JAMB Mathematics 2015 Past Questions & Answers

To find the percentage birth rate, you must express the number of births as a fraction of the total population and then multiply the result by 100. The formula is (Number of Births / Total Population) × 100%. For this question, the calculation is $(62 / 6250)$ × 100%, which equals 0.992%. This value, when rounded to one decimal place, is 1.0%.

Common mistake: A common mistake is to set up the fraction incorrectly by dividing the total population by the number of births $(6250 / 62)$ instead of dividing the number of births by the total population.

To solve this, we follow the order of operations, BODMAS. First, we simplify the expression in the bracket, which is (2 ÷ ¼ of 28). 'Of' means multiplication, so we calculate ¼ of 28 first, which gives 7. The bracket then becomes $(2 ÷ 7)$ or $2/7$. Now the problem is 1¼ ÷ $2/7$, which is $5/4$ ÷ $2/7$. To divide fractions, we invert the second fraction and multiply, giving $5/4$ × $7/2$ = $35/8$, which simplifies to the mixed number 4 $3/8$.

Common mistake: The most common mistake is to perform the division inside the bracket (2 ÷ ¼) before the 'of' operation (¼ of 28), which violates the BODMAS rule.

To factorize the quadratic expression x² + 9x + 20, we need to find two numbers that multiply to give the constant term (+20) and add up to give the coefficient of x (+9). The two numbers that satisfy these conditions are +5 and +4, because 5 × 4 = 20 and 5 + 4 = 9. Therefore, the expression can be written in its factored form as (x + 5)(x + 4).

Common mistake: Students often find the correct numbers (5 and 4) but use the wrong signs, forgetting that both numbers must be positive to add up to +9.

First, find the sum of the ratios of their ages, which is 18 + 20 + 22 = 60. To find the amount the second staff received, take their ratio (20), divide it by the total sum of the ratios (60), and multiply by the total amount of money (N120,000). This gives $(20/60)$ * N120,000, which simplifies to $(1/3)$ * N120,000, resulting in N40,000.

Common mistake: A common mistake is to calculate the share for the first staff $(18/60 * N120,000 = N36,000)$, forgetting that the question asks for the second staff's share.

To find the number of elements in the union of two sets, X and Y, you use the formula n(X ∪ Y) = n(X) + n(Y) - n(X ∩ Y). This formula works by adding the number of elements in both sets and then subtracting the number of elements in their intersection to avoid double-counting. Plugging in the given values, we have 15 + 12 - 7, which equals 27 - 7. Therefore, the number of elements in the union of X and Y is 20.

Common mistake: The most common mistake is to simply add the number of elements in both sets $(15 + 12 = 27)$ without subtracting the intersection, which leads to double-counting the elements common to both sets.

To find the intercepts, we use the boundary line equation 3x - z = 9. The x-intercept is the point where the line crosses the x-axis, so z = 0. Substituting z=0 gives 3x = 9, which means x = 3. The z-intercept is where the line crosses the z-axis, so x = 0. Substituting x=0 gives -z = 9, which means z = -9. Therefore, the x and z intercepts are 3 and -9.

Common mistake: Students often make sign errors, for instance, incorrectly solving -z = 9 to get z = 9, which leads to the wrong option B.

To simplify the expression, first apply the power law of logarithms to the middle term, so `3 log₁₀ 2` becomes `log₁₀ 2³`, which is `log₁₀ 8`. The expression is now `log₁₀ 1.5 + log₁₀ 8 − log₁₀ 0.3`. Using the addition and subtraction laws, we combine this into a single logarithm: `log₁₀ $((1.5 × 8) / 0.3)`$. This simplifies to `log₁₀ $(12 / 0.3)`$, which equals `log₁₀ 40`.

Common mistake: Students might incorrectly calculate the final division, `12 ÷ 0.3`, as `4` instead of `40`, leading them to choose option A.

The total surface area of a cylinder is the sum of the area of its two circular bases and its curved surface area. The formula is TSA = 2πr² + 2πrh, which can be simplified to 2πr(r + h). Using the given values, r = 5cm, h = 7cm, and π = 3.14, we get TSA = 2 * 3.14 * 5 * $(5 + 7)$. This becomes 31.4 * 12, which equals 376.8 cm², and this is rounded to 377.0 cm².

Common mistake: A common mistake is to calculate only the curved surface area (2πrh) and forget to include the area of the two circular ends (2πr²).

First, expand the bracket (sinx+siny)², which gives sin²x + 2sinxsiny + sin²y. When you subtract 2sinxsiny from this, you are left with sin²x + sin²y. Since x + y = 90°, it means y = 90° - x, so siny is the same as sin(90° - x), which equals cosx. The expression then becomes sin²x + cos²x, which is a fundamental trigonometric identity equal to 1.

