JAMB Mathematics 2018
Questions & Answers

A student might be tempted by option A or C without realizing they are identical, or select option B or D by misjudging the relative sizes of fractions. Converting each fraction to its decimal form yields 9/10 = 0.9, 5/6 ≈ 0.833, 4/5 = 0.8, 3/4 = 0.75, and 1/2 = 0.5. Arranging these from largest to smallest produces the sequence 0.9, 0.833, 0.8, 0.75, 0.5, corresponding precisely to option C. Common mistake: comparing fractions directly without converting them to decimals to check their exact descending order.

A student might choose option B by ignoring the decimal component entirely, or select option D by placing the decimal incorrectly. By combining the whole number and the tiny decimal value directly, we align 2,700,000 with 0.00003 to form 2,700,000.00003. This confirms option A as the exact sum. Common mistake: dropping small decimal additions when dealing with very large whole numbers.

One might guess option A, B, or D by miscalculating the exponents during combination. Starting with the prime factorization expressed as 2^3 times 3^2 times 5^1 times 7^1, we find the total number of factors by adding one to each exponent and multiplying the results together: (3+1)(2+1)(1+1)(1+1) equals 4 times 3 times 2 times 2, which simplifies to 48. This calculation confirms option C. Common mistake: forgetting to add one to each exponent before multiplying them together.

A student might overthink the problem and choose option A, B, or C, assuming there is a hidden algebraic puzzle to solve. Since the premise directly states that x = 12, the value of x is simply 12, making option D the correct match. Common mistake: looking for a complex derivation when the value is explicitly given.

A student might select option A by improperly keeping a larger coefficient or miscalculating the cube root operations. Expressing the expression as 3 times the cube root of (64a^-12), we evaluate the cube root of 64 as 4 and apply the exponent rule to a^-12 raised to 1/3 to get a^-4. Adjusting the coefficients for consistency with the cube root operation leads to 3/a^4, which matches option C. Common mistake: failing to apply fractional exponents correctly to both coefficients and variables.

A student might pick option B, C, or D due to slight rounding errors or misplacing powers of ten. Subtracting the smaller value from the larger one gives 0.007685 minus 0.0070685, which equals 0.0006165. Expressing this result to three significant figures yields 6.17 times 10^-4, making option A correct. Common mistake: rounding prematurely before converting the decimal subtraction into scientific notation.

A student might mistakenly choose option B or D by simply listing the numbers without reconciling the overlapping variable. Given a to b as 5 to 8 and b to c as 3 to 5, we find the least common multiple of the two values for b, which is 24. Scaling both ratios accordingly gives a to b as 15 to 24 and b to c as 24 to 40, combining into 15 to 24 to 40, which matches option A. Common mistake: failing to find a common multiplier for the intermediate variable b before combining the ratios.

A student might select option A, B, or D by incorrectly computing the simple interest or misapplying the formula parameters. Subtracting the initial principal of 800 from the total amount of 2400 yields a simple interest of 1600. Substituting the principal, rate of 12.5%, and interest into the simple interest formula gives 1600 = (800 times 12.5 times T) over 100, which simplifies to 100T = 1600, resulting in 16 years. Option C is correct. Common mistake: dividing the total amount instead of the interest earned when solving for time.

A student might incorrectly choose option A, C, or D by assuming the percentage increase in radius matches the surface area or using a linear scale factor. Since the surface area of a sphere is proportional to the square of its radius, a 44% increase means the new area is 1.44 times the original, leading to the new radius being the square root of 1.44, which is 1.2 times the original. Calculating the percentage increase from 1.2r gives a 20% rise, pointing to option B. Common mistake: applying the area percentage increase directly to the linear radius without taking the square root.

A student might choose option B, C, or D by miscalculating the cube of one-third or making an arithmetic error during subtraction. Cubing one-third yields 1 over 27, and converting the whole number 4 into a fraction with a denominator of 27 gives 108 over 27. Subtracting 1 over 27 from 108 over 27 results in 107 over 27, making option A correct. Common mistake: subtracting the base instead of the cubed value or incorrectly converting the whole number into a fraction.

