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.

A student might mistakenly choose option B or D by confusing the formula for the circumference of a circle with its area. Using the circumference formula 2 times pi times r, set equal to 44, allows for solving the radius r as 7 when substituting 22 over 7 for pi. Substituting this radius into the area formula pi times r squared gives 22 over 7 multiplied by 7 squared, which evaluates to 154 square centimeters. Common mistake: Using the diameter instead of the radius in the area calculation.

An examinee could select option B or C by inverting the trigonometric ratios for sine and cosine. Given that the sine of theta is 3 over 5, the opposite side length is 3 and the hypotenuse is 5. Applying the Pythagorean theorem determines the adjacent side length to be the square root of 5 squared minus 3 squared, which equals 4. The cosine of theta is then found by placing this adjacent side over the hypotenuse, yielding 4 over 5. Common mistake: Mixing up the adjacent and opposite sides when computing the cosine ratio.

A student might mistakenly pick option A, C, or D by rounding incorrectly or miscalculating the sum of the dataset. Adding all eight values together yields a total sum of 170. Dividing this sum by the total count of 8 students results in a mean value of 21.25, which rounds to 21. Common mistake: Incorrectly rounding decimal averages to the nearest whole integer.

One might select option A, C, or D by multiplying the values instead of dividing distance by time. Speed is calculated by dividing the total distance traveled by the time elapsed, which in this case is 120 kilometers divided by 2 hours. Performing this division yields a speed of 60 kilometers per hour. Common mistake: Multiplying the distance and time values together instead of dividing.

An examinee could mistakenly choose option B or C by failing to recognize the difference of two squares algebraic identity. Expressing the expression as the square of 3x minus the square of 4y reveals a standard difference of two squares pattern. Factoring this expression produces the product of the sum and difference of these terms, resulting in 3x minus 4y multiplied by 3x plus 4y. Common mistake: Treating the binomial as a perfect square trinomial instead of applying the difference of squares rule.

A student might choose option B, C, or D by incorrectly substituting the linear equations during simultaneous elimination. Rearranging the second equation gives x in terms of y as y plus 1. Substituting this expression into the first equation yields 2 times y plus 1 plus 3y equals 12, which simplifies to 5y plus 2 equals 12, giving y equal to 2. Substituting y back into the expression for x determines x to be 3. Common mistake: Forgetting to substitute the solved value back into the rearranged equation to find the other variable.

One might select option A, C, or D by using an incorrect number of sides or misapplying the polygon angle sum formula. The interior angle sum of any polygon is given by the expression n minus 2 multiplied by 180 degrees. For a decagon, the number of sides n is 10, meaning the calculation becomes 10 minus 2 multiplied by 180 degrees, which evaluates to 1440 degrees. Common mistake: Using the wrong number of sides for a decagon.

An examinee could mistakenly pick option B, C, or D by misidentifying prime numbers or using an incorrect count. Identifying all prime numbers between 1 and 50 yields 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47, totaling 15 primes. Placing this count over the total sample size of 50 creates the fraction 15 over 50, which simplifies to 3 over 10. Common mistake: Including 1 or other composite numbers in the count of primes.

A student might select option C or D by reversing the inequality sign during algebraic manipulation. Solving the inequality begins by adding 5 to both sides of 2x minus 5 less than 7, yielding 2x less than 12. Dividing both sides by 2 isolates x as less than 6. Common mistake: Flipping the inequality sign without multiplying or dividing by a negative number.

An examinee could choose option A, C, or D by calculating an incorrect number of terms or adding them improperly. Evaluating the given nth term formula for the first five values of n produces 1, 5, 9, 13, and 17. Summing these five individual terms together yields 1 plus 5 plus 9 plus 13 plus 17, which equals 45. Common mistake: Miscalculating individual sequence terms before summing them up.

A student might mistakenly pick option A, C, or D by calculating the gross revenue instead of the percentage profit. The total cost for purchasing 50 pens at 10 units each is 500, while the selling price for 50 pens at 15 units each is 750. Subtracting the cost from the selling price yields a profit of 250. Dividing this profit by the initial cost of 500 and multiplying by 100 results in a percentage profit of 50 percent. Common mistake: Dividing the profit by the selling price instead of the cost price.

Distractor check: Option B might be chosen by forgetting that sine and cosine must be added together, selecting only the individual trigonometric value, while Option D incorrectly mixes up standard radical values. Reasoning to the answer: An acute angle whose tangent equals 1 is necessarily 45 degrees. Evaluating the sine and cosine functions at 45 degrees yields 1 divided by the square root of 2 for both expressions. Summing these identical fractions together gives 2 divided by the square root of 2, which simplifies directly to the square root of 2. Common mistake: Forgetting to rationalize the combined fraction or incorrectly adding 1 over the square root of 2 to itself.

