WAEC Mathematics
Past Questions

AI Explanation

## The reasoning This is about **factoring before simplifying rational expressions**. First, look at the numerator: 3x² - 12. Notice both terms share a common factor of 3: 3x² - 12 = 3(x² - 4) Now, x² - 4 is a **difference of two squares** (a² - b² = (a+b)(a-b)), so: x² - 4 = (x + 2)(x - 2) Therefore: 3x² - 12 = 3(x + 2)(x - 2) Now substitute back: (3x² - 12)/(x - 2) = [3(x + 2)(x - 2)]/(x - 2) Cancel the common factor (x - 2): = 3(x + 2) ✓ ## Why the wrong options tempt you **B) 3(x - 2)** — You might cancel incorrectly and keep the wrong factor. **C) x + 2** — You forgot to carry the 3 from the factoring step. **D) 3x + 6** — This *equals* 3(x + 2), but the question asks you to simplify, and factored form is simpler. ## Quick takeaway Always factor completely *before* canceling—look for common factors first, then special patterns like difference of squares!

AI Explanation

## The reasoning The **volume of a cylinder formula** is what you need here: V = πr²h Let's substitute step by step: - radius (r) = 7 cm - height (h) = 10 cm - π = 22/7 V = (22/7) × 7² × 10 V = (22/7) × 49 × 10 V = 22 × 7 × 10 (the 7s cancel!) V = 1,540 cm³ The key is remembering to **square the radius first** (7² = 49), then multiply everything together. ## Why the wrong options tempt you **Option B (770)** — You probably forgot to square the radius and just did 22/7 × 7 × 10. Always square r! **Option C (440)** — Maybe you used 2πr instead of πr², mixing up circumference formula with volume. **Option D (220)** — You likely divided somewhere instead of multiplying, or forgot the height entirely. ## Quick takeaway For cylinder volume, think **"Pi-R-Squared-H"** — say it like a chant, and never forget to square that radius before multiplying by height!

AI Explanation

**The reasoning** The **mode** is simply the number that appears *most frequently* in a dataset. Let me count each number's appearances: - 2 appears **1 time** - 3 appears **3 times** ✓ - 5 appears **2 times** - 7 appears **1 time** - 9 appears **1 time** Since 3 shows up three times (more than any other number), **3 is the mode**. **Why the wrong options tempt you** **A) 2** — This is the smallest number (the minimum), not the most frequent. Don't confuse "first" with "most common." **C) 5** — It appears twice, which seems significant, but 3 appears even more often. You might pick this if you stop counting too soon. **D) 7** — This is near the middle value, but mode has nothing to do with position or average — only *frequency*. **Quick takeaway** Mode = Most Often: Count how many times each number appears, and the winner is your mode! (Tip: A dataset can even have *two* modes if two numbers tie for most frequent.)

AI Explanation

**The reasoning** This is a **linear equation** problem. Your goal: get all the x terms on one side and all the numbers on the other. Starting with: 4x - 3 = 2x + 7 **Step 1:** Subtract 2x from both sides (to collect x terms together) 4x - 2x - 3 = 2x - 2x + 7 2x - 3 = 7 **Step 2:** Add 3 to both sides (to isolate the x term) 2x - 3 + 3 = 7 + 3 2x = 10 **Step 3:** Divide both sides by 2 x = 5 **Check:** 4(5) - 3 = 20 - 3 = 17, and 2(5) + 7 = 10 + 7 = 17 ✓ **Why the wrong options tempt you** - **x = 2** comes from adding the coefficients wrongly (4 + 2 = 6, then 10 ÷ 6) - **x = 10** is what you get if you forget to divide by 2 at the end (stopping at 2x = 10) - **x = 4** appears if you subtract instead of add when moving -3 across (7 - 3 = 4, then forgetting the coefficient) **Quick takeaway** Always move x terms to one side and numbers to the other, then isolate x — and **verify your answer** by substituting back!

