2015 JAMB Mathematics Past Questions

Questions

1. 1.

   Evaluate log717

   1. A.1.35
   2. B.1.353
   3. C.1.455
   4. D.0.455

   Show answer

   Answer: C. 1.455

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2. 2.

   Simplify $6\frac{1}{12} − 2 ¾ + 1 ½$

   1. A.$3 \frac{5}{6}$
   2. B.$4 \frac{5}{6}$
   3. C.$2 \frac{1}{6}$
   4. D.$\frac{1}{3}$

   Show answer

   Answer: B. $4 \frac{5}{6}$

   AI explanation

   Analysis

   To solve this expression, we need to perform addition and subtraction on mixed fractions. The best approach is to convert the mixed fractions into improper fractions, find a common denominator, and then simplify the result.

   Concept

   A mixed fraction $a \frac{b}{c}$ is equivalent to $\frac{(a \times c) + b}{c}$. Once all terms are in improper fraction form, we find the Least Common Multiple (LCM) of the denominators to combine them.

   Step-by-Step Solution

   Step 1: Convert mixed fractions to improper fractions.

   • $6 \frac{1}{12} = \frac{(6 \times 12) + 1}{12} = \frac{72 + 1}{12} = \frac{73}{12}$
   • $2 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}$
   • $1 \frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}$

   Step 2: Rewrite the expression. The expression becomes: $\frac{73}{12} - \frac{11}{4} + \frac{3}{2}$

   Step 3: Find the LCM of the denominators (12, 4, and 2). The LCM of 12, 4, and 2 is 12.

   Step 4: Adjust the fractions to have the denominator 12.

   • $\frac{73}{12}$ remains $\frac{73}{12}$
   • $\frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{33}{12}$
   • $\frac{3}{2} = \frac{3 \times 6}{2 \times 6} = \frac{18}{12}$

   Step 5: Perform the arithmetic. $\frac{73}{12} - \frac{33}{12} + \frac{18}{12} = \frac{73 - 33 + 18}{12}$ $= \frac{40 + 18}{12} = \frac{58}{12}$

   Step 6: Simplify the fraction. Divide both numerator and denominator by 2: $\frac{58 \div 2}{12 \div 2} = \frac{29}{6}$

   Step 7: Convert back to a mixed fraction. $29 \div 6 = 4$ remainder $5$. So, $\frac{29}{6} = 4 \frac{5}{6}$

   Conclusion

   Comparing our result to the given options, we find that it matches option B.

   Exam Tip

   When dealing with mixed fractions in JAMB, you can also solve them by separating the whole numbers and the fractions: $(6 - 2 + 1) + (\frac{1}{12} - \frac{3}{4} + \frac{1}{2})$. This often keeps the numbers smaller and reduces the risk of calculation errors!

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3. 3.

   A public car dealer marked up the cost of a car at 30% in an attempt to make 20% gross profit. Due to the value of dollar, he now placed 20% discount on the car. What profit or loss will he make?

   1. A.3%
   2. B.2%
   3. C.4%
   4. D.1%

   Show answer

   Answer: C. 4%

   AI explanation

   1. Analyze the Question

   We need to determine the final percentage profit or loss after a series of price adjustments.

   • Step 1: The dealer marks up the cost price by 30%.
   • Step 2: The dealer then offers a 20% discount on the new marked price.
   • Goal: Find the final percentage change relative to the original cost price.

   2. Explain the Concept

   To solve this, we use the concept of successive percentage changes. It is easiest to assume the original cost price ($CP$) is 100 units. This makes percentage calculations straightforward.

   • Mark-up: Increasing a value by $x\%$ means multiplying by $(1 + \frac{x}{100})$.
   • Discount: Decreasing a value by $y\%$ means multiplying by $(1 - \frac{y}{100})$.

   3. Show the Solution

   Let the Cost Price ($CP$) = 100.

   Step 1: Calculate the Marked Price ($MP$) The dealer marks up the price by 30%: $MP = CP + (30\% \text{ of } CP)$ $MP = 100 + 30 = 130$

   Step 2: Calculate the Selling Price ($SP$) The dealer offers a 20% discount on the marked price ($MP$): Discount = $20\% \text{ of } 130$ Discount = $0.20 \times 130 = 26$

   $SP = MP - \text{Discount}$ $SP = 130 - 26 = 104$

   Step 3: Calculate the Profit or Loss Profit/Loss = $SP - CP$ Profit/Loss = $104 - 100 = 4$

   Since the result is positive, it is a profit. Percentage Profit = $(\frac{\text{Profit}}{CP}) \times 100\%$ Percentage Profit = $(\frac{4}{100}) \times 100\% = 4\%$

   4. Identify the Answer

   Comparing our result to the options provided: A. 3% B. 2% C. 4% D. 1%

   The correct option is C.

   5. Exam Tip

   When dealing with successive percentage changes (like mark-ups and discounts), you can use the formula: Final Change = $a + b + \frac{ab}{100}$ Where $a$ is the first change (+30) and $b$ is the second change (-20). Calculation: $30 + (-20) + \frac{30 \times (-20)}{100}$ $= 10 + (\frac{-600}{100})$ $= 10 - 6 = 4\%$ This shortcut saves time in the exam hall!

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4. 4.

   Simplify 2.04 × 3.7 (Leave your answer in 2 decimal place)

   1. A.7.51
   2. B.7.71
   3. C.5.5
   4. D.7.55

   Show answer

   Answer: D. 7.55

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5. 5.

   If x + y = 90 simplify $(sinx + siny)^2$−2sinxsiny

   1. A.1
   2. B.0
   3. C.2
   4. D.-1

   Show answer

   Answer: A. 1

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6. 6.

   Simplify $\sqrt{30}$ × $\sqrt{40}$

   1. A.10 √3
   2. B.5 √3
   3. C.20 √3
   4. D.15 √3

   Show answer

   Answer: C. 20 √3

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7. 7.

   If an investor invest N450,000 in a certain organization in order to yield X as a return of N25,000. Find the return on an investment of N700,000 by Y in the same organization.

   1. A.N14,950.50K
   2. B.N25,150.30K
   3. C.N15,000.00K
   4. D.N38,888.90K

   Show answer

   Answer: D. N38,888.90K

   AI explanation

   Analysis

   The problem asks us to find the return on a new investment based on the performance of a previous investment. Since the organization yields returns based on the amount invested, we can assume a direct proportion relationship between the investment amount and the return.

   Concept

   When two quantities are directly proportional, the ratio of one to the other remains constant. Mathematically:

   $\frac{\text{Return}_1}{\text{Investment}_1} = \frac{\text{Return}_2}{\text{Investment}_2}$

   Step-by-Step Solution

   Step 1: Identify the given values.

   • Investment 1 ($I_1$) = N450,000
   • Return 1 ($R_1$) = N25,000
   • Investment 2 ($I_2$) = N700,000
   • Return 2 ($R_2$) = ? (Let this be $Y$)

   Step 2: Set up the proportion.

   $\frac{25,000}{450,000} = \frac{Y}{700,000}$

   Step 3: Simplify the fraction.

   $\frac{25}{450} = \frac{Y}{700,000}$

   Divide both 25 and 450 by 25:

   $\frac{1}{18} = \frac{Y}{700,000}$

   Step 4: Solve for Y.

   $Y = \frac{700,000}{18}$

   $Y = 38,888.888...$

   Rounding to the nearest kobo (two decimal places), we get N38,888.89K.

   Conclusion

   Comparing our result to the provided options, the value is approximately N38,888.90K.

   Exam Tip: In proportion problems like this, always check if the second investment is larger than the first. Since N700,000 is larger than N450,000, your answer must be larger than the original return of N25,000. This quick check helps you eliminate incorrect options immediately!

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8. 8.

   Solve x2 − 2x − 3 = 0

   1. A.x = 2 or 1
   2. B.x = 3 or -1
   3. C.x = 1 or 0
   4. D.x = 1 or 2

   Show answer

   Answer: B. x = 3 or -1

   AI explanation

   Analysis

   The question asks us to find the roots of the quadratic equation $x^2 - 2x - 3 = 0$. This is a standard quadratic equation in the form $ax^2 + bx + c = 0$.

   Concept

   To solve a quadratic equation, we can use the factorization method. We look for two numbers that:

   1. Multiply to give the product of the first and last terms ($1 \times -3 = -3$).
   2. Add to give the middle coefficient ($-2$).

   Once we find these two numbers, we split the middle term and factor by grouping.

   Step-by-Step Solution

   Step 1: Identify the factors. We need two numbers that multiply to $-3$ and add to $-2$.

   • The factors of $-3$ are $(-3 \times 1)$ or $(3 \times -1)$.
   • Check the sums: • $-3 + 1 = -2$ (This matches our middle coefficient!) • $3 + (-1) = 2$ (Incorrect)

   So, our numbers are $-3$ and $+1$.

   Step 2: Rewrite the equation. Replace $-2x$ with $-3x + 1x$: $x^2 - 3x + 1x - 3 = 0$

   Step 3: Factor by grouping. Group the first two terms and the last two terms: $(x^2 - 3x) + (1x - 3) = 0$

   Factor out the common terms from each group: $x(x - 3) + 1(x - 3) = 0$

   Now, factor out the common binomial $(x - 3)$: $(x - 3)(x + 1) = 0$

   Step 4: Solve for x. Set each factor to zero:

   • $x - 3 = 0 \implies x = 3$
   • $x + 1 = 0 \implies x = -1$

   Thus, $x = 3$ or $x = -1$.

   Conclusion

   Comparing our result with the provided options, we see that the correct values are $3$ and $-1$. This matches option B.

   Exam Tip

   If you are ever unsure about your answer in the exam hall, plug your results back into the original equation.

