WAEC Mathematics 2019
Questions & Answers

Significant figures are the digits that contribute to the precision of a number, starting from the first non-zero digit. For 0.003597, the significant digits are 3, 5, 9, 7. To round to three significant figures, look at the fourth digit (7), which is 5 or greater, so round up the third digit (9) to 10, carrying over 1 to the second digit (5 becomes 6), resulting in 3.60 × 10^{-3}, or 0.00360. This ensures the number is represented with appropriate precision without unnecessary digits.

The expression (0.064)^{-1/3} is the reciprocal of the cube root of 0.064. First, compute the cube root: 0.064 = 64/1000 = (4/10)^3, so (0.064)^{1/3} = 0.4. Then, the reciprocal is 1/0.4 = 2.5, which is 5/2. Alternatively, express in scientific form: 0.064 = 6.4 × 10^{-2} = 2^6 × 10^{-2}, but the direct method confirms the result. This demonstrates the property of exponents: a^{-n} = 1 / a^n.

To solve the linear equation, eliminate the fractions by multiplying both sides by the least common multiple of the denominators, which is 6: 6 × [(y+1)/2 - (2y-1)/3] = 6 × 4. This simplifies to 3(y+1) - 2(2y-1) = 24. Distribute: 3y + 3 - 4y + 2 = 24, combine like terms: -y + 5 = 24, subtract 5: -y = 19, multiply by -1: y = -19. Verification: Substitute y = -19: (-18)/2 - (-39)/3 = -9 + 13 = 4, which holds true. This method ensures all terms are handled correctly in algebraic manipulation.

Recognize this as a difference of squares: a^2 - b^2 = (a - b)(a + b), where a = 27.63, b = 12.37. Compute a - b = 15.26, a + b = 40.00, product = 15.26 × 40 = 610.4. Direct calculation: 27.63^2 = 763.4169, 12.37^2 = 152.9769, difference = 610.44. To three significant figures, 610.4 rounds to 610 since the fourth digit is 4 < 5. This technique simplifies computation and reduces rounding errors.

Modular congruence means 7 + y leaves a remainder of 4 when divided by 8, so y ≡ 4 - 7 ≡ -3 ≡ 5 (mod 8), since -3 + 8 = 5. Thus, y = 8k + 5 for integer k. For 10 ≤ y ≤ 30, test k=1: 13; k=2: 21; k=3: 29. The smallest is 13. Verification: 7 + 13 = 20, 20 ÷ 8 = 2 remainder 4. This illustrates solving linear congruences and finding solutions in an interval.

The universal set μ = {1,2,3,4,5,6,7,8,9,10}. T (primes) = {2,3,5,7}, so T' (non-primes) = {1,4,6,8,9,10}. M (odds) = {1,3,5,7,9}, so M' (evens) = {2,4,6,8,10}. Intersection T' ∩ M' = elements in both: {4,6,8,10}. Note 1 is odd non-prime but not even, 2 is even prime but not in T'. This set operation identifies composite even numbers in the set.

Logarithms simplify using properties: log₃ 9 = log₃ (3^2) = 2, log₂ 8 = log₂ (2^3) = 3. Numerator: 2 - 3 = -1. Denominator: 2. Result: -1/2. The expression can be rewritten as 1 - (log₂ 8)/log₃ 9, but direct evaluation is straightforward. This highlights the importance of base-specific logarithm values and basic arithmetic.

Convert binary 1111₂ to decimal: 1×2^3 + 1×2^2 + 1×2^1 + 1×2^0 = 8 + 4 + 2 + 1 = 15. The base-y number 23_y = 2×y^1 + 3×y^0 = 2y + 3. Set equal: 2y + 3 = 15, 2y = 12, y = 6. Verify: 2×6 + 3 = 15. This problem tests conversion between positional numeral systems.

In an AP, the difference between consecutive terms is constant. For terms a, a+d, a+2d, here 6, P, 14, so 2P = 6 + 14 = 20, P = 10. Common difference d = 10 - 6 = 4, and 14 - 10 = 4. This property ensures equal spacing, fundamental to arithmetic sequences.

Simplify each radical: √28 = √(4×7) = 2√7, so 2×2√7 = 4√7; √50 = √(25×2) = 5√2, so 3×5√2 = 15√2; √72 = √(36×2) = 6√2. Combine: 4√7 - 15√2 + 6√2 = 4√7 - 9√2. Factoring out perfect squares reduces the expression to simplest radical form.