AI Explanation

**The reasoning** You're looking at a **difference of two squares**: a² − b². This special pattern always factorizes as (a − b)(a + b). Here, x² − 9 can be rewritten as x² − 3², where a = x and b = 3. Apply the formula: x² − 3² = (x − 3)(x + 3) To verify: expand (x − 3)(x + 3) = x² + 3x − 3x − 9 = x² − 9 ✓ **Why the wrong options tempt you** - **(x−3)²** expands to x² − 6x + 9, not x² − 9. Students see "−3" and think squaring gives the answer. - **(x+3)²** gives x² + 6x + 9. Same trap — squaring one bracket doesn't work here. - **(x−9)(x+1)** expands to x² − 8x − 9. This looks tempting because you see "9" in the bracket, but the pattern is wrong. **Quick takeaway** When you see **something² minus something²**, think: difference of two squares → (first − second)(first + second). The signs are always opposite!

AI Explanation

## The reasoning A regular pentagon has 5 equal sides and 5 equal angles. Here's the principle: **The sum of interior angles of any polygon = (n − 2) × 180°**, where n is the number of sides. For a pentagon: n = 5 Sum of interior angles = (5 − 2) × 180° = 3 × 180° = **540°** Since it's *regular*, all 5 angles are equal: Each angle = 540° ÷ 5 = **108°** ## Why the wrong options tempt you **120°** — This is each interior angle of a regular *hexagon* (6 sides). Students mix up their polygons! **144°** — This is for a regular *decagon* (10 sides). Way too many sides. **150°** — This is for a regular *dodecagon* (12 sides). The pattern tricks you because the numbers seem close. The trap: memorizing random angles instead of understanding the formula. When you forget which angle belongs to which shape, you'll pick wrongly. ## Quick takeaway **Pentagon = 5 sides → (5−2)×180° = 540° → divide by 5 = 108°.** Master the formula, and you'll handle *any* polygon they throw at you!

AI Explanation

**The reasoning** A square has **four equal sides**. The perimeter means the total distance around the shape – like fencing around a compound. So: Perimeter = side + side + side + side = 4 × side Given: side = 9 cm Perimeter = 4 × 9 = **36 cm** That's it! You're adding all four sides: 9 + 9 + 9 + 9 = 36 cm. **Why the wrong options tempt you** - **A) 18 cm** – This is 2 × 9. You only counted *two* sides, maybe thinking of diameter or confusing square with something else. - **B) 27 cm** – This is 3 × 9. You forgot one side! Easy mistake under exam pressure. - **D) 81 cm** – This is 9 × 9 = 9². You calculated the **area** instead of perimeter. Common mix-up! **Quick takeaway** Perimeter of a square = **4 × side** (distance *around*), while Area = side² (space *inside*) – don't let exam stress make you confuse the two!

AI Explanation

**The reasoning** sin 30° is a special angle value you need to memorize, but let's see *why* it's ½. Imagine an equilateral triangle with all sides = 2 units. All angles = 60°. Now cut it in half vertically. You get two right-angled triangles. Each has: - Hypotenuse = 2 (original side) - Base = 1 (half of the bottom) - Height = √3 (by Pythagoras: 2² − 1² = 3, so √3) - Angles: 30°, 60°, 90° Since sin = opposite/hypotenuse, and the side opposite the 30° angle is 1: sin 30° = 1/2 = **½** ✓ **Why the wrong options tempt you** **B) √3/2** — That's sin 60° or cos 30°. Easy to mix up the 30-60 pair! **C) 1** — That's sin 90°. Students sometimes guess the "nice" number. **D) 0** — That's sin 0°. Completely different angle. **Quick takeaway** For JAMB/WAEC: memorize the 30-60-90 triangle ratio (1 : √3 : 2) — sin 30° = ½, cos 30° = √3/2, and they swap for 60°.

