JAMB UTME Mathematics
Past Questions

AI Explanation

**The reasoning** We have two equations with two unknowns — this is a **system of simultaneous equations**. The smartest move? Use **substitution** or **elimination**. From equation 2: x - y = 1, so **x = y + 1** Substitute this into equation 1: 2(y + 1) + 3y = 12 2y + 2 + 3y = 12 5y + 2 = 12 5y = 10 **y = 2** Now find x: x = y + 1 = 2 + 1 = **3** ✓ **Why the wrong options tempt you** - **Option A (1):** You might get this if you rushed and swapped x and y values - **Option B (2):** This is actually the value of y, not x! Classic mix-up under exam pressure - **Option D (4):** Likely from arithmetic errors when expanding 2(y + 1) or combining like terms **Quick takeaway** When solving simultaneous equations, **label your answers clearly** — write "y = 2" and "x = 3" separately so you don't submit the wrong variable's value. Always substitute back to verify: Does 2(3) + 3(2) = 12? Yes! Does 3 - 2 = 1? Yes!

AI Explanation

## **The reasoning** A fraction is **undefined** when its denominator equals zero — because you can't divide by zero in mathematics. Here, the denominator is **(x - 2)**. Set it equal to zero: x - 2 = 0 x = 2 So when **x = 2**, the expression becomes (something)/**0**, which is undefined. Notice the numerator x² - 4x + 4 = (x - 2)² also equals zero when x = 2, giving us **0/0** — still undefined! The concept: *look at the denominator alone* to find where division breaks down. --- ## **Why the wrong options tempt you** **A) 0** — Students might think "zero makes things undefined," but substitute: (0 - 0 + 4)/(0 - 2) = 4/(-2) = -2. Perfectly defined! **C) -2** — Looks similar to 2, but check: (4 + 8 + 4)/(-2 - 2) = 16/(-4) = -4. Still defined. **D) 4** — Might come from solving the numerator, but (16 - 16 + 4)/(4 - 2) = 4/2 = 2. Also defined. --- ## **Quick takeaway** **A fraction dies when the bottom line hits zero** — always set the denominator to zero to find undefined values.

AI Explanation

**The reasoning** We're using the **power rule of differentiation**: when you differentiate xⁿ, you get n·xⁿ⁻¹. Let's break down (3x² + 2x - 5) term by term: - **3x²**: Bring down the power 2, multiply by 3 → 2 × 3x²⁻¹ = **6x** - **2x**: This is really 2x¹, so bring down the 1 → 1 × 2x⁰ = **2** (since x⁰ = 1) - **-5**: This is a constant. Constants vanish when differentiated → **0** Add them up: 6x + 2 + 0 = **6x + 2** **Why the wrong options tempt you** - **B (3x + 2)**: You forgot to multiply by the power when differentiating 3x². Easy slip! - **C (6x - 5)**: You differentiated 2x wrongly (maybe got confused) and kept the constant -5 instead of removing it. - **D (6x + 2 - 5)**: You kept the constant! Remember: constants always disappear during differentiation because their rate of change is zero. **Quick takeaway** When differentiating polynomials, bring down powers and reduce them by 1; constants always vanish because they don't change.

AI Explanation

## The reasoning The angles of any triangle always sum to **180°** — that's your foundation. If the angles are in the ratio 2:3:5, think of them as: - First angle = 2x - Second angle = 3x - Third angle = 5x (the largest) Now set up your equation: **2x + 3x + 5x = 180°** Combine: **10x = 180°** Solve: **x = 18°** So the actual angles are: - 2x = 2(18°) = 36° - 3x = 3(18°) = 54° - 5x = 5(18°) = **90°** The largest angle is **90°**. ## Why the wrong options tempt you **Options A, B, C** are actually the *other angles* in this same triangle! If you rush and just calculate one value without identifying which is largest, you'll pick one of these. The question specifically asks for the **largest**, which corresponds to the biggest ratio number (5). ## Quick takeaway In ratio problems, always find the multiplier first (x), then multiply by the *specific* ratio you need — here, the largest ratio was 5, giving 5 × 18° = 90°.

