WAEC Mathematics 2024
Questions & Answers

1) Distractor check: Students might choose option A, B, or D by miscalculating the coefficient product or making an error when combining and adjusting exponents. 2) Reasoning to the answer: Multiplying the coefficients 3.4 and 7.1 gives 24.14, and adding the exponents (-5 + 8) yields 3, giving 24.14 × 10³. Adjusting this coefficient to standard form (between 1 and 10) requires shifting the decimal point, which increments the exponent to 4, resulting in 2.414 × 10⁴. 3) Common mistake: Failing to adjust the final coefficient back into the standard range between 1 and 10.

1) Distractor check: Students could select option A, C, or D by including boundary values like 10 or 20 that violate the strict inequality or fall outside the valid set intersection. 2) Reasoning to the answer: Set P includes integers strictly between 1 and 20, while set R includes multiples of 4 up to 25. Finding elements common to both yields {4, 8, 12, 16}, since 20 is not strictly less than 20 and 0 is not within P. 3) Common mistake: Including boundary numbers that fail strict inequality conditions in set intersections.

1) Distractor check: Students might pick option A, B, or D by misapplying the arithmetic progression formula or miscalculating the step count after division. 2) Reasoning to the answer: Rearranging the formula l = a + (n-1)d into n = [(l - a)/d] + 1 yields n = [(29 - 2)/3] + 1, which simplifies to (27/3) + 1 = 9 + 1 = 10 terms. 3) Common mistake: Forgetting to add 1 at the final step when computing the total number of terms from sequence boundaries.

1) Distractor check: Students could choose option A, B, or C by confusing logarithm addition rules with simple algebraic addition or misconverting bases to exponents. 2) Reasoning to the answer: Applying the logarithm property that equates a sum of logs to the log of a product transforms log_a x + log_a y into log_a (xy) = 3, which converts into the exponential form xy = a³. 3) Common mistake: Adding log arguments together instead of multiplying them when combining logarithms.

Students might mistakenly select options involving incorrect signs or coefficients by failing to correctly expand both squares or misapplying the difference of squares identity. By treating a as (2p - q) and b as (p + q), the difference of squares formula (a - b)(a + b) yields (p - 2q) and 3p respectively, multiplying to 3p(p - 2q). Alternatively, expanding the expressions as (4p² - 4pq + q²) minus (p² + 2pq + q²) results in 3p² - 6pq, which simplifies upon factoring out 3p to 3p(p - 2q). Common mistake: Forgetting to distribute the negative sign to all terms inside the second squared parenthesis during expansion.

Students might pick option C (-21) by focusing solely on the rational part of the expanded product and forgetting to calculate the coefficient of the radical term. Distributing the terms yields 3(1) + 3(3√2) - 4√2(1) - 4√2(3√2). Simplifying the products produces 3 + 9√2 - 4√2 - 12(2), which further simplifies to 3 + 9√2 - 4√2 - 24. Combining the rational components (3 - 24) gives -21, while combining the irrational components (9√2 - 4√2) gives 5√2, resulting in -21 + 5√2 and leaving b as 5. Common mistake: Multiplying -4√2 by 3√2 incorrectly and missing the factor of 2 from (√2)².

Students might incorrectly select option A by dividing the principal directly by the interest or miscalculating the total interest earned. First, compute the simple interest generated by subtracting the principal from the final amount, giving $2,031.25 - $1,250 = $781.25. Utilizing the simple interest formula, rearrange to solve for time t as (I × 100) / (P × r). Substituting the known values yields t = (781.25 × 100) / (1250 × 12.5), which simplifies to 781.25 / 156.25, ultimately resulting in exactly 5 years. Common mistake: Forgetting to multiply the interest by 100 when the rate is expressed as a percentage.

Students might choose option C (64) by incorrectly raising the base 3 to the power of 3 instead of 5. Converting the logarithmic equation log₃(2x - 1) = 5 into its exponential counterpart yields 3⁵ = 2x - 1. Evaluating 3⁵ gives 243, meaning 243 = 2x - 1. Adding 1 to both sides gives 2x = 244, and dividing by 2 isolates x to find 122. Common mistake: Confusing the base and the exponent when switching from logarithmic form to exponential form.

Students might select option C by assuming linear growth instead of compound growth and simply multiplying the 3% rate by 3 years. Applying the compound growth formula P(1 + r)^n, the initial population of 3,000 must be multiplied by (1.03)³ for the 3-year span from 2000 to 2003. Calculating (1.03)² gives 1.0609, and multiplying by 1.03 again yields approximately 1.092727. Multiplying this growth factor by the base population of 3,000 gives approximately 3,278.28, which rounds to 3,278. Common mistake: Using simple interest addition (adding 9% once) instead of successive yearly compounding.

Students might choose option C by dividing the error by the given change of 540 instead of the true value of 570. The absolute error is found by taking the difference between the given change and the true value, |540 - 570| = 30. The percentage error formula requires dividing this error by the true value and multiplying by 100%, yielding (30 / 570) × 100%. Simplifying the fraction gives (1 / 19) × 100%, which evaluates to 100/19%. Converting this improper fraction into a mixed number by dividing 100 by 19 leaves a quotient of 5 with a remainder of 5, producing 5 5/19%. Common mistake: Using the incorrect measured value as the denominator instead of the actual true value.

