2018 JAMB Mathematics Past Questions

Questions

1. 1.

   Two sisters, Taiwo and Kehinde, own a store. The ratio of Taiwo's share to Kehinde's is 11:9. Later Kehinde sells $\frac{2}{3}$ of her share to Taiwo for N720.00. Find the value of the store

   1. A.N1,080.00
   2. B.N2,400.00
   3. C.N3,000.00
   4. D.N3,600.00

   Show answer

   Answer: B. N2,400.00

   AI explanation

   1. Analyze the question

   We are given the initial ratio of shares between Taiwo and Kehinde. We are told that Kehinde sells a specific fraction of her share to Taiwo, and the monetary value of that transaction is N720.00. We need to find the total value of the store.

   2. Explain the concept

   • Ratio to Fraction: If the ratio of Taiwo's share to Kehinde's share is $11:9$, then the total parts in the store are $11 + 9 = 20$ parts.
   • Fractional Value: Taiwo owns $\frac{11}{20}$ of the store, and Kehinde owns $\frac{9}{20}$ of the store.
   • Transaction: Kehinde sells $\frac{2}{3}$ of her share. This means the amount of money she received (N720) represents exactly $\frac{2}{3}$ of her total share of the store.

   3. Show the solution

   Step 1: Find the total value of Kehinde's share. Let $K$ be the total value of Kehinde's share. We know that $\frac{2}{3}$ of $K = N720$.

   $\frac{2}{3} \times K = 720$

   $2K = 720 \times 3$

   $2K = 2160$

   $K = \frac{2160}{2} = 1080$

   So, Kehinde's total share is worth N1,080.00.

   Step 2: Relate Kehinde's share to the total store value. From the ratio $11:9$, Kehinde's share is $\frac{9}{20}$ of the total store value ($V$).

   $\frac{9}{20} \times V = 1080$

   $9V = 1080 \times 20$

   $9V = 21600$

   $V = \frac{21600}{9}$

   $V = 2400$

   The total value of the store is N2,400.00.

   4. Identify the answer

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

   5. Tip

   When dealing with ratio problems involving money, always find the value of "one part" or the value of the specific person's share first. Once you have the monetary value of one person's share, you can easily scale it up to the total value of the store using the ratio fractions.

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2. 2.

   A trader realises 10x - x$^2$ Naira profit from the sale of x bags of corn. How many bags will give him the maximum profit?

   1. A.7
   2. B.6
   3. C.5
   4. D.4

   Show answer

   Answer: C. 5

   AI explanation

   1. Analyze the question

   The profit function is given by $P(x) = 10x - x^2$. We need to find the value of $x$ (the number of bags) that results in the maximum profit. This is a classic optimization problem involving quadratic functions.

   2. Explain the concept

   In calculus, to find the maximum or minimum value of a function, we use the first derivative.

   • If $P(x)$ is the profit function, the rate of change of profit with respect to the number of bags is $P'(x)$.
   • At the maximum point, the slope of the curve is zero. Therefore, we set the first derivative to zero ($P'(x) = 0$) and solve for $x$.
   • Alternatively, for a quadratic equation in the form $ax^2 + bx + c$, the maximum or minimum occurs at $x = -b / 2a$.

   3. Show the solution

   Method 1: Using Calculus (Differentiation) Given: $P(x) = 10x - x^2$

   1. Find the derivative of $P(x)$ with respect to $x$: $P'(x) = \frac{d}{dx}(10x - x^2)$ $P'(x) = 10 - 2x$

   2. Set the derivative to zero to find the stationary point: $10 - 2x = 0$ $10 = 2x$ $x = \frac{10}{2}$ $x = 5$

   Method 2: Using the Vertex Formula For a quadratic equation $ax^2 + bx + c$: Here, $a = -1$ and $b = 10$. The maximum occurs at $x = \frac{-b}{2a}$ $x = \frac{-10}{2(-1)}$ $x = \frac{-10}{-2}$ $x = 5$

   4. Identify the answer

   Both methods confirm that the trader will make the maximum profit when he sells 5 bags of corn. This corresponds to option C.