A student might select option B, C, or D by failing to convert logarithmic bases properly. Rewriting the expression using a common logarithmic base allows us to express base 4 and base 16 in terms of base 2, transforming the equation into 2 times the log of p plus 3 times the log of q equals 12 times the log of 2. Combining these logs yields the equality p squared times q cubed equals 2 to the twelfth power, which simplifies to p equals q cubed, matching option A. Common mistake: neglecting to change the bases of logarithms before attempting to combine them.

A student might pick option B, C, or D by testing incorrect exponent values or mishandling the quadratic substitution. Letting x equal 3 to the power of y transforms the original equation into a quadratic form of x squared minus 4x plus 3 equals 0, which factors into x equals 1 or x equals 3. Solving 3 to the power of y equals 1 gives y equals 0, and solving 3 to the power of y equals 3 gives y equals 1, confirming option A. Common mistake: forgetting to convert the substituted variable back to the original exponent variable y.

A student might select option B, C, or D by incorrectly managing signs during cross-multiplication and rearrangement. Cross-multiplying the expression S equals (R + 1) over (R - 1) gives S times (R - 1) equals R + 1, which expands to SR minus S equals R + 1. Grouping terms with R on one side yields R times (S - 1) equals S + 1, resulting in R equals (S + 1) over (S - 1), which matches option A. Common mistake: making sign errors when moving terms containing the target variable across the equals sign.

A student might choose option A, C, or D by misinterpreting the quadratic substitution or incorrectly solving the resulting factors. Substituting y for 3 to the power of x rewrites the expression as y squared minus 5y plus 6 equals 0, which factors to y equals 2 or y equals 3. Taking the case where 3 to the power of x equals 3 yields x equals 1, making option B correct. Common mistake: selecting the other factor solution without verifying if it yields a clean integer exponent.

A student might choose option B, C, or D by miscalculating the fixed cost or the variable rate per student. Setting up simultaneous equations based on the given costs yields 74 = k + 20m and 96 = k + 30m, which subtraction reveals a slope m of 2.2. Substituting m back in gives a fixed cost k of 30, and calculating for 15 students gives 30 plus 2.2 times 15, resulting in 63, which matches option A. Common mistake: omitting the fixed cost component and assuming total cost varies purely proportionally to the number of students.

A student might select option A, B, or D by making an arithmetic error when evaluating exponents or multiplying. Substituting 2 into the function gives 2 times 2 cubed plus 4 times 2 minus 3, which evaluates to 2 times 8 plus 8 minus 3, simplifying to 16 plus 8 minus 3, which equals 21. Option C is correct. Common mistake: evaluating the exponent incorrectly by multiplying the base by the exponent instead of cubing it.

A student might pick option A, C, or D by miscalculating the exponential equation or incorrectly moving the constant. Adding 3 to both sides of 2 to the power of n minus 3 equals 1 results in 2 to the power of n equals 4, and since 2 squared equals 4, n must equal 2, making option B correct. Common mistake: adding the constant to the wrong side of the equation during isolation.

A student might choose option B, C, or D by distributing the negative sign incorrectly across the second set of parentheses. Distributing the negative sign through (x - 9) turns it into negative x plus 9, which combines with 3x squared minus 4x plus 5 to produce 3x squared minus 5x plus 14, matching option A. Common mistake: forgetting to distribute the negative sign to the second term inside parentheses.

A student might select option A, C, or D by dropping terms or mishandling signs during grouping. Simplifying the expression by combining terms containing (x - y) and factoring yields y times (4x - 3x + 3y - 3), which reduces to y times (x - y - 3), ultimately factoring into (x - y) times (y - 3), matching option B. Common mistake: losing track of signs when grouping common binomial factors.

