GCE Mathematics
Past Questions

AI Explanation

**The reasoning** This is a simple linear equation. To find x, we need to isolate it on one side. We have: 5x = 35 Divide both sides by 5: x = 35 ÷ 5 = 7 **Principle: Inverse operations** — Whatever you do to one side of an equation, do to the other. Since x is multiplied by 5, we divide by 5 to "undo" it. **Why the wrong options tempt you** - **Option A (5):** You might confuse the coefficient (the number multiplying x) with the answer itself. Don't just pick numbers you see! - **Option B (6):** This could come from careless mental math — maybe rushing and thinking 5 × 6 = 30, then adjusting wrongly. - **Option D (8):** Another mental math slip, perhaps confusing it with 40 ÷ 5. The trap is **not verifying your answer**. Always check: Does 5 × 7 = 35? Yes! **Quick takeaway** To solve ax = b, divide both sides by a to get x = b/a — then always substitute back to confirm your answer works.

AI Explanation

**The reasoning** The formula for the area of a square is: **Area = side × side = side²** Since each side is 6 units: Area = 6 × 6 = **36 square units** This works because a square has all four sides equal, and area measures how many unit squares fit inside. When you multiply length by width (which are the same in a square), you get the total space covered. **Why the wrong options tempt you** - **Option A (12):** You might add instead of multiply: 6 + 6 = 12. But area needs multiplication, not addition! - **Option B (24):** This is the *perimeter* (adding all four sides: 6 + 6 + 6 + 6 = 24). Easy mix-up if you're rushing. - **Option D (48):** Maybe you multiplied 6 × 8 or doubled the perimeter by mistake. **Quick takeaway** For area of a square, always **square the side** (side × side), never add. Perimeter adds, area multiplies!

AI Explanation

**The reasoning** A rectangle has **two lengths** and **two widths**. The perimeter is the total distance around the shape. Formula: **Perimeter = 2(length + width)** Given: length = 6, width = 4 Perimeter = 2(6 + 4) = 2(10) = **20** Alternatively, add all four sides: 6 + 6 + 4 + 4 = 20 --- **Why the wrong options tempt you** **A) 10** — You only added length + width once (6 + 4), forgetting a rectangle has *four* sides, not two. **C) 24** — You multiplied 6 × 4, which gives the **area** (space inside), not the perimeter (distance around). **D) 30** — You might have added 6 + 6 + 6 + 6 + 6, treating all sides as 6, or made a calculation error. --- **Quick takeaway** **Perimeter = add all sides; Area = multiply length × width.** For rectangles, always remember you're going around *four* sides: two long, two short. Double-check which the question asks for!

AI Explanation

**The reasoning** Consecutive odd numbers follow a pattern: each one is 2 more than the previous. Starting from 3: - 1st odd number: 3 - 2nd odd number: 3 + 2 = 5 - 3rd odd number: 5 + 2 = 7 - 4th odd number: 7 + 2 = 9 Now sum them: 3 + 5 + 7 + 9 = **24** **Quick formula**: For consecutive odd numbers starting at *a*, the sum of 4 numbers = 4a + 12. Here: 4(3) + 12 = 24 ✓ **Why the wrong options tempt you** - **16 (A)**: You might have added only three numbers (3 + 5 + 7 = 15) or miscounted. - **20 (B)**: Common if you added consecutive *even* numbers instead (2 + 4 + 6 + 8 = 20) or made an arithmetic slip. - **28 (D)**: You may have started from 4 instead of 3, giving 4 + 6 + 8 + 10 = 28. **Quick takeaway** Consecutive odd numbers jump by 2 each time—list them carefully, then add: 3, 5, 7, 9 always gives 24.