Substitute ratio: let m = 2k, n = k. Numerator: 3(2k)^2 - 2(k)^2 = 3(4k²) - 2k² = 12k² - 2k² = 10k². Denominator: (2k)^2 + (2k)(k) = 4k² + 2k² = 6k². Ratio: 10k² / 6k² = 10/6 = 5/3. k cancels out, showing the result is independent of the scale, a key feature of ratios.

Direct variation with p and inverse with y² gives H = k p / y². Plug in values: 1 = k × 8 / 2² = k × 8 / 4 = 2k, so k = 1/2. Thus, H = (1/2) p / y² = p / (2 y²). This constant k is determined from given conditions, establishing the full relationship.

Quadratic formula: x = [16 ± √(256 - 240)] / 8 = [16 ± √16] / 8 = [16 ± 4]/8. So x = 20/8 = 5/2, x = 12/8 = 3/2. Discriminant positive indicates two real roots. Factoring: (4x - 6)(x - 5/2) wait, better (2x - 3)(2x - 5) = 4x² - 16x + 15, yes. Roots 3/2 and 5/2.

Significant digits: 4,5. Third digit 6 ≥5, round up 5 to 6: 4.6 × 10^{-4} = 0.00046. Leading zeros ignored.

Use logarithm properties: log 6 = log (2×3) = log2 + log3, 3 log3 = log3^3 = log27, 2/3 log27 = 2/3 log (3^3) = 2 log3. So expression: log2 + log3 - log27 + 2 log3 = log2 + log3 - 3 log3 + 2 log3 = log2 + (1 - 3 + 2) log3 = log2. The terms cancel, leaving log2, demonstrating how log properties simplify complex expressions to basic forms.

First, find cost price (CP): SP = CP × (1 + profit%) , 6900 = CP × 1.15, CP = 6900 / 1.15 = 6000. For new SP = 6600, profit = 6600 - 6000 = 600, percentage profit = (600 / 6000) × 100% = 10%. This uses the concept of percentage change relative to cost price, allowing comparison of profit margins under different selling prices.

From 3p = 4q, p = (4/3)q. Substitute into second: 9 [(4/3)q]^2 = 8q - 12, 9 (16/9 q²) = 8q - 12, 16 q² = 8q - 12, 16 q² - 8q + 12 = 0, divide by 4: 4 q² - 2q + 3 = 0? Wait, wait: 16 q² = 8q - 12, 16 q² - 8q + 12 = 0, divide by 4: 4 q² - 2q + 3 = 0, discriminant 4 - 48 = negative? Wait, error. From 9p² = 8q - 12, but p = 4q/3, 9 (16 q² / 9) = 16 q² = 8q - 12, yes, 16 q² - 8q + 12 = 0, divide by 4: 4 q² - 2q + 3 = 0, D = 4 - 48 = -44, no real. Wait, perhaps the second equation is 9p = 8q - 12. Assume that, 9 (4q/3) = 12 q = 8q - 12, 12q - 8q = -12, 4q = -12, q = -3, p = 4(-3)/3 = -4, pq = 12. Yes, likely transcription error, pq = 12. This simultaneous equation solution uses substitution to find product.

Take log: y log 0.25 = log 32. 0.25 = 1/4 = 2^{-2}, 32 = 2^5. So (2^{-2})^y = 2^5, 2^{-2y} = 2^5, -2y = 5, y = -5/2. Negative exponent indicates 32 is 1 / (0.25)^{-y}, confirming the calculation. Exponential equations are solved by equating bases or using logarithms.

Assuming random order, total people = 12, favorable = 8 boys. P(boy first) = 8/12 = 2/3. This is a basic probability of an event in a uniform sample space, where each person has equal chance to exit first.

Factor numerator: x² - 5x - 14 = (x - 7)(x + 2). Denominator: x² - 9x + 14 = (x - 7)(x - 2). Cancel common (x - 7), assuming x ≠ 7: (x + 2)/(x - 2). Factoring quadratic expressions allows simplification by canceling common factors, but note the restriction on domain.

The expression is undefined when denominator = 0: p² - p = p(p - 1) = 0, so p = 0 or p = 1. Among options, p = 1 makes it undefined. The numerator at p = 1 is 2 ≠ 0, so no removable discontinuity, just undefined. Rational functions are undefined at zeros of denominator.

Radius r = 3.5 cm. Total surface area = 2πr h + 2π r² = 2π r (h + r) = 165. 2π r = 2 × (22/7) × 3.5 = 22, so 22 (h + 3.5) = 165, h + 3.5 = 7.5, h = 4 cm. This formula includes curved surface and two bases, solved by isolating h.

√64 = 8 = 2^3, so a = 3. b = 3a = 9. a² + b² = 9 + 81 = 90. Exponential equations with same base simplify to equating exponents, then substitute into the ratio to find b.

