JUPEB Mathematics
Past Questions

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**The reasoning** When differentiating, we use the **power rule**: bring down the exponent as a coefficient, then reduce the exponent by 1. For y = 3x²: - The coefficient is 3 - The exponent is 2 - Bring down the 2: 3 × 2 = 6 - Reduce the exponent: 2 − 1 = 1 - Result: 6x¹ = **6x** The formula is: if y = axⁿ, then dy/dx = n·axⁿ⁻¹ **Why the wrong options tempt you** **A) 3x** — You forgot to multiply by the original exponent (2). You only reduced the power but didn't bring down the 2. **C) x³** — You *integrated* instead of differentiated! Integration raises the power; differentiation lowers it. **D) 6** — You differentiated twice by accident, or forgot that x¹ = x (you can't just drop the variable). **Quick takeaway** Power rule: **multiply by the exponent, then subtract 1 from the exponent** — always do both steps, and keep your variable unless it disappears naturally (like x⁰ = 1).

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**The reasoning** When you integrate 2x, you're finding the *antiderivative* — the function whose derivative gives you 2x back. Using the power rule for integration: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C So: ∫2x dx = 2∫x¹ dx = 2 · (x²/2) + C = **x² + C** The "+ C" is crucial! It's called the **constant of integration**. Since the derivative of any constant is zero, when we reverse differentiation, we must account for that "lost" constant. **Why the wrong options tempt you** **A) x²** — You did the math correctly but forgot the constant! In indefinite integrals, C is *always* needed because there are infinitely many antiderivatives (x², x²+5, x²−3, etc. all have derivative 2x). **C) 2** — This treats integration like a simple operation instead of finding the antiderivative. **D) 2x²** — You forgot to divide by the new power (2). **Quick takeaway** Every indefinite integral needs "+ C" — no C, no marks in JAMB!

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**The reasoning** This is basic **differentiation using the power rule**: when you have xⁿ, the derivative is n·xⁿ⁻¹. Here, y = x³, so n = 3. Apply the power rule: - Bring down the power as a coefficient: 3 - Reduce the power by 1: 3 − 1 = 2 So dy/dx = 3x² **Why the wrong options tempt you** **Option A (3x)**: You brought down the 3 correctly but reduced the power too much (went to x¹ instead of x²). The power decreases by just 1, not 2. **Option B (x²)**: You reduced the power correctly but forgot to bring down the original power (3) as a coefficient. Never forget that multiplier! **Option D (x³)**: You didn't differentiate at all—you just wrote the original function. Differentiation *always* changes the expression. **Quick takeaway** Power rule: bring the power down front, then subtract 1 from the power—so x³ becomes 3x². Practice this until it's automatic; it's the foundation for all calculus in JAMB!

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**The reasoning** Integration is the reverse of differentiation. When we integrate 6x with respect to x, we use the **power rule for integration**: ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C Here, we have 6x = 6x¹ So: ∫ 6x dx = 6 × (x¹⁺¹)/(1+1) + C = 6 × (x²/2) + C = **3x² + C** The **+ C is crucial** — it represents the constant of integration. Since the derivative of any constant is zero, when we reverse the process (integrate), we must account for all possible constants that could have been there originally. **Why the wrong options tempt you** - **Option A (3x²)**: You did the math correctly but forgot the +C. This is the most common mistake — never omit the constant in indefinite integrals! - **Option C (6x² + C)**: You forgot to divide by the new power (2). You kept the 6 but didn't apply the full power rule. - **Option D (6)**: You might have confused integration with differentiation. This would be if you differentiated 6x instead. **Quick takeaway** For indefinite integrals, **always add + C** — it's not optional, it's part of the complete answer!

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**The reasoning** The derivative tells us the **rate of change** of a function. For sin x, we're asking: "How fast is the sine value changing at any point x?" From first principles (or the standard derivative rules you must memorize): d/dx (sin x) = cos x This is a **fundamental derivative** — one of the building blocks of calculus. Just like how the derivative of x² is 2x, the derivative of sin x is cos x. No negative sign, no other trig function. Pure and simple. **Why the wrong options tempt you** - **−cos x** is the derivative of cos x, not sin x. Students mix these up! - **−sin x** is the derivative of cos x. The pairing gets confusing if you don't practice. - **tan x** looks trigonometric but has nothing to do with differentiating sine. The negative signs in options B and C are traps for students who half-remember the rules. **Quick takeaway** **d/dx (sin x) = cos x** and **d/dx (cos x) = −sin x** — memorize this pair like your name; they're the heartbeat of calculus.

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4x = 24 → x = 6.

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dy/dx: 3x² + 2.

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∫ 2x dx = 2 · x²/2 + C = x² + C.

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log₁₀ x = 2 → x = 10² = 100.

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(x − 2)(x − 3) = 0 → x = 2 or 3.

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sin 30° = 0.5.

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Sum = 40 ÷ 4 = 10.

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Coin has 2 equally-likely outcomes; head = 1 out of 2 = 1/2.

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3² + 4² = 9 + 16 = 25 = 5². Pythagoras confirms right-angled.

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x − 3 = 5 or x − 3 = −5 → x = 8 or x = −2.