AI Explanation

**The reasoning** This is a simple linear equation. To find x, we need to isolate it on one side. Starting with: x − 7 = 5 Add 7 to both sides (we're "undoing" the subtraction): x − 7 + 7 = 5 + 7 x = 12 **Principle**: Whatever you do to one side of an equation, do to the other to keep it balanced. **Why the wrong options tempt you** - **Option A (5)**: You might think x equals the number on the right side. Don't just copy what you see! - **Option C (7)**: You're tempted to pick the number being subtracted. That's the trap — 7 is what we're taking away *from* x, not x itself. - **Option D (0)**: Maybe you subtracted wrong (5 − 7)? Remember, we're solving for x, not calculating 5 − 7. **Quick takeaway** When solving "x − something = answer," always **add that 'something' to both sides** to find x. The answer is bigger than what's on the right!

AI Explanation

## The reasoning When expanding two brackets, we use the **FOIL method** (First, Outer, Inner, Last) or simply "multiply everything in the first bracket by everything in the second." Let's work through **(x + 2)(x + 3)**: - **First terms**: x × x = x² - **Outer terms**: x × 3 = 3x - **Inner terms**: 2 × x = 2x - **Last terms**: 2 × 3 = 6 Now combine: x² + 3x + 2x + 6 = **x² + 5x + 6** Notice the middle term (5x) comes from adding the two "x" terms together (3x + 2x), and the constant (6) is just multiplying the two numbers (2 × 3). ## Why the wrong options tempt you **Option B (x² + 6x + 5)**: You swapped the coefficient and constant — maybe you multiplied 2 × 3 = 6 but placed it with x instead. **Option C (x² + 5x + 5)**: You forgot to multiply the constants properly (2 × 3 = 6, not 5). **Option D (x² + 6)**: You completely missed the middle term! Never forget to add those outer and inner products. ## Quick takeaway **Middle term = sum of the numbers (2 + 3 = 5), constant = product of the numbers (2 × 3 = 6).**

AI Explanation

3x = 21 → x = 7.

AI Explanation

Area = πr² = 22/7 × 49 = 154 cm².

AI Explanation

Add: 3x = 9 → x = 3. Substitute: 3 − y = 1 → y = 2.

AI Explanation

log₁₀ 1000 = log₁₀ 10³ = 3.

AI Explanation

Sum = (n − 2) × 180 = 3 × 180 = 540°.

AI Explanation

26 red cards out of 52 = 1/2.

AI Explanation

dy/dx: derivative of 3x² = 6x; derivative of 5x = 5. Total: 6x + 5.

AI Explanation

Sum = 40 ÷ 5 = 8.

AI Explanation

If the cost price is CP, then selling price = CP × (1 + 20/100) = 1.2 × CP = 3600, so CP = 3600 ÷ 1.2 = ₦3,000. Option A (₦2,880) results from subtracting 20% of the selling price instead of the cost price, and D is the result of adding 20% to the selling price.

AI Explanation

Expanding 11011₂: (1×2⁴) + (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 16 + 8 + 0 + 2 + 1 = 27. Common errors include miscounting bit positions or omitting the zero-value term, leading to distractors like 25 or 31.

AI Explanation

In 0.003040, the leading zeros (0.00) are not significant. The digits 3, 0, 4, and the trailing zero after 4 are all significant, giving 4 significant figures. The trailing zero after the decimal point is significant because it indicates precision, but the leading zeros before 3 are merely placeholders.

AI Explanation

To write 0.000407 in standard form, move the decimal point 4 places to the right to get 4.07, giving 4.07 × 10⁻⁴. Option B incorrectly uses an exponent of -3, while C is not in proper standard form (coefficient must be between 1 and 10), and D has a positive exponent which would make the number large.

AI Explanation

Using the logarithm product rule: log₁₀ 8 + log₁₀ 125 = log₁₀ (8 × 125) = log₁₀ 1000 = log₁₀ 10³ = 3. The product 8 × 125 = 1000 = 10³, so the answer is 3.

AI Explanation

From log_x 64 = 3, we have x³ = 64. Taking the cube root of both sides: x = ∛64 = 4. Checking: 4³ = 64 ✓. Option A (2) gives 2³ = 8, not 64; option C (8) gives 8³ = 512, not 64.