AI Explanation

**The reasoning** The median is the **middle value** when you arrange numbers from smallest to largest. First, sort the numbers: 2, 3, 4, **5**, 7, 8, 9 Count them: there are 7 numbers (odd count). The middle position is the 4th number — that's **5**. **Formula reminder**: For *n* numbers, the median position = (n+1)/2 = (7+1)/2 = 4th position. **Why the wrong options tempt you** **A) 4** — You might pick this if you don't sort first, or if you miscount positions (it's actually the 3rd value). **C) 6** — This trap catches students who think "find the average of all numbers" instead of the middle. Also, 6 isn't even in the list! **D) 7** — Counting from the wrong end, or confusing median with mode or range. **Quick takeaway** **"Sort first, find middle" — the median is always about POSITION after arranging, not calculation.** With odd counts, it's one number; with even counts, average the two middle ones.

AI Explanation

**The reasoning** The key principle here is the **power rule of logarithms**: log(aⁿ) = n·log(a) Notice that 8 = 2³ (2 × 2 × 2 = 8) So: log₁₀ 8 = log₁₀(2³) Using the power rule: log₁₀(2³) = 3 × log₁₀ 2 Since log₁₀ 2 = 0.3010: log₁₀ 8 = 3 × 0.3010 = **0.9030** **Why the wrong options tempt you** - **Option A (0.6020)**: You multiplied by 2 instead of 3, possibly thinking 8 = 2². Remember: 2² = 4, not 8! - **Option C (0.7525)**: This looks like you tried averaging or did 2.5 × 0.3010. There's no mathematical basis for this. - **Option D (2.4080)**: You multiplied by 8 instead of recognizing that 8 is a power of 2. Don't confuse the number itself with its relationship to the base. **Quick takeaway** When finding log of a number, **always express it as a power of something you already know** — then multiply the log by that power. 8 = 2³, so log 8 = 3 × log 2. Simple!

AI Explanation

**The reasoning** When solving inequalities, treat them like equations — but with one golden rule: **preserve the inequality direction** (unless you multiply/divide by a negative number, which doesn't happen here). Starting with: 3x - 2 < 7 **Step 1:** Add 2 to both sides 3x - 2 + 2 < 7 + 2 3x < 9 **Step 2:** Divide both sides by 3 x < 3 The inequality sign stays as "less than" because we divided by a positive number. Simple as that! --- **Why the wrong options tempt you** - **Option B (x > 3):** You might flip the sign by mistake, thinking all inequalities reverse. They only reverse when multiplying/dividing by **negative** numbers. - **Options C & D (≤ and ≥):** The original has strict "<", not "≤". There's no equal sign anywhere in 3x - 2 < 7, so don't add one to your answer. --- **Quick takeaway** Solve inequalities like equations, keeping the inequality sign unless you multiply/divide by a negative — then flip it!

AI Explanation

**The reasoning** A perfect square trinomial has the form (x + a)² = x² + 2ax + a². Notice the pattern: the constant term is the **square of half the coefficient of x**. Here, our constant is 16, so a² = 16, meaning a = 4 (we take positive since b > 0). Now, the middle term must be 2ax = 2(4)(x) = 8x. Therefore, b = **8**. Check: (x + 4)² = x² + 8x + 16 ✓ **Why the wrong options tempt you** - **A) 4**: You might think "√16 = 4, so b = 4" — but you forgot to double it! The formula is 2a, not just a. - **B) 6**: Random guess, maybe averaging 4 and 8? No mathematical basis. - **D) 16**: You took the constant term itself, forgetting the relationship entirely. **Quick takeaway** For x² + bx + c to be a perfect square: find √c, then **double it** to get b. The middle coefficient is always twice the square root of the constant.

