent.

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## Section B: Equations and Formulae

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**9. Solve: $5(2x - 3) = 2(3x + 7)$ [2 marks]**

**Answer:** $x = \frac{29}{4} = 7.25$

**Working and Explanation:**
- Step 1: Expand both sides: $10x - 15 = 6x + 14$ [1 mark for correct expansion]
- Step 2: Rearrange: $10x - 6x = 14 + 15$ [bring $x$ to left, numbers to right; add 15 to both sides, subtract 6x from both sides]
- Step 3: $4x = 29$, so $x = \frac{29}{4} = 7.25$ [1 mark]

**Checking:** LHS = $5(2 \times 7.25 - 3) = 5(14.5 - 3) = 5 \times 11.5 = 57.5$; RHS = $2(3 \times 7.25 + 7) = 2(21.75 + 7) = 2 \times 28.75 = 57.5$ ✓

**Common mistake:** Sign error expanding to $10x - 15 = 6x + 14$ — wait, that's correct. Actually, common error is $10x - 15 = 6x + 7$ (forgetting to multiply 2 × 7 = 14).

---

**10. Make $t$ the subject of the formula: $v = u + at$ [2 marks]**

**Answer:** $t = \frac{v - u}{a}$ (where $a \neq 0$)

**Working and Explanation:**
- **Key concept:** "Make $t$ the subject" means rearrange so $t = $ [expression without $t$]. This is a physics formula relating velocity ($v$), initial velocity ($u$), acceleration ($a$), and time ($t$).
- Step 1: Subtract $u$ from both sides: $v - u = at$ [1 mark]
- Step 2: Divide both sides by $a$: $t = \frac{v - u}{a}$ [1 mark]

**Common mistake:** Dividing by $v$ or $u$ instead of isolating the $at$ term first. Always isolate the term containing the target variable *before* dividing.

---

**11. Solve the inequality $3(2y - 1) < 21$ and illustrate on the number line. [2 marks]**

**Answer:** $y < 4$

**Working and Explanation:**
- **Key concept:** Solving inequalities is like solving equations, EXCEPT when you multiply or divide by a negative number, you must reverse the inequality sign. Here we don't need to, so the method matches equation solving.
- Step 1: Expand: $6y - 3 < 21$ [0.5 mark]
- Step 2: Add 3: $6y < 24$ [0.5 mark]
- Step 3: Divide by 6: $y < 4$ [0.5 mark]
- Number line: Open circle at 4, arrow pointing left (towards smaller numbers) [0.5 mark]



> Image pending generation: number_line for Q11.



**Common mistake:** Using a closed circle (filled dot) at 4. Since the inequality is strict ($<$ not $\leq$), $y = 4$ is NOT a solution. The circle must be open.

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**12. The perimeter of a rectangle is 48 cm. If the length is $(3x + 2)$ cm and the breadth is $(2x - 1)$ cm, find $x$. [3 marks]**

**Answer:** $x = 4.6$ or $x = \frac{23}{5}$

**Working and Explanation:**
- **Key concept:** Perimeter of rectangle = $2 \times$ length + $2 \times$ breadth = $2(\text{length} + \text{breadth})$.
- Step 1: Set up equation: $2[(3x + 2) + (2x - 1)] = 48$ or $2(3x+2) + 2(2x-1) = 48$ [1 mark for correct set-up]
- Step 2: Simplify: $2(5x + 1) = 48$ or $10x + 2 = 48$ [1 mark]
- Step 3: Solve: $10x = 46$, so $x = \frac{46}{10} = 4.6 = \frac{23}{5}$ [1 mark]

**Verification:** Length = 3(4.6)+2 = 15.8 cm, Breadth = 2(4.6)-1 = 8.2 cm. Perimeter = 2(15.8) + 2(8.2) = 31.6 + 16.4 = 48. ✓

**Common mistake:** Forgetting to multiply both length and breadth by 2, writing $(3x+2)+(2x-1)=48$ directly. Also, sign errors with $2 \times (-1) = -2$.

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**13. Solve: $\frac{2x - 1}{3} = \frac{x + 5}{2}$ [2 marks]**

**Answer:** $x = 17$

**Working and Explanation:**
- **Key concept:** Eliminate fractions by cross-multiplying, or multiply both sides by the LCM of denominators (which is 6).
- Method — Cross-multiplication: $2(2x-1) = 3(x+5)$ [1 mark for correct cross-multiplication]
- Expand: $4x - 2 = 3x + 15$
- Rearrange: $4x - 3x = 15 + 2$
- Solve: $x = 17$ [1 mark]

**Checking:** LHS = $\frac{2(17)-1}{3} = \frac{33}{3} = 11$; RHS = $\frac{17+5}{2} = \frac{22}{2} = 11$ ✓

**Common mistake:** Incorrect cross-multiplication like $2(2x-1) = x+5$ (multiplying only one side by 2) or $(2x-1) \times (x+5) = 3 \times 2$ (multiplying numerators together and denominators together — this is wrong!).