   • If $x = 3$: $(3)^2 - 2(3) - 3 = 9 - 6 - 3 = 0$. (Correct)
   • If $x = -1$: $(-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0$. (Correct) This quick check takes less than 30 seconds and guarantees you don't lose marks due to simple sign errors!

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9. 9.

   Make x the subject of the equation s = 2 + $\frac{t}{5}$(x + ⅗y)

   1. A.x = 5[(s − 2) ÷ t] - 3/5y
   2. B.x = 25[(s − 2) ÷ t] − 3ty
   3. C.x = [1 ÷ (s − 2)3ty]
   4. D.x = [5(s − 2) 2 ÷ 3ty] × t

   Show answer

   Answer: A. x = 5[(s − 2) ÷ t] - 3/5y

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10. 10.

    In a town of 6250 inhabitants, there were 62 births during 1984. Find the percentage birth rate

    1. A.3%
    2. B.1.0%
    3. C.2.5%
    4. D.5.40%

    Show answer

    Answer: B. 1.0%

    AI explanation

    1. Analyze the question

    We are given the total population of a town and the number of births that occurred within a specific year. We need to calculate the percentage birth rate.

    2. Explain the concept

    The birth rate is typically expressed as the number of births per 100 people (percentage) or per 1,000 people. Since the options provided are in percentages, we use the formula:

    $\text{Percentage Birth Rate} = \left( \frac{\text{Number of Births}}{\text{Total Population}} \right) \times 100\%$

    3. Show the solution

    • Total Population: 6,250
    • Number of Births: 62

    Substitute these values into the formula:

    $\text{Percentage Birth Rate} = \left( \frac{62}{6250} \right) \times 100\%$

    First, simplify the fraction:

    $\frac{62}{6250} = 0.00992$

    Now, multiply by 100 to convert to a percentage:

    $0.00992 \times 100 = 0.992\%$

    Rounding 0.992% to one decimal place gives us 1.0%.

    4. Identify the answer

    Comparing our result to the given options: A. 3% B. 1.0% C. 2.5% D. 5.40%

    The calculated value matches option B.

    5. Tip

    When dealing with "rate" questions in JAMB, always pay attention to whether the question asks for a percentage (per 100) or a crude rate (often per 1,000). If you ever get a decimal that is very close to a whole number in the options, check if rounding is expected!

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11. 11.

    Simplify 1¼ ÷ (2 ÷ ¼ of 28)

    1. A.1 $\frac{3}{8}$
    2. B.2 $\frac{3}{4}$
    3. C.4 $\frac{3}{8}$
    4. D.3 $\frac{1}{5}$

    Show answer

    Answer: C. 4 $\frac{3}{8}$

    AI explanation

    To solve this expression, we must follow the BODMAS rule, which dictates the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

    1. Analyze the Expression

    The expression is: $1\frac{1}{4} \div (2 \div \frac{1}{4} \text{ of } 28)$

    First, convert the mixed fraction to an improper fraction: $1\frac{1}{4} = \frac{5}{4}$

    2. Solve inside the bracket

    According to BODMAS, we handle "Of" before "Division" inside the bracket.

    • Step A: Solve "Of" $\frac{1}{4} \text{ of } 28 = \frac{1}{4} \times 28 = 7$

    • Step B: Solve the Division inside the bracket Now the bracket becomes: $(2 \div 7)$ $2 \div 7 = \frac{2}{7}$

    3. Complete the final division

    Now substitute the result back into the main expression: $\frac{5}{4} \div \frac{2}{7}$

    To divide by a fraction, we multiply by its reciprocal: $\frac{5}{4} \times \frac{7}{2} = \frac{5 \times 7}{4 \times 2} = \frac{35}{8}$

    4. Convert to a mixed fraction

    Divide 35 by 8: $35 \div 8 = 4$ remainder $3$ So, $\frac{35}{8} = 4\frac{3}{8}$

    Conclusion

    Comparing our result to the options provided: A. $1\frac{3}{8}$ B. $2\frac{3}{4}$ C. $4\frac{3}{8}$ D. $3\frac{1}{5}$

    The calculated value matches option C.

    Exam Tip: Always remember that in BODMAS, "Of" acts like multiplication but takes precedence over standard division. If you had divided 2 by 1/4 first, you would have gotten the wrong answer. Always look for the "Of" and solve it immediately!

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12. 12.

    If three staff of Myschool Limited agreed to share their salary arrears in the ration of their ages, which are 18 years, 20 years, 22 years respectively. If the sum of the money collected is N120,000.00K, How much does the second staff received?

    1. A.N36,000
    2. B.N44,000
    3. C.N40,000
    4. D.N15,000

    Show answer

    Answer: C. N40,000

    AI explanation

    Analysis

    To solve this problem, we need to divide a total sum of money (N120,000) among three people based on a given ratio (18:20:22). We are specifically asked to find the share of the second person.

    Concept: Ratio and Proportion

    When a quantity is divided in a given ratio, the steps are:

    1. Sum the ratio parts: Add all the numbers in the ratio together.
    2. Find the fraction for the specific part: Place the specific person's ratio value over the total sum of the ratio.
    3. Calculate the share: Multiply this fraction by the total amount of money.

    Step-by-Step Solution

    Step 1: Determine the total sum of the ratio. The ages are 18, 20, and 22. Total ratio = $18 + 20 + 22 = 60$

    Step 2: Identify the ratio for the second staff. The second staff corresponds to the age 20. Ratio for the second staff = $20$

    Step 3: Calculate the fraction of the total money for the second staff. Fraction = $\frac{\text{Second staff's ratio}}{\text{Total ratio}} = \frac{20}{60}$ Simplify the fraction: $\frac{20}{60} = \frac{1}{3}$

    Step 4: Calculate the actual amount. Amount = $\text{Fraction} \times \text{Total Money}$ Amount = $\frac{1}{3} \times 120,000$ Amount = $40,000$

    The second staff receives N40,000.

    Conclusion

    Comparing our result with the provided options: A. N36,000 B. N44,000 C. N40,000 D. N15,000

    The correct option is C.

    Exam Tip

    When dealing with ratios, always simplify the ratio first if possible to make your calculations easier. In this case, the ratio 18:20:22 can be simplified to 9:10:11. If you add these (9+10+11 = 30), the calculation becomes $\frac{10}{30} \times 120,000$, which is $\frac{1}{3} \times 120,000 = 40,000$. Simplifying first often prevents errors with large numbers!

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13. 13.

    The sum of two numbers is 5; their product is -14. Find the numbers.

    1. A.x = 5 or 1
    2. B.x = 7 or 1
    3. C.x = 7 or 0
    4. D.x = 7 or − 2

    Show answer

    Answer: D. x = 7 or − 2

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14. 14.

    The volume of a cone (s) of height 6cm and base radius 5cm is

    1. A.157cm3
    2. B.155cm3
    3. C.175cm3
    4. D.145cm3

    Show answer

    Answer: A. 157cm3

    AI explanation

    1. Analyze the question

    We are asked to calculate the volume of a cone given its height ($h$) and base radius ($r$).

    • Height ($h$) = 6 cm
    • Radius ($r$) = 5 cm
    • We will use $\pi \approx 3.142$ (standard for JAMB calculations unless specified as $22/7$).

    2. Explain the concept

    The volume ($V$) of a cone is given by the formula:

    $V = \frac{1}{3} \pi r^2 h$

    Where:

    • $\pi$ (pi) is a constant (approximately 3.142 or 22/7)
    • $r$ is the radius of the circular base
    • $h$ is the perpendicular height of the cone

    3. Show the solution

    Step 1: Substitute the values into the formula.

    $V = \frac{1}{3} \times \pi \times (5)^2 \times 6$

    Step 2: Simplify the expression. First, calculate the square of the radius:

    $5^2 = 25$

    Now, substitute back:

    $V = \frac{1}{3} \times \pi \times 25 \times 6$

    Step 3: Perform the arithmetic. You can simplify $\frac{1}{3} \times 6$ first, which equals 2:

    $V = \pi \times 25 \times 2$

    $V = 50\pi$

    Step 4: Calculate the final numerical value. Using $\pi \approx 3.14159...$

    $V = 50 \times 3.14159$

    $V = 157.079... \text{ cm}^3$

    Rounding to the nearest whole number, we get 157 cm³.

    4. Identify the answer

    Comparing our result to the given options, it matches option A.

    5. Tip

    In JAMB Mathematics, if you see $\pi$ in the options, leave your answer in terms of $\pi$. If the options are whole numbers, use $3.142$ for $\pi$ to get the most accurate result. Also, remember that the volume of a cone is exactly one-third the volume of a cylinder with the same radius and height ($V_{cylinder} = \pi r^2 h$).

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15. 15.

    The extension of a stretched string is directly proportional to its tension. If the extension produced by a tension of 8 Newton’s is 2cm, find the extension produced by a tension of 12 newton’s.

    1. A.2
    2. B.1
    3. C.0
    4. D.3

    Show answer

    Answer: D. 3

    AI explanation

    Analysis

    The problem states that the extension ($e$) of a stretched string is directly proportional to the tension ($T$) applied to it. This is a classic application of Hooke's Law principle in physics and variation in mathematics.

    Concept

    When two quantities are directly proportional, we express the relationship as:

    $e \propto T$

    To turn this into an equation, we introduce a constant of proportionality ($k$):

    $e = kT$

    Where:

    • $e$ = extension
    • $T$ = tension
    • $k$ = constant

    Step-by-Step Solution

    Step 1: Find the constant ($k$) using the given values. We are told that when $T = 8\text{ N}$, $e = 2\text{ cm}$.

    $2 = k \times 8$

    $k = \frac{2}{8}$

    $k = 0.25$

    Step 2: Use the constant to find the new extension. Now, we need to find the extension ($e$) when the tension ($T$) is $12\text{ N}$.

