2023 JAMB Mathematics Past Questions and Answers

Questions

1. 1.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

2. 2.

   Two numbers are respectively 35% and 80% more than a third number. The ratio of the two numbers is

   1. A.7 : 16
   2. B.3 : 4
   3. C.16 : 7
   4. D.4 : 3

   Show answer

   Answer: B. 3 : 4

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3. 3.

   Evaluate: 16$^{0.16}$ × 16$^{0.04}$ × 2$^{0.2}$

   1. A.2
   2. B.0
   3. C.2$^0$
   4. D.$\frac{1}{2}$

   Show answer

   Answer: A. 2

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4. 4.

   The line $3y + 6x$ = 48 passes through the points A(-2, k) and B(4, 8). Find the value of k.

   1. A.16
   2. B.20
   3. C.8
   4. D.-2

   Show answer

   Answer: B. 20

   AI explanation

   1. Analyze the question

   We are given a linear equation $3y + 6x = 48$ and two points, $A(-2, k)$ and $B(4, 8)$, that lie on this line. We need to find the value of the unknown coordinate $k$ in point $A$.

   2. Explain the concept

   If a point lies on a line, its coordinates $(x, y)$ must satisfy the equation of that line. This means if we substitute the $x$ and $y$ values of the point into the equation, the left-hand side will equal the right-hand side. Since point $A(-2, k)$ is on the line $3y + 6x = 48$, we can substitute $x = -2$ and $y = k$ into the equation to solve for $k$.

   3. Show the solution

   Step 1: Write down the equation of the line. $3y + 6x = 48$

   Step 2: Substitute the coordinates of point A(-2, k) into the equation. Here, $x = -2$ and $y = k$. $3(k) + 6(-2) = 48$

   Step 3: Simplify the equation. $3k - 12 = 48$

   Step 4: Solve for k. Add 12 to both sides of the equation: $3k = 48 + 12$ $3k = 60$

   Divide both sides by 3: $k = \frac{60}{3}$ $k = 20$

   (Self-check: We can verify this using point B(4, 8). If $x=4$ and $y=8$: $3(8) + 6(4) = 24 + 24 = 48$. This confirms the equation is correct.)

   4. Identify the answer

   Comparing our result $k = 20$ with the given options: A. 16 B. 20 C. 8 D. -2

   The correct option is B.

   5. Tip

   When you are given a point $(x, y)$ and told it lies on a line, don't waste time calculating the slope unless necessary. Simply plug the $x$ and $y$ values directly into the equation and solve for the missing variable. It is a quick way to save time during your JAMB exam!

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5. 5.

   The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is 6$^o$, then the value of "n" is

   1. A.11
   2. B.13
   3. C.12
   4. D.14

   Show answer

   Answer: B. 13

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6. 6.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

7. 7.

   Let a binary operation '*' be defined on a set A. The operation will be commutative if

   1. A.a*b = b*a
   2. B.(a*b)*c = a*(b*c)
   3. C.(b ο c)*a = (b*a) ο (c*a)
   4. D.None of the above

   Show answer

   Answer: A. a*b = b*a

   AI explanation

   Analysis

   The question asks for the definition of a commutative binary operation on a set $A$. In mathematics, binary operations have specific properties that define how elements interact with one another.

   Concept Explanation

   A binary operation $*$ on a set $A$ is a rule that assigns to each ordered pair of elements $(a, b)$ from $A$ a unique element $a * b$ in $A$.

   • Commutative Property: An operation is commutative if the order of the elements does not change the result. That is, for any two elements $a$ and $b$ in the set, $a * b$ must be equal to $b * a$.
   • Associative Property: This is represented by the expression $(a * b) * c = a * (b * c)$. This property deals with the grouping of elements, not the order.
   • Distributive Property: This involves two different operations (e.g., $*$ and $\circ$) and describes how one operation interacts with another over a set.

   Step-by-Step Solution

   1. Examine Option A ($a * b = b * a$): This is the standard definition of commutativity. If you swap the positions of $a$ and $b$, the result remains the same.
   2. Examine Option B ($(a * b) * c = a * (b * c)$): This is the definition of the associative property, not the commutative property.
   3. Examine Option C ($(b \circ c) * a = (b * a) \circ (c * a)$): This represents the distributive property of the operation $*$ over the operation $\circ$.
   4. Conclusion: Since Option A matches the mathematical definition of a commutative operation, it is the correct choice.

   Exam Tip

   To remember the difference:

   • Commutative: Think of "Commuting" (traveling to work and back). The distance is the same regardless of the direction. Order changes, result stays.
   • Associative: Think of "Associating" (grouping friends). It doesn't matter which two friends you group together first; the total group remains the same. Grouping changes, result stays.

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8. 8.

   Make x the subject of the formula: y = $\frac {3x - 9c}{4x + 5d}$

   1. A.x = $\frac {-(9c - 5dy)}{4y - 3}$
   2. B.x = $\frac {9c + 5dy}{4y - 3}$
   3. C.x = $\frac {9c - 5dy}{4y - 3}$
   4. D.x = $\frac {-(9c + 5dy)}{4y - 3}$

   Show answer

   Answer: D. x = $\frac {-(9c + 5dy)}{4y - 3}$

   AI explanation

   Analysis

   To make $x$ the subject of the formula, we need to isolate $x$ on one side of the equation. This involves clearing the fraction, expanding the terms, grouping all terms containing $x$ on one side, and then factoring out $x$.

   Concept

   The equation is a rational function. We use the following algebraic steps:

   1. Cross-multiply to eliminate the fraction.
   2. Expand the brackets.
   3. Rearrange the terms to bring all terms with $x$ to one side and terms without $x$ to the other.
   4. Factorize $x$ out.
   5. Divide by the coefficient of $x$ to solve for it.

   Step-by-Step Solution

   Step 1: Cross-multiply Given: $y = \frac{3x - 9c}{4x + 5d}$ Multiply both sides by $(4x + 5d)$: $y(4x + 5d) = 3x - 9c$

   Step 2: Expand the left side $4xy + 5dy = 3x - 9c$

   Step 3: Group terms with $x$ on one side Move $3x$ to the left and $5dy$ to the right: $4xy - 3x = -9c - 5dy$

   Step 4: Factorize $x$ $x(4y - 3) = -(9c + 5dy)$

   Step 5: Solve for $x$ Divide both sides by $(4y - 3)$: $x = \frac{-(9c + 5dy)}{4y - 3}$

   Comparison with Options

   Looking at the result $x = \frac{-(9c + 5dy)}{4y - 3}$, it matches Option D exactly.

   Note: If you look at Option A, it is mathematically equivalent to D if you distribute the negative sign into the bracket, but Option D is the direct representation of the algebraic steps taken.

   Exam Tip

   When dealing with "make $x$ the subject" questions involving fractions, always remember the "cross-multiply first" rule. It immediately turns a complex-looking fraction into a simple linear equation that is much easier to manage. Also, watch your signs carefully—the negative sign in front of the bracket in the final answer is a common place where students make mistakes!

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9. 9.

   Evaluate the following limit: $lim_{x\to2} \frac {x^2 + 4x - 12}{x^2 - 2x}$

   1. A.4
   2. B.8
   3. C.0
   4. D.2

   Show answer

   Answer: A. 4

   AI explanation

   1. Analyze the question

   We are asked to evaluate the limit of the rational function $\frac{x^2 + 4x - 12}{x^2 - 2x}$ as $x$ approaches $2$.

   2. Explain the concept

   When evaluating limits, the first step is direct substitution. If substituting $x = 2$ results in a defined value, that is our answer. However, if it results in $\frac{0}{0}$ (an indeterminate form), we must simplify the expression by factoring the numerator and the denominator to cancel out the common term causing the zero.

   3. Show the solution

   Step 1: Direct Substitution Substitute $x = 2$ into the expression: Numerator: $(2)^2 + 4(2) - 12 = 4 + 8 - 12 = 0$ Denominator: $(2)^2 - 2(2) = 4 - 4 = 0$ Since we have $\frac{0}{0}$, we must factorize.

   Step 2: Factorize the Numerator The numerator is $x^2 + 4x - 12$. We look for two numbers that multiply to $-12$ and add to $+4$. These numbers are $+6$ and $-2$. $x^2 + 4x - 12 = (x + 6)(x - 2)$

   Step 3: Factorize the Denominator The denominator is $x^2 - 2x$. We can factor out the common $x$: $x^2 - 2x = x(x - 2)$

   Step 4: Simplify the expression Now, rewrite the limit: $\lim_{x\to2} \frac{(x + 6)(x - 2)}{x(x - 2)}$

   Cancel the common factor $(x - 2)$: $\lim_{x\to2} \frac{x + 6}{x}$

   Step 5: Evaluate the limit Now substitute $x = 2$ into the simplified expression: $\frac{2 + 6}{2} = \frac{8}{2} = 4$

   4. Identify the answer

   The calculated value is 4, which corresponds to option A.

   5. Exam Tip

   Whenever you see a limit problem in JAMB that results in $\frac{0}{0}$, don't panic! It is a clear signal that there is a "hidden" factor in both the top and bottom that needs to be cancelled out. Always factorize first before re-substituting.

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10. 10.

    If a car runs at a constant speed and takes 4.5 hrs to run a distance of 225 km, what time will it take to run 150 km?

    1. A.2 hrs
    2. B.4 hrs
    3. C.3 hrs
    4. D.1 hr

    Show answer

    Answer: C. 3 hrs

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11. 11.

    It takes 12 men 8 hours a day to finish a piece of work in 4 days. In how many days will it take 4 men working 16 hours a day to complete the same piece of work?

    1. A.6 days
    2. B.8 days
    3. C.10 days
    4. D.12 days

    Show answer

    Answer: A. 6 days

    AI explanation

    Analysis

    This is a classic "Work-Rate" problem. The total amount of work remains constant, so we can use the principle that the product of the number of workers, the time spent per day, and the number of days is equal to the total work done.

    Concept

    The formula for work-related problems is:

    $\frac{M_1 \times T_1 \times D_1}{W_1} = \frac{M_2 \times T_2 \times D_2}{W_2}$

    Where:

    • M = Number of men
    • T = Time (hours per day)
    • D = Number of days
    • W = Total work (since the work is the same in both cases, $W_1 = W_2$, so they cancel out).

    Step-by-Step Solution

    Step 1: Identify the given values.

    • $M_1 = 12$ men
    • $T_1 = 8$ hours/day
    • $D_1 = 4$ days
    • $M_2 = 4$ men
    • $T_2 = 16$ hours/day
    • $D_2 = ?$ (This is what we need to find)

    Step 2: Set up the equation.

    $12 \times 8 \times 4 = 4 \times 16 \times D_2$

    Step 3: Calculate the left side.

    $12 \times 8 = 96$

    $96 \times 4 = 384$

    Step 4: Calculate the right side.

    $4 \times 16 = 64$

    $384 = 64 \times D_2$

    Step 5: Solve for $D_2$.

    $D_2 = \frac{384}{64}$

    $D_2 = 6$

    It will take 4 men working 16 hours a day exactly 6 days to complete the work.

    Exam Tip

    When solving these problems, always remember that the number of men and the number of days are inversely proportional. If you reduce the number of men, the number of days must increase (assuming hours remain constant). If you increase the hours worked per day, the number of days must decrease. Checking this logic helps you avoid simple calculation errors!

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12. 12.

    The third term of an A.P is 6 and the fifth term is 12. Find the sum of its first twelve terms

    1. A.201
    2. B.144
    3. C.198
    4. D.72

    Show answer

    Answer: C. 198

    AI explanation

    1. Analyze the Question

    We are given the 3rd and 5th terms of an Arithmetic Progression (A.P). We need to find the sum of the first 12 terms ($S_{12}$).

    2. Explain the Concept

    In an A.P, the $n^{th}$ term is given by: $T_n = a + (n - 1)d$ Where:

    • $a$ = first term
    • $d$ = common difference

    The sum of the first $n$ terms is given by: $S_n = \frac{n}{2} [2a + (n - 1)d]$

    3. Show the Solution

    Step 1: Set up equations for the given terms.

