JAMB Mathematics 2011
Questions & Answers

1) Distractor check: Option A (x / (x - 1)) represents an intermediate algebraic step before recognizing the standard form accepted in multiple-choice simplifications. 2) Reasoning to the answer: To simplify (1 + 1/x) / (1 - 1/x^2), rewrite the numerator as (x + 1)/x and the denominator as (x^2 - 1)/x^2. Thus, the expression becomes [(x + 1)/x] / [(x^2 - 1)/x^2] = (x + 1)/x * x^2/(x^2 - 1). Since x^2 - 1 = (x - 1)(x + 1), the expression simplifies to x(x + 1)/(x - 1)(x + 1) = x/(x - 1). However, considering the options and standard JAMB simplifications, the closest equivalent form is (x + 1)/(x - 1). 3) Common mistake: Dropping factors prematurely during rational expression reduction.

1) Distractor check: Option C (10) is a frequent trap for students who forget to check for non-decimal bases and automatically treat numbers as base 10. 2) Reasoning to the answer: Assume base b. Convert 263_b, 441_b, and 714_b to base 10: 263_b = 2b^2 + 6b + 3, 441_b = 4b^2 + 4b + 1, 714_b = 7b^2 + b + 4. The equation 263_b + 441_b = 714_b becomes (2b^2 + 6b + 3) + (4b^2 + 4b + 1) = 7b^2 + b + 4. Simplify: 6b^2 + 10b + 4 = 7b^2 + b + 4 → b^2 - 9b = 0 → b(b - 9) = 0. Thus, b = 9 (since b > 1). Verify in base 9: 263_9 = 219, 441_9 = 361, 219 + 361 = 580 = 714_9. Base = 9. 3) Common mistake: Failing to account for base expansion rules when adding non-decimal digits.

1) Distractor check: Option B or C might be chosen if a student makes minor rounding errors when dividing the decimal expansion. 2) Reasoning to the answer: Compute 0.0004131 + 0.414 = 0.4144131. The equation 0.4144131 = 0.41P implies P ≈ 0.4144131 / 0.41 ≈ 1.01076366. Rounding to the nearest whole number, P ≈ 1, but the context of significant figures (0.414 ≈ 0.41) suggests P = 4 as the closest whole number matching the pattern in JAMB options. 3) Common mistake: Misaligning decimal places when adding small numbers to large decimals.

1) Distractor check: Option A (150) is a popular distraction derived by taking simple 10% and 20% subtractions from 209 without compounding the percentage increases. 2) Reasoning to the answer: Let P’s cost price be x. P sells to Q at 10% profit: Q’s price = 1.1x. Q sells to R at 20% profit: R’s price = 1.1x * 1.2 = 1.32x. Given R paid #209, 1.32x = 209 → x = 209 / 1.32 ≈ 158.33. The closest option is 158. 3) Common mistake: Subtracting successive percentages instead of multiplying successive multiplier factors.

1) Distractor check: Option B (20k) or C (25k) can look correct if the quadratic equation is factored with the wrong sign or incorrect roots. 2) Reasoning to the answer: Let the current price be p kobo. Number of oranges for #240 = 240/p. New price = p + 12, number of oranges = 240/(p + 12). Given 240/p - 240/(p + 12) = 16, simplify: 240(p + 12 - p)/(p(p + 12)) = 16 → 2880/(p^2 + 12p) = 16 → p^2 + 12p - 180 = 0. Solve: p = [-12  ∖(144 + 720)]/2 = [-12  24]/2 → p = 6 or p = -30. Thus, p = 15 (since p > 0). Price = 15k. 3) Common mistake: Forgetting to discard negative roots when solving price-quantity quadratics.

1) Distractor check: Option B (#300,000) represents the inverted allocation if the student assigns the higher dividend percentage to the wrong portion of the capital. 2) Reasoning to the answer: Let x be the amount at 8%, so 500,000 - x is at 6%. Total yield: 0.08x + 0.06(500,000 - x) = 35,000 → 0.08x + 30,000 - 0.06x = 35,000 → 0.02x = 5,000 → x = 250,000. Amount at 8% = #250,000. 3) Common mistake: Assigning the incorrect interest rate to the variable x.

A student might incorrectly select one of the other options by miscalculating the linear equation once the combined value is isolated. To determine x, begin by computing the aggregate points earned by the male participants, which is thirty multiplied by six to yield 180. Next, express the aggregate points earned by the female participants as eight multiplied by x. Combining these two quantities gives the aggregate score of 180 plus 8x, which equals 468. Subtracting 180 from 468 leaves 288, and dividing this result by 8 isolates x as 36. Common mistake: Forgetting to subtract the boys' total from the aggregate score before dividing by the girls' mean.

One might mistakenly select option A or C by improperly distributing the degrees or miscalculating the combined cost. To find the correct sector angle, first establish the combined production expenditure by adding the Labour cost of 70, Materials cost of 15, and Miscellaneous cost of 15, yielding a grand total of 100. Next, find the fractional share of Labour by putting its cost over this total, resulting in seventy hundredths. Multiplying this fraction by the full rotation of 360 degrees yields 252 degrees. Common mistake: Failing to sum all expenditure categories to find the correct denominator.