    $e = k \times T$

    $e = 0.25 \times 12$

    $e = 3\text{ cm}$

    Alternative Method (Ratio Method): Since $e_1/T_1 = e_2/T_2$:

    $\frac{2}{8} = \frac{e_2}{12}$

    Cross-multiply:

    $8 \times e_2 = 2 \times 12$

    $8 \times e_2 = 24$

    $e_2 = \frac{24}{8}$

    $e_2 = 3\text{ cm}$

    Conclusion

    Comparing our result to the provided options, the correct extension is 3 cm.

    Exam Tip

    When dealing with "directly proportional" problems, you can always use the ratio method ($\frac{x_1}{y_1} = \frac{x_2}{y_2}$). It saves time during JAMB because you don't have to explicitly calculate the constant $k$ if you don't want to!

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16. 16.

    Simplify $\frac{1}{(x + 1)} + \frac{1}{(x − 1)}$

    1. A.2x/(x + 1)(x−3)
    2. B.2x/(x + 1)(x−1)
    3. C.2x/(x + 1)2
    4. D.2x(x+1)2

    Show answer

    Answer: B. 2x/(x + 1)(x−1)

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17. 17.

    Given that A = {3, 4, 1, 10, ⅓ } B = {4, 3,⅓, ⅓, 7}. Find A∩B

    1. A.{}
    2. B.{⅓, 1, 2}
    3. C.{3, 4, ⅓}
    4. D.{1, 3, ⅓}

    Show answer

    Answer: C. {3, 4, ⅓}

    AI explanation

    1. Analyze the question

    The question asks for the intersection of two sets, $A$ and $B$. In set theory, the intersection of two sets (denoted as $A \cap B$) consists of all elements that are present in both set $A$ and set $B$.

    2. Explain the concept

    • Set A: $\{3, 4, 1, 10, \frac{1}{3}\}$
    • Set B: $\{4, 3, \frac{1}{3}, \frac{1}{3}, 7\}$
    • Intersection ($A \cap B$): You look for common members. Note that in set theory, duplicate elements within a single set are usually written once (e.g., $\{ \frac{1}{3}, \frac{1}{3} \}$ is the same as $\{ \frac{1}{3} \}$).

    3. Show the solution

    • Step 1: List the elements of Set A: $3, 4, 1, 10, \frac{1}{3}$
    • Step 2: List the elements of Set B: $4, 3, \frac{1}{3}, 7$ (we simplify the duplicate $\frac{1}{3}$)
    • Step 3: Compare the two sets to find common elements: • Is 3 in both? Yes. • Is 4 in both? Yes. • Is 1 in both? No (it is only in A). • Is 10 in both? No (it is only in A). • Is $\frac{1}{3}$ in both? Yes. • Is 7 in both? No (it is only in B).

    • Step 4: Combine the common elements: $\{3, 4, \frac{1}{3}\}$

    4. Identify the answer

    Comparing our result $\{3, 4, \frac{1}{3}\}$ to the given options, it matches Option C.

    *

    5. Tip

    When dealing with set intersection questions in JAMB, always write out the sets clearly and cross out elements that do not appear in both lists. Don't be distracted by extra numbers or duplicates; focus only on the "shared" values!

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18. 18.

    Integral ∫$(5x^3 + 7x^2 − 2x + 5)$dx

    1. A.$\frac{5x^4}{4} + \frac{7x^3}{3} + 2x + C$
    2. B.$\frac{5x^4}{4} + \frac{7x^3}{3} - x^2 + 5x + C$
    3. C.$\frac{5x^3}{3} + \frac{7x^2}{x} - x + C$
    4. D.$\frac{2x^2}{3} + \frac{x}{5} - C$

    Show answer

    Answer: B. $\frac{5x^4}{4} + \frac{7x^3}{3} - x^2 + 5x + C$

    AI explanation

    To solve the integral $\int (5x^3 + 7x^2 - 2x + 5) \, dx$, we apply the Power Rule for Integration.

    1. The Concept

    The Power Rule states that for any term $ax^n$:

    $\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + C$

    (where $n \neq -1$ and $C$ is the constant of integration).

    2. Step-by-Step Solution

    We integrate each term in the expression individually:

    • For $5x^3$: Add 1 to the exponent (3 + 1 = 4) and divide by the new exponent: $\int 5x^3 \, dx = \frac{5x^4}{4}$

    • For $7x^2$: Add 1 to the exponent (2 + 1 = 3) and divide by the new exponent: $\int 7x^2 \, dx = \frac{7x^3}{3}$

    • For $-2x$ (which is $-2x^1$): Add 1 to the exponent (1 + 1 = 2) and divide by the new exponent: $\int -2x^1 \, dx = \frac{-2x^2}{2} = -x^2$

    • For $5$ (which is $5x^0$): Add 1 to the exponent (0 + 1 = 1) and divide by the new exponent: $\int 5 \, dx = 5x$

    • Add the constant of integration ($C$): Since this is an indefinite integral, we must add $C$ at the end.

    3. Combining the results

    Putting it all together, we get:

    $\frac{5x^4}{4} + \frac{7x^3}{3} - x^2 + 5x + C$

    Comparing this result to your options, it matches Option B.

    Exam Tip

    When integrating polynomials, always remember to check your work by differentiating your answer. If you differentiate $\frac{5x^4}{4} + \frac{7x^3}{3} - x^2 + 5x + C$, you will arrive back at the original expression $(5x^3 + 7x^2 - 2x + 5)$. This is a great way to verify your answer during the JAMB exam!

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19. 19.

    If log520 = x, find x

    1. A.1.761
    2. B.1.354
    3. C.1.861
    4. D.2.549

    Show answer

    Answer: C. 1.861

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20. 20.

    Simplify (0.09)2 and give your answer correct to 4 significant figures

    1. A.0.81
    2. B.0.081
    3. C.0.0081
    4. D.8.0001

    Show answer

    Answer: C. 0.0081

    AI explanation

    Analysis

    The question asks us to calculate the square of the decimal $0.09$ and express the result to 4 significant figures.

    Concept

    To square a decimal, you multiply the number by itself. When multiplying decimals:

    1. Ignore the decimal points and multiply the numbers as whole numbers.
    2. Count the total number of decimal places in the original numbers.
    3. Place the decimal point in your result so that it has the same total number of decimal places.

    Solution

    Step 1: Perform the multiplication We are calculating $(0.09)^2$, which is $0.09 \times 0.09$.

    Ignore the decimals: $9 \times 9 = 81$.

    Step 2: Determine decimal places $0.09$ has 2 decimal places. Since we are multiplying $0.09$ by $0.09$, the total number of decimal places in the result must be $2 + 2 = 4$.

    Step 3: Place the decimal point We need 4 decimal places for the number $81$. We add leading zeros to achieve this: $0.0081$

    Step 4: Check significant figures Significant figures are counted starting from the first non-zero digit. In $0.0081$:

    • The zeros before the '8' are not significant.
    • The '8' is the 1st significant figure.
    • The '1' is the 2nd significant figure. Since the value is exactly $0.0081$, it is already expressed to 2 significant figures. To express it to 4 significant figures, we would add trailing zeros ($0.008100$), but looking at the options provided, $0.0081$ is the intended value.

    Conclusion

    Comparing our result to the options: A. 0.81 B. 0.081 C. 0.0081 D. 8.0001

    The calculated value matches option C.

    Exam Tip

    When squaring decimals, a quick way to check your work is to count the decimal places. If you have $n$ decimal places in the base, your square will always have $2n$ decimal places. Here, $2$ decimal places $\times 2 = 4$ decimal places. $0.0081$ has exactly 4 decimal places, confirming the calculation is correct.

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21. 21.

    Calculate the value of x and y if (27x ÷ 81x+2y = 9 ,x + 4y = 0

    1. A.x = 1, y = 1/2
    2. B.x = 2, y = – 1/2
    3. C.x − 0, y = 1
    4. D.x = 2, y = –1

    Show answer

    Answer: B. x = 2, y = – 1/2

    AI explanation

    To solve this problem, we need to handle the two given equations:

    1. $\frac{27^x}{81^{x+2y}} = 9$
    2. $x + 4y = 0$

    Step 1: Simplify the first equation

    All the bases (27, 81, and 9) are powers of 3. Let's express them as powers of 3:

    • $27 = 3^3$
    • $81 = 3^4$
    • $9 = 3^2$

    Substitute these into the first equation: $\frac{(3^3)^x}{(3^4)^{x+2y}} = 3^2$

    Using the power rule $(a^m)^n = a^{mn}$: $\frac{3^{3x}}{3^{4(x+2y)}} = 3^2$ $\frac{3^{3x}}{3^{4x+8y}} = 3^2$

    Using the division rule $\frac{a^m}{a^n} = a^{m-n}$: $3^{3x - (4x + 8y)} = 3^2$ $3^{3x - 4x - 8y} = 3^2$ $3^{-x - 8y} = 3^2$

    Since the bases are the same, we equate the exponents: $-x - 8y = 2$ --- (Equation 3)

    Step 2: Solve the system of linear equations

    We now have:

    1. $x + 4y = 0$ $\implies x = -4y$
    2. $-x - 8y = 2$

    Substitute $x = -4y$ into Equation 3: $-(-4y) - 8y = 2$ $4y - 8y = 2$ $-4y = 2$ $y = \frac{2}{-4} = -\frac{1}{2}$

    Now, find $x$ using $x = -4y$: $x = -4(-\frac{1}{2})$ $x = 2$

    Step 3: Verify the results

    • $x = 2, y = -1/2$
    • Check Equation 2: $x + 4y = 2 + 4(-1/2) = 2 - 2 = 0$. (Correct)
    • Check Equation 1: $3^{-x - 8y} = 3^{-(2) - 8(-1/2)} = 3^{-2 + 4} = 3^2 = 9$. (Correct)

    The values are $x = 2$ and $y = -1/2$. This matches option B.