AI Explanation

## The reasoning The **range** is the difference between the **highest** and **lowest** values in a dataset. It tells you how "spread out" the numbers are. **Step 1:** Identify the smallest number = **5** **Step 2:** Identify the largest number = **22** **Step 3:** Calculate: Range = Largest − Smallest = 22 − 5 = **17** So the answer is **C) 17**. --- ## Why the wrong options tempt you **A) 5** — This is the *smallest value*, not the range. Don't confuse the lowest number with the spread. **B) 12** — This is the *middle value* (median), but that's a different measure entirely. **D) 22** — This is the *largest value*. Again, the range isn't just one extreme — it's the gap between both extremes. --- ## Quick takeaway **Range = Biggest − Smallest.** You're measuring the *distance* between the two ends, not picking one of them!

AI Explanation

**The reasoning** When you have a power raised to another power, like (2³)², you **multiply the exponents**. This is the **Power of a Power Rule**: (aᵐ)ⁿ = aᵐⁿ So (2³)² = 2³ˣ² = 2⁶ = 64 You can also think of it step-by-step: - First: 2³ = 8 - Then: (2³)² = 8² = 64 Either way, you get **64**. **Why the wrong options tempt you** - **16 (A)**: You might mistakenly do 2³⁺¹ = 2⁴ = 16, adding instead of multiplying the exponents. - **32 (B)**: This is 2⁵, which you'd get if you added the exponents wrong (3+2=5). - **128 (D)**: This is 2⁷. You might have made an arithmetic error or confused the operation entirely. The trap is **adding exponents** when you should **multiply** them for powers of powers. **Quick takeaway** Power to a power? **Multiply the exponents**: (aᵐ)ⁿ = aᵐⁿ — it's like saying "three, but twice" = six times total!

AI Explanation

**The reasoning** To find 15% of 200, convert the percentage to a decimal or fraction, then multiply: 15% = 15/100 = 0.15 So: 0.15 × 200 = 30 Or think of it this way: 10% of 200 = 20, and 5% of 200 = 10. Add them: 20 + 10 = **30**. The principle here is **percentage as "parts per hundred"** — 15% means 15 out of every 100. **Why the wrong options tempt you** - **Option A (15)** tricks you if you just write down the percentage number itself without actually calculating. - **Option C (45)** might catch you if you mistakenly multiply 15 × 3 or confuse your working. - **Option D (150)** is the classic trap — you might have divided 200 by 15 backwards, or found what percentage 200 is of something else. **Quick takeaway** Always convert the percentage to decimal (move the point two places left), then multiply — 15% means "multiply by 0.15," not by 15!

AI Explanation

**The reasoning** When you see 0.25, read it as "twenty-five hundredths" because the last digit (5) is in the hundredths place. So: 0.25 = 25/100 Now simplify by finding the highest common factor (HCF) of 25 and 100. Both divide by 25: 25 ÷ 25 = 1 100 ÷ 25 = 4 Therefore: **0.25 = 1/4** **Why the wrong options tempt you** - **A (1/2):** This is 0.5, not 0.25. Students confuse "half" with "quarter" when rushing. - **C (1/5):** This equals 0.2 or 0.20. Close, but not quite 0.25. - **D (25/100):** This is *correct* but **not in lowest terms**! The question asks for the simplified form. Always reduce your fractions. **Quick takeaway** Write the decimal as a fraction over the place value (hundredths = /100), then simplify by dividing top and bottom by their HCF. The decimal 0.25 is exactly one quarter—picture splitting ₦1 into four equal parts!