   5. Tip

   Whenever you see a quadratic expression (where the highest power is 2) and you are asked to find the "maximum" or "minimum" value, you can always use the formula $x = -b/2a$ to save time during your JAMB exam. It is much faster than performing full differentiation!

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3. 3.

   Tanθ is positive and Sinθ is negative. In which quadrant does θ lies

   1. A.Second only
   2. B.Third only
   3. C.Fourth only
   4. D.First and third only

   Show answer

   Answer: B. Third only

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4. 4.

   A room is 12m long, 9m wide and 8m high. Find the cosine of the angle which a diagonal of the room makes with the floor of the room.

   1. A.$\frac{15}{17}$
   2. B.$\frac{9}{17}$
   3. C.$\frac{8}{15}$
   4. D.$\frac{12}{17}$

   Show answer

   Answer: A. $\frac{15}{17}$

   AI explanation

   1. Analyze the question

   We are given a rectangular room with dimensions:

   • Length ($L$) = 12m
   • Width ($W$) = 9m
   • Height ($H$) = 8m

   We need to find the cosine of the angle ($\theta$) between the space diagonal of the room and the floor.

   2. Explain the concept

   • The Floor Diagonal: The floor is a rectangle with length 12m and width 9m. The diagonal of the floor ($d$) forms a right-angled triangle with the length and width. Using Pythagoras' theorem: $d = \sqrt{L^2 + W^2}$.
   • The Space Diagonal: The diagonal of the room connects a bottom corner to the opposite top corner. This forms a right-angled triangle where the base is the floor diagonal ($d$), the height is the room height ($H$), and the hypotenuse is the space diagonal ($D$).
   • The Angle ($\theta$): The angle $\theta$ is between the floor diagonal ($d$) and the space diagonal ($D$). In this right-angled triangle, $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{d}{D}$.

   3. Show the solution

   Step 1: Calculate the floor diagonal ($d$)

   $d = \sqrt{12^2 + 9^2}$

   $d = \sqrt{144 + 81}$

   $d = \sqrt{225} = 15\text{m}$

   Step 2: Calculate the space diagonal ($D$) The space diagonal $D$ is the hypotenuse of the triangle with base $d$ (15m) and height $H$ (8m):

   $D = \sqrt{d^2 + H^2}$

   $D = \sqrt{15^2 + 8^2}$

   $D = \sqrt{225 + 64}$

   $D = \sqrt{289} = 17\text{m}$

   Step 3: Calculate $\cos(\theta)$

   $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{d}{D}$

   $\cos(\theta) = \frac{15}{17}$

   4. Identify the answer

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

   5. Tip

   When dealing with 3D geometry problems in JAMB, always visualize the two-step Pythagoras process: first, find the diagonal of the base, then use that result to find the space diagonal. Also, keep an eye out for common Pythagorean triples like (8, 15, 17) to save time during calculations!

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5. 5.

   A man stands on a tree 150cm high and sees a boat at an angle of depression of 74°. Find the distance of the boat from the base of the tree.

   1. A.52cm
   2. B.43cm
   3. C.40cm
   4. D.15cm

   Show answer

   Answer: B. 43cm

   AI explanation

   1. Analyze the question

   We are given the height of a tree ($h = 150\text{ cm}$) and the angle of depression from the top of the tree to a boat on the ground ($\theta = 74^\circ$). We need to find the horizontal distance ($d$) from the base of the tree to the boat.

   2. Explain the concept

   In trigonometry, the angle of depression from an observer to an object is equal to the angle of elevation from the object to the observer (due to alternate angles).

   We can visualize this as a right-angled triangle where:

   • The height of the tree is the opposite side to the angle of elevation.
   • The distance from the base of the tree is the adjacent side.
   • We use the tangent ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

   3. Show the solution

   • Given: • Height (Opposite) = $150\text{ cm}$ • Angle ($\theta$) = $74^\circ$ • Distance (Adjacent) = $d$

   • Formula: $$\tan(74^\circ) = \frac{150}{d}$$

   • Rearrange to solve for $d$: $$d = \frac{150}{\tan(74^\circ)}$$

   • Calculation: Using a calculator, $\tan(74^\circ) \approx 3.4874$ $$d = \frac{150}{3.4874}$$ $$d \approx 43.01\text{ cm}$$

   Rounding to the nearest whole number, we get 43 cm.