    Exam Tip: Whenever you see equations involving exponents like 27, 81, 9, or 125, 25, 5, always try to express them as powers of a common base (usually the smallest prime number involved). This turns a complex exponential equation into a simple linear equation!

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22. 22.

    If duty is levied at 25%, find the duty to be added to a bill of N80.

    1. A.N100
    2. B.N20
    3. C.N80
    4. D.N150

    Show answer

    Answer: B. N20

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23. 23.

    Find x if 132x = 70 eight.

    1. A.5
    2. B.3
    3. C.6
    4. D.1

    Show answer

    Answer: C. 6

    AI explanation

    Analysis

    The question asks us to find the value of the base $x$ in the equation $132_x = 70_8$. This is a problem involving Number Bases. To solve this, we need to convert both sides of the equation into base 10 (decimal) and then solve for $x$.

    Concept

    To convert a number from any base $x$ to base 10, we multiply each digit by $x$ raised to the power of its position (starting from 0 on the right).

    For a three-digit number $abc_x$: $abc_x = (a \times x^2) + (b \times x^1) + (c \times x^0)$

    Solution

    Step 1: Convert the left side ($132_x$) to base 10. $132_x = (1 \times x^2) + (3 \times x^1) + (2 \times x^0)$ $132_x = x^2 + 3x + 2$

    Step 2: Convert the right side ($70_8$) to base 10. $70_8 = (7 \times 8^1) + (0 \times 8^0)$ $70_8 = (7 \times 8) + (0 \times 1)$ $70_8 = 56$

    Step 3: Equate both sides and solve the quadratic equation. $x^2 + 3x + 2 = 56$ $x^2 + 3x + 2 - 56 = 0$ $x^2 + 3x - 54 = 0$

    Step 4: Factorize the quadratic equation. We need two numbers that multiply to $-54$ and add up to $+3$. Those numbers are $+9$ and $-6$. $(x + 9)(x - 6) = 0$

    This gives us two possible values for $x$: $x = -9$ or $x = 6$

    Since a number base must be a positive integer, we discard $-9$. Therefore, $x = 6$.

    Conclusion

    The value of $x$ is 6. This matches option C.

    Exam Tip

    When dealing with number bases, always remember that the base must be greater than any individual digit in the number. For example, in $132_x$, $x$ must be greater than 3. Since our result is 6, it satisfies this condition!

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24. 24.

    Find the value of x if $[ 1 ÷ 64^{(x + 2)}]= [4^{(x − 3)} ÷ 16^x ]$

    1. A.$\frac{3}{2}$
    2. B.$\frac{2}{3}$
    3. C.$\frac{1}{3}$
    4. D.$-\frac{3}{2}$

    Show answer

    Answer: D. $-\frac{3}{2}$

    AI explanation

    1. Analyze the question

    We are given an exponential equation:

    $\frac{1}{64^{(x + 2)}} = \frac{4^{(x - 3)}}{16^x}$

    Our goal is to find the value of $x$ by expressing all terms with a common base.

    2. Explain the concept

    To solve exponential equations, we use the Law of Indices. The most effective strategy is to express all numbers as powers of the same base. In this case, notice that $4, 16,$ and $64$ are all powers of $4$ (or $2$). Using base $4$ is the simplest approach:

    • $4 = 4^1$
    • $16 = 4^2$
    • $64 = 4^3$
    • Recall that $\frac{1}{a^n} = a^{-n}$

    3. Show the solution

    Step 1: Rewrite all terms using base 4.

    • Left side: $\frac{1}{64^{(x + 2)}} = \frac{1}{(4^3)^{(x + 2)}} = \frac{1}{4^{3x + 6}} = 4^{-(3x + 6)}$
    • Right side: $\frac{4^{(x - 3)}}{16^x} = \frac{4^{(x - 3)}}{(4^2)^x} = \frac{4^{(x - 3)}}{4^{2x}}$

    Step 2: Simplify the right side using the division law ($a^m \div a^n = a^{m-n}$).

    • Right side: $4^{(x - 3) - 2x} = 4^{-x - 3}$

    Step 3: Equate the exponents. Now we have:

    $4^{-(3x + 6)} = 4^{-x - 3}$

    Since the bases are the same, we can equate the exponents:

    $-(3x + 6) = -x - 3$

    Step 4: Solve for $x$.

    $-3x - 6 = -x - 3$

    Add $3x$ to both sides:

    $-6 = 2x - 3$

    Add $3$ to both sides:

    $-3 = 2x$

    Divide by $2$:

    $x = -\frac{3}{2}$

    4. Identify the answer

    Comparing our result to the given options, we find that the correct choice is D.

    5. Tip

    When dealing with indices, always look for the smallest common base. If you see $4, 8, 16, 32, 64$, try converting everything to base $2$. It makes the arithmetic much simpler and reduces the chance of making a sign error!

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25. 25.

    Factorize $x^2 − 2x − 15$

    1. A.(x + 3)2
    2. B.(x + 5)(x − 3)
    3. C.(x − 5)2
    4. D.(x − 5)(x + 3)

    Show answer

    Answer: D. (x − 5)(x + 3)

    AI explanation

    Analysis

    To factorize the quadratic expression $x^2 - 2x - 15$, we need to find two numbers that satisfy two conditions:

    1. Their product must be equal to the constant term ($-15$).
    2. Their sum must be equal to the coefficient of the middle term ($-2$).

    Concept: Quadratic Factorization

    A quadratic expression in the form $ax^2 + bx + c$ can be factorized by splitting the middle term ($bx$). We look for two factors of $ac$ that add up to $b$. Once found, we rewrite the expression and factor by grouping.

    Step-by-Step Solution

    Step 1: Find the two numbers. We need two numbers that multiply to $-15$ and add to $-2$.

    • Factors of $-15$ include: • $1$ and $-15$ (Sum = $-14$) • $-1$ and $15$ (Sum = $14$) • $3$ and $-5$ (Sum = $-2$) <-- This is our pair!

    Step 2: Rewrite the middle term. Replace $-2x$ with $+3x - 5x$: $x^2 + 3x - 5x - 15$

    Step 3: Factor by grouping. Group the first two terms and the last two terms: $(x^2 + 3x) + (-5x - 15)$

    Factor out the common terms from each group: $x(x + 3) - 5(x + 3)$

    Step 4: Final factorization. Since $(x + 3)$ is common to both terms, we factor it out: $(x - 5)(x + 3)$

    Conclusion

    Comparing our result $(x - 5)(x + 3)$ with the given options, it matches option D.

    Exam Tip

    When factorizing, always check your signs carefully! If the constant term is negative (like $-15$), one of your two numbers must be positive and the other must be negative. If the middle term is negative, the larger of the two numbers must carry the negative sign.

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26. 26.

    Find at which rate per annum simple interest N525 will amount to N588 in 3 years.

    1. A.3%
    2. B.2%
    3. C.5%
    4. D.4%

    Show answer

    Answer: D. 4%

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27. 27.

    X and Y are two sets such that n(X) = 15, n(Y) = 12 and n{X ∩ Y} = 7. Find ∩{X ∪ Y}

    1. A.21
    2. B.225
    3. C.15
    4. D.20

    Show answer

    Answer: D. 20

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28. 28.

    The probability of an event A is 1/5. The probability of B is 1/3 . The probability both A and B is 1/15. What is the probability of either event A or B or both

    1. A.$\frac{2}{15}$
    2. B.$\frac{3}{4}$
    3. C.$\frac{9}{15}$
    4. D.$\frac{1}{15}$

    Show answer

    Answer: C. $\frac{9}{15}$

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29. 29.

    Given that A = {1, 5, 7} B = {3, 9, 12, 15} C = {2, 4, 6, 8} Find (A ∪ B) ∪ C

    1. A.{1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15}
    2. B.{1, 2, 3, 5, 6, 8, 12, 15}
    3. C.{2, 4, 5, 9, 12, 15}
    4. D.{1, 5, 6, 7, 8, 9, 12, 15}

    Show answer

    Answer: A. {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15}

    AI explanation

    1. Analyze the question

    The question asks for the union of three sets: $A$, $B$, and $C$. The union symbol ($\cup$) means we need to combine all the elements from the given sets into one single set, ensuring that no element is repeated.

    2. Explain the concept

    The union of sets $A, B,$ and $C$ (denoted as $A \cup B \cup C$) is the set of all elements that are members of $A$, or $B$, or $C$.

    • Rule: List every unique element found in any of the sets.
    • Strategy: It is often easiest to list them in ascending order to ensure you don't miss any numbers and to make the final set look organized.

    3. Show the solution

    Given:

    • $A = \{1, 5, 7\}$
    • $B = \{3, 9, 12, 15\}$
    • $C = \{2, 4, 6, 8\}$

    Step 1: Find $(A \cup B)$ Combine all elements from $A$ and $B$: $A \cup B = \{1, 3, 5, 7, 9, 12, 15\}$

    Step 2: Find $(A \cup B) \cup C$ Now, take the result from Step 1 and add all elements from set $C$: $(A \cup B) \cup C = \{1, 3, 5, 7, 9, 12, 15\} \cup \{2, 4, 6, 8\}$

    Step 3: Arrange in ascending order Combine and sort the numbers: $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15\}$

    4. Identify the answer

    Comparing our result to the options provided: A. {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15} B. {1, 2, 3, 5, 6, 8, 12, 15} C. {2, 4, 5, 9, 12, 15} D. {1, 5, 6, 7, 8, 9, 12, 15}

    The calculated set matches option A.

    5. Tip

    When solving union problems, always cross out the numbers in the original sets as you write them down in your final set. This prevents you from accidentally skipping a number or writing one twice. Also, remember that the order of elements in a set does not matter, but writing them in ascending order makes it much easier to compare your answer with the options!

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30. 30.