AI Explanation

## The reasoning This is a **factorization problem**. We need two numbers that **multiply to give +6** (the constant term) and **add up to give -5** (the coefficient of x). Think: What two numbers multiply to 6? Could be 1×6 or 2×3. Now which pair adds to -5? That's **-2 and -3** (because -2 + -3 = -5, and -2 × -3 = +6). So we rewrite: x² - 5x + 6 = (x - 2)(x - 3) = 0 For this to equal zero, either: - x - 2 = 0 → **x = 2**, OR - x - 3 = 0 → **x = 3** ## Why the wrong options tempt you **Option A (1 or 6):** These multiply to 6, but they add to 7, not -5. You forgot to check the sum. **Option B (-2 or -3):** You found the factors correctly but forgot the final step! If x - 2 = 0, then x = +2, not -2. **Option D (-1 or -6):** Wrong factor pair entirely—these multiply to 6 but add to -7. ## Quick takeaway **Find two numbers that multiply to c and add to b, then set each bracket equal to zero and solve for x—watch your signs!**

AI Explanation

## The reasoning This is a **factorization problem**. The numerator a² - b² is a special pattern called **difference of two squares**. The key formula to remember: **a² - b² = (a + b)(a - b)** So let's rewrite the fraction: (a² - b²) / (a + b) = [(a + b)(a - b)] / (a + b) Now, (a + b) appears in both numerator and denominator, so they cancel out (as long as a ≠ -b): = (a - b) ## Why the wrong options tempt you **Option A (a + b)**: You might think "the denominator stays", but that's backwards — we're canceling the denominator, not keeping it. **Options C & D (a²b² and a²/b²)**: These look "mathy" but come from randomly multiplying or dividing the letters without using the factorization rule. Pure distraction. ## Quick takeaway Whenever you see **a² - b²**, immediately think **(a + b)(a - b)** — it's the most tested factorization pattern in JAMB/WAEC algebra!

AI Explanation

**The reasoning** The formula for the area of a circle is **A = πr²**, where r is the radius. Given: r = 7 cm and π = 22/7 Let's substitute: A = πr² A = (22/7) × 7² A = (22/7) × 49 Now simplify: The 7 in the denominator cancels with 49: A = 22 × 7 = **154 cm²** The key principle: **Area of a circle = πr²** (remember: radius *squared*, then multiply by π) **Why the wrong options tempt you** - **Option A (22 cm²)**: You forgot to square the radius and just did 22/7 × 7 = 22. Classic mistake! - **Option B (44 cm²)**: You might have calculated circumference (2πr = 2 × 22/7 × 7 = 44) instead of area. Wrong formula! - **Option D (308 cm²)**: You probably used diameter instead of radius, calculating π × 14² ÷ 4, or made an arithmetic error. **Quick takeaway** For circle area, always **square the radius first** before multiplying by π — it's A = πr², not A = πr!

AI Explanation

**The reasoning** When you need the diagonal of a rectangle, think **Pythagoras' Theorem** — the diagonal splits the rectangle into a right-angled triangle. The two sides (length and width) are the two shorter sides, and the diagonal is the hypotenuse. Using the formula: d² = l² + w² d² = 12² + 8² d² = 144 + 64 d² = 208 d = √208 d ≈ **14.42 cm** So the diagonal is approximately 14.42 cm. **Why the wrong options tempt you** - **Option A (4 cm):** You might subtract 12 − 8 = 4. But diagonals are *longer* than sides, not shorter! - **Option B (√208 cm):** This is technically correct but left unsimplified. The question wants the *value*, not the surd form. They gave you 14.42 cm for practicality. - **Option D (20 cm):** Classic trap — you added 12 + 8 instead of using Pythagoras. Addition doesn't work for diagonals! **Quick takeaway** Rectangle diagonal? Always **Pythagoras**: d² = length² + width², then find the square root — and the diagonal is always *longer* than any single side.