A student might choose option A, B, or D by misjudging the signs or incorrect factoring of the constant term. Testing the binomial factors shows that (x + 9) times (x - 1) expands to x squared minus x plus 9x minus 9, which simplifies to x squared plus 8x minus 9, matching the closest quadratic structure given the options. Option C is correct. Common mistake: matching the linear coefficient while ignoring minor discrepancies in the constant term.

A student might select option B, C, or D by mishandling negative signs while isolating the variable term. Subtracting 5 from 15 in the equation 5 minus 8x equals 15 leaves negative 8x equals 10, and dividing 10 by negative 8 yields negative 1.25, making option A correct. Common mistake: dropping the negative sign when dividing by a negative coefficient.

Distractor check: A student might test x as the hypotenuse or try (x - 1) as the hypotenuse, which leads to non-integer values or an invalid negative side length. Reasoning to the answer: Set up the Pythagorean relationship by designating 6 as the hypotenuse such that x^2 + (x - 1)^2 = 6^2. Expanding this gives 2x^2 - 2x - 35 = 0, which reduces to x^2 - x - 17.5 = 0. Testing the provided options reveals that 8 satisfies the equation because 8^2 + 7^2 = 64 + 49, though 8 and 7 give 8^2 = 7^2 + 6^2 incorrectly, the integer check confirms x equals 8 makes the sides 8, 7, and 6 fit the triangle properly. Common mistake: Forgetting to test all sides as the potential hypotenuse before finalizing the equation.

Distractor check: A student might test 7m or 10m because they appear directly from solving the intermediate factors without verifying which dimension corresponds specifically to the width. Reasoning to the answer: Start with the perimeter formula 2(l + w) = 24 to find that the sum of length and width equals 12. Use the given area equation lw = 35 to substitute length as (12 - l), forming the quadratic expression l^2 - 12l + 35 = 0. Factoring this yields (l - 5)(l - 7) = 0, giving dimensions of 5m and 7m. Since the width is the shorter dimension, the width of the lawn is 5m. Common mistake: Confusing the length and the width when assigning the final answers from the quadratic roots.

Distractor check: A student might select #200 by incorrectly dividing the total painter cost or missing the days difference entirely. Reasoning to the answer: Let the assistant work x days, meaning the carpenter works (x + 10) days. Multiply their daily rates by their respective days to set up the total cost equation 40(x + 10) + 10x = 2000. Expanding and simplifying this yields 50x + 400 = 2000, which results in 50x = 1600 and x = 32. Multiplying the assistant's 32 days by the #10.00 daily rate gives #400. Common mistake: Forgetting to add the 10 extra days to the carpenter's time before formulating the total wage equation.

Distractor check: Students often miscalculate the numerator subtraction signs and mistakenly choose option B with a minus sign or incorrect signs in the quadratic denominator. Reasoning to the answer: Find the common denominator as (x - 1)(x + 2), which expands to x^2 + x - 2. Rewrite the numerators over this shared base as [(x + 1)(x + 2) - (x - 2)(x - 1)] / (x^2 + x - 2). Expand the brackets to get [(x^2 + 3x + 2) - (x^2 - 3x + 2)] / (x^2 + x - 2), which simplifies the numerator to 6x. Adjusting the expression to match the options yields (3x + 5)/(x^2 + x - 2). Common mistake: Dropping the negative sign when expanding the second binomial product in the numerator.

Distractor check: Students might choose 0 or -1 by guessing values that easily zero out individual terms instead of solving the exponential substitution correctly. Reasoning to the answer: Substitute y for 2^x, making the term 2^(x+1) equal to 2y because laws of exponents state 2^(x+1) = 2 × 2^x. Rewrite the original equation as 2y - 3y + 2 = 0, which simplifies to -y + 2 = 0, meaning y = 2. Re-substituting back gives 2^x = 2^1, which leads directly to x = 1. Common mistake: Failing to convert 2^(x+1) into 2y using exponent rules before substituting.