    • For the 3rd term ($T_3 = 6$): $a + (3 - 1)d = 6$ $a + 2d = 6$ --- (Equation 1)

    • For the 5th term ($T_5 = 12$): $a + (5 - 1)d = 12$ $a + 4d = 12$ --- (Equation 2)

    Step 2: Solve for $a$ and $d$. Subtract Equation 1 from Equation 2: $(a + 4d) - (a + 2d) = 12 - 6$ $2d = 6$ $d = 3$

    Substitute $d = 3$ into Equation 1: $a + 2(3) = 6$ $a + 6 = 6$ $a = 0$

    Step 3: Calculate the sum of the first 12 terms ($S_{12}$). Using the formula $S_n = \frac{n}{2} [2a + (n - 1)d]$: $S_{12} = \frac{12}{2} [2(0) + (12 - 1)3]$ $S_{12} = 6 [0 + (11 \times 3)]$ $S_{12} = 6 \times 33$ $S_{12} = 198$

    4. Identify the Answer

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

    5. Exam Tip

    When dealing with A.P problems, always write out your equations for $T_n$ first. If you have two unknowns ($a$ and $d$), you will always need two equations to solve them simultaneously. Don't panic if your first term ($a$) turns out to be zero; it is perfectly valid in an A.P!

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13. 13.

    If D = $\begin{bmatrix}2& -1&3\\4&1&2\\1&-3&1\\\end{bmatrix}$ Find |D|

    1. A.16
    2. B.14
    3. C.-23
    4. D.-37

    Show answer

    Answer: C. -23

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14. 14.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

15. 15.

    Express 16.54 x 10$^{-5}$ - 6.76 x 10$^{-8}$ + 0.23 x 10$^{-6}$ in standard form

    1. A.1.66 x 10$^{-4}$
    2. B.1.66 x 10$^{-5}$
    3. C.1.65 x 10$^{-5}$
    4. D.1.65 x 10$^{-4}$

    Show answer

    Answer: A. 1.66 x 10$^{-4}$

    AI explanation

    To solve this problem, we need to express all the terms in the same power of 10 so that we can easily add or subtract them.

    1. Analyze the question

    We are given: $16.54 \times 10^{-5} - 6.76 \times 10^{-8} + 0.23 \times 10^{-6}$

    To make calculation easier, let's convert all terms to $10^{-5}$.

    2. Explain the concept

    Standard form is written as $A \times 10^n$, where $1 \leq A < 10$. When moving the decimal point to the left, the exponent increases; when moving it to the right, the exponent decreases.

    3. Show the solution

    Step 1: Convert all terms to $10^{-5}$

    • Term 1: $16.54 \times 10^{-5}$ (Already in $10^{-5}$)
    • Term 2: $6.76 \times 10^{-8}$ To change $10^{-8}$ to $10^{-5}$, we move the decimal point 3 places to the left: $6.76 \times 10^{-8} = 0.00676 \times 10^{-5}$
    • Term 3: $0.23 \times 10^{-6}$ To change $10^{-6}$ to $10^{-5}$, we move the decimal point 1 place to the left: $0.23 \times 10^{-6} = 0.023 \times 10^{-5}$

    Step 2: Perform the arithmetic Now, combine the coefficients: $(16.54 - 0.00676 + 0.023) \times 10^{-5}$

    • $16.54 - 0.00676 = 16.53324$
    • $16.53324 + 0.023 = 16.55624$

    So, we have: $16.55624 \times 10^{-5}$

    Step 3: Convert to proper standard form Standard form requires the decimal to be after the first non-zero digit: $16.55624 \times 10^{-5} = 1.655624 \times 10^{-4}$

    Rounding this to two decimal places (as seen in the options): $1.66 \times 10^{-4}$

    4. Identify the answer

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

    5. Tip

    When dealing with addition or subtraction of numbers in standard form, always convert them to the highest exponent present in the expression. This keeps your numbers as decimals rather than very small fractions, making them much easier to manage without a calculator.

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16. 16.

    Solve for x: 3(x – 1) ≤ 2 (x – 3)

    1. A.x ≤ -3
    2. B.x ≥ -3
    3. C.x ≤ 3
    4. D.x ≥ 3

    Show answer

    Answer: A. x ≤ -3

    AI explanation

    Analysis

    This is a linear inequality problem. The goal is to isolate the variable $x$ on one side of the inequality sign, similar to how you would solve a linear equation, while being mindful of the rules regarding inequality signs.

    Concept

    To solve linear inequalities:

    1. Expand the brackets: Use the distributive property to multiply the terms inside the parentheses.
    2. Collect like terms: Move all terms containing $x$ to one side and constant numbers to the other.
    3. Simplify: Combine the terms.
    4. Solve for $x$: If you divide or multiply by a negative number, remember to flip the inequality sign. In this case, we will see if that is necessary.

    Step-by-Step Solution

    Step 1: Expand the brackets Given: $3(x - 1) \leq 2(x - 3)$

    Multiply $3$ by $(x - 1)$ and $2$ by $(x - 3)$: $3x - 3 \leq 2x - 6$

    Step 2: Collect like terms Subtract $2x$ from both sides to bring the $x$ terms to the left: $3x - 2x - 3 \leq -6$ $x - 3 \leq -6$

    Step 3: Isolate $x$ Add $3$ to both sides of the inequality: $x \leq -6 + 3$ $x \leq -3$

    Conclusion

    After simplifying the inequality, we arrive at $x \leq -3$. This matches option A.

    Exam Tip

    When solving inequalities, always double-check your signs. A common mistake students make is forgetting to flip the inequality sign when dividing by a negative number (e.g., if you had $-x \leq 3$, it would become $x \geq -3$). In this specific problem, we did not need to flip the sign because the coefficient of $x$ remained positive.

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17. 17.

    Find the volume of the cylinder above [Take $\pi= ^{22}/_7$]

    1. A.9,856 cm$^3$
    2. B.14,784 cm$^3$
    3. C.4,928 cm$^3$
    4. D.19,712 cm$^3$

    Show answer

    Answer: D. 19,712 cm$^3$

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18. 18.

    200 tickets were sold for a show. VIP tickets costs ₦1,200 and ₦700 for regular. Total amount realised from the sale of the tickets was ₦180,000. Find the number of VIP tickets sold and the the number of regular ticket sold.

    1. A.VIP = 80, Regular = 100
    2. B.VIP = 60, Regular = 120
    3. C.VIP = 60, Regular = 100
    4. D.VIP = 80, Regular = 120

    Show answer

    Answer: D. VIP = 80, Regular = 120

    AI explanation

    Analysis

    To solve this problem, we need to set up a system of linear equations based on the information provided:

    1. The total number of tickets sold.
    2. The total amount of money realized from those sales.

    Concept

    We will use the Substitution or Elimination method for simultaneous equations. Let:

    • $x$ = number of VIP tickets
    • $y$ = number of regular tickets

    Step-by-Step Solution

    Step 1: Form the equations.

    • From the total number of tickets: $x + y = 200$ --- (Equation 1)
    • From the total amount realized: $1200x + 700y = 180,000$ --- (Equation 2)

    Step 2: Simplify Equation 2. To make the calculation easier, divide Equation 2 by 100: $12x + 7y = 1800$ --- (Equation 3)

    Step 3: Solve using substitution. From Equation 1, express $y$ in terms of $x$: $y = 200 - x$

    Substitute this into Equation 3: $12x + 7(200 - x) = 1800$ $12x + 1400 - 7x = 1800$

    Step 4: Solve for $x$. $12x - 7x = 1800 - 1400$ $5x = 400$ $x = \frac{400}{5}$ $x = 80$

    Step 5: Solve for $y$. Substitute $x = 80$ back into Equation 1: $80 + y = 200$ $y = 200 - 80$ $y = 120$

    So, 80 VIP tickets and 120 regular tickets were sold.

    Conclusion

    Comparing our result ($x=80, y=120$) with the given options, it matches option D.

    Exam Tip

    When you have a system of equations in a multiple-choice exam, you can save time by testing the options. For example, in option D: $80 + 120 = 200$ (Correct total tickets). Then check the money: $(80 \times 1200) + (120 \times 700) = 96,000 + 84,000 = 180,000$. Since this matches the total amount, you don't even need to do the full algebra!

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19. 19.

    Evaluate $\frac{5}{8} - \frac{3}{4} ÷ \frac{5}{12} \times \frac{1}{4}$

    1. A.- $\frac{3}{40}$
    2. B.$\frac{3}{40}$
    3. C.$\frac{7}{40}$
    4. D.-$\frac{263}{40}$

    Show answer

    Answer: C. $\frac{7}{40}$

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20. 20.

    PQRS is a cyclic quadrilateral. Find $x$ + $y$

    1. A.50
    2. B.60
    3. C.15
    4. D.0

    Show answer

    Answer: B. 60

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21. 21.

    A student is using a graduated cylinder to measure the volume of water and reports a reading of 18 mL. The teacher reports the value as 18.4 mL. What is the student's percent error?

    1. A.2.17%
    2. B.1.73%
    3. C.2.23%
    4. D.1.96%

    Show answer

    Answer: A. 2.17%

    AI explanation

    1. Analyze the Question

    We are asked to calculate the percent error of a measurement.

    • Measured Value (Student's reading): 18 mL
    • Accepted Value (Teacher's reading): 18.4 mL

    2. Explain the Concept

    Percent error is a measure of how far off a measured value is from the accepted (true) value. The formula is:

    $\text{Percent Error} = \left| \frac{\text{Measured Value} - \text{Accepted Value}}{\text{Accepted Value}} \right| \times 100\%$

    Note: The vertical bars indicate the absolute value, meaning the result should always be positive.

    3. Show the Solution

    Step 1: Find the absolute error.

    $\text{Absolute Error} = |18 - 18.4|$

    $\text{Absolute Error} = |-0.4| = 0.4 \text{ mL}$

    Step 2: Divide the absolute error by the accepted value.

    $\text{Relative Error} = \frac{0.4}{18.4}$

    To simplify this, multiply both numerator and denominator by 10:

    $\text{Relative Error} = \frac{4}{184}$

    $\text{Relative Error} = \frac{1}{46} \approx 0.021739...$

    Step 3: Convert to a percentage.

    $\text{Percent Error} = 0.021739 \times 100\%$

    $\text{Percent Error} \approx 2.1739\%$

    Rounding to two decimal places, we get 2.17%.

    4. Identify the Answer

    Comparing our result to the options provided, the correct choice is A.

    5. Exam Tip

    When calculating percent error, always remember that the denominator is the Accepted Value (the "true" value provided by the teacher or the standard), not the measured value. A common mistake students make is using their own measurement as the denominator—avoid this to ensure you get the correct answer!

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22. 22.

    The population of a village decreased from 1,230 to 1,040 due to breakout of an epidemic. What is the percentage decrease in the population?

    1. A.15.44%
    2. B.15.43%
    3. C.15.42%
    4. D.15.45%

    Show answer

    Answer: D. 15.45%

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23. 23.

    Differentiate the function y = $\sqrt[3]{x^2}(2x - x^2)$

    1. A.$\frac {dy}{dx} = \frac {10x^{5/3}}{3} - \frac {8x^{2/3}}{3}$
    2. B.$\frac {dy}{dx} = \frac {10x^{2/3}}{3} - \frac {8x^{5/3}}{3}$
    3. C.$\frac {dy}{dx} = \frac {10x^{5/3}}{3} - \frac {8x^{5/3}}{3}$
    4. D.$\frac {dy}{dx} = \frac {10x^{2/3}}{3} - \frac {8x^{2/3}}{3}$

    Show answer

    Answer: B. $\frac {dy}{dx} = \frac {10x^{2/3}}{3} - \frac {8x^{5/3}}{3}$

    AI explanation

    1. Analyze the question

    We are asked to find the derivative ($\frac{dy}{dx}$) of the function $y = \sqrt[3]{x^2}(2x - x^2)$. To solve this, we need to simplify the expression into a polynomial form (using indices) before applying the power rule of differentiation.

    2. Explain the concept

    • Indices: Recall that $\sqrt[n]{x^m} = x^{m/n}$. Therefore, $\sqrt[3]{x^2} = x^{2/3}$.
    • Power Rule: If $y = ax^n$, then $\frac{dy}{dx} = n \cdot ax^{n-1}$.
    • Distributive Law: Multiply the term outside the bracket by each term inside the bracket.

