JAMB Mathematics 2009 Past Questions & Answers

To solve this, you first need to convert the numbers from base x to base 10. The number 55 in base x is (5 * x¹) + (5 * x⁰), which equals 5x + 5. Similarly, 52 in base x is (5 * x¹) + (2 * x⁰), which equals 5x + 2. Now, add them together and set the sum equal to 77: (5x + 5) + (5x + 2) = 77. This simplifies to 10x + 7 = 77, and solving for x gives 10x = 70, so x = 7.

Common mistake: Students often make the mistake of adding the numbers on the left as if they are in base 10 $(55 + 52 = 107)$, instead of first converting each number from base x to base 10.

First, convert all the mixed fractions to improper fractions: 7 $1/12$ becomes $85/12$, 4 $3/4$ becomes $19/4$, and 2 $1/2$ becomes $5/2$. Find the lowest common denominator (LCM) for 12, 4, and 2, which is 12. The expression becomes $(85/12)$ - $(57/12)$ + $(30/12)$. Combine the numerators following the order of operations: $(85 - 57 + 30)$ / 12 = $58/12$. Simplifying this improper fraction gives $29/6$, which when converted back to a mixed number is 4 with a remainder of 5, resulting in $4(5/6)$.

Common mistake: A common mistake is to correctly calculate the improper fraction as $58/12$ but then make a simple arithmetic error during the final division to convert it to a mixed number, getting a remainder of 2 instead of 10, which leads to the incorrect answer $4(1/6)$.

First, add the numbers in the numerator: 81.81 + 99.44 = 181.25. Next, add the numbers in the denominator: 20.09 + 36.16 = 56.25. Then, divide the result of the numerator by the result of the denominator, which is 181.25 ÷ 56.25 = 3.2222... To correct this to 3 significant figures, we take the first three digits (3, 2, and 2), and since the next digit is 2 (which is less than 5), the answer remains 3.22.

Common mistake: A common mistake is to approximate the individual numbers before performing the addition and division, which can introduce rounding errors and lead to an incorrect final answer.

To find the selling price, you must first determine the total cost price, which is the initial cost plus the servicing cost (N34,000 + N2,000 = N36,000). Next, calculate the profit, which is 15% of this total cost (0.15 x N36,000 = N5,400). The selling price is the total cost price plus the profit, so N36,000 + N5,400 gives N41,400.

Common mistake: The most common mistake is calculating the 15% profit on the initial buying price of N34,000 alone, instead of on the total cost price which includes the servicing fee.

First, find the fraction of the allowance remaining after buying books, which is 1 - $1/5$ = $4/5$. The student then spent $1/2$ of this remainder on food, meaning the other $1/2$ of the remainder was kept for contingencies. Therefore, the fraction kept is $1/2$ of $4/5$, which equals $4/10$ and simplifies to $2/5$.

Common mistake: The most common mistake is to calculate the remainder after buying books $(4/5)$ and incorrectly select it as the final answer, forgetting to account for the money spent on food from that remainder.

To solve this, you need to use the laws of logarithms. First, express 280 as a product of numbers we know, like 2, 7, and 10, which is 280 = 2² x 7 x 10. Using the logarithm product rule, log₁₀280 becomes log₁₀(2²) + log₁₀7 + log₁₀10. This simplifies to 2log₁₀2 + log₁₀7 + 1, and substituting the given values gives 2(0.3010) + 0.8451 + 1, which equals 0.6020 + 0.8451 + 1 = 2.4471.

Common mistake: Students often choose C by correctly calculating 2log₁₀2 + log₁₀7 but forgetting to add the value of log₁₀10, which is 1.

To simplify this fraction, we must remove the square root from the denominator by a process called 'rationalizing the denominator'. We achieve this by multiplying both the numerator and the denominator by the conjugate of the denominator, which is $(3-√7)$. This results in $[(5+√7)(3-√7)]$ / $[(3+√7)(3-√7)]$, which simplifies to $(15 - 5√7 + 3√7 - 7)$ / $(9 - 7)$. The expression then becomes $(8 - 2√7)$ / 2, and after dividing the terms in the numerator by 2, the final answer is 4 - $√7$.

Common mistake: Students often make calculation errors when expanding the brackets in the numerator, for example, getting the signs wrong when multiplying the terms, which leads to an incorrect final answer.

First, list the elements of set X by substituting n = 1, 2, 3, 4, and 5 into the formula n² + 1, which gives X = 2, 5, 10, 17, 26. Next, list the elements of set Y by substituting n = 1, 2, 3, 4, and 5 into the formula 5n, which gives Y = 5, 10, 15, 20, 25. The symbol '∩' means intersection, which are the elements common to both sets. By comparing set X and set Y, we can see that the common elements are 5 and 10.

Common mistake: Students may mistakenly choose option B by including 15 from set Y without verifying that it is also an element of set X.

The condition S⊂T⊂W means that set S is completely inside set T, and set T is completely inside set W. For statement I, the intersection (∩) finds what is common to all three sets; since S is the innermost set, the part common to all is S itself, making statement I true. For statement II, the union (∪) combines all elements; since S and T are already contained within the largest set W, combining everything just gives you W, making statement II true. Statement III is false because the intersection of T and W is T, not S.

Common mistake: Students often mistakenly think statement III is true, forgetting that the intersection of a set and its subset (T and W) is the smaller of the two sets (T), not a different subset within it (S).

To make 'r' the subject of the formula, first remove the square root by squaring both sides of the equation, which gives p² = (rs³)/t. Next, multiply both sides by 't' to clear the fraction, resulting in p²t = rs³. Finally, isolate 'r' by dividing both sides by s³, which gives the final answer r = p²t/s³.

Common mistake: Students might mistakenly choose option B by incorrectly cubing 'p' because they see the 's³' term in the equation, instead of correctly squaring 'p' to remove the square root.