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**14. Make $a$ the subject of: $A = \frac{1}{2}(a + b)h$ [3 marks]**

**Answer:** $a = \frac{2A}{h} - b$ or $a = \frac{2A - bh}{h}$ (where $h \neq 0$)

**Working and Explanation:**
- **Key concept:** When the target variable appears inside brackets, we need to "unpeel" the layers of operations surrounding it. Work from the outside in.
- Step 1: Multiply both sides by 2: $2A = (a + b)h$ [1 mark]
- Step 2: Divide both sides by $h$: $\frac{2A}{h} = a + b$ [1 mark]
- Step 3: Subtract $b$ from both sides: $a = \frac{2A}{h} - b$ [1 mark]

**Common mistake:** 
- Dividing by $h$ before multiplying by 2, getting $\frac{2A}{h} = \frac{1}{2}(a+b)$ which is wrong.
- Forgetting that $(a+b)$ is a single quantity being multiplied by $h$; cannot divide individual terms by $h$ separately.

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## Section C: Problem Solving and Applications

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**15. A number is multiplied by 4, then 7 is subtracted. The result is 5 more than twice the original number. Find the original number. [2 marks]**

**Answer:** The original number is 4.

**Working and Explanation:**
- **Key concept:** "Let $x$ be the unknown" is the standard approach. Read the words carefully and translate phrase by phrase.
- Step 1: Let the number be $x$ [0.5 mark for defining variable]
- Step 2: Translate: "multiplied by 4, then 7 is subtracted" → $4x - 7$ [0.5 mark]
- Step 3: Translate: "5 more than twice the original number" → $2x + 5$ [0.5 mark]
- Step 4: Set equal: $4x - 7 = 2x + 5$
- Solve: $2x = 12$, so $x = 4$ [0.5 mark]

**Checking:** LHS = $4(4) - 7 = 16 - 7 = 9$; RHS = $2(4) + 5 = 8 + 5 = 13$? Wait, let me recheck...

Actually: $4(4) - 7 = 16 - 7 = 9$. And $2(4) + 5 = 13$. These don't match! Let me re-solve.

$4x - 7 = 2x + 5$
$4x - 2x = 5 + 7$
$2x = 12$
$x = 6$

Let me verify: $4(6) - 7 = 24 - 7 = 17$. And $2(6) + 5 = 17$. ✓

**Corrected Answer:** The original number is 6.

---

**16. Simplify $\frac{2x^2 - 8}{x + 2}$ for $x \neq -2$. [2 marks]**

**Answer:** $2x - 4$ or $2(x - 2)$

**Working and Explanation:**
- **Key concept:** Factorise the numerator first, then look for common factors with the denominator that cancel.
- Step 1: Factorise numerator by taking out common factor of 2: $2(x^2 - 4)$ [0.5 mark]
- Step 2: Recognise difference of squares: $x^2 - 4 = (x + 2)(x - 2)$ [0.5 mark]
- Step 3: So numerator = $2(x + 2)(x - 2)$ [0.5 mark]
- Step 4: Cancel $(x + 2)$ with denominator: $\frac{2(x+2)(x-2)}{x+2} = 2(x-2) = 2x - 4$ [0.5 mark]

**Checking with $x = 3$:** Original = $\frac{18-8}{5} = 2$. Simplified = $2(3)-4 = 2$. ✓

**Common mistake:** Trying to split the fraction as $\frac{2x^2}{x} + \frac{-8}{2}$ — this is completely wrong! You cannot split a fraction across addition in the numerator like that. Must factorise first.

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**17. The sum of three consecutive even numbers is 78. If the smallest number is $n$, write an equation and solve for the three numbers. [3 marks]**

**Answer:** The three numbers are 24, 26, and 28.

**Working and Explanation:**
- **Key concept:** Consecutive even numbers differ by 2, not by 1. If the smallest is $n$, the next is $n + 2$, and the largest is $n + 4$.
- Step 1: Express three consecutive even numbers: $n$, $(n + 2)$, $(n + 4)$ [0.5 mark]
- Step 2: Set up equation: $n + (n + 2) + (n + 4) = 78$ [1 mark]
- Step 3: Simplify: $3n + 6 = 78$ [0.5 mark]
- Step 4: Solve: $3n = 72$, so $n = 24$ [0.5 mark]
- Step 5: Three numbers: 24, 26, 28 [0.5 mark]

**Checking:** $24 + 26 + 28 = 78$ ✓

**Common mistake:** Using $n$, $n+1$, $n+2$ (consecutive integers, not even numbers). Also, solving for $n$ but not stating all three numbers as the final answer.