    3. Show the solution

    Step 1: Rewrite the function using indices $y = x^{2/3}(2x - x^2)$

    Step 2: Expand the expression Multiply $x^{2/3}$ by each term inside the bracket: $y = (x^{2/3} \cdot 2x^1) - (x^{2/3} \cdot x^2)$

    Using the law of indices ($x^a \cdot x^b = x^{a+b}$):

    • For the first term: $x^{2/3} \cdot 2x^1 = 2x^{(2/3 + 1)} = 2x^{5/3}$
    • For the second term: $x^{2/3} \cdot x^2 = x^{(2/3 + 2)} = x^{(2/3 + 6/3)} = x^{8/3}$

    So, $y = 2x^{5/3} - x^{8/3}$

    Step 3: Differentiate with respect to $x$ Apply the power rule $\frac{d}{dx}(ax^n) = nax^{n-1}$:

    • Derivative of $2x^{5/3}$: $\frac{5}{3} \cdot 2x^{(5/3 - 1)} = \frac{10}{3}x^{2/3}$

    • Derivative of $x^{8/3}$: $\frac{8}{3}x^{(8/3 - 1)} = \frac{8}{3}x^{5/3}$

    Combining these: $\frac{dy}{dx} = \frac{10x^{2/3}}{3} - \frac{8x^{5/3}}{3}$

    4. Identify the answer

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

    5. Tip

    When you see roots (like $\sqrt{x}$ or $\sqrt[3]{x^2}$) in a differentiation problem, always convert them to fractional exponents first. It makes the power rule much easier to apply and prevents errors with the chain rule or product rule.

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24. 24.

    Find the volume of the composite solid above.

    1. A.2048 cm$^3$
    2. B.2568 cm$^3$
    3. C.2672 cm$^3$
    4. D.1320 cm$^3$

    Show answer

    Answer: B. 2568 cm$^3$

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25. 25.

    What is the general term of the sequence 3, 8, 13, 18, ...?

    1. A.5n - 2
    2. B.5n + 2
    3. C.5
    4. D.5n

    Show answer

    Answer: A. 5n - 2

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26. 26.

    Find the value of y, if log (y + 8) + log (y - 8) = 2log 3 + 2log 5

    1. A.y = ±5
    2. B.y = ±10
    3. C.y = ±17
    4. D.y = ±13

    Show answer

    Answer: C. y = ±17

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27. 27.

    A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increase by 68 cm$^2$ if we add 2 cm to each side of the rectangle. Find the length of the shorter side.

    1. A.15 cm
    2. B.19 cm
    3. C.13 cm
    4. D.21 cm

    Show answer

    Answer: C. 13 cm

    AI explanation

    1. Analyze the question

    We are dealing with a rectangle where the sides have a specific relationship. We need to find the length of the shorter side by setting up an algebraic equation based on the change in area when the dimensions are increased.

    2. Explain the concept

    • Area of a rectangle: $Area = \text{length} \times \text{width}$
    • Let the shorter side be $x$.
    • Since the other side is 6 cm longer, the longer side is $(x + 6)$.
    • Original Area = $x(x + 6)$
    • When we add 2 cm to each side: • New shorter side = $(x + 2)$ • New longer side = $(x + 6 + 2) = (x + 8)$ • New Area = $(x + 2)(x + 8)$

    3. Show the solution

    Step 1: Set up the equation. The problem states that the new area is 68 cm² larger than the original area:

    $\text{New Area} = \text{Original Area} + 68$

    $(x + 2)(x + 8) = x(x + 6) + 68$

    Step 2: Expand both sides. Left side: $x^2 + 8x + 2x + 16 = x^2 + 10x + 16$ Right side: $x^2 + 6x + 68$

    Step 3: Solve for $x$.

    $x^2 + 10x + 16 = x^2 + 6x + 68$

    Subtract $x^2$ from both sides:

    $10x + 16 = 6x + 68$

    Subtract $6x$ from both sides:

    $4x + 16 = 68$

    Subtract 16 from both sides:

    $4x = 52$

    Divide by 4:

    $x = 13$

    The shorter side is 13 cm.

    4. Identify the answer

    Comparing our result to the options provided: A. 15 cm B. 19 cm C. 13 cm D. 21 cm

    The correct option is C.

    5. Tip

    When dealing with "area increase" problems, always write out the "Before" and "After" expressions for the area separately before equating them. This prevents errors in expansion. Also, remember that adding 2 cm to each side means adding 2 to both the length and the width!

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28. 28.

    Divide 1101001$_{two}$ by 101$_{two}$

    1. A.11101$_{two}$
    2. B.111$_{two}$
    3. C.10111$_{two}$
    4. D.10101$_{two}$

    Show answer

    Answer: D. 10101$_{two}$

    AI explanation

    To solve this division problem in base two (binary), we use the long division method, similar to how we divide in base ten.

    1. Analyze the question

    We are dividing $1101001_2$ by $101_2$.

    2. Explain the concept

    In binary division, we look at the divisor ($101_2$) and see how many times it fits into the leading digits of the dividend ($1101001_2$). We use the same rules of binary subtraction:

    • $0 - 0 = 0$
    • $1 - 0 = 1$
    • $1 - 1 = 0$
    • $10 - 1 = 1$ (borrowing 1 from the next column)

    3. Show the solution

    Let's perform the long division:

    • Step 1: How many times does $101$ go into $110$? It goes 1 time. • $1 \times 101 = 101$ • $110 - 101 = 001$
    • Step 2: Bring down the next digit ($1$). Now we have $011$. $101$ goes into $011$ 0 times. • $0 \times 101 = 000$ • $011 - 000 = 011$
    • Step 3: Bring down the next digit ($0$). Now we have $110$. $101$ goes into $110$ 1 time. • $1 \times 101 = 101$ • $110 - 101 = 001$
    • Step 4: Bring down the final digit ($1$). Now we have $011$. $101$ goes into $011$ 0 times? Wait, let's re-check the alignment.

    Let's re-calculate carefully: $1101001 \div 101$

    1. $110 \div 101 = 1$ remainder $1$.
    2. Bring down $1 \rightarrow 11$. $11 \div 101 = 0$.
    3. Bring down $0 \rightarrow 110$. $110 \div 101 = 1$ remainder $1$.
    4. Bring down $1 \rightarrow 11$. $11 \div 101 = 0$.

    Actually, the standard long division for $1101001_2 \div 101_2$ is: $1101001_2 = 105_{10}$ $101_2 = 5_{10}$ $105 \div 5 = 21_{10}$

    Now, convert $21_{10}$ back to binary: $21 \div 2 = 10$ rem $1$ $10 \div 2 = 5$ rem $0$ $5 \div 2 = 2$ rem $1$ $2 \div 2 = 1$ rem $0$ $1 \div 2 = 0$ rem $1$ Reading from bottom to top, we get $10101_2$.

    4. Identify the answer

    Comparing our result ($10101_2$) to the given options, it matches option D.

    5. Tip

    When you are stuck with binary arithmetic, convert the numbers to base ten (decimal), perform the operation (addition, subtraction, multiplication, or division), and then convert your final answer back to binary. It is often much faster and less prone to errors!

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29. 29.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

30. 30.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

31. 31.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

32. 32.

    Calculate the mean deviation of the first five prime numbers.

    1. A.2.72
    2. B.5.6
    3. C.5.25
    4. D.13.6

    Show answer

    Answer: A. 2.72

    AI explanation

    1. Analyze the question

    We need to find the mean deviation of the first five prime numbers. To do this, we must first identify the numbers, calculate their arithmetic mean, and then find the average of the absolute differences between each number and that mean.

    2. Explain the concept

    • Prime Numbers: Numbers greater than 1 that have only two factors: 1 and themselves.
    • Arithmetic Mean ($\bar{x}$): The sum of the values divided by the number of values.
    • Mean Deviation: The average of the absolute differences between each data point ($x$) and the mean ($\bar{x}$). The formula is: $$\text{Mean Deviation} = \frac{\sum |x - \bar{x}|}{n}$$

    3. Show the solution

    Step 1: Identify the first five prime numbers. The first five prime numbers are: 2, 3, 5, 7, and 11. Number of values ($n$) = 5.

    Step 2: Calculate the mean ($\bar{x}$).

    $\bar{x} = \frac{2 + 3 + 5 + 7 + 11}{5} = \frac{28}{5} = 5.6$

    Step 3: Calculate the absolute deviations from the mean $|x - \bar{x}|$.

    • $|2 - 5.6| = |-3.6| = 3.6$
    • $|3 - 5.6| = |-2.6| = 2.6$
    • $|5 - 5.6| = |-0.6| = 0.6$
    • $|7 - 5.6| = |1.4| = 1.4$
    • $|11 - 5.6| = |5.4| = 5.4$

    Step 4: Calculate the mean of these deviations.

    $\text{Mean Deviation} = \frac{3.6 + 2.6 + 0.6 + 1.4 + 5.4}{5}$

    $\text{Mean Deviation} = \frac{13.6}{5} = 2.72$

    4. Identify the answer

    Comparing our result to the given options, the correct value is 2.72. This matches option A.

    5. Tip

    When calculating mean deviation, remember that the result of $|x - \bar{x}|$ must always be positive. If you get a negative number, you have made a mistake in your subtraction. Also, ensure you don't confuse "Mean Deviation" with "Standard Deviation"—they are calculated using different formulas!

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33. 33.

    A coin is thrown 3 times. What is the probability that atleast one head is obtained?

    1. A.$\frac {7}{8}$
    2. B.$\frac {3}{8}$
    3. C.None the above
    4. D.$\frac {1}{8}$

    Show answer

    Answer: A. $\frac {7}{8}$

    AI explanation

    1. Analyze the question

    We are tossing a fair coin 3 times. We need to find the probability of getting at least one head.

    2. Explain the concept

    When dealing with "at least one" problems in probability, it is often much faster to use the Complement Rule. The sum of the probability of an event occurring and the probability of it not occurring is always 1.

    • $P(\text{at least one head}) = 1 - P(\text{no heads})$

    "No heads" is the same as getting "all tails" (TTT).

    3. Show the solution

    • Step 1: Find the total number of possible outcomes. Since a coin has 2 sides (Head or Tail) and it is tossed 3 times, the total outcomes = $2^3 = 2 \times 2 \times 2 = 8$. The sample space is: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.

    • Step 2: Identify the outcome for "no heads". The only way to get no heads is to get tails on all three tosses: {TTT}. There is only 1 such outcome.

    • Step 3: Calculate the probability of no heads. $P(\text{no heads}) = \frac{\text{Number of "no head" outcomes}}{\text{Total outcomes}} = \frac{1}{8}$

    • Step 4: Apply the complement rule. $P(\text{at least one head}) = 1 - P(\text{no heads})$ $P(\text{at least one head}) = 1 - \frac{1}{8}$ $P(\text{at least one head}) = \frac{8}{8} - \frac{1}{8} = \frac{7}{8}$

    4. Identify the answer

    Comparing our result to the options provided, the correct choice is A.

    5. Exam Tip

    Whenever you see the phrase "at least one" in a probability question, immediately think: "1 minus the probability of none." It saves you from having to list out all the combinations for 1 head, 2 heads, and 3 heads separately!

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34. 34.

    If A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find (A – B) ⋃ (B – A).

    1. A.{1, 3, 5, 8}
    2. B.{8}
    3. C.{1, 2, 3, 4, 5, 6, 8}
    4. D.{1, 3, 5}

    Show answer

    Answer: A. {1, 3, 5, 8}

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35. 35.

    A committee of 5 people is to be chosen from a group of 6 men and 4 women. How many committees are possible if there is to be a majority of women?

    1. A.60
    2. B.15
    3. C.66
    4. D.4

    Show answer

    Answer: C. 66

    AI explanation

    Analysis

    The problem asks us to form a committee of 5 people from a total group of 10 (6 men and 4 women). The condition is that there must be a majority of women.

    Concept

    Since the committee size is 5, a "majority of women" means there must be more women than men. The possible compositions for the committee are:

    1. 3 women and 2 men
    2. 4 women and 1 man

    We use the combination formula, denoted as $^nC_r = \frac{n!}{r!(n-r)!}$, where $n$ is the total number of items and $r$ is the number of items to be chosen.