AI Explanation

## The reasoning The formula for the sum of interior angles of any polygon is **(n − 2) × 180°**, where n is the number of sides. A hexagon has **6 sides**, so: - Sum = (6 − 2) × 180° - Sum = 4 × 180° - Sum = **720°** **Why this formula works:** Imagine drawing lines from one vertex to all other vertices in the hexagon. You create 4 triangles. Since each triangle has angles summing to 180°, the total is 4 × 180° = 720°. ## Why the wrong options tempt you **A) 360°** — This is the sum of *exterior* angles of any polygon, not interior angles. Easy mix-up! **B) 540°** — This is correct for a *pentagon* (5 sides): (5−2) × 180° = 540°. Students sometimes miscount the sides. **D) 1080°** — This would be for an octagon (8 sides). Way too many sides! ## Quick takeaway **"Take the number of sides, subtract 2, then multiply by 180° — hexagon has 6 sides, so (6−2) × 180° = 720°."** Memorize the formula **(n − 2) × 180°** and you'll crack any polygon question!

AI Explanation

**The reasoning** You're given sin θ = 3/5. Think of this as a right triangle where the **opposite side = 3** and the **hypotenuse = 5**. To find cos θ, you need the adjacent side. Use **Pythagoras' theorem**: hyp² = opp² + adj² 5² = 3² + adj² 25 = 9 + adj² adj² = 16 adj = 4 So cos θ = adjacent/hypotenuse = **4/5** ✓ **Why the wrong options tempt you** - **A) 3/4** — You might flip the fraction wrongly or divide opposite by adjacent (that's actually tan θ!) - **C) 5/4** — Putting hypotenuse in the numerator breaks the rule; cosine can never be greater than 1 - **D) 4/3** — Again, this is greater than 1, which is impossible for any trig ratio involving the hypotenuse as denominator **Quick takeaway** Draw the triangle! SOH-CAH-TOA means cos θ is always **Adjacent/Hypotenuse**, and when you know sin, Pythagoras gives you the missing side. Your cosine answer must always be ≤ 1 for acute angles.

AI Explanation

## The reasoning The tangent of an angle is the ratio: **tan θ = opposite/adjacent** (or sin θ/cos θ). For the special angle 45°, picture an **isosceles right triangle** where both legs are equal. Let's say both legs = 1 unit. tan 45° = opposite/adjacent = 1/1 = **1** You can also remember: sin 45° = cos 45° = √2/2, so tan 45° = (√2/2)/(√2/2) = 1. **Key principle:** In a 45-45-90 triangle, the two legs are always equal, making their ratio exactly 1. ## Why the wrong options tempt you **A) 0** — This is tan 0°, not tan 45°. Students confuse small angles. **B) 1/2** — Some guess this looks "reasonable" or confuse it with sin 30° (which is 1/2). **D) √3** — This is tan 60°! It's easy to mix up the special angles 30°, 45°, and 60°. ## Quick takeaway **45° gives you equality**: tan 45° = 1 because you're dividing equal sides. Memorize: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

AI Explanation

## The reasoning To convert degrees to radians, use the fundamental relationship: **180° = π radians** This means 1° = π/180 radians So for 540°: 540° = 540 × (π/180) Simplify the fraction: = (540π)/180 = 3π radians **Alternative check:** Notice that 540° = 3 × 180°. Since each 180° equals π radians, we get 3π directly! ## Why the wrong options tempt you **A) π** — You might mistakenly think 180° = π, so 540° = π. But you forgot to multiply: 540° is *three times* 180°. **B) 2π** — Common confusion! 360° (a full circle) = 2π. But 540° goes beyond a full rotation. **D) 5π/2** — This equals 450°. You might have miscalculated the simplification of 540/180. ## Quick takeaway **Remember: 180° = π, so divide the degrees by 180 and multiply by π** — or just remember that every 180° adds another π to your answer!

AI Explanation

**The reasoning** The gradient of a curve at any point is found using **differentiation**. When you differentiate, you're finding how steep the curve is at that exact spot. Given: y = x² + 3x **Step 1:** Differentiate using the power rule (bring down the power, reduce the power by 1): - dy/dx = 2x + 3 **Step 2:** Substitute x = 2: - dy/dx = 2(2) + 3 = 4 + 3 = **7** That's your gradient at x = 2. **Why the wrong options tempt you** - **Option A (5):** You likely forgot to multiply the 2 in front of x, doing just 2 + 3 instead of 2(2) + 3. - **Option C (10):** You probably found y itself at x = 2 (which is 4 + 6 = 10) instead of finding dy/dx. Remember: gradient ≠ y-value! - **Option D (12):** Maybe you multiplied 2 × 2 × 3 incorrectly or confused the operations. **Quick takeaway** Gradient = differentiate first, then substitute the x-value — never substitute before differentiating!