Common mistake: A common mistake is failing to use the given condition x + y = 90° to substitute y = 90° - x, which prevents the use of the identity sin(90° - x) = cosx to simplify the expression.

To find the income tax, first determine the taxable income. This is calculated by subtracting the allowances from the annual salary, which gives N2000 - N600 = N1400. Then, calculate 5% of this taxable income to find the tax expense. The income tax is $5/100$ multiplied by N1400, which equals N70.

Common mistake: A common mistake is to calculate the tax on the gross salary (N2000) or the total income including allowances (N2600), instead of first finding the taxable income by subtracting the allowance.

To solve this quadratic equation, we use the 'almighty formula', x = $[−b ± √(b² − 4ac)]$ / 2a. From the equation 3x² − 4x − 5 = 0, we have a = 3, b = −4, and c = −5. Substituting these values gives x = $[−(−4) ± √((−4)² − 4(3)(−5))]$ / 2(3), which simplifies to x = $[4 ± √(16 + 60)]$ / 6. This becomes x = $[4 ± √76]$ / 6, and since $√76$ is about 8.72, the two values for x are $(4 + 8.72)/6$ ≈ 2.12 and $(4 - 8.72)/6$ ≈ -0.79.

Common mistake: Students often make sign errors, especially when calculating the value of b² − 4ac, for instance calculating it as 16 - 60 instead of the correct 16 + 60.

To solve this, use the formula for the sum of an Arithmetic Progression (AP) when the first and last terms are known, which is Sn = n/2 * (a + l). We are given the sum (Sn) is 40, the first term (a) is 6, and the last term (l) is 10. Substituting these values gives 40 = n/2 * $(6 + 10)$, which simplifies to 40 = n/2 * 16, or 40 = 8n. Dividing both sides by 8, we find that the number of terms (n) is 5.

Common mistake: A common mistake is to incorrectly divide the sum (40) by the last term (10) to get 4, instead of using the correct sum of AP formula.

The equation of a straight line is given by y = mx + c, where 'm' is the gradient and 'c' is the y-intercept. Since the line passes through the origin (0,0), the y-intercept 'c' is 0, so the equation is y = mx. The gradient 'm' is calculated using the two points (0,0) and (-3, -4) as (change in y) / (change in x), which is m = $(-4 - 0)$ / $(-3 - 0)$ = -4/-3 = $4/3$. Substituting the gradient back into the equation gives y = $(4/3)x$, which is the same as y = 4x/3.

Common mistake: Students often mix up the formula for the gradient, calculating it as (change in x) / (change in y) which would give (-3)/(-4) = $3/4$, leading to the incorrect option A.

To find the amount, we use the compound interest formula A = P(1 + r/100)ⁿ. Substitute the given values: P = 126, r = 4, and n = 2. This gives A = $126(1 + 4/100)²$. First, calculate the value in the bracket: 1 + $4/100$ = 1 + 0.04 = 1.04. Then, square this value: (1.04)² = 1.0816. Finally, multiply by the principal: A = 126 × 1.0816 = 136.2816, which is approximately N136.30.

Common mistake: Students might mistakenly calculate simple interest instead of compound interest, which would involve multiplying the rate by the number of years before adding to one, leading to an incorrect answer.

To make x the subject of the formula, first isolate the term containing x by subtracting 2 from both sides to get s − 2 = t/5(x + ⅗y). Next, multiply both sides by the reciprocal of t/5, which is 5/t, to get 5(s − 2)/t = x + ⅗y. Finally, subtract ⅗y from both sides to leave x by itself, which gives the final answer x = 5(s − 2)/t - ⅗y.

Common mistake: A common mistake is to expand the bracket first, which creates more complex fractions to deal with, instead of isolating the bracketed term as the first main step.

To solve for x in log₅20 = x, we apply the change of base rule for logarithms. This rule lets us convert the logarithm to a common base, like base 10, which can be found on a calculator. The formula is log₅20 = log₁₀20 / log₁₀5. Using a calculator, log₁₀20 is approximately 1.3010 and log₁₀5 is approximately 0.6990. Dividing these two values $(1.3010 / 0.6990)$ gives approximately 1.861.

Common mistake: A common mistake is to simply find the value of log₁₀20 (which is about 1.301) and pick the closest option, forgetting that the change of base rule requires dividing by log₁₀5.

First, calculate the simple interest (I) earned by subtracting the principal (P) from the amount (A): I = N588 - N525 = N63. Then, use the simple interest formula I = (P x R x T) / 100, where R is the rate and T is the time. Substitute the values: 63 = (525 x R x 3) / 100. Solving for R gives R = (63 x 100) / (525 x 3), which simplifies to R = 6300 / 1575 = 4%.