    Step-by-Step Solution

    Case 1: 3 women and 2 men

    • Number of ways to choose 3 women from 4: $^4C_3 = \frac{4!}{3!(4-3)!} = 4$
    • Number of ways to choose 2 men from 6: $^6C_2 = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15$
    • Total for Case 1: $4 \times 15 = 60$

    Case 2: 4 women and 1 man

    • Number of ways to choose 4 women from 4: $^4C_4 = 1$
    • Number of ways to choose 1 man from 6: $^6C_1 = 6$
    • Total for Case 2: $1 \times 6 = 6$

    Total Committees To get the final answer, add the results of the two cases: $60 + 6 = 66$

    Conclusion

    The total number of possible committees with a majority of women is 66.

    Exam Tip

    When you see "majority" in committee problems, always list out the possible scenarios first. If the committee size is odd (like 5), you don't have to worry about a tie, but always check if the number of available people in each category is enough to satisfy the condition (e.g., you couldn't have 5 women because there are only 4 available).

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36. 36.

    Determine the area of the region bounded by y = $2x^2$ + 10 and Y = $4x + 16$.

    1. A.18
    2. B.$\frac {-10}{3}$
    3. C.$\frac {44}{3}$
    4. D.$\frac {64}{3}$

    Show answer

    Answer: D. $\frac {64}{3}$

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37. 37.

    At simple interest, a man made a deposit of some money in the bank. The amount in his bank account after 10 years is three times the money deposited. If the interest rate stays the same, after how many years will the amount be five times the money deposited?

    1. A.15 years
    2. B.25 years
    3. C.20 years
    4. D.30 years

    Show answer

    Answer: C. 20 years

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38. 38.

    Evaluate $\int_0^1 4x - 6\sqrt[3] {x^2}dx$

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

    Show answer

    Answer: B. - $\frac{8}{5}$

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39. 39.

    Bello buys an old bicycle for ₦9,200.00 and spends ₦1,500.00 on its repairs. If he sells the bicycle for ₦13,400.00, his gain percent is

    1. A.25.23%
    2. B.31.34%
    3. C.88.81%
    4. D.42.54%

    Show answer

    Answer: A. 25.23%

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40. 40.

    How many different 8 letter words are possible using the letters of the word SYLLABUS?

    1. A.(8 - 1)!
    2. B.$\frac{8!}{2!}$
    3. C.$\frac{8!}{2! 2!}$
    4. D.8!

    Show answer

    Answer: C. $\frac{8!}{2! 2!}$

    AI explanation

    Analysis

    To solve this problem, we need to determine the number of distinct arrangements (permutations) of the letters in the word SYLLABUS.

    Concept: Permutation of Multisets

    When arranging objects where some are identical, we use the formula for permutations of a multiset:

    $P = \frac{n!}{n_1! \times n_2! \times \dots \times n_k!}$

    Where:

    • $n$ is the total number of items.
    • $n_1, n_2, \dots$ are the frequencies of the repeating items.

    Step-by-Step Solution

    1. Count the total number of letters ($n$): The word S-Y-L-L-A-B-U-S has 8 letters. So, $n = 8$.

    2. Identify the repeating letters: • S appears 2 times. • Y appears 1 time. • L appears 2 times. • A appears 1 time. • B appears 1 time. • U appears 1 time.

    3. Apply the formula: Since 'S' repeats twice and 'L' repeats twice, we divide the total factorial ($8!$) by the factorial of the frequency of each repeating letter: $$\text{Number of arrangements} = \frac{8!}{2! \times 2!}$$

    Conclusion

    The calculation $\frac{8!}{2! 2!}$ accounts for the fact that swapping the two 'S's or the two 'L's does not create a new unique word. This matches option C.

    Exam Tip

    Whenever you see a word arrangement question, always write out the letters and their counts first. If no letters repeat, the answer is simply $n!$. If letters repeat, always remember to divide by the factorial of the number of times each letter repeats!

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41. 41.

    Tickets for the school play were priced at ₦520.00 each for adults and ₦250.00 each for kids. How many kids' tickets were sold if the total sales were ₦171,000.00 and there were 5 times as many adult tickets sold as children's tickets?

    1. A.20
    2. B.300
    3. C.50
    4. D.60

    Show answer

    Answer: D. 60

    AI explanation

    Analysis

    To solve this problem, we need to translate the word problem into algebraic equations. We are looking for the number of kids' tickets sold, given the prices, the total revenue, and the relationship between the number of adult and kids' tickets.

    Concept

    We can use a system of linear equations. Let:

    • $x$ = the number of kids' tickets sold.
    • $5x$ = the number of adult tickets sold (since there were 5 times as many adult tickets as kids' tickets).

    The total revenue is the sum of (price of adult tickets × number of adult tickets) and (price of kids' tickets × number of kids' tickets).

    Step-by-Step Solution

    1. Set up the equation: • Price of adult ticket = ₦520 • Price of kids' ticket = ₦250 • Total revenue = ₦171,000

    The equation is: $(520 \times 5x) + (250 \times x) = 171,000$

    2. Simplify the equation: • $2600x + 250x = 171,000$ • $2850x = 171,000$

    3. Solve for $x$: • $x = \frac{171,000}{2850}$ • $x = \frac{17,100}{285}$ • $x = 60$

    4. Verify: • If kids' tickets ($x$) = 60, then adult tickets ($5x$) = 300. • Revenue from kids = $60 \times 250 = 15,000$ • Revenue from adults = $300 \times 520 = 156,000$ • Total = $15,000 + 156,000 = 171,000$. The calculation is correct.

    Conclusion

    The number of kids' tickets sold is 60. This matches option D.

    Exam Tip

    When you see "5 times as many X as Y," always write it as $X = 5Y$. A common mistake is to write $5X = Y$, which flips the ratio. Always do a quick mental check: if there are more adults, the adult count should be the larger number!

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42. 42.

    A bag contains 8 red balls and some white balls. If the probability of drawing a white ball is half of the probability of drawing a red ball then find the probability of drawing a red ball and a white ball if the balls are drawn without replacement.

    1. A.$\frac {1}{3}$
    2. B.$\frac {2}{9}$
    3. C.$\frac {2}{3}$
    4. D.$\frac {8}{33}$

    Show answer

    Answer: D. $\frac {8}{33}$

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43. 43.

    The perimeter of an isosceles right-angled triangle is 2 meters. Find the length of its longer side.

    1. A.2 - $\sqrt2$
    2. B.-4 + 3$\sqrt2$
    3. C.It cannot be determined
    4. D.-2 + 2$\sqrt2$ m

    Show answer

    Answer: D. -2 + 2$\sqrt2$ m

    AI explanation

    1. Analyze the question

    We are given an isosceles right-angled triangle with a perimeter of 2 meters. We need to find the length of the "longer side," which in this case refers to the hypotenuse.

    2. Explain the concept

    • Isosceles Right-Angled Triangle: This triangle has two equal sides (let's call their length $x$) and one hypotenuse ($h$).
    • Pythagoras Theorem: For a right-angled triangle, $a^2 + b^2 = c^2$. Here, $x^2 + x^2 = h^2$, which simplifies to $2x^2 = h^2$, so $h = x\sqrt{2}$.
    • Perimeter: The sum of all sides is $x + x + h = 2$.

    3. Show the solution

    Step 1: Set up the perimeter equation. $2x + h = 2$ Since $h = x\sqrt{2}$, we substitute $h$ into the equation: $2x + x\sqrt{2} = 2$

    Step 2: Solve for $x$ (the equal sides). Factor out $x$: $x(2 + \sqrt{2}) = 2$ $x = \frac{2}{2 + \sqrt{2}}$

    Step 3: Rationalize the denominator. Multiply the numerator and denominator by the conjugate $(2 - \sqrt{2})$: $x = \frac{2(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})}$ $x = \frac{4 - 2\sqrt{2}}{4 - 2}$ $x = \frac{4 - 2\sqrt{2}}{2}$ $x = 2 - \sqrt{2}$

    Step 4: Find the hypotenuse ($h$). We know $h = x\sqrt{2}$. Substitute the value of $x$ we just found: $h = (2 - \sqrt{2}) \times \sqrt{2}$ $h = 2\sqrt{2} - (\sqrt{2} \times \sqrt{2})$ $h = 2\sqrt{2} - 2$ Rearranging this gives: $-2 + 2\sqrt{2}$

    4. Identify the answer

    Comparing our result with the options provided: A. $2 - \sqrt{2}$ B. $-4 + 3\sqrt{2}$ C. It cannot be determined D. $-2 + 2\sqrt{2}$ m

    The calculated value matches option D.

    5. Tip

    When dealing with surds in geometry, always remember to rationalize the denominator early. Also, remember that in an isosceles right-angled triangle, the ratio of the sides is always $1 : 1 : \sqrt{2}$. If you know the perimeter, you can quickly find the sides by dividing the perimeter by $(2 + \sqrt{2})$.

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44. 44.

    Solve the following quadratic inequality: $x^2 - x$ - 4 ≤ 2

    1. A.$-3 < x < 2$
    2. B.$-2 ≤ x ≤ 3$
    3. C.$x ≤ -2, x ≤ 3$
    4. D.$-2 < x < 3$

    Show answer

    Answer: B. $-2 ≤ x ≤ 3$

    AI explanation

    Analysis

    To solve the quadratic inequality $x^2 - x - 4 \leq 2$, we need to move all terms to one side to set the inequality to zero, find the roots of the corresponding quadratic equation, and then determine the interval that satisfies the condition.

    Concept

    A quadratic inequality in the form $ax^2 + bx + c \leq 0$ is solved by:

    1. Finding the roots of the equation $ax^2 + bx + c = 0$.
    2. Testing the intervals created by these roots to see which one satisfies the inequality.
    3. Since the inequality is "$\leq$", the solution will include the roots themselves (using $\leq$ or $\geq$ signs).

    Step-by-Step Solution

    Step 1: Rearrange the inequality Subtract 2 from both sides to set the right side to zero: $x^2 - x - 4 - 2 \leq 0$ $x^2 - x - 6 \leq 0$

    Step 2: Factor the quadratic expression We need two numbers that multiply to $-6$ and add up to $-1$. These numbers are $-3$ and $+2$. $(x - 3)(x + 2) \leq 0$

    Step 3: Find the critical points Set each factor to zero: $x - 3 = 0 \implies x = 3$ $x + 2 = 0 \implies x = -2$

    Step 4: Determine the interval We are looking for where the product $(x - 3)(x + 2)$ is less than or equal to zero (negative or zero).

    • If $x < -2$ (e.g., $x = -3$): $(-3-3)(-3+2) = (-6)(-1) = 6$ (Positive)
    • If $-2 \leq x \leq 3$ (e.g., $x = 0$): $(0-3)(0+2) = (-3)(2) = -6$ (Negative)
    • If $x > 3$ (e.g., $x = 4$): $(4-3)(4+2) = (1)(6) = 6$ (Positive)

    Since we want the expression to be $\leq 0$, the valid range is between the two roots, inclusive. Thus, $-2 \leq x \leq 3$.

    Conclusion

    The solution to the inequality is $-2 \leq x \leq 3$. This matches option B.

    Exam Tip: Whenever you have a quadratic inequality in the form $(x - a)(x - b) \leq 0$ (where $a < b$), the solution is almost always the interval between the roots: $a \leq x \leq b$. If the sign were $\geq 0$, the solution would be the "outer" regions: $x \leq a$ or $x \geq b$.

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45. 45.

    A man sells different brands of an items. $^1/_9$ of the items he has in his shop are from Brand A, $^5/_8$ of the remainder are from Brand B and the rest are from Brand C. If the total number of Brand C items in the man's shop is 81, how many more Brand B items than Brand C does the shop has?

    1. A.243
    2. B.108
    3. C.54
    4. D.135

    Show answer

    Answer: C. 54

    AI explanation

    1. Analyze the question

    We need to find the total number of items in the shop first. We are given:

    • Brand A = $1/9$ of the total.
    • Brand B = $5/8$ of the remainder.
    • Brand C = 81 items.
    • Goal: Find (Brand B - Brand C).

    2. Explain the concept

    Let the total number of items be $x$.

    • If Brand A is $1/9$ of $x$, then the remainder is $x - 1/9x = 8/9x$.
    • Brand B is $5/8$ of that remainder ($8/9x$).
    • Brand C is what is left after taking out Brand A and Brand B.

    3. Show the solution

    Step 1: Find the number of items for each brand in terms of $x$.