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**The reasoning** Integration is the reverse of differentiation. We use the power rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + c For ∫(2x + 3) dx, integrate each term separately: - ∫2x dx = 2 · (x²/2) = x² - ∫3 dx = 3x (because 3 is just 3x⁰, so we get 3x¹) Therefore: ∫(2x + 3) dx = **x² + 3x + c** This is called **integration by the power rule**. Always add the constant c because there are infinitely many antiderivatives. **Why the wrong options tempt you** **Option B (2x² + 3x + c)**: You forgot to divide by 2 after raising x to power 2. The coefficient 2 doesn't just carry forward—you must divide by the new power! **Option C (x² + 3 + c)**: You treated the constant 3 incorrectly. The integral of 3 is 3x, not just 3. **Option D**: Completely mixed up the terms and coefficients. **Quick takeaway** When integrating xⁿ, increase the power by 1, then divide by that new power—and don't forget: the integral of a constant k is kx, not k!

AI Explanation

**The reasoning** To find f'(1), we need to differentiate f(x) first, then substitute x = 1. Using the **power rule of differentiation**: if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹ Given f(x) = x³ - 2x + 5 Differentiating term by term: - x³ becomes 3x² - -2x becomes -2 (since x¹ becomes 1x⁰ = 1) - 5 becomes 0 (constants vanish when differentiated) So f'(x) = 3x² - 2 Now substitute x = 1: f'(1) = 3(1)² - 2 = 3(1) - 2 = 3 - 2 = **1** **Why the wrong options tempt you** **A) 0** — You might get this if you forgot to subtract 2, or mixed up f'(1) with finding where f'(x) = 0. **C) 2** — This comes from forgetting the "-2" entirely, giving just 3(1)² - 1 = 2. **D) 4** — You probably substituted into the original function f(1) = 1 - 2 + 5 = 4, instead of the derivative. **Quick takeaway** Differentiate *first* using the power rule, *then* substitute the value — never reverse the order!

AI Explanation

**The reasoning** The **mean** (or average) is the sum of all values divided by how many values you have. Here's the working: - Add all the numbers: 12 + 15 + 18 + 20 + 25 = 90 - Count how many numbers: 5 numbers - Divide: Mean = 90 ÷ 5 = **18** This is the **principle of arithmetic mean** — you're finding the central value that would result if you distributed the total equally. **Why the wrong options tempt you** - **17** might catch you if you miscounted the total or divided by 6 instead of 5 - **19** could come from adding incorrectly (maybe getting 95 instead of 90) - **20** is tempting because it's the middle number when you arrange them in order (that's the *median*, not the mean!) Students often confuse mean with median — don't fall for it! **Quick takeaway** Mean = Add everything up, then divide by how many items you counted — it's about fair sharing, not middle position.

AI Explanation

## The reasoning This is a **permutation problem** — you're arranging distinct items where order matters. Think of filling 4 positions on the shelf, one by one: - **1st position:** You can pick any of the 4 books - **2nd position:** 3 books left to choose from - **3rd position:** 2 books remaining - **4th position:** Only 1 book left So total arrangements = 4 × 3 × 2 × 1 = **24 ways** This is called **4 factorial**, written as **4!** ## Why the wrong options tempt you **Option A (4):** You might think "4 books = 4 ways" — but that's forgetting that *each* first choice opens up more possibilities. **Option B (12):** Maybe you multiplied 4 × 3 and stopped early, forgetting the last two positions. **Option C (16):** This looks like 4², but squaring doesn't apply here — arrangements follow the factorial pattern. ## Quick takeaway **When arranging *n* different items in a line, the answer is always *n*! (n factorial) — multiply all whole numbers from *n* down to 1.**