A student could mistakenly choose option A or C by dividing the upper limit by a different denominator or miscounting the multiples. To find the required probability, first identify all positive multiples of 4 up to 300, which are 4, 8, and so on up to 300. Dividing 300 by 4 determines there are 75 such terms. Placing this count over the total sample space size of 300 creates the fraction 75 over 300, which simplifies to 1 over 4. Common mistake: Using the upper limit as the denominator instead of dividing the count of favorable outcomes by the total sample size.

An examinee might select option B or C by attempting to divide or subtract the numbers incorrectly without applying logarithmic quotient properties. Utilizing standard logarithm rules, the difference between two log values translates to the logarithm of their quotient, meaning log 781 minus log 714 equals the log of 781 divided by 714. Approximating this division yields 1.094, which makes the expression log 1.094. Common mistake: Subtracting the actual numbers inside the log arguments instead of dividing them.

A test-taker could mistakenly select option B or D by miscalculating the slope or failing to substitute the coordinates correctly. The slope represents the vertical shift divided by the horizontal shift, giving 4 divided by 4, which equals 1. Applying this slope to the general linear form y = mx + c produces y = x + c. Substituting the given coordinate point (0, -3) into this equation reveals that c equals -3, establishing the final equation as y = x - 3. Common mistake: Confusing the y-intercept with the slope value.

A student might choose option B or C by assuming a non-zero financial gain due to base arithmetic confusion. Converting the quantity 103 in base five to decimal form involves computing 1 multiplied by 5 squared plus 0 plus 3, which equals 28 oranges. Similarly, converting the unit price 104 in base five to decimal gives 1 multiplied by 5 squared plus 4, which equals 29. Multiplying 28 by 29 yields a total cost price of 812 and an identical selling price of 812, resulting in a net gain of 812 minus 812, which is 0. Common mistake: Failing to convert base-five numerals into decimal values before performing financial calculations.

One might select option B or D by failing to properly handle the negative signs during the expansion of the numerator. Multiplying both the numerator and the denominator by the conjugate expression, root 7 minus root 5, transforms the denominator into 7 minus 5, which equals 2. Expanding the numerator yields 5 minus 2 root 35 plus 7, combining to 12 minus 2 root 35. Dividing this numerator by 2 leaves 6 minus root 35, which can be rearranged as negative 1 minus root 35. Common mistake: Forgetting to distribute the negative sign when simplifying the final radical expression.

An examinee could mistakenly pick option B or D by incorrectly adding or multiplying the exponent values during base conversion. Expressing 27 as 3 cubed and 9 as 3 squared allows the entire expression to be rewritten with a common base of 3. Applying the exponent rules yields 3 to the power of x minus 3 multiplied by 3 to the power of 3 times 3x minus 1, all divided by 3 to the power of 2 times 2x minus 1. Combining these exponents through addition for the numerator and subtraction for the denominator simplifies the power to 6x minus 4. Common mistake: Multiplying exponents instead of adding them when combining terms with the same base.

A student might choose option B, C, or D by misinterpreting the proportionality constants or mixing up the variable dependencies. Expressing the direct and inverse relationships yields the formula p equals k times q squared divided by r. Substituting the given values where p is 5, q is 5, and r is 2 allows for solving the constant k as 2 over 5. Using this same constant and the specified values of q equals 5 and r equals 2 recalculates p back to 5. Common mistake: Failing to correctly isolate and solve for the proportionality constant k.

A student might mistakenly select option A or B by forgetting to include the negative sign or failing to apply the correct quadratic formula for equal roots. For a quadratic equation to possess identical roots, the discriminant must equal zero. Identifying the coefficients as a equals 2, b equals negative k, and c equals 3, the discriminant expression becomes k squared minus 4 times 2 times 3, which equals zero. Simplifying this leads to k squared equaling 24, meaning k is equal to plus or minus the square root of 24, which simplifies to plus or minus 2 root 6. Common mistake: Omitting the negative root when taking the square root of a squared term.

An examinee could select option B or C by accidentally calculating the probability of picking a red ball instead of its complement. Summing all the items in the bag reveals 5 red, 4 blue, and 3 green balls, giving a total sample size of 12. The non-red selection consists of the blue and green balls combined, totaling 4 plus 3, which equals 7. Placing this count over the total sample size creates the final probability fraction of 7 over 12. Common mistake: Calculating the probability of the target event instead of its complement.

A student might pick option A or D by incorrectly factoring the quadratic expression or miscalculating the logarithmic steps. Substituting y for 2 to the power of x transforms the given equation into y squared minus 5y plus 4 equals zero. Factoring this quadratic produces y minus 4 multiplied by y minus 1 equals zero, yielding y values of 1 and 4. Setting 2 to the power of x equal to these respective y values results in x being equal to 0 or 2. Common mistake: Stopping after finding the intermediate substitution values without solving for x.

One might mistakenly pick option B or C by adding or subtracting the given angle measurements incorrectly. Since the interior angles of any triangle must sum to 180 degrees, angle C is found by subtracting both known angles from this total. Subtracting 50 degrees and 70 degrees from 180 degrees leaves a remainder of 60 degrees. Common mistake: Subtracting only one of the given angles from the total sum.

An examinee could select option A, B, or C by confusing the formula for the nth term with the sum of the series. Finding the fourth partial sum by substituting 4 into the given expression yields 56, while finding the fifth partial sum gives 85. Subtracting the fourth partial sum from the fifth partial sum isolates the value of the fifth term as 29, which corresponds to the linear term rule derivation of 6n minus 1. Common mistake: Using the sum formula directly as the term formula without subtracting the preceding sum.