    • Brand A: $\frac{1}{9}x$
    • Remainder after A: $x - \frac{1}{9}x = \frac{8}{9}x$
    • Brand B: $\frac{5}{8} \times (\frac{8}{9}x) = \frac{5}{9}x$
    • Brand C: The total minus (Brand A + Brand B): $x - (\frac{1}{9}x + \frac{5}{9}x) = x - \frac{6}{9}x = x - \frac{2}{3}x = \frac{1}{3}x$

    Step 2: Solve for $x$ using the given value for Brand C. We know Brand C = 81. $\frac{1}{3}x = 81$ $x = 81 \times 3 = 243$

    Step 3: Calculate the number of items for Brand B. Brand B = $\frac{5}{9}x$ Brand B = $\frac{5}{9} \times 243$ $243 \div 9 = 27$ $5 \times 27 = 135$

    Step 4: Find the difference between Brand B and Brand C. Difference = Brand B - Brand C Difference = $135 - 81 = 54$

    4. Identify the answer

    Comparing our result to the options provided: A. 243 B. 108 C. 54 D. 135

    The correct option is C.

    5. Tip

    When dealing with "remainder" problems, always calculate the remainder first before applying the next fraction. A common mistake is to apply the second fraction ($5/8$) to the total instead of the remainder. Also, notice how the fractions ($1/9$ and $5/9$) sum up to $6/9$, leaving $3/9$ (or $1/3$) for the final category—this is a quick way to verify your work!

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46. 46.

    A rectangular plot of land has sides with lengths of 38 m and 52 m corrected to the nearest m. Find the range of the possible values of the area of the rectangle

    1. A.1931.25 m$^2$ ≤ A < 2021.25 m$^2$
    2. B.1950 m$^2$ ≤ A < 2002 m$^2$
    3. C.1957 m$^2$ ≤ A < 1995 m$^2$
    4. D.1931.25 m$^2$ ≥ A > 2021.25 m$^2$

    Show answer

    Answer: A. 1931.25 m$^2$ ≤ A < 2021.25 m$^2$

    AI explanation

    1. Analyze the Question

    We are given the dimensions of a rectangular plot as 38 m and 52 m, both corrected to the nearest meter. We need to find the range of possible values for the area ($A$) of the rectangle.

    2. Explain the Concept

    When a measurement is given to the nearest whole number (1 m), the maximum error is half of that unit.

    • Maximum error = $\frac{1}{2} \times 1\text{ m} = 0.5\text{ m}$.
    • The lower bound of a measurement is: $\text{Value} - 0.5$.
    • The upper bound of a measurement is: $\text{Value} + 0.5$.

    For a rectangle with length $L$ and width $W$, the area $A = L \times W$.

    • The minimum area is calculated using the lower bounds: $A_{min} = L_{min} \times W_{min}$.
    • The maximum area is calculated using the upper bounds: $A_{max} = L_{max} \times W_{max}$.

    3. Show the Solution

    Step 1: Determine the bounds for the sides.

    • For the side 38 m: • Lower bound ($L_1$) = $38 - 0.5 = 37.5\text{ m}$ • Upper bound ($U_1$) = $38 + 0.5 = 38.5\text{ m}$
    • For the side 52 m: • Lower bound ($L_2$) = $52 - 0.5 = 51.5\text{ m}$ • Upper bound ($U_2$) = $52 + 0.5 = 52.5\text{ m}$

    Step 2: Calculate the minimum and maximum area.

    • Minimum Area ($A_{min}$) = $37.5 \times 51.5$ • $37.5 \times 51.5 = 1931.25\text{ m}^2$
    • Maximum Area ($A_{max}$) = $38.5 \times 52.5$ • $38.5 \times 52.5 = 2021.25\text{ m}^2$

    Step 3: Express the range. The area $A$ must be greater than or equal to the minimum area and strictly less than the maximum area (since the upper bound is the limit the value approaches). Range: $1931.25\text{ m}^2 \leq A < 2021.25\text{ m}^2$

    4. Identify the Answer

    Comparing our result with the given options, it matches Option A.

    5. Tip

    Always remember the "Half-Unit Rule": If a value is given to the nearest $n$, the range is always $\pm \frac{n}{2}$. If you are asked for the area, never just multiply the given numbers; always calculate the bounds first!

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47. 47.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

48. 48.

    The area A of a circle is increasing at a constant rate of 1.5 cm$^2s^{-1}$. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm$^2$.

    1. A.0.200 cms$^{-1}$
    2. B.0.798 cms$^{-1}$
    3. C.0.300 cms$^{-1}$
    4. D.0.299 cms$^{-1}$

    Show answer

    Answer: D. 0.299 cms$^{-1}$

    AI explanation

    Analysis

    We are given the rate of change of the area of a circle with respect to time ($\frac{dA}{dt} = 1.5 \text{ cm}^2\text{s}^{-1}$) and asked to find the rate of change of the radius ($\frac{dr}{dt}$) at the specific moment when the area $A = 2 \text{ cm}^2$.

    Concept

    The area of a circle is given by the formula:

    $A = \pi r^2$

    To find the relationship between the rates of change, we differentiate both sides with respect to time ($t$) using the chain rule:

    $\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$

    $\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$

    Step-by-Step Solution

    Step 1: Find the radius $r$ when $A = 2 \text{ cm}^2$ Using $A = \pi r^2$:

    $2 = \pi r^2$

    $r^2 = \frac{2}{\pi}$

    $r = \sqrt{\frac{2}{\pi}} \approx 0.79788 \text{ cm}$

    Step 2: Substitute the known values into the differentiated formula We know $\frac{dA}{dt} = 1.5$ and $r \approx 0.79788$.

    $1.5 = 2 \cdot \pi \cdot (0.79788) \cdot \frac{dr}{dt}$

    Step 3: Solve for $\frac{dr}{dt}$

    $\frac{dr}{dt} = \frac{1.5}{2 \cdot \pi \cdot 0.79788}$

    $\frac{dr}{dt} = \frac{1.5}{5.01325}$

    $\frac{dr}{dt} \approx 0.2992 \text{ cms}^{-1}$

    Rounding to 3 significant figures, we get 0.299 cms$^{-1}$.

    Conclusion

    Comparing our result to the given options, the correct value is 0.299. This matches option D.

    Exam Tip

    When dealing with "related rates" problems in JAMB, always write down the geometric formula first (like $A = \pi r^2$ or $V = \frac{4}{3}\pi r^3$), differentiate it with respect to time $t$, and then plug in your specific values. Don't plug in the numbers before differentiating, or you will lose the variable you are trying to solve for!

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49. 49.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

50. 50.

    How many students scored at least 25%

    1. A.16
    2. B.19
    3. C.3
    4. D.8

    Show answer

    Answer: A. 16

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51. 51.

    Let '*' and '^' be two binary operations such that a * b = a$^2$b and a ^ b = 2a + b. Find (-4 * 2) ^ (7 * -1).

    1. A.-49
    2. B.64
    3. C.113
    4. D.15

    Show answer

    Answer: D. 15

    AI explanation

    To solve this problem, we need to evaluate the expression step-by-step using the definitions provided for the binary operations.

    1. Analyze the definitions

    • Operation (*): $a * b = a^2b$
    • Operation (^): $a \wedge b = 2a + b$

    We need to find the value of: $(-4 * 2) \wedge (7 * -1)$

    2. Step-by-step calculation

    Step 1: Solve the first bracket $(-4 * 2)$ Using the rule $a * b = a^2b$: $(-4 * 2) = (-4)^2 \times 2$ $(-4 * 2) = 16 \times 2$ $(-4 * 2) = 32$

    Step 2: Solve the second bracket $(7 * -1)$ Using the rule $a * b = a^2b$: $(7 * -1) = (7)^2 \times (-1)$ $(7 * -1) = 49 \times (-1)$ $(7 * -1) = -49$

    Step 3: Combine the results using the (^) operation Now we have $(32) \wedge (-49)$. Using the rule $a \wedge b = 2a + b$: $32 \wedge -49 = 2(32) + (-49)$ $32 \wedge -49 = 64 - 49$ $32 \wedge -49 = 15$

    3. Conclusion

    Following the order of operations and the definitions provided, the result is 15. This matches option D.

    Exam Tip: When dealing with binary operations, always treat the brackets as separate mini-problems. Solve the contents of the brackets first before applying the final operator. Also, be very careful with negative signs when squaring numbers (e.g., $(-4)^2$ is $+16$, not $-16$).

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52. 52.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

53. 53.

    An article when sold for ₦230.00 makes a 15% profit. Find the profit or loss % if it was sold for ₦180.00

    1. A.10% gain
    2. B.10% loss
    3. C.12% loss
    4. D.12% gain

    Show answer

    Answer: B. 10% loss

    AI explanation

    1. Analyze the Question

    We are given the selling price of an item and the profit percentage made at that price. We need to determine the profit or loss percentage if the selling price is reduced to a new value.

    2. Explain the Concept

    • Cost Price (CP): The original amount the item was bought for.
    • Selling Price (SP): The amount the item is sold for.
    • Profit % Formula: $\text{Profit \%} = \frac{\text{SP} - \text{CP}}{\text{CP}} \times 100$
    • Relationship: If an item is sold at a 15% profit, the Selling Price is $115\%$ of the Cost Price ($100\% + 15\%$).

    3. Show the Solution

    Step 1: Find the Cost Price (CP) We know that $115\%$ of the Cost Price = ₦230.00. Let $CP = x$. $1.15 \times x = 230$ $x = \frac{230}{1.15}$ $x = 200$ So, the Cost Price of the article is ₦200.00.

    Step 2: Calculate the result for the new Selling Price The new Selling Price is ₦180.00. Since the new Selling Price (₦180) is less than the Cost Price (₦200), we have a loss.

    Step 3: Calculate the loss percentage $\text{Loss} = \text{CP} - \text{New SP}$ $\text{Loss} = 200 - 180 = 20$

    $\text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100$ $\text{Loss \%} = \frac{20}{200} \times 100$ $\text{Loss \%} = 0.1 \times 100 = 10\%$

    4. Identify the Answer

    The calculation shows a 10% loss. This matches option B.

    5. Tip

    When calculating profit or loss, always find the Cost Price first. It is the "anchor" for all your calculations. If your new Selling Price is lower than your Cost Price, you are guaranteed to have a loss; if it is higher, you have a gain.

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54. 54.

    The interior angle of a regular polygon is five times the size of its exterior angle. Identify the polygon.

    1. A.dodecagon
    2. B.enneadecagon
    3. C.icosagon
    4. D.hendecagon

    Show answer

    Answer: A. dodecagon

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55. 55.

    In a group of 500 people, 350 people can speak English, and 400 people can speak French. Find how many people can speak both languages.

    1. A.750
    2. B.850
    3. C.250
    4. D.150

    Show answer

    Answer: C. 250

    AI explanation

    Analysis

    This is a classic set theory problem involving two sets. We are given the total number of people (the universal set) and the number of people in two overlapping categories (English speakers and French speakers). We need to find the intersection of these two sets (those who speak both).

    Concept

    We use the principle of inclusion-exclusion for two sets:

    $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

    Where:

    • $n(A \cup B)$ is the total number of people (Universal set).
    • $n(A)$ is the number of people who speak English.
    • $n(B)$ is the number of people who speak French.
    • $n(A \cap B)$ is the number of people who speak both languages.

    Solution

    Step 1: Identify the given values.

    • Total number of people, $n(A \cup B) = 500$
    • Number of English speakers, $n(E) = 350$
    • Number of French speakers, $n(F) = 400$

    Step 2: Apply the formula.

    $500 = 350 + 400 - n(E \cap F)$

    Step 3: Simplify the equation.

    $500 = 750 - n(E \cap F)$

    Step 4: Solve for $n(E \cap F)$.

    $n(E \cap F) = 750 - 500$

    $n(E \cap F) = 250$

    The number of people who speak both languages is 250.

    Exam Strategy

    When dealing with set theory questions in JAMB, always check if the total given is the sum of the individual sets. If the sum of the individual sets ($350 + 400 = 750$) is greater than the total number of people ($500$), the difference ($750 - 500 = 250$) represents the people who were counted twice because they belong to both groups. This is a quick way to verify your answer!

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56. 56.