   4. Identify the answer

   Comparing our result to the given options: A. 52cm B. 43cm C. 40cm D. 15cm

   The calculated value matches option B.

   5. Exam Tip

   When dealing with angles of depression, always remember that the angle is measured from the horizontal line of sight downwards. If you draw a diagram, the angle of depression is outside the triangle at the top, but it is equal to the angle inside the triangle at the base of the boat. This simple "Z-angle" rule (alternate angles) will save you from making mistakes in your setup!

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6. 6.

   If the simple interest on a sum of money invested at 3% per annum for 2$\frac{1}{2}$  years is N123, find the principal.

   1. A.N676.50
   2. B.N820
   3. C.N1,640
   4. D.N4,920

   Show answer

   Answer: C. N1,640

   AI explanation

   Analysis

   To solve this problem, we need to use the standard formula for Simple Interest. We are given the interest rate, the time, and the total interest earned, and we are asked to find the principal amount that was invested.

   Concept

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

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

   Where:

   • $I$ = Simple Interest (N123)
   • $P$ = Principal (the amount we need to find)
   • $R$ = Rate per annum (3%)
   • $T$ = Time in years ($2\frac{1}{2}$ years, which is 2.5 years)

   Step-by-Step Solution

   Step 1: Rearrange the formula to make $P$ the subject. Since $I = \frac{P \times R \times T}{100}$, we can multiply both sides by 100 and divide by $(R \times T)$:

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

   Step 2: Substitute the given values into the formula.

   • $I = 123$
   • $R = 3$
   • $T = 2.5$

   $P = \frac{123 \times 100}{3 \times 2.5}$

   Step 3: Perform the calculation.

   • First, calculate the denominator: $3 \times 2.5 = 7.5$
   • Now, divide the numerator by the denominator:

   $P = \frac{12300}{7.5}$

   To make the division easier, multiply both numerator and denominator by 10:

   $P = \frac{123000}{75}$

   Dividing 123,000 by 75: $123000 \div 75 = 1640$

   The Principal is N1,640.

   Conclusion

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

   Exam Tip

   When dealing with mixed fractions like $2\frac{1}{2}$ in interest calculations, always convert them to decimals (2.5) or improper fractions ($\frac{5}{2}$) before plugging them into your formula. This prevents errors during multiplication. Also, remember that "per annum" simply means "per year," so ensure your time ($T$) is always in years!

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7. 7.

   Evaluate log$_2$ 8 – log$_3$ $\frac{1}{9}$

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

   Show answer

   Answer: D. 5

   AI explanation

   To solve this expression, we need to apply the fundamental laws of logarithms. Let's break it down into two parts.

   1. Analyze the question

   We are asked to evaluate: $\log_2 8 - \log_3 \frac{1}{9}$

   2. Explain the concepts

   • Logarithm Definition: $\log_b a = x$ means $b^x = a$.
   • Power Rule: $\log_b (x^n) = n \cdot \log_b x$.
   • Negative Exponent Rule: $\frac{1}{x^n} = x^{-n}$.

   3. Show the solution

   Step 1: Evaluate $\log_2 8$ We know that $8$ can be written as a power of $2$: $8 = 2 \times 2 \times 2 = 2^3$ So, $\log_2 8 = \log_2 (2^3)$ Using the power rule, this becomes: $3 \cdot \log_2 2$ Since $\log_b b = 1$, then $3 \cdot 1 = \mathbf{3}$.

   Step 2: Evaluate $\log_3 \frac{1}{9}$ We know that $9 = 3^2$. Therefore, $\frac{1}{9} = \frac{1}{3^2}$. Using the negative exponent rule, $\frac{1}{3^2} = 3^{-2}$. So, $\log_3 \frac{1}{9} = \log_3 (3^{-2})$ Using the power rule, this becomes: $-2 \cdot \log_3 3$ Since $\log_3 3 = 1$, then $-2 \cdot 1 = \mathbf{-2}$.

