JAMB Mathematics 2013 Past Questions & Answers

To convert the number 27 from base 10 to base 3, we use the method of repeated division by 3 and record the remainders. First, 27 divided by 3 is 9 with a remainder of 0. Next, 9 divided by 3 is 3 with a remainder of 0. Then, 3 divided by 3 is 1 with a remainder of 0. Finally, 1 divided by 3 is 0 with a remainder of 1. Reading the remainders from the bottom up gives us 1000, so 27 in base 10 is equal to 1000 in base 3.

Common mistake: A common mistake is to read the remainders from top to bottom instead of bottom to top, or to make a simple error in the division process.

The ratio of the shares is 5:3:2. The highest share corresponds to the largest part of the ratio, which is 5. If 5 parts equal 40 apples, then we can find the value of one part by dividing 40 by 5, which gives us 8. The smallest share corresponds to the smallest part of the ratio, which is 2, so we multiply this value of one part (8) by 2 to get 16 apples.

Common mistake: A common mistake is to calculate the middle share (3 parts) instead of the smallest share, which would lead to 3 times 8, giving 24 (Option B).

First, evaluate the expression 1.25 × 0.025 / 0.05. To simplify the calculation, you can multiply the numerator and the denominator by 1000, which gives $(1.25 × 25)$ / 50. This simplifies to 1.25 × $(1/2)$, which equals 0.625. The question asks for the answer correct to 1 decimal place, so we look at the second decimal digit, which is 2. Since 2 is less than 5, we round down, and the final answer is 0.6.

Common mistake: Students may misplace the decimal point during calculation, for example getting a result like 6.25, and then incorrectly rounding up to 6.3.

The formula for simple interest is I = (P x R x T) / 100, where I is the interest, P is the principal, R is the rate, and T is the time. We are given I = N600, P = N3000, and R = 8%. Substituting these values gives 600 = (3000 x 8 x T) / 100. This simplifies to 600 = 240T, and solving for T gives T = 600 / 240, which is 2.5 years or $2(1/2)$ years.

Common mistake: A common mistake is a calculation error when dividing 600 by 240, which might lead a student to incorrectly choose 3 years instead of the correct 2.5 years.

First, express all the numbers (9 and 27) as powers of the smallest base, which is 3. So, 9 becomes 3² and 27 becomes 3³. The expression then becomes 3⁻⁵ⁿ / (3²)¹⁻ⁿ × (3³)ⁿ⁺¹. Using the laws of indices, simplify the powers to get 3⁻⁵ⁿ / 3²⁻²ⁿ × 3³ⁿ⁺³. Finally, combine the exponents by subtracting for division and adding for multiplication: -5n - $(2-2n)$ + (3n+3) = -5n - 2 + 2n + 3n + 3 = 1. Therefore, the simplified expression is 3¹, which is 3.

Common mistake: Students often make mistakes with the signs when combining the exponents, especially forgetting to distribute the negative sign from the division, like calculating -5n - $(2-2n)$ as -5n - 2 - 2n.

To solve this, you need to use the power rule of logarithms, which states that log $a^b$ is the same as b * log a. Applying this rule to the question, log₁₀ 4¹/³ becomes $(1/3)$ * log₁₀ 4. Since the question gives us the value of log₁₀ 4 as 0.6021, we just need to calculate $(1/3)$ * 0.6021. Dividing 0.6021 by 3 gives 0.2007, which is the correct answer.

Common mistake: A common mistake is to multiply the given value by 3 instead of $1/3$, leading to option C, by misinterpreting the power rule for logarithms.

To simplify the expression, first break down the surds inside the bracket into their simplest forms. $√147$ simplifies to $√(49 x 3)$ which is $7√3$, and $√12$ simplifies to $√(4 x 3)$ which is $2√3$. The expression in the bracket becomes $(7√3 - 2√3)$ which equals $5√3$. Now, multiply this by $√5$ in the numerator to get $5√15$, and finally, divide $5√15$ by the $√15$ in the denominator, which gives the answer 5.

Common mistake: Students often make the mistake of subtracting the numbers under the surds directly $(i.e., calculating √147 - √12 as √135)$, which is mathematically incorrect and leads to a wrong answer.

First, list the elements of each set based on the given rules. Set P contains all odd numbers greater than -1 and less than or equal to 20, which gives P = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. Set Q contains all prime numbers greater than -2 and less than or equal to 25, which gives Q = 2, 3, 5, 7, 11, 13, 17, 19, 23. The intersection, P ∩ Q, consists of elements that are present in both sets. Comparing the two sets, the common elements are 3, 5, 7, 11, 13, 17, 19.