AI Explanation

## The reasoning When you toss two coins, there are **four equally likely outcomes**: - HH (both heads) - HT (first head, second tail) - TH (first tail, second head) - TT (both tails) Only **one** of these four outcomes gives you two heads (HH). So probability = Number of favorable outcomes ÷ Total possible outcomes = **1/4** This follows the **basic probability principle**: P(event) = (ways to get what you want) ÷ (all possible ways) ## Why the wrong options tempt you **1/2 (Option B)** tricks you if you think "each coin is 1/2, so the answer must be 1/2" — but you need *both* to show heads simultaneously, which is rarer. **3/4 (Option C)** might catch you if you counted "at least one head" instead of "two heads." **1 (Option D)** would mean it's certain — clearly wrong since you could get tails! ## Quick takeaway **List all outcomes first, then count your target** — two coins give you 4 possibilities, and only 1 shows two heads, so it's always 1/4.

AI Explanation

**The reasoning** Binary (base 2) uses only 0s and 1s, where each position represents a power of 2. Reading from **right to left**: 1101₂ = (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) Let me break it down: - 1 × 2³ = 1 × 8 = 8 - 1 × 2² = 1 × 4 = 4 - 0 × 2¹ = 0 × 2 = 0 - 1 × 2⁰ = 1 × 1 = 1 Add them up: 8 + 4 + 0 + 1 = **13** **Why the wrong options tempt you** **A) 11** — You might have just counted the number of 1s (three 1s) plus something random, or added the digits: 1+1+0+1 = 3... this doesn't work for base conversion! **C) 15** — This is 1111₂. Easy to confuse if you misread the question. **D) 21** — Maybe you treated it like base 10 and subtracted 8? Total confusion. **Quick takeaway** Remember **"Powers of 2 from the right"**: each position is worth 1, 2, 4, 8, 16... Multiply each binary digit by its position value, then sum everything up!

AI Explanation

**The reasoning** HCF (Highest Common Factor) means the **largest number that divides both numbers exactly** without remainder. Let's find all factors: - **24:** 1, 2, 3, 4, 6, 8, **12**, 24 - **36:** 1, 2, 3, 4, 6, 9, **12**, 18, 36 The common factors are: 1, 2, 3, 4, 6, 12 The **highest** among them? **12** ✓ *Alternative method (prime factorization):* - 24 = 2³ × 3¹ - 36 = 2² × 3² HCF = take the **lowest power** of each common prime = 2² × 3¹ = 4 × 3 = **12** **Why the wrong options tempt you** **A) 6** — This IS a common factor, but not the *highest* one. Don't stop when you find one that works! **B) 8** — 8 divides 24, but check: does 36 ÷ 8 work? No! (36 ÷ 8 = 4.5). Always verify both numbers. **D) 72** — That's the LCM (Lowest Common Multiple), not HCF. Students often confuse these two concepts. **Quick takeaway** HCF is the **biggest** number that divides both—always smaller than or equal to the smallest number given (here, 24).

AI Explanation

**The reasoning** To find the LCM (Lowest Common Multiple), we need the **smallest number that all three numbers divide into evenly**. First, break each number into prime factors: - 12 = 2² × 3 - 15 = 3 × 5 - 20 = 2² × 5 Now, take the **highest power** of each prime that appears: - Highest power of 2: 2² (from 12 or 20) - Highest power of 3: 3¹ (from 12 or 15) - Highest power of 5: 5¹ (from 15 or 20) Multiply them: 2² × 3 × 5 = 4 × 3 × 5 = **60** Check: 60÷12 = 5 ✓, 60÷15 = 4 ✓, 60÷20 = 3 ✓ **Why the wrong options tempt you** **A) 30** – You might think "common multiple" and pick the first one you see, but 30÷20 = 1.5 (not a whole number!). **B) 45** – Looks decent, but 45÷20 doesn't work either. **D) 120** – This IS a common multiple, but not the *lowest*. You probably multiplied all three: 12×15×20 ÷ something, or just picked a "safe" big number. **Quick takeaway** LCM = use prime factorization, then multiply the **highest powers** together—not just any common multiple!