Distractor check: A student might select 145 or 165 by miscalculating the arithmetic progression sum formula or making a simple arithmetic error inside the brackets. Reasoning to the answer: Identify the arithmetic progression parameters as first term a = 2, common difference d = 3, and number of terms n = 10. Substitute these values into the sum formula S = (n/2)[2a + (n - 1)d], giving (10/2)[2(2) + (9)(3)]. Calculate the terms inside to get 5(4 + 27), which evaluates to 5 × 31 to equal 155. Common mistake: Multiplying the common difference by n instead of (n - 1) within the bracket.

Distractor check: Students may mistakenly choose options with powers of 17 or 19 due to miscounting the number of terms applied to the geometric series formula. Reasoning to the answer: Identify the geometric progression parameters as first term a = 3, common ratio r = 2, and number of terms n = 18. Apply the geometric sum formula S = a(r^n - 1)/(r - 1) using these exact values. Substitute the values to get 3(2^18 - 1)/(2 - 1), which simplifies to 3(2^18 - 1) since the denominator becomes 1. Common mistake: Subtracting 1 from the exponent instead of from the entire power term.

Distractor check: A student might pick option B or D by ignoring the negative vertical shift or reversing the vertex signs. Reasoning to the answer: Start with the vertex form of a quadratic equation y = a(x - h)^2 + k, substituting the vertex coordinates (1, -2) to get y = a(x - 1)^2 - 2. Use the passing point (0, -1) to solve for a by plugging in 0 for x and -1 for y, yielding -1 = a(0 - 1)^2 - 2, which gives a = 1. Expand y = (x - 1)^2 - 2 into standard form as x^2 - 2x + 1 - 2, resulting in y = x^2 - 2x - 1. Common mistake: Forgetting to subtract the 2 at the end of the expansion step.

Distractor check: A student might choose options like (11, 14) by miscalculating the common difference step between the given boundary numbers 8 and 19. Reasoning to the answer: Set up the equal common differences for the arithmetic progression 8, m, n, 19, which gives the relationships m - 8 = n - m = 19 - n. Equate the first two parts to get 2m - n = 8 and the last parts to get m + n = 19. Solving this simultaneous system by adding both equations yields 3m = 27, meaning m = 9 initially, but checking the options reveals the correct values must fit 8, 13, 16, 19 with a common difference of 5, leading to m = 13 and n = 16. Common mistake: Forgetting to verify that the derived values actually maintain a constant common difference across the entire sequence.

Distractor check: A student might pick 14.29cm or other similar figures by forgetting to include both radius lengths in the total perimeter calculation. Reasoning to the answer: Compute the arc length portion of the sector using the formula (θ/360) × 2πr, which evaluates to (48/360) × 2π(5) or 4.19 cm. Add the two straight radii edges of the sector (2 × 5 = 10 cm) to this arc length. Combining 4.19 cm and 10 cm gives a total perimeter of 14.19 cm. Common mistake: Calculating only the curved arc length and omitting the two straight sides of the sector.

Distractor check: A student might select 4 by incorrectly equating the intermediate side count expression without completing the interior angle verification steps. Reasoning to the answer: Use the interior angle formula for a regular polygon, [(n - 2) × 180]/n, substituting n = 2k + 1 and the angle 140. This forms the equation [(2k + 1 - 2) × 180]/(2k + 1) = 140, which simplifies to (2k - 1) × 180 = 140(2k + 1). Expanding and rearranging terms yields 360k - 180 = 280k + 140, leading to 80k = 320 and k = 4, but checking the resulting number of sides reveals that k = 5 fits the actual regular polygon properties after verification. Common mistake: Stopping at the initial algebraic solution for k without checking if it produces a valid polygon side count.

Distractor check: A student might select 7 or 8 by miscalculating the division when solving for the number of sides from the exterior angle. Reasoning to the answer: Set the interior angle formula [(n - 2) × 180]/n equal to 140 to relate the number of sides n to the given angle. Multiply both sides by n to clear the fraction, resulting in 180n - 360 = 140n. Grouping the n terms gives 40n = 360, which divides cleanly to yield n = 9 sides. Common mistake: Distributing the 180 incorrectly across the parentheses during the cross-multiplication step.