Common mistake: Students may mistakenly choose option D by including the number 2, forgetting that although 2 is a prime number in set Q, it is not an odd number and therefore cannot be in set P.

The expression inside the square root, t² - 4t + 4, is a perfect square which factorizes to (t - 2)². Therefore, the equation becomes S = $√((t - 2)²)$. The square root and the square cancel out, leaving S = t - 2. To make t the subject of the formula, add 2 to both sides, which results in t = S + 2.

Common mistake: Students may correctly simplify to S = t - 2 but then make a sign error when making 't' the subject of the formula, incorrectly choosing S - 2.

According to the factor theorem, if x - 4 is a factor of the polynomial $x^2$ - x - k, then the polynomial will be equal to zero when you substitute x = 4. So, we set up the equation: $(4)^2$ - (4) - k = 0. This simplifies to 16 - 4 - k = 0, which gives 12 - k = 0. Therefore, k must be equal to 12.

Common mistake: The most common mistake is substituting x = -4 instead of x = 4, which comes from incorrectly handling the sign and leads to the wrong answer of k = 20.

This question is solved using the Remainder Theorem, which states that if a polynomial f(p) is divided by (p - a), the remainder is f(a). Here, the polynomial is 6p³ - p² - 47p + 30 and the divisor is p - 3, so we set p = 3. Substituting p = 3 into the polynomial gives 6(3)³ - (3)² - 47(3) + 30. This calculates to 6(27) - 9 - 141 + 30, which simplifies to 162 - 9 - 141 + 30 = 42.

Common mistake: The most likely mistake is a sign error, for example, incorrectly substituting p = -3 instead of p = 3, or making an arithmetic error during the final addition and subtraction steps.

First, establish the relationship as an equation, P = (kmu)/q, where k is the constant of variation. Using the initial values p=4, m=3, u=2, and q=1, we find the constant k by solving 4 = $(k * 3 * 2)/1$, which gives k = $4/6$ = $2/3$. Finally, use this constant with the new values m=6, u=4, and $q=8/5$ to find the new value of p: p = $((2/3) * 6 * 4)$ / $(8/5)$ = 16 / $(8/5)$ = 16 * $(5/8)$ = 10.

Common mistake: Students often forget to calculate the constant of variation (k) and mistakenly assume it is 1, which leads to an incorrect answer of 15.

The statement 'r varies inversely as the square root of s and t' is written as `r = k / $√(st)`$, where k is the constant of variation. To find how s varies, we need to make 's' the subject of the formula. Rearranging gives $`√(st)$ = k / r`, and squaring both sides gives `st = k² / r²`. Finally, dividing by 't' gives `s = k² / (r² * t)`, which means s varies inversely as r² and t.

Common mistake: A common mistake is incorrectly squaring the terms when rearranging the equation, such as forgetting to square the 'r' in the denominator, which leads to a wrong relationship for 's'.

To solve an absolute value inequality like |x - 2| < 3, you can rewrite it as a compound inequality: -3 < x - 2 < 3. To find x, you need to get rid of the '-2' in the middle by adding 2 to all three parts of the inequality. This gives you -3 + 2 < x - 2 + 2 < 3 + 2, which simplifies to -1 < x < 5.

Common mistake: A common mistake is to only solve for the positive case (x - 2 < 3), which gives x < 5, and forget to consider the negative case which establishes the lower bound of -1.

To find the nth term, treat the numerators and denominators as separate sequences. The numerators (4, 7, 10, 13, ...) form an Arithmetic Progression (AP) with first term a=4 and common difference d=3, so its nth term is a + (n-1)d = 4 + (n-1)3 = 3n + 1. The denominators (2, 3, 4, 5, ...) also form an AP with first term a=2 and common difference d=1, so its nth term is a + (n-1)d = 2 + (n-1)1 = n + 1. Therefore, the nth term of the entire progression is the numerator's nth term over the denominator's nth term, which is (3n + 1)/(n+1).

Common mistake: Students may correctly identify the denominator's progression but make a small calculation error in the numerator's formula, leading them to incorrectly choose (3n - 1)/(n+1).

To solve this, we first evaluate the expression inside the bracket, which is $(3 * 4)$. Using the rule x * y = x + 2y, we substitute x=3 and y=4 to get 3 + 2(4) = 3 + 8 = 11. Now, the original problem becomes 2 * 11. We apply the rule again with x=2 and y=11, which gives us 2 + 2(11) = 2 + 22 = 24.

Common mistake: A common mistake is to ignore the brackets and calculate from left to right, finding $(2 * 3)$ first, which would incorrectly lead to the answer 16.