Common mistake: A common mistake is to incorrectly use the final amount (N588) as the interest in the formula, instead of first calculating the actual interest earned (N63).

The symbol '∪' represents the union of sets, which means you combine all the elements from the sets involved without repeating any. First, find the union of A and B (A ∪ B) by listing all elements in both sets: 1, 3, 5, 7, 9, 12, 15. Next, find the union of this new set with set C, which means combining all its elements with the elements of C: 2, 4, 6, 8. The final result is 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, which contains every unique element from all three sets.

Common mistake: A common mistake is to confuse the union symbol (∪) with the intersection symbol (∩), which would lead to finding elements common to the sets instead of combining them.

Since the extension is directly proportional to the tension, we can set up a ratio such that Extension1 / Tension1 = Extension2 / Tension2. Plugging in the given values, we get 2 / 8 = Extension2 / 12. To find the new extension, we solve for it: Extension2 = $(2 × 12)$ / 8 = 24 / 8 = 3cm.

Common mistake: A student might find that the tension increases by 50% (from 8 to 12) and correctly calculate that the extension also increases by 50% (which is 1cm), but then mistakenly choose 1 as the answer instead of adding the increase to the original 2cm to get the final extension of 3cm.

To factorize the expression x² − 2x − 15, you need to find two numbers that multiply to give the constant term (-15) and add up to the coefficient of x (-2). The two numbers that satisfy these conditions are -5 and +3, because (-5) × (+3) = -15 and -5 + 3 = -2. Therefore, the expression can be written as the product of two brackets: (x − 5)(x + 3).

Common mistake: A common mistake is choosing option B, where students correctly identify the numbers 5 and 3 but mix up the signs, which results in a middle term of +2x instead of the required -2x.

To find the midpoint of a line segment, you use the formula ( (x₁ + x₂)/2 , (y₁ + y₂)/2 ). Given the points A(-3, 5) and B(4, -1), we substitute the values into the formula. The x-coordinate is $(-3 + 4)/2$ = $1/2$, and the y-coordinate is (5 + (-1))/2 = $4/2$ = 2. Therefore, the coordinate of the midpoint is (½, 2).

Common mistake: A common mistake is swapping the x and y coordinates in the final answer, leading to the incorrect choice of (2, ½).

To find the simple interest, use the formula I = PRT, where I is Interest, P is Principal, R is Rate, and T is Time. The Principal is N325, the Rate is 3% $(or 3/100)$, and the Time is 5 years. So, the interest is N325 x $(3/100)$ x 5. This calculation gives (325 x 3 x 5) / 100 = 4875 / 100, which equals N48.75.

Common mistake: A common mistake is to make an arithmetic error during the multiplication of the principal, rate, and time, which can lead to selecting an incorrect option like D.

To simplify (0.09) squared, you multiply 0.09 by itself. First, multiply the numbers as if they were whole numbers: 9 x 9 = 81. Next, count the total number of decimal places in the numbers you are multiplying, which is two for the first 0.09 and two for the second 0.09, giving a total of four decimal places. Therefore, your final answer must have four decimal places, making it 0.0081.

Common mistake: A common mistake is misplacing the decimal point after multiplying 9 by 9, for instance, writing 0.81 which has only two decimal places instead of the correct four.

First, factorize the quadratic expressions in the denominators to get 1/[(x+1)(x+2)] + 1/[(x+2)(x+3)]. The lowest common multiple (LCM) of the denominators is (x+1)(x+2)(x+3). Expressing the sum over this LCM gives [(x+3) + (x+1)] / [(x+1)(x+2)(x+3)], which simplifies to 2(x+2) / [(x+1)(x+2)(x+3)]. After cancelling the common factor (x+2) from the numerator and denominator, the result is 2/[(x+1)(x+3)].

Common mistake: The most likely mistake is incorrectly finding the LCM of the denominators, for example by just using the factors of the first term, which can lead to an incorrect result like option B.

The probability of drawing a blue ball from the first bag is 3 out of 8 total balls $(3/8)$. The probability of drawing a blue ball from the second bag is 2 out of 6 total balls $(2/6)$. Since these are independent events, we multiply the probabilities to find the chance of both happening, which gives $(3/8)$ * $(2/6)$ = $6/48$, simplifying to $3/24$.

Common mistake: A common mistake is to miscalculate the total number of balls in the second bag as 7 instead of 6, leading to an incorrect calculation of $(3/8)$ * $(2/7)$ = $3/28$.