Distractor check: A student might pick Option D by incorrectly multiplying the side length by 3 instead of cubing it, or select Option B through random guessing without taking the cube root. Reasoning to the answer: Cubing the side length yields the total volume, meaning that raising the unknown side to the third power equals 64 cubic centimeters. Extracting the cube root of 64 identifies the exact side dimension as 4 centimeters. Common mistake: Mistaking volume calculations for perimeter or area formulas by multiplying or dividing by 3.

Distractor check: Option D could be chosen if a student makes a sign error while distributing negative values or forgets to include the constant term. Reasoning to the answer: Substituting negative 1 for every instance of x in the quadratic expression requires computing 2 times negative 1 squared, subtracting 3 times negative 1, and adding 1. Squaring negative 1 produces positive 1, which multiplied by 2 gives 2. Subtracting the product of 3 and negative 1 adds 3, and adding the final constant brings the total sum to 6. Common mistake: Failing to handle the order of operations and negative sign distribution properly during substitution.

Distractor check: Option A could be mistakenly selected by focusing solely on getting a single specific outcome like heads-heads, whereas Option B might be chosen by miscounting the total outcomes that fulfill the condition. Reasoning to the answer: Listing the complete sample space for tossing two fair coins reveals four total outcomes: both heads, head-tail, tail-head, and both tails. Evaluating the condition of achieving at least one head includes every combination except both tails, leaving three favorable outcomes. Dividing the favorable outcomes by the total sample space results in a probability of 3/4. Common mistake: Counting the sample space incorrectly or misunderstanding the phrase 'at least one' as meaning 'exactly one'.

Distractor check: Option D is a common trap from confusing reciprocal directions with a simple straight-line reversal, while Option C represents the original angle without applying the directional offset. Reasoning to the answer: Finding the reverse bearing requires adjusting the initial angle by either adding or subtracting 180 degrees depending on whether the original measurement is below or above 180 degrees. Since the given bearing of 060 degrees is less than 180 degrees, adding 180 degrees to it yields the correct return direction of 240 degrees. Common mistake: Adding or subtracting 180 degrees incorrectly when the initial angle crosses the 180-degree threshold.

Distractor check: Option D represents the largest share allocated to the child with the 5 parts, which a student might pick by misreading which share was requested. Reasoning to the answer: Determining the total number of proportional parts involves summing the ratio values 2, 3, and 5 to get 10. Calculating the portion for the child with the smallest share means taking 2 parts out of the total 10 parts and multiplying this fraction by the total shared amount of 120,000, which results in 24,000. Common mistake: Calculating the largest share instead of the smallest share by picking the wrong ratio numerator.

Distractor check: Option A or C might be chosen from arithmetic errors when subtracting coordinates or mixing up the numerator and denominator positions. Reasoning to the answer: Finding the gradient of a line passing through two coordinate points requires computing the change in the y-values divided by the change in the x-values. Subtracting the first y-coordinate from the second gives 7 minus 3, which equals 4, while subtracting the corresponding x-coordinates gives 4 minus 2, which equals 2. Dividing the vertical change of 4 by the horizontal change of 2 yields a gradient of 2. Common mistake: Inverting the slope formula by putting the change in x over the change in y.

Distractor check: Option D might be chosen by forgetting to multiply the final bracketed result by the first term, while Option B represents an intermediate calculation step. Reasoning to the answer: Applying the geometric series sum formula requires using the first term of 2, common ratio of 3, and 10 terms. Raising 3 to the 10th power gives 59,049, and subtracting 1 leaves 59,048 in the numerator. Dividing by the denominator of 3 minus 1, which is 2, and then multiplying by the leading factor of 2 cancels the denominator to leave 59,048. Common mistake: Incorrectly canceling terms in the geometric series formula numerator and denominator.

Distractor check: Option B or D might be selected by confusing the coefficient of the linear term directly with the constant term k. Reasoning to the answer: Analyzing a quadratic equation of the standard form reveals that the sum of its roots equals the negation of the linear coefficient divided by the leading coefficient, which matches the given sum of 5. Using the relationship between roots and coefficients allows setting up a specific example where roots 2 and 3 multiply to give a product of 6. Because the constant term k represents the product of the roots, evaluating this product yields a value of 6. Common mistake: Assuming the linear coefficient itself is the unknown constant k without using the root relationships.