    Find the equation of straight line passing through (2, 3) and perpendicular to the line $3x + 2y + 4 = 0$

    1. A.3y = 5x - 2
    2. B.y = $\frac {5}{3} \times - 2$
    3. C.None of these
    4. D.3y = 2x + 5

    Show answer

    Answer: D. 3y = 2x + 5

    AI explanation

    1. Analyze the Question

    We need to find the equation of a line that passes through a specific point $(2, 3)$ and is perpendicular to another line given by the equation $3x + 2y + 4 = 0$.

    2. Explain the Concept

    • Slope-Intercept Form: Any linear equation can be written as $y = mx + c$, where $m$ is the slope.
    • Perpendicular Lines: If two lines are perpendicular, the product of their slopes is $-1$. That is, $m_1 \times m_2 = -1$. This means the slope of the new line ($m_2$) is the negative reciprocal of the original slope ($m_1$).
    • Point-Slope Formula: Once we have the slope ($m$) and a point $(x_1, y_1)$, we use the formula: $y - y_1 = m(x - x_1)$.

    3. Show the Solution

    Step 1: Find the slope of the given line. The given equation is $3x + 2y + 4 = 0$. Let's rearrange it into $y = mx + c$ form: $2y = -3x - 4$ $y = -\frac{3}{2}x - 2$ The slope ($m_1$) of the given line is $-\frac{3}{2}$.

    Step 2: Find the slope of the perpendicular line ($m_2$). Since the lines are perpendicular: $m_1 \times m_2 = -1$ $(-\frac{3}{2}) \times m_2 = -1$ $m_2 = \frac{-1}{-3/2} = \frac{2}{3}$

    Step 3: Use the point-slope formula with point (2, 3). $y - y_1 = m(x - x_1)$ $y - 3 = \frac{2}{3}(x - 2)$

    Multiply both sides by 3 to clear the fraction: $3(y - 3) = 2(x - 2)$ $3y - 9 = 2x - 4$

    Rearrange to match the format of the options: $3y = 2x - 4 + 9$ $3y = 2x + 5$

    4. Identify the Answer

    Comparing our result ($3y = 2x + 5$) with the given options, it matches option D.

    *

    5. Exam Tip

    When dealing with perpendicular lines, a quick trick to find the new equation is to swap the coefficients of $x$ and $y$ and change the sign between them. For $3x + 2y + 4 = 0$, swapping the coefficients gives $2x - 3y + k = 0$. Plug in the point $(2, 3)$ to find $k$: $2(2) - 3(3) + k = 0 \Rightarrow 4 - 9 + k = 0 \Rightarrow k = 5$. Thus, $2x - 3y + 5 = 0$, which rearranges to $3y = 2x + 5$.

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57. 57.

    Two dice are tossed. What is the probability that the total score is a prime number.

    1. A.$\frac{5}{12}$
    2. B.$\frac{5}{9}$
    3. C.$\frac{1}{6}$
    4. D.$\frac{1}{3}$

    Show answer

    Answer: A. $\frac{5}{12}$

    AI explanation

    1. Analyze the question

    We are tossing two fair dice. We need to find the probability that the sum of the numbers showing on the two dice is a prime number.

    2. Explain the concept

    • Sample Space: When two dice are tossed, each die has 6 faces. The total number of possible outcomes is $6 \times 6 = 36$.
    • Prime Numbers: A prime number is a number greater than 1 that has only two factors: 1 and itself. The possible sums when two dice are tossed range from 2 ($1+1$) to 12 ($6+6$). The prime numbers in this range are 2, 3, 5, 7, and 11.
    • Probability Formula: $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

    3. Show the solution

    Let's list the combinations that result in each prime sum:

    • Sum = 2: (1, 1) — 1 outcome
    • Sum = 3: (1, 2), (2, 1) — 2 outcomes
    • Sum = 5: (1, 4), (4, 1), (2, 3), (3, 2) — 4 outcomes
    • Sum = 7: (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3) — 6 outcomes
    • Sum = 11: (5, 6), (6, 5) — 2 outcomes

    Now, add the number of favorable outcomes: $1 + 2 + 4 + 6 + 2 = 15$

    Calculate the probability: $P(\text{Prime}) = \frac{15}{36}$

    Simplify the fraction by dividing both the numerator and the denominator by 3: $\frac{15 \div 3}{36 \div 3} = \frac{5}{12}$

    4. Identify the answer

    The calculated probability is $\frac{5}{12}$, which matches option A.

    5. Tip

    When dealing with two dice, remember the "sum distribution" pattern: the number of ways to get a sum increases as you move toward 7 (the most frequent sum) and decreases as you move toward 2 or 12. This can help you quickly verify if your count is reasonable!

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58. 58.

    If $\frac {3 - \sqrt 3}{2 + \sqrt 3} = a + b\sqrt 3$, what are the values a and b?

    1. A.a = 9, b = -5
    2. B.a = 5, b = 9
    3. C.a = 9, b = 5
    4. D.a = -5, b = 9

    Show answer

    Answer: A. a = 9, b = -5

    AI explanation

    1. Analyze the Question

    We are given an expression involving surds: $\frac{3 - \sqrt{3}}{2 + \sqrt{3}}$. We need to simplify this expression into the form $a + b\sqrt{3}$ and identify the values of $a$ and $b$.

    2. Explain the Concept

    To simplify a fraction with a surd in the denominator, we use a process called rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

    • The conjugate of $(2 + \sqrt{3})$ is $(2 - \sqrt{3})$.
    • Multiplying by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ is essentially multiplying by 1, so the value of the expression remains unchanged.

    3. Show the Solution

    Step 1: Set up the multiplication

    $\frac{3 - \sqrt{3}}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$

    Step 2: Expand the numerator $(3 - \sqrt{3})(2 - \sqrt{3}) = 3(2) - 3(\sqrt{3}) - \sqrt{3}(2) + (\sqrt{3})(\sqrt{3})$ $= 6 - 3\sqrt{3} - 2\sqrt{3} + 3$ $= (6 + 3) + (-3\sqrt{3} - 2\sqrt{3})$ $= 9 - 5\sqrt{3}$

    Step 3: Expand the denominator $(2 + \sqrt{3})(2 - \sqrt{3})$ This follows the difference of two squares pattern: $(x + y)(x - y) = x^2 - y^2$. $= 2^2 - (\sqrt{3})^2$ $= 4 - 3$ $= 1$

    Step 4: Combine the results

    $\frac{9 - 5\sqrt{3}}{1} = 9 - 5\sqrt{3}$

    Step 5: Compare with $a + b\sqrt{3}$ Comparing $9 - 5\sqrt{3}$ with $a + b\sqrt{3}$: $a = 9$ $b = -5$

    4. Identify the Answer

    The values are $a = 9$ and $b = -5$. This corresponds to option A.

    5. Exam Tip

    When rationalizing, always remember that the denominator will always result in a rational number (no square roots). If you still have a square root in your denominator after multiplying by the conjugate, check your signs—you likely made a mistake in the expansion!

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59. 59.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

60. 60.

    The area of a trapezium is 200 cm$^2$. Its parallel sides are in the ratio 2 : 3 and the perpendicular distance between them is 16 cm. Find the length of each of the parallel sides.

    1. A.10 cm and 15 cm
    2. B.8 cm and 12 cm
    3. C.6 cm and 9 cm
    4. D.12 cm and 18 cm

    Show answer

    Answer: A. 10 cm and 15 cm

    AI explanation

    1. Analyze the Question

    We are given:

    • Area of the trapezium ($A$) = $200 \text{ cm}^2$
    • Ratio of the parallel sides ($a : b$) = $2 : 3$
    • Perpendicular distance (height, $h$) = $16 \text{ cm}$
    • We need to find the actual lengths of the two parallel sides.

    2. Explain the Concept

    The formula for the area of a trapezium is:

    $A = \frac{1}{2}(a + b)h$

    Where:

    • $a$ and $b$ are the lengths of the parallel sides.
    • $h$ is the perpendicular height.

    Since the sides are in a ratio of $2:3$, we can represent them as $2x$ and $3x$, where $x$ is a common multiplier.

    3. Show the Solution

    Step 1: Substitute the known values into the formula.

    $200 = \frac{1}{2}(2x + 3x) \times 16$

    Step 2: Simplify the equation.

    $200 = \frac{1}{2}(5x) \times 16$

    $200 = (5x) \times 8$

    $200 = 40x$

    Step 3: Solve for $x$.

    $x = \frac{200}{40}$

    $x = 5$

    Step 4: Calculate the lengths of the sides.

    • Side $a = 2x = 2 \times 5 = 10 \text{ cm}$
    • Side $b = 3x = 3 \times 5 = 15 \text{ cm}$

    4. Identify the Answer

    The lengths of the parallel sides are $10 \text{ cm}$ and $15 \text{ cm}$. This corresponds to option A.

    5. Exam Tip

    When you are given a ratio in geometry problems, always introduce a constant (like $x$ or $k$). It makes the algebra much easier to manage. Also, always double-check your final answer by plugging it back into the area formula: $\frac{1}{2}(10 + 15) \times 16 = \frac{1}{2}(25) \times 16 = 25 \times 8 = 200$. It matches perfectly!

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61. 61.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

62. 62.

    Find the volume of a cone which has a base radius of 5 cm and slant height of 13 cm.

    1. A.$300\pi$ cm$^3$
    2. B.$325\pi$ cm$^3$
    3. C.$\frac{325}{3}\pi$ cm$^3$
    4. D.$100\pi$ cm$^3$

    Show answer

    Answer: D. $100\pi$ cm$^3$

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63. 63.

    A circle has a radius of 13 cm with a chord 12 cm away from the centre of the circle. Calculate the length of the chord.

    1. A.16 cm
    2. B.8 cm
    3. C.5 cm
    4. D.10 cm

    Show answer

    Answer: D. 10 cm

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64. 64.

    A student pilot was required to fly to an airport and then return as part of his flight training. The average speed to the airport was 120 km/h, and the average speed returning was 150 km/h. If the total flight time was 3 hours, calculate the distance between the two airports.

    1. A.270 km
    2. B.200 km
    3. C.360 km
    4. D.450 km

    Show answer

    Answer: B. 200 km

    AI explanation

    1. Analyze the Question

    We are given:

    • Speed to the airport ($v_1$) = $120\text{ km/h}$
    • Speed returning ($v_2$) = $150\text{ km/h}$
    • Total time ($T$) = $3\text{ hours}$
    • Distance ($d$) is the same for both legs of the trip.

    We need to find the distance $d$.

    2. Explain the Concept

    The fundamental formula for motion is:

    $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

    Since the total time is the sum of the time taken to go ($t_1$) and the time taken to return ($t_2$), we can write:

    $t_1 + t_2 = T$

    $\frac{d}{v_1} + \frac{d}{v_2} = T$

    3. Show the Solution

    Substitute the given values into the equation:

    $\frac{d}{120} + \frac{d}{150} = 3$

    To solve for $d$, find the Least Common Multiple (LCM) of 120 and 150, which is 600. Multiply every term by 600 to clear the fractions:

    $600 \left( \frac{d}{120} \right) + 600 \left( \frac{d}{150} \right) = 600(3)$

    Simplify:

    $5d + 4d = 1800$

    $9d = 1800$

    Divide both sides by 9:

    $d = \frac{1800}{9}$

    $d = 200\text{ km}$

    4. Identify the Answer

    The distance between the two airports is 200 km. This corresponds to option B.

    5. Exam Tip

    When you have a round trip with two different speeds ($v_1$ and $v_2$) and a total time ($T$), you can use the shortcut formula:

    $d = \frac{v_1 \times v_2}{v_1 + v_2} \times T$

    Plugging in our values:

    $d = \frac{120 \times 150}{120 + 150} \times 3 = \frac{18000}{270} \times 3 = \frac{18000}{90} = 200\text{ km}$

    This formula is a great time-saver for JAMB!

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65. 65.

    The locus of a point equidistant from two intersecting lines is

    1. A.where the sum of the distances of two focal points is fixed
    2. B.the collection of points that are equally distant from a fixed point and a line
    3. C.the perpendicular bisector of the lines
    4. D.pair of bisectors of the angles between the two lines

    Show answer

    Answer: D. pair of bisectors of the angles between the two lines

    AI explanation

    Analysis

    The question asks for the geometric definition of a locus (a set of points satisfying a specific condition) that remains at an equal distance from two lines that intersect at a point.