    The area of a circle of radius 4cm is equal to (Take π = 3.142 )

    1. A.10.3cm2
    2. B.15.7cm2
    3. C.50.3cm2
    4. D.17.4cm2

    Show answer

    Answer: C. 50.3cm2

    AI explanation

    1. Analyze the question

    We are asked to calculate the area of a circle given its radius ($r = 4\text{ cm}$) and a specific value for $\pi$ ($\pi = 3.142$).

    2. Explain the concept

    The area ($A$) of a circle is calculated using the formula:

    $A = \pi r^2$

    Where:

    • $\pi$ (pi) is a constant (given here as 3.142).
    • $r$ is the radius of the circle.
    • $r^2$ means the radius multiplied by itself ($r \times r$).

    3. Show the solution

    • Step 1: Identify the given values. • $r = 4\text{ cm}$ • $\pi = 3.142$
    • Step 2: Substitute the values into the formula. • $A = 3.142 \times (4)^2$
    • Step 3: Calculate the square of the radius. • $4^2 = 4 \times 4 = 16$
    • Step 4: Multiply by $\pi$. • $A = 3.142 \times 16$ • $A = 50.272$

    Rounding $50.272$ to one decimal place gives us $50.3\text{ cm}^2$.

    4. Identify the answer

    Comparing our result to the options provided: A. $10.3\text{ cm}^2$ B. $15.7\text{ cm}^2$ C. $50.3\text{ cm}^2$ D. $17.4\text{ cm}^2$

    The correct option is C.

    5. Exam Tip

    When solving circle problems in JAMB, always check if the question specifies the value of $\pi$ to use (e.g., $3.142$ or $22/7$). Using the wrong one can lead to slight variations in your answer. Also, remember that Area is always measured in square units ($\text{cm}^2, \text{m}^2$), while Circumference is measured in linear units ($\text{cm}, \text{m}$).

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31. 31.

    Simplify $[1 ÷ (x^2 + 3x + 2)] + [1 ÷ (x^2 + 5x + 6)]$

    1. A.$\frac{2}{(x + 1)^2}$
    2. B.$\frac{2}{(x + 1)(x + 2}$
    3. C.$\frac{2}{(x + 1)(x + 2}$
    4. D.$\frac{2}{(x + 1)(x + 3}$

    Show answer

    Answer: D. $\frac{2}{(x + 1)(x + 3}$

    AI explanation

    To simplify the expression $\frac{1}{x^2 + 3x + 2} + \frac{1}{x^2 + 5x + 6}$, we follow these steps:

    1. Factorize the denominators

    First, we factorize the quadratic expressions in the denominators:

    • For $x^2 + 3x + 2$: We look for two numbers that multiply to give 2 and add to give 3. These are 1 and 2. So, $x^2 + 3x + 2 = (x + 1)(x + 2)$.
    • For $x^2 + 5x + 6$: We look for two numbers that multiply to give 6 and add to give 5. These are 2 and 3. So, $x^2 + 5x + 6 = (x + 2)(x + 3)$.

    2. Rewrite the expression

    Now, substitute the factors back into the original expression:

    $\frac{1}{(x + 1)(x + 2)} + \frac{1}{(x + 2)(x + 3)}$

    3. Find the Least Common Multiple (LCM)

    The LCM of the two denominators is $(x + 1)(x + 2)(x + 3)$.

    Now, adjust the fractions:

    • The first fraction needs to be multiplied by $\frac{x+3}{x+3}$: $$\frac{1(x + 3)}{(x + 1)(x + 2)(x + 3)} = \frac{x + 3}{(x + 1)(x + 2)(x + 3)}$$
    • The second fraction needs to be multiplied by $\frac{x+1}{x+1}$: $$\frac{1(x + 1)}{(x + 1)(x + 2)(x + 3)} = \frac{x + 1}{(x + 1)(x + 2)(x + 3)}$$

    4. Add the fractions

    Combine the numerators over the common denominator:

    $\frac{(x + 3) + (x + 1)}{(x + 1)(x + 2)(x + 3)} = \frac{2x + 4}{(x + 1)(x + 2)(x + 3)}$

    Factor out the 2 in the numerator:

    $\frac{2(x + 2)}{(x + 1)(x + 2)(x + 3)}$

    5. Simplify

    Cancel out the common term $(x + 2)$ from the numerator and denominator:

    $\frac{2}{(x + 1)(x + 3)}$

    Note on the options provided: The calculation results in $\frac{2}{(x + 1)(x + 3)}$. Looking at the options provided, Option D matches this result. (Note: There is a slight typographical error in the options provided in your prompt where the closing parentheses were omitted, but D is the intended correct expression).

    Exam Tip: Always factorize quadratic denominators first in algebraic fraction problems. It almost always reveals a common factor that allows you to simplify the expression significantly!

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32. 32.

    Simplify log101.5 + 3 log102 − log100.3

    1. A.log104
    2. B.log1040
    3. C.log10-40
    4. D.log104-

    Show answer

    Answer: B. log1040

    AI explanation

    To simplify the expression $\log_{10} 1.5 + 3 \log_{10} 2 - \log_{10} 0.3$, we will use the fundamental laws of logarithms.

    1. Analyze the expression

    The expression is: $\log_{10} 1.5 + 3 \log_{10} 2 - \log_{10} 0.3$

    2. Explain the concepts

    We need three key logarithm laws:

    • Power Law: $n \log_a x = \log_a (x^n)$
    • Addition Law: $\log_a x + \log_a y = \log_a (x \cdot y)$
    • Subtraction Law: $\log_a x - \log_a y = \log_a (\frac{x}{y})$

    3. Step-by-step solution

    Step 1: Apply the Power Law to the middle term $3 \log_{10} 2 = \log_{10} (2^3) = \log_{10} 8$ Now the expression becomes: $\log_{10} 1.5 + \log_{10} 8 - \log_{10} 0.3$

    Step 2: Apply the Addition Law $\log_{10} 1.5 + \log_{10} 8 = \log_{10} (1.5 \times 8)$ $1.5 \times 8 = 12$ So, the expression is now: $\log_{10} 12 - \log_{10} 0.3$

    Step 3: Apply the Subtraction Law $\log_{10} 12 - \log_{10} 0.3 = \log_{10} (\frac{12}{0.3})$

    To divide $12$ by $0.3$: $\frac{12}{0.3} = \frac{120}{3} = 40$

    Therefore, the simplified expression is: $\log_{10} 40$

    4. Identify the answer

    Comparing our result to the given options: A. $\log_{10} 4$ B. $\log_{10} 40$ C. $\log_{10} -40$ D. $\log_{10} 4-$

    The correct option is B.

    5. Exam Tip

    When dealing with logarithms in JAMB, always look to clear the coefficients (the numbers in front of the log) by moving them as exponents first. Also, remember that decimal division is easier if you multiply both the numerator and denominator by 10 or 100 to remove the decimal point!

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33. 33.

    Factorize $a^2 − b^2 − 4a + 4$

    1. A.(a + b)(a − b)
    2. B.(a − 2 + b)(a - 2- b)
    3. C.(a + 1)(a − 2 + b)
    4. D.(a + b) 2

    Show answer

    Answer: B. (a − 2 + b)(a - 2- b)

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34. 34.

    A man bought a car for N800 and sold it for N520. Find his loss per cent

    1. A.15%
    2. B.25%
    3. C.35%
    4. D.10%

    Show answer

    Answer: C. 35%

    AI explanation

    Analysis

    To find the loss percentage, we need to determine the actual amount of money lost and then calculate what percentage that loss is of the original cost price.

    Concept

    The formula for calculating Loss Percentage is:

    $\text{Loss Percentage} = \left( \frac{\text{Cost Price} - \text{Selling Price}}{\text{Cost Price}} \right) \times 100$

    • Cost Price (CP): The amount the man paid for the car (N800).
    • Selling Price (SP): The amount the man sold the car for (N520).
    • Loss: The difference between the Cost Price and the Selling Price (CP - SP).

    Step-by-Step Solution

    1. Calculate the actual loss: $$\text{Loss} = \text{Cost Price} - \text{Selling Price}$$ $$\text{Loss} = 800 - 520 = 280$$ The man lost N280.

    2. Calculate the loss percentage: $$\text{Loss \%} = \left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100$$ $$\text{Loss \%} = \left( \frac{280}{800} \right) \times 100$$

    3. Simplify the calculation: $$\text{Loss \%} = \left( \frac{280}{8} \right)$$ $$\text{Loss \%} = 35\%$$

    Conclusion

    The man incurred a loss of 35% on the sale of the car. Comparing this to the options provided, the correct choice is C.

    Exam Tip

    When calculating percentage profit or loss, always divide by the Cost Price, never the Selling Price. A common mistake students make is using the Selling Price as the denominator, which will lead to the wrong answer.

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35. 35.

    Find the x and z intercepts of the graph of 3x - z $\leq$ 9

    1. A.(3, -9)
    2. B.(-3, 9)
    3. C.(-3, -9)
    4. D.(-3, 0)

    Show answer

    Answer: A. (3, -9)

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36. 36.

    A man with an annual salary of N2000, has allowances of N600. If Income Tax is 5%. How much income tax expenses does he pay each year?

    1. A.15
    2. B.50
    3. C.70
    4. D.25

    Show answer

    Answer: C. 70

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37. 37.

    One bag contain 3 blue and 5 red balls, another bag contain 2 blue and 4 red balls respectively. One ball is drawn for each bag. What is the probability both balls are blue

    1. A.$\frac{2}{15}$
    2. B.$\frac{3}{24}$
    3. C.$\frac{3}{21}$
    4. D.$\frac{3}{28}$

    Show answer

    Answer: B. $\frac{3}{24}$

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38. 38.

    The area of an ellipse is 132cm$^2$.The length of its major axis is 14cm.Find the length of its minor axis

    1. A.10.5cm
    2. B.5cm
    3. C.10cm
    4. D.12cm

    Show answer

    Answer: D. 12cm

    AI explanation

    1. Analyze the question

    We are given the area of an ellipse ($132 \text{ cm}^2$) and the length of its major axis ($14 \text{ cm}$). We need to find the length of the minor axis.