The ladder, pole height, and ground form a right triangle with legs 6 m and 8 m, hypotenuse is ladder length. By Pythagoras theorem: √(6² + 8²) = √(36 + 64) = √100 = 10 m. This classic application verifies the relationship in right-angled triangles.

8 = 2^3, so log_x 8 = log_x (2^3) = 3 log_x 2 = 3 × 0.3 = 0.9. The power rule of logarithms allows breaking down the argument into powers, multiplying the log value by the exponent.

0.008 = 8 × 10^{-3} = (2)^3 × 10^{-3}, cube root: 2 × 10^{-1} = 0.2. Verify: 0.2^3 = 0.008.

lb = (a + b)h / 2. Multiply both sides by 2: 2lb = (a + b)h. Expand: 2lb = ah + bh. Rearrange: 2lb - bh = ah, b (2l - h) = ah, b = ah / (2l - h). This algebraic rearrangement isolates b, useful for formula manipulation in physics or geometry contexts like trapezoid area.

Received = SP - 8% SP = 92% SP. So SP = received / 0.92 = 117760 / 0.92. Calculate 0.92 = 92/100 = 23/25, but 117760 ÷ 0.92 = 117760 × (100/92) = 117760 × (25/23) = (117760 / 23) × 25. 23 × 5120 = 117760, 5120 × 25 = 128000. Yes, SP = $128,000. Percentage commission is deducted from SP to find net receipt, reversed to find gross.

θ = (arc length / circumference) × 360° = [22 / (2π × 15)] × 360 = [22 / (30π)] × 360 = (22 × 360) / (30 × 22/7) = (7920) / (660/7) = 7920 × 7 / 660 = 55440 / 660 = 84°. The proportion of the arc to full circle gives the central angle in degrees.

Original area = length × width = 15x cm². Doubling sides: new length = 30 cm, new width = 2x cm, new area = 30 × 2x = 60x cm². Since area scales with square of linear dimensions, (2)^2 × 15x = 4 × 15x = 60x, confirming the scaling factor for similar figures.

Subtract 3: 2x = 10. Divide by 2: x = 5. Verify: 2(5) + 3 = 13.

Factor common terms: first term 2(x + y)(x - y), second 2(x - y)(x + y). So 2(x + y)(x - y) + 2(x - y)(x + y) = 2(x - y)(x + y) (1 + 1) = 4(x - y)(x + y). Recognizing common binomial factors allows full factorization into difference of squares form.

Difference of squares: (25 - 15)(25 + 15) = 10 × 40 = 400. Direct: 625 - 225 = 400.

Total balls = 5. With replacement, independent events. P(different) = P(W then B) + P(B then W) = (2/5)(3/5) + (3/5)(2/5) = 6/25 + 6/25 = 12/25. Replacement keeps probabilities constant, allowing multiplication rule for sequential events.

Point-slope form: y - y1 = m (x - x1), y + 5 = (3/4)(x - 1). Multiply by 4 to clear fraction: 4y + 20 = 3x - 3, rearrange: 3x - 4y - 23 = 0. Verify: at (1,-5), 3(1) - 4(-5) - 23 = 3 + 20 - 23 = 0. Slope = -A/B for Ax + By + C = 0, but here 3x - 4y = 23, slope 3/4.

Tan x = opposite/adjacent = 3/4, hypotenuse = √(9 + 16) = 5. Sin x = 3/5, cos x = 4/5. Cos x / (2 sin x) = (4/5) / (2 × 3/5) = (4/5) / (6/5) = 4/6 = 2/3. Trigonometric identities in right triangles allow finding all ratios from one.

k ≡ 1 - 4 ≡ -3 ≡ 2 (mod 5), add 5 to -3. Verify: 4 + 2 = 6 ≡ 1 mod 5.

Arc length = (θ/360) × 2π r = (70/360) × 2 × (22/7) × 18. Simplify 70/360 = 7/36, 2 × 22/7 × 18 = 792/7, so (7/36) × (792/7) = (1/36) × 792 = 22 cm. Exact calculation using fraction simplification yields the precise length.

Union: all unique elements from A and B: {1,2,3,4,5}. Includes overlap 3 once.

By alternate segment theorem, the angle between tangent and chord QR is equal to the angle in the alternate segment, so angle QRT = angle QPR = 52°. In triangle PQR, angles sum to 180°, angle PQR = 70°, angle QPR = 52°, so angle PRQ = 180 - 70 - 52 = 58°. The theorem links tangent angles to inscribed angles, aiding geometry proofs.