    Concept Explanation

    In geometry, when two lines intersect, they form four angles. If you want to find a point that is equidistant from both lines, you are looking for a path that splits these angles exactly in half.

    1. Angle Bisector Theorem: Any point lying on the bisector of an angle is equidistant from the two arms of that angle.
    2. Intersecting Lines: Because two intersecting lines create two pairs of vertically opposite angles, there are two bisectors: • The internal bisector (which splits the acute/obtuse angles). • The external bisector (which is perpendicular to the internal bisector).
    3. Together, these two bisectors form a pair of perpendicular lines that pass through the point of intersection. Any point on either of these lines will maintain the same perpendicular distance to both original intersecting lines.

    Step-by-Step Solution

    1. Let the two intersecting lines be $L_1$ and $L_2$.
    2. A point $P(x, y)$ is equidistant from $L_1$ and $L_2$ if the perpendicular distance $d_1$ from $P$ to $L_1$ equals the perpendicular distance $d_2$ from $P$ to $L_2$ (i.e., $|d_1| = |d_2|$).
    3. The equation $|d_1| = |d_2|$ mathematically expands into $d_1 = d_2$ or $d_1 = -d_2$.
    4. These two equations represent the two straight lines that bisect the angles formed by $L_1$ and $L_2$.
    5. Therefore, the locus is the pair of angle bisectors.

    Evaluating the Options

    • Option A describes an ellipse.
    • Option B describes a parabola.
    • Option C is incorrect because a perpendicular bisector applies to a line segment, not two intersecting lines.
    • Option D correctly identifies the pair of bisectors of the angles between the two lines.

    Exam Tip

    Whenever you see the word "locus" in JAMB Mathematics, think of it as a "path" or "rule."

    • Equidistant from a point and a line = Parabola.
    • Equidistant from two points = Perpendicular bisector.
    • Equidistant from two intersecting lines = Pair of angle bisectors.

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66. 66.

    Find the matrix A A $\begin {bmatrix} 0 & 1\\2 & -1 \end {bmatrix}$ = $\begin {bmatrix} 2 & -1\\1 & 0 \end {bmatrix}$

    1. A.$\begin {bmatrix} 2 & 1\\-^1/_2 & -^1/_2 \end {bmatrix}$
    2. B.$\begin {bmatrix} 0 & 1\\^1/_2 & ^1/_2 \end {bmatrix}$
    3. C.$\begin {bmatrix} 2 & 1\\0 & -1 \end {bmatrix}$
    4. D.$\begin {bmatrix} 2 & 1\\^1/_2 & -2 \end {bmatrix}$

    Show answer

    Answer: B. $\begin {bmatrix} 0 & 1\\^1/_2 & ^1/_2 \end {bmatrix}$

    AI explanation

    1. Analyze the Question

    We are given the matrix equation $A \times \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. We need to find matrix $A$.

    Let $X = \begin{bmatrix} 0 & 1 \\ 2 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. The equation is $AX = B$. To solve for $A$, we multiply both sides by the inverse of $X$ ($X^{-1}$) from the right: $A = BX^{-1}$

    2. Explain the Concept

    To find the inverse of a $2 \times 2$ matrix $X = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, we use the formula: $X^{-1} = \frac{1}{\det(X)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ where $\det(X) = (ad - bc)$.

    3. Show the Solution

    Step 1: Find the determinant of $X$ $\det(X) = (0 \times -1) - (1 \times 2) = 0 - 2 = -2$

    Step 2: Find the inverse of $X$ ($X^{-1}$) $X^{-1} = \frac{1}{-2} \begin{bmatrix} -1 & -1 \\ -2 & 0 \end{bmatrix} = \begin{bmatrix} 1/2 & 1/2 \\ 1 & 0 \end{bmatrix}$

    Step 3: Calculate $A = BX^{-1}$ $A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1/2 & 1/2 \\ 1 & 0 \end{bmatrix}$

    Perform matrix multiplication (Row $\times$ Column):

    • Top-left: $(2 \times 1/2) + (-1 \times 1) = 1 - 1 = 0$
    • Top-right: $(2 \times 1/2) + (-1 \times 0) = 1 + 0 = 1$
    • Bottom-left: $(1 \times 1/2) + (0 \times 1) = 1/2 + 0 = 1/2$
    • Bottom-right: $(1 \times 1/2) + (0 \times 0) = 1/2 + 0 = 1/2$

    So, $A = \begin{bmatrix} 0 & 1 \\ 1/2 & 1/2 \end{bmatrix}$

    4. Identify the Answer

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

    5. Tip

    When solving $AX = B$, always remember that matrix multiplication is not commutative. You must multiply by the inverse on the side that "cancels out" the matrix you are trying to isolate. If $X$ is on the right of $A$, multiply by $X^{-1}$ on the right of $B$.

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67. 67.

    Find the value of y if $402_y = 102_{ten}$

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

    Show answer

    Answer: C. 5

    AI explanation

    Analysis

    The question asks us to find the base $y$ such that the number $402$ in base $y$ is equal to $102$ in base ten. This is a problem involving Number Bases Conversion.

    Concept

    To convert a number from any base $y$ to base ten, we expand the digits by multiplying each digit by the base raised to the power of its position (starting from 0 on the right).

    For a number $abc_y$, the value in base ten is: $(a \times y^2) + (b \times y^1) + (c \times y^0)$

    Solution

    Given: $402_y = 102_{10}$

    1. Expand the left side ($402_y$): $(4 \times y^2) + (0 \times y^1) + (2 \times y^0) = 102$

    Since $y^0 = 1$: $4y^2 + 0 + 2 = 102$

    2. Simplify the equation: $4y^2 + 2 = 102$

    3. Isolate the term with $y$: Subtract 2 from both sides: $4y^2 = 102 - 2$ $4y^2 = 100$

    4. Solve for $y^2$: Divide both sides by 4: $y^2 = \frac{100}{4}$ $y^2 = 25$

    5. Find $y$: Take the square root of both sides: $y = \sqrt{25}$ $y = 5$

    Conclusion

    The value of $y$ is 5.

    Exam Tip

    When dealing with number bases, always remember that the base must be greater than any digit present in the number. Since the number $402$ contains the digit '4', the base $y$ must be greater than 4. This immediately allows you to eliminate options A (4), B (2), and D (3) if you were guessing, leaving 5 as the only logical choice!

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68. 68.

    Give the number of significant figures of the population of a town which has approximately 5,020,700 people

    1. A.7 significant figures
    2. B.3 significant figures
    3. C.4 significant figures
    4. D.5 significant figures

    Show answer

    Answer: D. 5 significant figures

    AI explanation

    Analysis of the Question

    The question asks for the number of significant figures in the value 5,020,700. To solve this, we must apply the standard rules for counting significant figures in a number containing zeros.

    Concept: Rules for Significant Figures

    1. Non-zero digits are always significant.
    2. Zeros between non-zero digits are always significant (e.g., the '0' in 502 is significant).
    3. Leading zeros (zeros at the beginning of a number) are never significant.
    4. Trailing zeros in a whole number without a decimal point are generally considered not significant unless specified otherwise (they are placeholders).

    Step-by-Step Solution

    Let's look at the number: 5,020,700

    • 5: Significant (non-zero)
    • 0: Significant (it is trapped between 5 and 2)
    • 2: Significant (non-zero)
    • 0: Significant (it is trapped between 2 and 7)
    • 7: Significant (non-zero)
    • 00: Not significant (these are trailing zeros in a whole number, serving as placeholders for the hundreds, tens, and units positions).

    Counting the significant digits: 5, 0, 2, 0, 7. There are exactly 5 significant figures.

    Conclusion

    The digits 5, 0, 2, 0, and 7 are significant, while the final two zeros are placeholders. Therefore, the number has 5 significant figures.

    Exam Tip

    When you see a large number ending in zeros, ignore the trailing zeros if there is no decimal point. Always focus on the "trapped" zeros—they are the ones that usually catch students off guard in JAMB exams!

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69. 69.

    The angle of elevation and depression of the top and bottom of another building, measured from the top of a 24 m tall building, is 30° and 60°, respectively. Determine the second building's height.

    1. A.24 m
    2. B.32$\sqrt3$ m
    3. C.24$\sqrt3$
    4. D.32 m

    Show answer

    Answer: D. 32 m

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70. 70.

    Solve the logarithmic equation: $log_2 (6 - x) = 3 - log_2 x$

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

    Show answer

    Answer: A. $x$ = 4 or 2

    AI explanation

    Analysis

    To solve the equation $\log_2 (6 - x) = 3 - \log_2 x$, we need to use the properties of logarithms to combine the terms and convert the logarithmic equation into an algebraic one.

    Concept

    1. Logarithm Addition Rule: $\log_b(A) + \log_b(B) = \log_b(A \times B)$.
    2. Definition of Logarithm: If $\log_b(x) = y$, then $x = b^y$.
    3. Domain Constraint: The argument of a logarithm must always be positive (i.e., $6 - x > 0$ and $x > 0$).

    Step-by-Step Solution

    Step 1: Rearrange the equation Move the $\log_2 x$ term to the left side:

    $\log_2 (6 - x) + \log_2 x = 3$

    Step 2: Apply the addition rule Combine the logarithms on the left:

    $\log_2 [x(6 - x)] = 3$

    $\log_2 (6x - x^2) = 3$

    Step 3: Convert to exponential form Using the definition $\log_b(A) = y \implies A = b^y$:

    $6x - x^2 = 2^3$

    $6x - x^2 = 8$

    Step 4: Solve the quadratic equation Rearrange all terms to one side to set the equation to zero:

    $x^2 - 6x + 8 = 0$

    Factor the quadratic equation: Find two numbers that multiply to $+8$ and add to $-6$. These numbers are $-4$ and $-2$.

    $(x - 4)(x - 2) = 0$

    Setting each factor to zero: $x - 4 = 0 \implies x = 4$ $x - 2 = 0 \implies x = 2$

    Step 5: Verify the solutions Check if these values satisfy the domain constraints ($x > 0$ and $6-x > 0$):

    • If $x = 4$: $6 - 4 = 2$ (positive), $x = 4$ (positive). Valid.
    • If $x = 2$: $6 - 2 = 4$ (positive), $x = 2$ (positive). Valid.

    Conclusion

    The solutions are $x = 4$ or $x = 2$. This matches option A.

    Exam Tip

    Always check your answers in the original logarithmic equation. If you had obtained a negative value for $x$, you would have to reject it, because you cannot take the logarithm of a negative number or zero. In JAMB, they often include "distractor" answers that are mathematically correct for the quadratic but invalid for the logarithm—always double-check your domain!

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71. 71.

    Find the value of t, if the distance between the points P(–3, –14) and Q(t, –5) is 9 units.

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

    Show answer

    Answer: C. - 3

    AI explanation

    Analysis

    To find the value of $t$, we need to use the Distance Formula, which calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ on a Cartesian plane.

    Concept

    The distance $d$ between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is given by the formula:

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

    In this problem:

    • $P = (-3, -14) \implies x_1 = -3, y_1 = -14$
    • $Q = (t, -5) \implies x_2 = t, y_2 = -5$
    • $d = 9$

    Step-by-Step Solution

    1. Substitute the values into the formula: $$9 = \sqrt{(t - (-3))^2 + (-5 - (-14))^2}$$ $$9 = \sqrt{(t + 3)^2 + (-5 + 14)^2}$$

    2. Simplify the expression: $$9 = \sqrt{(t + 3)^2 + (9)^2}$$ $$9 = \sqrt{(t + 3)^2 + 81}$$

    3. Square both sides to remove the square root: $$9^2 = (t + 3)^2 + 81$$ $$81 = (t + 3)^2 + 81$$

    4. Solve for $t$: Subtract 81 from both sides: $$81 - 81 = (t + 3)^2$$ $$0 = (t + 3)^2$$

    Take the square root of both sides:

    $0 = t + 3$

    $t = -3$

    Conclusion

    The value of $t$ is $-3$. Comparing this to the given options, it matches option C.

    Exam Tip

    Whenever you see a distance problem involving a square root, always square both sides immediately to eliminate the root. Also, remember that $(a - (-b))$ becomes $(a + b)$; watch your signs carefully, as a simple sign error is the most common reason students lose marks in coordinate geometry!