    2. Explain the concept

    • The formula for the area of an ellipse is: Area = $\pi ab$ • Where $a$ is the semi-major axis (half of the major axis). • Where $b$ is the semi-minor axis (half of the minor axis).
    • The length of the major axis is $2a$, and the length of the minor axis is $2b$.
    • We will use $\pi \approx \frac{22}{7}$ as is standard for JAMB questions involving multiples of 7 or 11.

    3. Show the solution

    Step 1: Find the semi-major axis ($a$). The major axis is $14 \text{ cm}$. $2a = 14$ $a = \frac{14}{2} = 7 \text{ cm}$

    Step 2: Use the area formula to find the semi-minor axis ($b$). Area = $\pi ab$ $132 = \frac{22}{7} \times 7 \times b$

    The $7$ in the numerator and denominator cancel out: $132 = 22 \times b$ $b = \frac{132}{22}$ $b = 6 \text{ cm}$

    Step 3: Find the length of the minor axis ($2b$). Minor axis = $2 \times b$ Minor axis = $2 \times 6 = 12 \text{ cm}$

    4. Identify the answer

    Comparing our result ($12 \text{ cm}$) to the given options, the correct choice is D.

    5. Tip

    When dealing with circles or ellipses in JAMB, always check if the area or radius involves multiples of 7 or 11. If it does, using $\pi = \frac{22}{7}$ will make your calculation much faster and cleaner than using $3.142$.

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39. 39.

    The probability of an outcome A is 1/6 . The probability of the B outcome is 1/4 . If the probability of A or B or both is 1/12 . What is the probability of both outcomes A and B?

    1. A.1/2
    2. B.1/3
    3. C.2/5
    4. D.3/4

    Show answer

    Answer: B. 1/3

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40. 40.

    Find the equation of a line which is form origin and passes through the point (−3, −4)

    1. A.y = $\frac{3x}{4}$
    2. B.y = $\frac{4x}{3}$
    3. C.y = $\frac{2x}{3}$
    4. D.y = $\frac{x}{2}$

    Show answer

    Answer: B. y = $\frac{4x}{3}$

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41. 41.

    A man sells his new brand car for N420,000 at a gain of 15%. What did it cost him?

    1. A.N410,000
    2. B.N365, 217
    3. C.N157, 250
    4. D.N257,000

    Show answer

    Answer: B. N365, 217

    AI explanation

    Analysis

    To solve this problem, we need to understand the relationship between Cost Price (CP), Selling Price (SP), and Percentage Gain.

    When an item is sold at a gain, the Selling Price is the sum of the Cost Price and the profit made. Since the gain is expressed as a percentage of the Cost Price, we use the following formula: Selling Price = Cost Price + (Percentage Gain × Cost Price) Or simply: Selling Price = Cost Price × (1 + Gain%)

    Concept

    • Cost Price (CP): The original amount the man paid for the car.
    • Selling Price (SP): The amount he sold it for (N420,000).
    • Gain Percentage: 15% (which is 0.15 in decimal form).

    Step-by-Step Solution

    1. Set up the equation: Let the Cost Price be $x$. $x + 0.15x = 420,000$

    2. Combine the terms: $1.15x = 420,000$

    3. Solve for $x$: $x = \frac{420,000}{1.15}$

    4. Perform the division: $x = 365,217.391...$

    Rounding to the nearest whole number, we get N365,217.

    Conclusion

    Comparing our result to the given options, it matches option B.

    Exam Tip

    When dealing with "gain" or "loss" percentage questions, always remember that the percentage is calculated on the Cost Price, not the Selling Price. A common mistake students make is calculating 15% of N420,000 and subtracting it; avoid this! Always set up your equation as $CP \times (1 \pm \text{rate}) = SP$.

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42. 42.

    The volume of a cylinder whose height is 4cm and whose radius 5cm is equal to (π = 3.14)

    1. A.3.13cm2
    2. B.145cm2
    3. C.314cm2
    4. D.214cm2

    Show answer

    Answer: C. 314cm2

    AI explanation

    1. Analyze the question

    We are asked to calculate the volume of a cylinder given its height ($h$) and radius ($r$).

    • Height ($h$) = 4 cm
    • Radius ($r$) = 5 cm
    • Constant ($\pi$) = 3.14

    2. Explain the concept

    The volume ($V$) of a cylinder is the amount of space it occupies. The formula to calculate this is:

    $V = \pi r^2 h$

    Where:

    • $\pi$ (pi) is a constant.
    • $r$ is the radius of the circular base.
    • $h$ is the height of the cylinder.

    3. Show the solution

    Step 1: Identify the given values. $r = 5$ $h = 4$ $\pi = 3.14$

    Step 2: Substitute the values into the formula. $V = 3.14 \times (5)^2 \times 4$

    Step 3: Calculate the square of the radius. $5^2 = 5 \times 5 = 25$

    Step 4: Multiply the values together. $V = 3.14 \times 25 \times 4$ $V = 3.14 \times 100$ $V = 314$

    Since we are calculating volume, the unit is in cubic centimeters ($\text{cm}^3$). Note: While the options provided in your question list the unit as $\text{cm}^2$ (which is technically for area), the numerical value corresponds to the volume calculation.

    4. Identify the answer

    Comparing our result of 314 to the given options, the correct choice is C.

    5. Tip

    When dealing with volume problems involving $\pi$, always square the radius before multiplying by the height and $\pi$. Also, remember that volume is always in cubic units ($\text{cm}^3$, $\text{m}^3$), while area is in square units ($\text{cm}^2$, $\text{m}^2$). Don't let the unit label in the question distract you from the correct numerical calculation!

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43. 43.

    Given that Z = {1,2,4,5} what is the power of set Z?

    1. A.16
    2. B.8
    3. C.10
    4. D.12

    Show answer

    Answer: A. 16

    AI explanation

    Analysis

    The question asks for the "power of set Z," which is mathematically referred to as the Power Set of Z. Specifically, it asks for the number of elements in the power set of Z.

    Concept

    The Power Set of a set is the set of all possible subsets of that set, including the empty set ($\emptyset$) and the set itself.

    To find the number of elements in a power set, we use the formula:

    $n(P(Z)) = 2^n$

    Where:

    • $n$ is the number of elements in the original set $Z$.
    • $n(P(Z))$ is the total number of subsets (the power of the set).

    Solution

    1. Identify the number of elements in set Z: The set $Z = \{1, 2, 4, 5\}$. Counting the elements, we see there are 4 elements. Therefore, $n = 4$.

    2. Apply the formula: Substitute $n = 4$ into the formula $2^n$: $$2^4 = 2 \times 2 \times 2 \times 2$$

    3. Calculate the result: $2 \times 2 = 4$ $4 \times 2 = 8$ $8 \times 2 = 16$

    The total number of subsets (the power of the set) is 16.

    Conclusion

    Comparing our result to the given options: A. 16 B. 8 C. 10 D. 12

    The correct option is A.

    Exam Tip

    Whenever you see a question asking for the "power of a set," don't get confused by the terminology. Just count the number of items in the curly brackets and raise 2 to the power of that count. If the set has 3 items, the answer is $2^3 = 8$; if it has 5 items, the answer is $2^5 = 32$. Always remember: 2 raised to the power of the number of elements.

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44. 44.

    Find the area of the curved surface of a cone whose base radius is 3cm and whose height is 4cm (π = 3.14)

    1. A.17.1cm2
    2. B.27.2cm2
    3. C.47.1cm2
    4. D.37.3cm2

    Show answer

    Answer: C. 47.1cm2

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45. 45.

    How many sides has a regular polygon whose interior angles are 120°each

    1. A.4
    2. B.5
    3. C.6
    4. D.3

    Show answer

    Answer: C. 6

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46. 46.

    The amount A to which a principal P amounts at r% compound interest for n years is given by the formula A = P(1 + (r ÷ 100)$^n$. Find A, if P = 126, r = 4 and n = 2.

    1. A.N132.50K
    2. B.N136.30K
    3. C.N125.40K
    4. D.N257.42K

    Show answer

    Answer: B. N136.30K

    AI explanation

    Analysis

    The question asks us to calculate the final amount ($A$) using the compound interest formula. We are given the principal ($P$), the rate ($r$), and the time period ($n$).

    Concept

    The formula for compound interest is:

    $A = P\left(1 + \frac{r}{100}\right)^n$

    Where:

    • $P$ = Principal (the initial amount)
    • $r$ = Interest rate per annum
    • $n$ = Number of years

    Step-by-Step Solution

    Step 1: Identify the given values

    • $P = 126$
    • $r = 4$
    • $n = 2$

    Step 2: Substitute the values into the formula

    $A = 126 \left(1 + \frac{4}{100}\right)^2$

    Step 3: Simplify the expression inside the bracket

    $\frac{4}{100} = 0.04$

    $A = 126(1 + 0.04)^2$

    $A = 126(1.04)^2$

    Step 4: Calculate the square of 1.04

    $1.04 \times 1.04 = 1.0816$

    Step 5: Multiply by the principal

    $A = 126 \times 1.0816$

    $A = 136.2816$

    Rounding this to two decimal places (as is standard for currency in Naira and Kobo), we get N136.28.

    Evaluation of Options

    The calculated value is approximately N136.28. Looking at the provided options: A. N132.50 B. N136.30 C. N125.40 D. N257.42

    While our precise calculation is 136.28, option B is the closest approximation to the result.

    Exam Tip

    When dealing with compound interest questions in JAMB, always calculate the decimal value carefully. If your answer is very close to one of the options (like 136.28 vs 136.30), it is likely due to rounding differences in the exam board's calculation process. Always choose the option that is closest to your result.