AI Explanation

## The reasoning To connect a:c, we need a common "bridge" — and that bridge is **b**. Right now: - a:b = 3:5 means a = 3 parts, b = 5 parts - b:c = 4:7 means b = 4 parts, c = 7 parts **Problem:** b has different values (5 in the first ratio, 4 in the second). We need to make them equal. Find the **LCM of 5 and 4 = 20**. Scale the first ratio so b becomes 20: - a:b = 3:5 → multiply by 4 → **a:b = 12:20** Scale the second ratio so b also becomes 20: - b:c = 4:7 → multiply by 5 → **b:c = 20:35** Now b matches! So: **a:b:c = 12:20:35** Therefore: **a:c = 12:35** ## Why the wrong options tempt you **B) 15:28** — You might've added wrongly or used an incorrect LCM. **C) 3:7** — Classic trap: multiplying first numbers (3×1) and last numbers (5×7 ÷ 5). That's not how ratios link! **D) 5:7** — Just copying the second part of each ratio without connecting through b. ## Quick takeaway **Always equalize the "bridge" value using LCM when connecting ratios.**

AI Explanation

## The reasoning This is a **linear equation** — we need to get all the x's on one side and all the numbers on the other. Starting with: 4x − 3 = 2x + 7 **Step 1:** Subtract 2x from both sides to collect the x terms together: 4x − 2x − 3 = 2x − 2x + 7 2x − 3 = 7 **Step 2:** Add 3 to both sides to isolate the x term: 2x − 3 + 3 = 7 + 3 2x = 10 **Step 3:** Divide both sides by 2: x = 5 **Check it:** 4(5) − 3 = 20 − 3 = 17, and 2(5) + 7 = 10 + 7 = 17 ✓ ## Why the wrong options tempt you - **x = 2**: You might get this if you subtracted 4x instead of 2x, or mixed up the signs - **x = 10**: This is what 2x equals, not x itself — forgetting that final division step - **x = 4**: Probably from arithmetic errors when adding/subtracting the constants ## Quick takeaway **Always move variables to one side, constants to the other, then solve — and CHECK your answer by plugging it back into the original equation!**

AI Explanation

**The reasoning** You're looking at a **difference of two squares**: x² − 9. Notice that 9 = 3², so you can rewrite this as: x² − 3² There's a beautiful formula for this pattern: **a² − b² = (a − b)(a + b)** Here, a = x and b = 3, so: x² − 9 = x² − 3² = (x − 3)(x + 3) You can verify: expand (x − 3)(x + 3) and you get x² + 3x − 3x − 9 = x² − 9 ✓ **Why the wrong options tempt you** - **A & C** look familiar because of the ±3, but squaring (x−3)² gives you x² − 6x + 9, not x² − 9. Perfect squares always have that middle term! - **D** is random—multiplying it out gives x² − 8x − 9, totally different. **Quick takeaway** Whenever you see **something² minus something²**, think *difference of two squares* and split it into **(first − second)(first + second)**—the middle terms will always cancel perfectly.

AI Explanation

**The reasoning** For any polygon, the sum of interior angles = (n − 2) × 180°, where n is the number of sides. A pentagon has 5 sides, so: Sum = (5 − 2) × 180° = 3 × 180° = 540° Since it's a **regular** pentagon (all angles equal), divide by 5: Each angle = 540° ÷ 5 = **108°** **Why the wrong options tempt you** **120°** — This is the interior angle of a regular *hexagon* (6 sides). Easy mix-up if you're rushing. **144°** — Interior angle of a regular *decagon* (10 sides). Looks "neat" so students guess it. **150°** — Interior angle of a regular *dodecagon* (12 sides). Another polygon trap. The examiners know students confuse polygons, so they deliberately include angles from other common shapes. **Quick takeaway** Remember the formula: **(n − 2) × 180° ÷ n** — pentagon has 5 sides, so (3 × 180°) ÷ 5 = 108°. Don't guess; calculate every time!