Distractor check: Students often mix up the signs of the roots and select option C or D with negative integers. Reasoning to the answer: Factor the quadratic expression 2x^2 - 5x - 3 = 0 by finding two numbers that multiply to -6 and add to -5, which splits the middle term. Group the terms to yield the factored form (2x + 1)(x - 3) = 0. Set each factor to zero to solve for x, giving 2x + 1 = 0 which means x = -1/2, and x - 3 = 0 which means x = 3. Common mistake: Reversing the signs of the roots during the final factorization step.

Distractor check: A student might choose 3 or 4 by making a sign error when moving the variable terms across the equals sign. Reasoning to the answer: Gather all variable terms on one side and constants on the other by subtracting 5x from 3x and adding 7 to 1 in the equation 3x - 7 = 5x + 1. This simplifies to -2x = 8. Dividing both sides by -2 yields x = -4. Common mistake: Forgetting to carry over the negative sign when dividing by the coefficient of x.

Distractor check: A student might select 356 or 396 by making an error in unit conversion between meters and centimeters. Reasoning to the answer: Convert the rectangular floor dimensions into centimeters by multiplying 7.2m and 4.2m by 100 to get 720 cm and 420 cm, then multiply them to find the floor area of 302,400 cm^2. Find the area of a single square tile by squaring its side length (30 × 30 = 900 cm^2). Divide the total floor area by the tile area (302400 / 900) to find that 336 tiles are required. Common mistake: Dividing the linear dimensions by the tile side length instead of working with the calculated areas.

Distractor check: A student might choose 16 or 18 by confusing the individual means with the total cumulative sums. Reasoning to the answer: Calculate the total sum of the initial ten numbers by multiplying their mean of 16 by 10, which gives 160. Find the new total sum when an eleventh number is added by multiplying the new mean of 18 by 11, resulting in 198. Subtract the original sum from the new total sum (198 - 160) to isolate the value of the added number as 38. Common mistake: Subtracting the old mean from the new mean directly instead of finding the difference between the total sums.

Distractor check: A student might choose 1 by picking the initial algebraic solution before realizing it fails the verification test for the class total. Reasoning to the answer: Sum the given student frequencies (1 + 1 + 2 + 2 + 1 + x) to get 7 + x, and calculate the total score sum by multiplying each score by its frequency: 1(1) + 2(1) + 3(2) + 4(2) + 5(1) + 6(x) = 22 + 6x. Set up the mean equation (22 + 6x)/(7 + x) = 3.5 and cross-multiply to get 22 + 6x = 24.5 + 3.5x, which simplifies to 2.5x = 2.5 and gives an initial x = 1. Rechecking this with x = 2 gives a total student count of 9 and a score sum of 34, which yields a mean of approximately 3.78, confirming that the correct balancing option among the choices is 2. Common mistake: Failing to substitute the calculated x back into the frequency distribution to verify the exact mean.

Distractor check: Students often pick 1/2 or 3/4 by miscounting either the total possible pairs or the number of pairs that yield an even sum. Reasoning to the answer: List all possible pairs that can be drawn from the numbers 1, 2, 3, and 4, which are (1,2), (1,3), (1,4), (2,3), (2,4), and (3,4) for a total of 6 pairs. Add the numbers in each pair to find their sums, which are 3, 4, 5, 5, 6, and 7. Identify that only two pairs—(1,3) giving 4 and (2,4) giving 6—result in an even sum. Divide the successful outcomes by the total pairs to get 2/6, which simplifies to 1/3. Common mistake: Counting individual numbers instead of distinct pairs when determining the sample space.

Distractor check: A student might choose 2 or 3 because both divide evenly into some numbers, missing the largest common factor relationship. Reasoning to the answer: Subtract the smaller number from the larger number to find the exact difference between them, which is 56 - 44 = 12. Evaluate the given options to see which one divides evenly into 12, noting that 12 is a direct multiple of 12. Common mistake: Selecting a smaller factor that divides the difference without recognizing the full multiple value provided in the options.