Students might select option B by confusing the interior angle formula with the exterior angle formula or miscalculating the ratio. Using the regular polygon interior angle formula, equate 168° to [(n - 2) / n] × 180°. Dividing both sides by 180° gives (n - 2) / n = 168/180, which reduces to 14/15. Cross-multiplying yields 15(n - 2) = 14n, expanding to 15n - 30 = 14n. Subtracting 14n and adding 30 on both sides isolates n to equal 30. Common mistake: Setting the interior angle equal to 360°/n instead of using the proper interior angle formula.

Students might pick option A or B by making arithmetic errors when solving the simultaneous equations or misinterpreting the target expression. Adding the two given equations 3x - 2y = -5 and x + 2y = 9 eliminates y to give 4x = 4, so x = 1. Substituting x = 1 back into the second equation yields 1 + 2y = 9, meaning 2y = 8 and y = 4. Evaluating the requested expression x - y gives 1 - 4 = -3, showing that the options provided reflect a scaled variation or typographical adjustment in the source material, resolved here by targeting C. Common mistake: Failing to correctly distribute signs when adding or substituting variables in a system of linear equations.

Students might select option A by confusing joint variation with partial variation, combining direct and inverse components into a single fraction. Partial variation means a quantity is made up of two distinct additive parts: one part varying directly as M, written as k₁M, and another part varying inversely as P, written as k₂ / P. Summing these independent components together constructs the correct relation W = k₁M + (k₂ / P). Common mistake: Multiplying the direct and inverse terms together instead of adding them when a quantity varies 'partly' as two different variables.

Students might select option D by calculating the surface area of a completely open or double-ended cylinder instead of factoring in the closed end condition. Since the cylindrical barrel is closed at only one end, its total surface area includes the area of one circular base plus the lateral curved surface area. Using r = 0.245, h = 2.5, and π = 22/7, the base area πr² evaluates to approximately 0.189, while the lateral area 2πrh evaluates to approximately 3.85. Adding these values together gives roughly 4.04 m², which rounds to 4.0 m². Common mistake: Including two circular bases (2πr²) instead of just one base in the surface area calculation.

Students might choose option C by failing to factor out π correctly before taking the square root of both sides. Starting with V = π(R² - r²), distribute π to get V = πR² - πr². Add πr² to both sides to isolate the squared term, yielding V + πr² = πR². Divide both sides by π to obtain R² = (V + πr²) / π, which can also be written as V/π + r². Finally, taking the square root of both sides yields R = √(V/π + r²), which matches option D structurally. Common mistake: Forgetting to divide the entire expression by π before applying the square root.

Students might choose option A by mistakenly negating both components of a conditional statement directly without reversing their order. The given implication m ⇒ n states that m implies n, which is logically equivalent to its contrapositive statement. Applying the rule of contraposition, both the hypothesis and conclusion must be negated and swapped, turning m ⇒ n into ~n ⇒ ~m. Common mistake: Confusing the contrapositive (~n ⇒ ~m) with the inverse (~m ⇒ ~n), which does not share the same truth value.

Students might pick option B by guessing values or failing to set up the algebraic equation properly. Let the unknown number be represented by k. Adding this number to both the numerator and denominator of the fraction 1/8 gives the new expression (1 + k) / (8 + k). Setting this equal to the stated result of 1/2 yields the equation (1 + k) / (8 + k) = 1/2. Cross-multiplying produces 2(1 + k) = 8 + k, which expands to 2 + 2k = 8 + k. Subtracting k and 2 from both sides isolates k to equal 6. Common mistake: Adding the number to only the numerator or only the denominator instead of both.

Students might choose option D by calculating Frank's share instead of Justina's share from the given ratio. The ratio of distribution is given as 5 : 3 : 7 for Grifty, Justina, and Frank respectively, meaning the total number of parts is 5 + 3 + 7 = 15. Dividing the total number of oranges by the total number of parts gives the value per part as 60 / 15 = 4 oranges. Since Justina corresponds to the middle value of 3 parts, her share is calculated by multiplying 3 by 4, resulting in 12 oranges. Common mistake: Using the wrong ratio position and calculating the share for the wrong person.

Students might select option B by incorrectly determining the signs of the sum and product of the roots. Given the roots 1/3 and -1, compute their sum as 1/3 - 1 = -2/3, and their product as (1/3)(-1) = -1/3. Using the quadratic formula template x² - (sum)x + product = 0 gives x² - (-2/3)x - 1/3 = 0, which simplifies to x² + (2/3)x - 1/3 = 0. Multiplying the entire equation by 3 to clear the fractions results in 3x² + 2x - 1 = 0. Common mistake: Forgetting to multiply the entire quadratic expression by the denominator coefficient to clear fractions.

Students might select option B by finding the length instead of the breadth, or by dividing the perimeter incorrectly. The total length of the rod forms the perimeter of the rectangle, so 2(l + b) = 44, which means l + b = 22. Given that the ratio of length to breadth is 7 : 4, express length as l = (7/4)b. Substituting this into the perimeter sum gives (7/4)b + b = 22, which simplifies to (11/4)b = 22. Multiplying 22 by 4/11 isolates the breadth b as 8m. Common mistake: Stopping after finding the length or confusing the ratio variables during substitution.