To find the derivative of y = x sin x, we must use the product rule because it is a product of two functions, u=x and v=sin x. The product rule states that dy/dx = u(dv/dx) + v(du/dx). The derivative of u=x is 1, and the derivative of v=sin x is cos x. Substituting these into the formula gives dy/dx = (x)(cos x) + (sin x)(1), which simplifies to x cos x + sin x.

Common mistake: Students may choose option B by incorrectly remembering the product rule, perhaps by mixing up the terms or using a minus sign instead of a plus.

To find the derivative of y = (2x + 2)³, we use the chain rule. First, differentiate the outer power by bringing the power (3) down and reducing it by one, which gives 3(2x + 2)². Then, you must multiply this result by the derivative of the expression inside the bracket (2x + 2), which is 2. Therefore, the final answer is 2 multiplied by 3(2x + 2)², resulting in 6(2x + 2)².

Common mistake: Students often forget to apply the full chain rule, only differentiating the outer power to get 3(2x + 2)² (Option A) without multiplying by the derivative of the inner bracket.

This is a related rates problem. First, write down the formula for the area of a circle, A = πr². To find the rate at which the area is increasing (dA/dt), we differentiate this formula with respect to time 't', which gives dA/dt = 2πr * (dr/dt). We are given that the radius r = 7cm and its rate of increase dr/dt = 0.02cms⁻¹. Substituting these values gives dA/dt = 2 * $(22/7)$ * 7 * 0.02, which simplifies to 44 * 0.02 = 0.88cm²s⁻¹.

Common mistake: A common mistake is to forget to apply the chain rule correctly, for instance by just calculating the area (πr²) or circumference (2πr) and then not knowing how to use the given rate of change.

To find the mean, you first add all the given terms together and then divide by the count of the terms. Here, we have four terms: (t + 2), (2t - 4), (3t + 2), and 2t. Summing them up gives (t + 2t + 3t + 2t) + $(2 - 4 + 2)$, which simplifies to 8t + 0, or just 8t. Since there are four terms, we divide the total sum by 4, which gives 8t / 4 = 2t.

Common mistake: Students often make calculation errors when summing the algebraic terms, for instance, incorrectly adding the constants $(2 - 4 + 2)$ to get 4 instead of 0, which leads to the incorrect option C.

First, find the total sum of the seven numbers by multiplying the mean by the count: 7 × 10 = 70. Next, sum the six given numbers: 2 + 4 + 8 + 14 + 16 + 18 = 62. The missing seventh number is the difference between the total sum and the sum of the six numbers, which is 70 - 62 = 8. The complete list of numbers is now 2, 4, 8, 8, 14, 16, 18, and the mode (the most frequent number) is 8.

Common mistake: A common mistake is to try to find the mode from only the six given numbers, but since no number repeats, students might get confused and guess an incorrect option.

First, find the total number of people by adding the frequencies from the table: 3 + 5 + 1 + 1 + 2 + 3 = 15. The median is the middle value, and its position is found using the formula (N+1)/2, which gives $(15+1)/2$ = the 8th position. By adding the frequencies cumulatively $(3, then 3+5=8)$, we see that the 8th person falls into the group with age 25, making it the median age.

Common mistake: A common mistake is to find the middle value of the 'Age' row from the table (between 30 and 35) without considering the frequencies (the number of people).

The question asks for the standard deviation, given that the variance is 4. The relationship between standard deviation (SD) and variance is that the standard deviation is the positive square root of the variance. Therefore, to find the standard deviation, you simply need to calculate the square root of the given variance. Since the variance is 4, the standard deviation is $√4$, which equals 2.

Common mistake: A common mistake is to get distracted by the extra information about the mean and the data points containing 'x', leading to complicated and unnecessary calculations instead of using the direct relationship between variance and standard deviation.

The range of a set of data is the difference between the highest value and the lowest value. From the frequency table provided in the image, the highest score obtained by a student is 10. The lowest score obtained is 3. Therefore, the range is calculated as Highest Score - Lowest Score, which is 10 - 3 = 7.

Common mistake: A common mistake is to calculate the range of the frequencies (highest frequency 7 minus lowest non-zero frequency 1, which gives 6) instead of the range of the actual scores.

This question is about combinations because the order in which the student selects the subjects does not matter. The formula for selecting 'r' items from a total of 'n' items is nCr = n! / (r!(n-r)!). In this case, we are selecting 2 subjects from 5, so n=5 and r=2. Plugging these values into the formula gives 5! / $(2!(5-2)!)$, which simplifies to 5! / (2!3!).

Common mistake: Students often confuse combinations with permutations and incorrectly use the permutation formula nPr = n!/(n-r)!, which would lead them to choose option A.