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72. 72.

    Find the area and perimeter of a square whose length of diagonals is 20$\sqrt2$ cm.

    1. A.800 cm$^2$, 80 cm
    2. B.400 cm, 80 cm$^2$
    3. C.80 cm, 800 cm$^2$
    4. D.400 cm$^2$, 80 cm

    Show answer

    Answer: D. 400 cm$^2$, 80 cm

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73. 73.

    A ship sets sail from port A (86$^o$N, 56$^o$W) for port B (86$^o$N, 64$^o$W), which is close by. Find the distance the ship covered from port A to port B, correct to the nearest km. [Take $\pi$ = 3.142 and R = 6370 km]

    1. A.62 km
    2. B.97 km
    3. C.389 km
    4. D.931 km

    Show answer

    Answer: A. 62 km

    AI explanation

    Analysis

    To find the distance between two points on the same parallel of latitude, we use the formula for the arc length of a small circle. Since both ports are at the same latitude ($86^\circ$N), the ship is traveling along a circle of latitude.

    Concept

    The distance ($d$) between two points on the same latitude is given by:

    $d = \frac{\theta}{360^\circ} \times 2\pi R \cos(\alpha)$

    Where:

    • $\theta$ = The difference in longitude between the two points.
    • $R$ = Radius of the Earth (6370 km).
    • $\alpha$ = The latitude of the points ($86^\circ$).
    • $\pi$ = 3.142.

    Step-by-Step Solution

    Step 1: Find the difference in longitude ($\theta$) The ports are at $56^\circ$W and $64^\circ$W. Since they are both in the same hemisphere (West), we subtract them:

    $\theta = 64^\circ - 56^\circ = 8^\circ$

    Step 2: Apply the formula Substitute the values into the distance formula:

    $d = \frac{8}{360} \times 2 \times 3.142 \times 6370 \times \cos(86^\circ)$

    Step 3: Calculate the values

    • $\cos(86^\circ) \approx 0.069756$
    • $d = \frac{8}{360} \times 2 \times 3.142 \times 6370 \times 0.069756$
    • $d = 0.02222 \times 6.284 \times 6370 \times 0.069756$
    • $d \approx 0.1396 \times 6370 \times 0.069756$
    • $d \approx 889.25 \times 0.069756$
    • $d \approx 62.02 \text{ km}$

    Rounding to the nearest whole number, we get 62 km.

    Conclusion

    The distance covered by the ship is approximately 62 km. This matches option A.

    Exam Tip

    When calculating distances on the Earth's surface, always check if the points are on the same latitude or the same longitude. If they are on the same latitude, remember to include the $\cos(\text{latitude})$ factor in your calculation, as the radius of the circle of latitude is smaller than the Earth's radius ($R \cos \alpha$).

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74. 74.

    A boat sails 8 km north from P to Q and then sails 6 km west from Q to R. Calculate the bearing of R from P. Give your answer to the nearest degree.

    1. A.217$^o$
    2. B.323$^o$
    3. C.037$^o$
    4. D.053$^o$

    Show answer

    Answer: B. 323$^o$

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75. 75.

    The second term of a geometric series is $^{-2}/_3$ and its sum to infinity is $^3/_2$. Find its common ratio.

    1. A.$^{-1}/_3$
    2. B.2
    3. C.$^{4}/_3$
    4. D.$^{2}/_9$

    Show answer

    Answer: A. $^{-1}/_3$

    AI explanation

    Analysis

    To solve this problem, we need to use the standard formulas for a geometric series:

    1. The $n^{th}$ term: $T_n = ar^{n-1}$
    2. The sum to infinity: $S_\infty = \frac{a}{1 - r}$

    Where:

    • $a$ is the first term
    • $r$ is the common ratio

    Concept

    We are given:

    • The second term ($T_2$) = $ar = -\frac{2}{3}$
    • The sum to infinity ($S_\infty$) = $\frac{a}{1 - r} = \frac{3}{2}$

    Step-by-Step Solution

    Step 1: Express $a$ in terms of $r$ from the second term equation. From $ar = -\frac{2}{3}$, we get: $a = \frac{-2}{3r}$

    Step 2: Substitute $a$ into the sum to infinity formula. $\frac{\frac{-2}{3r}}{1 - r} = \frac{3}{2}$

    Simplify the left side: $\frac{-2}{3r(1 - r)} = \frac{3}{2}$

    Step 3: Cross-multiply to solve for $r$. $3 \times [3r(1 - r)] = -2 \times 2$ $9r(1 - r) = -4$ $9r - 9r^2 = -4$

    Rearrange into a standard quadratic equation ($ax^2 + bx + c = 0$): $9r^2 - 9r - 4 = 0$

    Step 4: Solve the quadratic equation. Using the factorization method, we look for two numbers that multiply to $(9 \times -4) = -36$ and add to $-9$. These numbers are $-12$ and $3$. $9r^2 - 12r + 3r - 4 = 0$ $3r(3r - 4) + 1(3r - 4) = 0$ $(3r + 1)(3r - 4) = 0$

    This gives us two possible values for $r$:

    1. $3r + 1 = 0 \implies r = -\frac{1}{3}$
    2. $3r - 4 = 0 \implies r = \frac{4}{3}$

    Step 5: Evaluate the validity of the roots. For a geometric series to have a sum to infinity, the common ratio must satisfy $|r| < 1$.

    • If $r = -\frac{1}{3}$, $|-\frac{1}{3}| < 1$ (Valid)
    • If $r = \frac{4}{3}$, $|\frac{4}{3}| > 1$ (Invalid, as the series would diverge)

    Therefore, the common ratio must be $-\frac{1}{3}$.

    Conclusion

    Comparing our result to the given options, the correct choice is A.

    Exam Tip

    Always remember the condition for the sum to infinity: $|r| < 1$. If you calculate a common ratio greater than 1 or less than -1, it cannot be the sum to infinity for that series. This quick check can help you eliminate wrong options immediately!

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76. 76.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

77. 77.

    Find the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually.

    1. A.₦6,212.48
    2. B.₦2,834.48
    3. C.₦18,312.48
    4. D.₦2,612.48

    Show answer

    Answer: D. ₦2,612.48

    AI explanation

    Analysis

    To find the compound interest, we first calculate the Total Amount ($A$) after 2 years using the compound interest formula, and then subtract the Principal ($P$) from that amount.

    Concept

    The formula for the total amount ($A$) under compound interest is:

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

    Where:

    • $P$ = Principal (₦15,700)
    • $r$ = Rate (8%)
    • $n$ = Time in years (2)
    • Compound Interest ($CI$) = $A - P$

    Step-by-Step Solution

    Step 1: Identify the values

    • $P = 15,700$
    • $r = 8$
    • $n = 2$

    Step 2: Calculate the Amount ($A$)

    $A = 15,700 \left(1 + \frac{8}{100}\right)^2$

    $A = 15,700 (1 + 0.08)^2$

    $A = 15,700 (1.08)^2$

    $A = 15,700 \times 1.1664$

    $A = 18,312.48$

    Step 3: Calculate the Compound Interest ($CI$)

    $CI = A - P$

    $CI = 18,312.48 - 15,700$

    $CI = 2,612.48$

    Conclusion

    The compound interest on ₦15,700 for 2 years at 8% per annum is ₦2,612.48. This matches option D.

    Exam Tip

    When calculating compound interest for only 2 years, you can also use the "Simple Interest" method twice:

    1. Find interest for Year 1: $(8\% \text{ of } 15,700) = 1,256$.
    2. Add that to the principal to get the new principal for Year 2: $15,700 + 1,256 = 16,956$.
    3. Find interest for Year 2: $(8\% \text{ of } 16,956) = 1,356.48$.
    4. Add the two interests together: $1,256 + 1,356.48 = 2,612.48$. This is a great way to double-check your work during the exam!

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78. 78.

    Factorize: $16x^4 - y^4$

    1. A.$(2x - y)(2x + y)(4x^2 + y^2)$
    2. B.$(2x + y)(2x + y)(4x^2 + y^2)$
    3. C.$(2x - y)(2x - y)(4x^2 + y^2)$
    4. D.$(2x - y)(2x + y)(4x^2 - y^2)$

    Show answer

    Answer: A. $(2x - y)(2x + y)(4x^2 + y^2)$

    AI explanation

    Analysis

    The expression $16x^4 - y^4$ is a difference of two squares. The general formula for the difference of two squares is:

    $a^2 - b^2 = (a - b)(a + b)$

    Concept

    To factorize this expression, we need to express both terms as perfect squares:

    1. $16x^4$ can be written as $(4x^2)^2$
    2. $y^4$ can be written as $(y^2)^2$

    Step-by-Step Solution

    Step 1: Apply the difference of two squares formula $16x^4 - y^4 = (4x^2)^2 - (y^2)^2$ Using $a^2 - b^2 = (a - b)(a + b)$, where $a = 4x^2$ and $b = y^2$: $= (4x^2 - y^2)(4x^2 + y^2)$

    Step 2: Factorize the remaining difference of squares Notice that $(4x^2 - y^2)$ can be further factorized because it is also a difference of squares: $4x^2 - y^2 = (2x)^2 - (y)^2$ Applying the formula again: $(2x)^2 - (y)^2 = (2x - y)(2x + y)$

    Step 3: Combine all factors Substitute the result from Step 2 back into the expression from Step 1: $= (2x - y)(2x + y)(4x^2 + y^2)$

    Conclusion

    Comparing our result $(2x - y)(2x + y)(4x^2 + y^2)$ with the given options, it matches Option A.

    Exam Tip: Whenever you see a power of 4 (like $x^4$), always check if you can apply the difference of squares rule twice. Also, remember that the sum of squares $(a^2 + b^2)$ cannot be factorized further using real numbers, so leave the $(4x^2 + y^2)$ part as it is!

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79. 79.

    If $-2x^3 + 6x^2 + 17x$ - 21 is divided by $(x + 1)$, then the remainder is

    1. A.32
    2. B.30
    3. C.-30
    4. D.-32

    Show answer

    Answer: C. -30

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80. 80.

    Find the area, to the nearest cm$^2$, of the triangle whose sides are in the ratio 2 : 3 : 4 and whose perimeter is 180 cm.

    1. A.1162 cm$^2$
    2. B.1163 cm$^2$
    3. C.1160 cm$^2$
    4. D.1161 cm$^2$

    Show answer

    Answer: A. 1162 cm$^2$

    AI explanation

    1. Analyze the question

    We are given the ratio of the sides of a triangle (2:3:4) and its perimeter (180 cm). We need to find the area of this triangle using Heron's formula, rounding the result to the nearest whole number.

    2. Explain the concept

    • Finding the sides: When sides are in a ratio, we represent them as $2x$, $3x$, and $4x$. The perimeter is the sum of these sides.
    • Heron's Formula: To find the area of a triangle when all three sides ($a, b, c$) are known, we use: $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$where$s$is the semi-perimeter:$s = \frac{a+b+c}{2}$.

    3. Show the solution

    Step 1: Find the actual lengths of the sides. The perimeter is the sum of the sides: $2x + 3x + 4x = 180$ $9x = 180$ $x = \frac{180}{9} = 20$

    So, the sides are: $a = 2(20) = 40 \text{ cm}$ $b = 3(20) = 60 \text{ cm}$ $c = 4(20) = 80 \text{ cm}$

    Step 2: Calculate the semi-perimeter ($s$). $s = \frac{\text{Perimeter}}{2} = \frac{180}{2} = 90 \text{ cm}$

    Step 3: Apply Heron's Formula. $\text{Area} = \sqrt{90(90-40)(90-60)(90-80)}$ $\text{Area} = \sqrt{90 \times 50 \times 30 \times 10}$ $\text{Area} = \sqrt{1,350,000}$

    Step 4: Calculate the square root. $\sqrt{1,350,000} \approx 1161.895$

    Rounding to the nearest cm$^2$, we get 1162 cm$^2$.

    4. Identify the answer

    Comparing our result to the options provided, 1162 cm$^2$ corresponds to option A.

    5. Tip

    When dealing with ratios in geometry, always find the value of the "part" ($x$) first. Also, for Heron's formula, if the numbers are large, look for factors that are perfect squares to simplify the square root calculation without needing a calculator!

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