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47. 47.

    Solve the equation $5^{(x − 2)} = (1 ÷ 125)^{(x +3)}$

    1. A.3/2
    2. B.− 7/4
    3. C.4/7
    4. D.7/4

    Show answer

    Answer: B. − 7/4

    AI explanation

    Analysis

    To solve this equation, we need to express both sides of the equation with the same base. Since the left side has a base of 5, we should aim to express the right side as a power of 5 as well.

    Concept

    The laws of indices are essential here:

    1. Negative Exponents: $\frac{1}{a^n} = a^{-n}$
    2. Power of a Power: $(a^m)^n = a^{m \times n}$
    3. Equality of Bases: If $a^x = a^y$, then $x = y$ (provided $a > 0$ and $a \neq 1$).

    Step-by-Step Solution

    Step 1: Rewrite the right side of the equation. The equation is $5^{(x - 2)} = (\frac{1}{125})^{(x + 3)}$. We know that $125 = 5^3$. Therefore, $\frac{1}{125} = \frac{1}{5^3} = 5^{-3}$.

    Step 2: Substitute this back into the equation. $5^{(x - 2)} = (5^{-3})^{(x + 3)}$

    Step 3: Simplify the exponents on the right side. Using the power of a power rule $(a^m)^n = a^{mn}$: $5^{(x - 2)} = 5^{-3(x + 3)}$ $5^{(x - 2)} = 5^{(-3x - 9)}$

    Step 4: Equate the exponents. Since the bases are now the same, we can equate the exponents: $x - 2 = -3x - 9$

    Step 5: Solve for $x$. Add $3x$ to both sides: $x + 3x - 2 = -9$ $4x - 2 = -9$

    Add 2 to both sides: $4x = -9 + 2$ $4x = -7$

    Divide by 4: $x = -\frac{7}{4}$

    Conclusion

    Comparing our result to the given options, we find that $x = -\frac{7}{4}$ matches option B.

    Exam Tip: Whenever you see fractions like $1/25, 1/125,$ or $1/625$ in an exponential equation, immediately think of negative powers of 5. It will save you time during the exam!

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48. 48.

    100112 + *****2 + 111002 + 1012 = 10011112

    1. A.11112
    2. B.110112
    3. C.101112
    4. D.110012

    Show answer

    Answer: B. 110112

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49. 49.

    Factorize x2 + 9x + 20

    1. A.(x − 5) 2
    2. B.(x + 5)(x + 4)
    3. C.(x – 5)(x + 3)
    4. D.(x + 3) 2

    Show answer

    Answer: B. (x + 5)(x + 4)

    AI explanation

    Analysis

    To factorize the quadratic expression $x^2 + 9x + 20$, we need to find two numbers that satisfy two conditions:

    1. Their product must be equal to the constant term ($20$).
    2. Their sum must be equal to the coefficient of the middle term ($9$).

    Concept: The AC Method (Factorization of Quadratics)

    For a quadratic expression in the form $ax^2 + bx + c$, we look for two factors of $ac$ that add up to $b$. In this case, $a=1$, $b=9$, and $c=20$. We need two numbers that multiply to give $20$ and add to give $9$.

    Step-by-Step Solution

    1. List the factors of 20: • $1 \times 20$ (Sum = $21$) • $2 \times 10$ (Sum = $12$) • $4 \times 5$ (Sum = $9$)

    2. Select the correct pair: The pair $4$ and $5$ satisfies both conditions (product is $20$, sum is $9$).

    3. Rewrite the middle term: Replace $9x$ with $4x + 5x$: $x^2 + 4x + 5x + 20$

    4. Factor by grouping: Group the first two terms and the last two terms: $(x^2 + 4x) + (5x + 20)$

    5. Factor out the common terms: $x(x + 4) + 5(x + 4)$

    6. Final grouping: Since $(x + 4)$ is common to both terms, we factor it out: $(x + 5)(x + 4)$

    Conclusion

    Comparing our result to the given options, the correct factorization is $(x + 5)(x + 4)$.

    Exam Tip: When the constant term ($c$) is positive, the two numbers you are looking for must have the same sign (both positive or both negative). Since the middle term ($9x$) is positive, you know immediately that both numbers must be positive!

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50. 50.

    Determine the third term of a geometrical progression whose first and second term are 2 and 54 respectively

    1. A.1458
    2. B.1485
    3. C.1345
    4. D.1258

    Show answer

    Answer: A. 1458

    AI explanation

    1. Analyze the question

    We are given a Geometric Progression (GP) where:

    • The first term ($a$) = 2
    • The second term ($ar$) = 54
    • We need to find the third term ($ar^2$).

    2. Explain the concept

    In a Geometric Progression, each term is obtained by multiplying the previous term by a constant called the common ratio ($r$).

    • The first term is $a$.
    • The second term is $ar$.
    • The third term is $ar^2$.

    To find the third term, we first need to find the common ratio ($r$) by dividing the second term by the first term.

    3. Show the solution

    Step 1: Find the common ratio ($r$) Since the second term is $ar = 54$ and the first term $a = 2$:

    $r = \frac{\text{second term}}{\text{first term}}$

    $r = \frac{54}{2}$

    $r = 27$

    Step 2: Find the third term ($ar^2$) The third term is the second term multiplied by the common ratio ($r$):

    $\text{Third term} = \text{second term} \times r$

    $\text{Third term} = 54 \times 27$

    Let's calculate $54 \times 27$: $54 \times 20 = 1080$ $54 \times 7 = 378$ $1080 + 378 = 1458$

    4. Identify the answer

    Comparing our result to the given options: A. 1458 B. 1485 C. 1345 D. 1258

    The correct option is A.

    5. Tip

    In a GP, remember the relationship: (Second term)² = (First term) × (Third term). You can use this to verify your answer: $54^2 = 2916$, and $2 \times 1458 = 2916$. Since they match, you can be confident in your result!

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51. 51.

    Evaluate log5($y^2x^5 ÷ 125b½)$

    1. A.2 log5y + 5log5 y2 − 3
    2. B.log5 y2 + 5log5 x + 3
    3. C.25logy 5 + 3
    4. D.2log5y + 5log5x − ½ log5b −3

    Show answer

    Answer: D. 2log5y + 5log5x − ½ log5b −3

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52. 52.

    Given that tan x = $\frac{2}{3}$, where 0o d" x d" 90o, Find the value of 2sinx.

    1. A.$\frac{2\sqrt{13}}{13}$
    2. B.$\frac{3\sqrt{13}}{13}$
    3. C.$\frac{4\sqrt{13}}{13}$
    4. D.$\frac{6\sqrt{13}}{13}$

    Show answer

    Answer: C. $\frac{4\sqrt{13}}{13}$

    AI explanation

    1. Analyze the Question

    We are given $\tan x = \frac{2}{3}$ in the first quadrant ($0^\circ \leq x \leq 90^\circ$). We need to find the value of $2\sin x$.

    2. Explain the Concept

    In a right-angled triangle, trigonometric ratios are defined by the mnemonic SOH CAH TOA:

    • SOH: $\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}}$
    • TOA: $\tan x = \frac{\text{Opposite}}{\text{Adjacent}}$

    Since $\tan x = \frac{2}{3}$, we can treat the opposite side as $2$ and the adjacent side as $3$. We must first find the hypotenuse using Pythagoras' theorem:

    $\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2$

    3. Show the Solution

    Step 1: Find the hypotenuse.

    $\text{Hypotenuse} = \sqrt{2^2 + 3^2}$

    $\text{Hypotenuse} = \sqrt{4 + 9} = \sqrt{13}$

    Step 2: Find $\sin x$. Using the definition $\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}}$:

    $\sin x = \frac{2}{\sqrt{13}}$

    To rationalize the denominator:

    $\sin x = \frac{2}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{2\sqrt{13}}{13}$

    Step 3: Calculate $2\sin x$.

    $2\sin x = 2 \times \left( \frac{2\sqrt{13}}{13} \right)$

    $2\sin x = \frac{4\sqrt{13}}{13}$

    4. Identify the Answer

    Comparing our result to the given options, the correct value is $\frac{4\sqrt{13}}{13}$. This corresponds to option C.

    5. Exam Tip

    Whenever you are given one trigonometric ratio (like $\tan x$), always sketch a quick right-angled triangle. Label the sides based on the ratio, calculate the missing side using Pythagoras' theorem, and you can easily find any other ratio ($\sin x, \cos x, \sec x$, etc.) without getting confused!

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53. 53.

    Find the distance between the points (-2,-3) and (-2,4)

    1. A.3m
    2. B.2.4m
    3. C.3.2m
    4. D.7.0m

    Show answer

    Answer: D. 7.0m

    AI explanation

    1. Analyze the question

    We are asked to find the distance between two points on a Cartesian plane: $P_1(-2, -3)$ and $P_2(-2, 4)$.

    2. Explain the concept

    The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the distance formula:

    $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

    Observation: Notice that the $x$-coordinates for both points are the same ($x_1 = -2$ and $x_2 = -2$). When the $x$-coordinates are identical, the line connecting the points is a vertical line. In this case, the distance is simply the absolute difference between the $y$-coordinates:

    $d = |y_2 - y_1|$

    3. Show the solution

    Using the coordinates $P_1(-2, -3)$ and $P_2(-2, 4)$:

    • $x_1 = -2, y_1 = -3$
    • $x_2 = -2, y_2 = 4$

    Method 1 (Using the formula):

    $d = \sqrt{(-2 - (-2))^2 + (4 - (-3))^2}$

    $d = \sqrt{(0)^2 + (4 + 3)^2}$

    $d = \sqrt{0^2 + 7^2}$

    $d = \sqrt{49}$

    $d = 7$

    Method 2 (Vertical distance shortcut): Since the $x$-coordinates are the same, the distance is the vertical change:

    $d = |4 - (-3)|$

    $d = |4 + 3|$

    $d = 7$

    4. Identify the answer

    Comparing our result to the given options, the distance is 7.0 units. Note that while the options include "m" (meters), the calculation yields a scalar value of 7.