AI Explanation

**The reasoning** A square has **four equal sides**. That's the key property you must remember. If one side = 9 cm, then all four sides = 9 cm each. **Perimeter** means the total distance around the shape — you add up all the sides. So: Perimeter = 9 + 9 + 9 + 9 = **36 cm** Or use the formula: **Perimeter of square = 4 × side = 4 × 9 = 36 cm** --- **Why the wrong options tempt you** - **18 cm (A):** You only added two sides (9 + 9). A square has *four* sides, not two! - **27 cm (B):** You multiplied by 3 instead of 4. Maybe confusion with triangles? Squares have 4 sides. - **81 cm (D):** You calculated the **area** (9 × 9 = 81 cm²), not the perimeter. Area uses multiplication of length × width; perimeter uses addition of all sides. --- **Quick takeaway** Perimeter = distance *around* = **add all sides**. For a square with side *s*: **P = 4s**. Never confuse it with area (s²)!

AI Explanation

**The reasoning** The sine of an angle in a right-angled triangle is the ratio: **opposite side ÷ hypotenuse**. For 30°, imagine a special triangle (the 30-60-90 triangle). It has sides in the ratio 1 : √3 : 2. When the angle is 30°: - Opposite side = 1 - Hypotenuse = 2 So: sin 30° = 1/2 This is a **standard value** you must memorize. It appears constantly in JAMB, WAEC, and Post-UTME! **Why the wrong options tempt you** - **B) √3/2** — This is sin 60° or cos 30°. Easy to mix up the special angles if you don't practice. - **C) 1** — That's sin 90°. Students sometimes think "30 is close to zero, maybe sin is 1?" Wrong direction! - **D) 0** — That's sin 0°. Confusing which angle gives which value. **Quick takeaway** Burn this into memory: **sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3**. The 30-60-90 triangle is your best friend in trigonometry!

AI Explanation

**The reasoning** Range means the **spread** of your data — how far apart the smallest and largest values are. The formula is simple: **Range = Highest value − Lowest value** From your data: 5, 7, 12, 15, 22 - Highest value = 22 - Lowest value = 5 Range = 22 − 5 = **17** That's it! You're measuring the "distance" between the extremes. --- **Why the wrong options tempt you** - **A) 5** — This is the lowest value itself, not the range. Don't confuse the starting point with the spread. - **B) 12** — This is the middle value (median). Range has nothing to do with the center; it's about the edges. - **D) 22** — This is the highest value. Again, you need the *difference*, not just one endpoint. --- **Quick takeaway** Range = Biggest number − Smallest number. Always subtract; never just pick one extreme!

AI Explanation

**The reasoning** To find a percentage of a number, convert the percentage to a decimal and multiply: 15% = 15/100 = 0.15 So: 15% of 200 = 0.15 × 200 = **30** Or use the faster method: (15 × 200) ÷ 100 = 3000 ÷ 100 = **30** This is the **percentage principle**: "of" means multiply, and "percent" means "per hundred." **Why the wrong options tempt you** - **A) 15** — You might just write down the percentage number itself, forgetting to actually multiply by 200. - **C) 45** — A careless calculation error, maybe mixing up numbers during multiplication. - **D) 150** — This traps students who confuse percentages with fractions. You might accidentally think "15% is close to 1/2" or forget to divide by 100, doing 15 × 200 ÷ 2 instead. **Quick takeaway** To find any percentage: multiply the number by the percent, then divide by 100 — or just move the decimal two places left first (15% → 0.15), then multiply!

AI Explanation

**The reasoning** To convert a decimal to a fraction, count the decimal places. 0.25 has **two decimal places**, so write it as 25/100 (25 over "one hundred" because of those two zeros). Now simplify to lowest terms by finding the highest common factor (HCF) of 25 and 100. Both divide by 25: 25 ÷ 25 = 1 100 ÷ 25 = 4 So 0.25 = **1/4** **Why the wrong options tempt you** - **A) 1/2** — This is 0.5, not 0.25. Students confuse "two decimal places" with "two in the numerator." - **C) 1/5** — This equals 0.2. Close, but you dropped a digit! - **D) 25/100** — This is technically *correct* but **not simplified**. JAMB/WAEC always want lowest terms, so this loses marks. **Quick takeaway** "Count decimal places to make your fraction, then simplify by dividing top and bottom by their HCF" — 0.25 has 2 places → 25/100 → divide by 25 → **1/4**. Done! ✨