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**18. Given that $T = 2\pi\sqrt{\frac{L}{g}}$, find the value of $L$ when $T = 4$, $\pi = 3$, and $g = 10$. [2 marks]**

**Answer:** $L = \frac{40}{9} \approx 4.44$ (or exactly $\frac{40}{9}$)

**Working and Explanation:**
- **Key concept:** Substitute given values, then rearrange to solve for the unknown. This is a physics formula for period of a pendulum.
- Step 1: Substitute: $4 = 2(3)\sqrt{\frac{L}{10}} = 6\sqrt{\frac{L}{10}}$ [0.5 mark]
- Step 2: Divide by 6: $\frac{4}{6} = \frac{2}{3} = \sqrt{\frac{L}{10}}$ [0.5 mark]
- Step 3: Square both sides: $\frac{4}{9} = \frac{L}{10}$ [0.5 mark]
- Step 4: Solve: $L = \frac{40}{9} = 4\frac{4}{9} \approx 4.44$ [0.5 mark]

**Common mistake:** Squaring before isolating the square root, getting confused with order of operations. Also, calculator error: $\frac{4}{6} = 0.666...$, and $0.666...^2 = 0.444...$, then $L = 4.444...$

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**19. Factorise $x^2 - 5x + 6$ and hence solve $x^2 - 5x + 6 = 0$. [2 marks]**

**Answer:** $(x - 2)(x - 3) = 0$; solutions $x = 2$ or $x = 3$

**Working and Explanation:**
- **Key concept:** For quadratic $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.
- Step 1: Find factor pair of 6 that adds to $-5$: $-2$ and $-3$ (since $-2 \times -3 = 6$ and $-2 + (-3) = -5$) [0.5 mark]
- Step 2: Factorise: $(x - 2)(x - 3)$ [0.5 mark]
- Step 3: Solve using null factor law: $x - 2 = 0$ or $x - 3 = 0$ [0.5 mark]
- Step 4: Solutions: $x = 2$ or $x = 3$ [0.5 mark]

**Checking:** $(2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$ ✓; $(3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0$ ✓

**Common mistake:** Writing $(x + 2)(x + 3)$ or $(x - 6)(x + 1)$ — check that your factors multiply to give the original expression. Also, forgetting to write "or" between solutions; they are separate answers, not $x = 2 = 3$!

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**20. Rectangle with quarter-circle removed. [3 marks total]**

**(a) Find an expression for the area of the shaded region. [1 mark]**

**Answer:** $(2x + 5)(x + 3) - \frac{1}{4}\pi x^2$ or $2x^2 + 11x + 15 - \frac{\pi x^2}{4}$

**Working and Explanation:**
- **Key concept:** Area of shaded region = Area of rectangle − Area of quarter-circle.
- Step 1: Area of rectangle = length × breadth = $(2x + 5)(x + 3)$ [0.5 mark]
- Step 2: Area of quarter-circle = $\frac{1}{4} \times \pi \times r^2 = \frac{1}{4}\pi x^2$ [0.5 mark]
- Shaded area = $(2x + 5)(x + 3) - \frac{\pi x^2}{4}$

If expanded: $(2x + 5)(x + 3) = 2x^2 + 6x + 5x + 15 = 2x^2 + 11x + 15$

**(b) Find the area when $x = 2$ and $\pi = 3.14$. [2 marks]**

**Answer:** Approximately 37.86 cm²

**Working and Explanation:**
- Step 1: Substitute $x = 2$ into rectangle area: $(2(2) + 5)(2 + 3) = (4 + 5)(5) = 9 \times 5 = 45$ cm² [0.5 mark]
- Step 2: Substitute into quarter-circle area: $\frac{1}{4} \times 3.14 \times (2)^2 = \frac{1}{4} \times 3.14 \times 4 = 3.14$ cm² [0.5 mark]
- Step 3: Subtract: $45 - 3.14 = 41.86$? Let me recheck...

Wait, let me recalculate: $(2(2)+5)(2+3) = (9)(5) = 45$. And $\frac{1}{4} \times 3.14 \times 4 = 3.14$.

Actually: $45 - 3.14 = 41.86$. Hmm, but let me verify with the expression from part (a).

Actually I had $2x^2 + 11x + 15 - \frac{\pi x^2}{4}$:
- $2(4) + 11(2) + 15 - \frac{3.14 \times 4}{4} = 8 + 22 + 15 - 3.14 = 45 - 3.14 = 41.86$

**Answer:** 41.86 cm² (accept 41.9 or 42 if rounded, or exact forms)

[1 mark for correct substitution, 1 mark for final answer with units]

**Common mistake:** 
- Using diameter instead of radius for the circle ($r = x = 2$, not $2x = 4$).
- Forgetting to take quarter of the circle area (using full circle area $\pi r^2$ instead).
- Arithmetic errors with substitution — write out each step clearly.

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*Syllabus reference: N1.3, N2.1-N2.2, A1.1-A1.6, A2.1-A2.3, A3.1-A3.2, A4.1-A4.3 (Lower Secondary G3 Mathematics Syllabus 2020)*

*Note: This answer key provides detailed explanations suitable for student self-study or teacher-led review. Mark allocations follow standard Secondary 1 assessment conventions where method marks are awarded for correct working even if final answer contains arithmetic errors.*