    The correct option is D.

    5. Exam Tip

    Whenever you see two points with the same $x$-coordinate, don't waste time with the full square root formula! Just subtract the $y$-values. If the $y$-coordinates were the same, you would just subtract the $x$-values. This saves precious time during your JAMB exam.

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54. 54.

    Solve the equation $3x^2 − 4x − 5 = 0$

    1. A.x = 1.75 or − 0.15
    2. B.x = 2.12 or − 0.79
    3. C.x = 1.5 or − 0.34
    4. D.x = 2.35 or −1.23

    Show answer

    Answer: B. x = 2.12 or − 0.79

    AI explanation

    Analysis

    To solve the quadratic equation $3x^2 - 4x - 5 = 0$, we need to find the values of $x$ that satisfy the equation. Since this equation does not factorize easily into simple integers, we use the Quadratic Formula.

    Concept: The Quadratic Formula

    For any quadratic equation in the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

    In your equation ($3x^2 - 4x - 5 = 0$):

    • $a = 3$
    • $b = -4$
    • $c = -5$

    Step-by-Step Solution

    1. Substitute the values into the formula:

    $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-5)}}{2(3)}$

    2. Simplify the terms:

    • $-(-4) = 4$
    • $(-4)^2 = 16$
    • $-4 \times 3 \times -5 = 60$
    • $2 \times 3 = 6$

    So, the equation becomes:

    $x = \frac{4 \pm \sqrt{16 + 60}}{6}$

    $x = \frac{4 \pm \sqrt{76}}{6}$

    3. Calculate the square root: $\sqrt{76} \approx 8.7178$

    4. Solve for the two possible values of $x$:

    • Case 1 (Addition): $x = \frac{4 + 8.7178}{6} = \frac{12.7178}{6} \approx 2.1196 \approx \mathbf{2.12}$

    • Case 2 (Subtraction): $x = \frac{4 - 8.7178}{6} = \frac{-4.7178}{6} \approx -0.7863 \approx \mathbf{-0.79}$

    Conclusion

    Comparing our results ($x \approx 2.12$ or $-0.79$) with the given options, we find that they match option B.

    Exam Tip

    When you see a quadratic equation in a JAMB exam, always check the discriminant ($b^2 - 4ac$) first. If it is not a perfect square, you know immediately that the answer will involve decimals, and you should reach for the quadratic formula rather than trying to factorize by grouping!

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55. 55.

    Find the simple interest on N325 in 5years at 3% per annum.

    1. A.N48.75K
    2. B.N50.10K
    3. C.N75.50K
    4. D.N15.75K

    Show answer

    Answer: A. N48.75K

    AI explanation

    Analysis

    To solve this problem, we need to calculate the Simple Interest using the standard financial formula. We are given the principal amount, the time period, and the annual interest rate.

    Concept

    The formula for Simple Interest ($I$) is:

    $I = \frac{P \times R \times T}{100}$

    Where:

    • $P$ (Principal) = The initial amount borrowed or invested = N325
    • $R$ (Rate) = The interest rate per annum = 3%
    • $T$ (Time) = The duration in years = 5 years

    Solution

    1. Identify the values: $P = 325$ $R = 3$ $T = 5$

    2. Substitute the values into the formula: $$I = \frac{325 \times 3 \times 5}{100}$$

    3. Perform the multiplication: First, multiply $3 \times 5 = 15$. Then, multiply $325 \times 15$: $325 \times 10 = 3250$ $325 \times 5 = 1625$ $3250 + 1625 = 4875$

    4. Divide by 100: $$I = \frac{4875}{100} = 48.75$$

    The simple interest is N48.75K.

    Conclusion

    Comparing our result to the given options, the correct value is N48.75K.

    Exam Tip: When calculating simple interest, always ensure your time ($T$) is in years. If the question gives you time in months, divide the number of months by 12 before plugging it into the formula!

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56. 56.

    Given that S and T are sets of real numbers such that S = {x : 0 $\leq$ x $\leq$ 5} and T = {x : − 2 < x < 3} Find S $\cup$ T

    1. A.−3 < x < $\leq$3
    2. B.−2< x < $\leq$5
    3. C.2< x < $\geq$ −5
    4. D.−1< x $\geq$ $\leq$2

    Show answer

    Answer: B. −2< x < $\leq$5

    AI explanation

    1. Analyze the Question

    We are given two sets of real numbers:

    • Set S: $\{x : 0 \leq x \leq 5\}$ (This includes all numbers from 0 to 5, including both 0 and 5).
    • Set T: $\{x : -2 < x < 3\}$ (This includes all numbers between -2 and 3, excluding -2 and 3).
    • Goal: Find the Union ($S \cup T$). The union of two sets contains all elements that are in either S, or T, or both.

    2. Explain the Concept

    • Union ($\cup$): Think of this as "combining" the two sets. You look for the lowest possible value covered by either set and the highest possible value covered by either set.
    • Interval Notation: • $S = [0, 5]$ • $T = (-2, 3)$
    • When you combine these on a number line, the range starts from the lowest value in $T$ (which is $-2$) and ends at the highest value in $S$ (which is $5$).

    3. Show the Solution

    • Step 1: Visualize the number line. • Set $T$ starts just after $-2$ and goes up to $3$. • Set $S$ starts at $0$ and goes up to $5$.
    • Step 2: Combine the ranges. • The combined range starts at $-2$. Since $-2$ is not included in $T$ (it is a "less than" sign), it remains excluded: $-2 < x$. • The combined range ends at $5$. Since $5$ is included in $S$ (it is a "less than or equal to" sign), it remains included: $x \leq 5$.
    • Step 3: Write the union. • Combining these, we get: $-2 < x \leq 5$.

    4. Identify the Answer

    Comparing our result to the given options: A. $-3 < x \leq 3$ B. $-2 < x \leq 5$ C. $2 < x \geq -5$ D. $-1 < x \geq \leq 2$

    The correct option is B.

    5. Exam Tip

    When dealing with inequalities, always draw a quick sketch of the number line. Use an open circle ($\circ$) for "less than" ($<$) or "greater than" ($>$), and a shaded circle ($\bullet$) for "less than or equal to" ($\leq$) or "greater than or equal to" ($\geq$). This makes it much easier to see where the combined set begins and ends!

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57. 57.

    If (159.75)10 = (x)6. Find x

    1. A.x6 = 123.346
    2. B.x6 = 424.56
    3. C.x6 = 122.436
    4. D.x6 = 124.456

    Show answer

    Answer: B. x6 = 424.56

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58. 58.

    If A = (−3,5) and B = (4,−1) find the co-ordinate of the mid point

    1. A.2, ½
    2. B.½ , 2
    3. C.1, ½
    4. D.0, 2

    Show answer

    Answer: B. ½ , 2

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59. 59.

    If √ (2x + 2) − √x =1 ,find x.

    1. A.x = 1 twice
    2. B.x = 0 or 1
    3. C.x = 3 or 1
    4. D.x = 2 twice

    Show answer

    Answer: A. x = 1 twice

    AI explanation

    Analysis

    To solve the equation $\sqrt{2x + 2} - \sqrt{x} = 1$, we need to isolate the radical terms and square both sides to eliminate the square roots.

    Concept: Solving Radical Equations

    When dealing with equations involving square roots, the standard procedure is:

    1. Isolate one radical on one side of the equation.
    2. Square both sides.
    3. If a radical remains, repeat the process.
    4. Always check your solutions in the original equation, as squaring can sometimes introduce "extraneous" (false) roots.

    Step-by-Step Solution

    Step 1: Isolate one radical. $\sqrt{2x + 2} = 1 + \sqrt{x}$

    Step 2: Square both sides. $(\sqrt{2x + 2})^2 = (1 + \sqrt{x})^2$ $2x + 2 = 1 + 2\sqrt{x} + x$

    Step 3: Simplify and isolate the remaining radical. Subtract $x$ and $1$ from both sides: $(2x - x) + (2 - 1) = 2\sqrt{x}$ $x + 1 = 2\sqrt{x}$

    Step 4: Square both sides again to remove the final radical. $(x + 1)^2 = (2\sqrt{x})^2$ $x^2 + 2x + 1 = 4x$

    Step 5: Solve the resulting quadratic equation. $x^2 + 2x - 4x + 1 = 0$ $x^2 - 2x + 1 = 0$

    Factor the quadratic: $(x - 1)(x - 1) = 0$ $(x - 1)^2 = 0$ $x = 1$

    Step 6: Verify the solution. Substitute $x = 1$ into the original equation: $\sqrt{2(1) + 2} - \sqrt{1} = 1$ $\sqrt{4} - 1 = 1$ $2 - 1 = 1$ $1 = 1$ (The solution is valid).

    Since the quadratic equation $(x-1)^2 = 0$ yields $x=1$ as a repeated root, we say $x=1$ twice.

    Conclusion

    This matches our algebraic result.

    Exam Tip

    Whenever you see a radical equation in JAMB, always remember to check your answers. Sometimes, squaring both sides creates a value that doesn't actually work in the original equation. In this case, $x=1$ works perfectly!

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60. 60.

    Rationalize 5 ÷ (2 − √3)

    1. A.3(2 + √5)
    2. B.2(3 + √5)
    3. C.5(2 + √3)
    4. D.3 √2 + 1

    Show answer

    Answer: C. 5(2 + √3)

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61. 61.

    The first and last term of a linear sequence (AP) are 6 and 10 respectively. If the sum of the sequence is 40. Find the number of terms

    1. A.nth = 3
    2. B.nth = 4
    3. C.nth = 5
    4. D.nth = 6

    Show answer

    Answer: C. nth = 5

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