AI Explanation

**The reasoning** Area of a rectangle = length × width You have: - Length = 8 cm - Width = 5 cm So: Area = 8 cm × 5 cm = **40 cm²** That's it! The principle here is **area formula for rectangles** — you multiply the two dimensions together. Notice the unit becomes cm² (square centimeters) because you're multiplying cm by cm. **Why the wrong options tempt you** - **Option A (13 cm²)**: This is 8 + 5. The trap? You *added* instead of multiplied. Addition gives you the perimeter's half (or just... nothing useful here), not area. - **Option B (26 cm²)**: This is 2(8 + 5) = perimeter, not area. You calculated the distance *around* the rectangle instead of the space *inside* it. - **Option D (80 cm²)**: Probably 8 × 10 or some careless multiplication error. Always double-check your arithmetic. **Quick takeaway** Area means "space inside" — always **multiply** length by width; perimeter means "distance around" — that's when you add and multiply by 2.

AI Explanation

**The reasoning** The mean (or average) is the sum of all values divided by how many values there are. Step 1: Add all the numbers together: 4 + 6 + 8 + 10 + 12 = 40 Step 2: Count how many numbers you have: There are **5 numbers** Step 3: Divide the sum by the count: Mean = 40 ÷ 5 = **8** This is called the **arithmetic mean** — it tells you the "central" value if everything were equally distributed. **Why the wrong options tempt you** - **Option A (6)** tricks you if you accidentally pick the second number instead of calculating - **Option B (7)** might come from dividing by 6 instead of 5 (counting error) - **Option D (9)** could happen if you mistakenly add wrong (like getting 45) or use the wrong divisor **Quick takeaway** Mean = **Sum ÷ Count** — always double-check you've counted all numbers and added correctly before dividing!

AI Explanation

## The reasoning To solve **3x = 12**, we need to isolate x (get x alone on one side). The principle here is **inverse operations** — we undo multiplication by dividing both sides by the same number. Since x is being multiplied by 3, we divide both sides by 3: 3x ÷ 3 = 12 ÷ 3 x = 4 **Quick check:** Does 3 × 4 = 12? Yes! ✓ --- ## Why the wrong options tempt you - **x = 3** tricks you if you subtract instead of divide (12 − 3 = 9... wait, that doesn't even give 3!) - **x = 6** catches you if you mistakenly do 12 ÷ 2 instead of 12 ÷ 3 - **x = 9** tempts you if you subtract 3 from 12, forgetting this is multiplication, not addition The trap is confusing which operation to reverse. Since 3 is **multiplying** x, you must **divide** to undo it. --- ## Quick takeaway **Whatever's being done to x, do the opposite to both sides** — multiply becomes divide, add becomes subtract.

AI Explanation

## **The reasoning** Think of a straight line as a perfectly flat path — no bending, no curves. When we talk about "angles on a straight line," we mean the **total measure of angles that sit side-by-side along that line**. Imagine standing at one end of a stick and turning to face the other end. You've made a **half turn** — that's exactly **180°**. This is the fundamental property: **angles on a straight line always sum to 180°**. For example, if two angles sit on a straight line and one is 110°, the other must be 70° because 110° + 70° = 180°. ## **Why the wrong options tempt you** - **90°** catches students thinking of a right angle (one corner of a square) — but that's angles at a perpendicular, not along a straight line. - **270°** and **360°** relate to turning further: 270° is three-quarter turn, 360° is a full circle/complete turn. They're correct for *those* situations, but not for a straight line. ## **Quick takeaway** **Straight line = half turn = 180° always.** Picture flipping a ruler end-to-end — that's your 180°!