   Step 3: Combine the results Now, substitute these values back into the original expression: $\log_2 8 - \log_3 \frac{1}{9} = 3 - (-2)$ $3 + 2 = 5$

   4. Identify the answer

   Comparing our result to the given options: A. -1 1/2 B. -1 C. 1 D. 5

   The correct option is D.

   5. Exam Tip

   When you see fractions in logarithms (like $1/9$), immediately think of negative exponents. Whenever you see $\log_b (\frac{1}{x})$, it is almost always going to result in a negative value. Also, always remember that $\log_b b = 1$; it is the most useful shortcut in JAMB mathematics!

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

9. 9.

   What is the loci of a distance 4cm from a given point P?

   1. A.A straight line of length 4cm
   2. B.a circle of radius 4cm
   3. C.perpendicular to point P at 4cm
   4. D.a circle of diameter 4cm

   Show answer

   Answer: B. a circle of radius 4cm

   AI explanation

   Analysis

   The question asks for the "locus" (the path traced by a point) that maintains a constant distance of 4cm from a fixed point $P$.

   Concept Explanation

   In geometry, a locus is the set of all points that satisfy a specific condition.

   • When you have a fixed point (let's call it the center) and you look for all points that are a fixed distance (the radius) away from it, you are describing a circle.
   • Imagine holding one end of a 4cm string at point $P$ and moving the other end around in every possible direction while keeping the string taut. The path traced by the tip of the string will be a circle with a radius of 4cm.

   Step-by-Step Solution

   1. Identify the fixed point: Point $P$ acts as the center of the shape.
   2. Identify the fixed distance: The distance of 4cm acts as the radius ($r = 4\text{cm}$).
   3. Apply the definition: By definition, a circle is the locus of points in a plane that are at a constant distance (radius) from a fixed point (center).
   4. Evaluate the options: • A straight line would not maintain a constant distance from a single point in all directions. • A circle of radius 4cm perfectly matches our definition. • "Perpendicular" describes an angle, not a locus of points. • A circle of diameter 4cm would have a radius of only 2cm, which is incorrect.

   Conclusion

   The locus of points at a constant distance from a fixed point is a circle where the fixed point is the center and the constant distance is the radius. Therefore, the locus is a circle of radius 4cm.

   Exam Tip: Whenever you see the word "locus" in a JAMB question, think of it as a "rule" or "path." If the rule involves a fixed distance from a point, it is always a circle. If it involves a fixed distance from a line, it is a pair of parallel lines.

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10. 10.

    In how many ways can the letters LEADER be arranged?

    1. A.72
    2. B.144
    3. C.360
    4. D.720

    Show answer

    Answer: C. 360

    AI explanation

    Analysis

    The question asks for the number of ways to arrange the letters in the word LEADER. This is a problem of permutations of objects where some are identical.

    Concept

    When arranging $n$ objects where some objects are repeated, we use the formula:

    $\text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \dots \times n_k!}$

    Where:

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

    Step-by-Step Solution

    1. Count the total number of letters ($n$): The word LEADER has 6 letters. So, $n = 6$.

    2. Identify the frequency of each letter: • L: 1 • E: 2 (appears twice) • A: 1 • D: 1 • R: 1

    3. Apply the formula: Since the letter 'E' repeats 2 times, we divide the total permutations ($6!$) by the factorial of the number of repetitions ($2!$): $$\text{Arrangements} = \frac{6!}{2!}$$

    4. Calculate the values: • $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ • $2! = 2 \times 1 = 2$

    $\text{Arrangements} = \frac{720}{2} = 360$

    Conclusion

    The total number of ways to arrange the letters is 360.

    Exam Tip

    Always count the total letters first, then immediately look for repeats. A common mistake is to forget to divide by the factorial of the repeated letters. If you see a word with repeated letters, remember: "Total factorial divided by the factorial of the repeats."

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11. 11.

    Simplify 25$^{(\frac{1}{2}}$ × 8$^{(\frac{-2}{3}}$

    1. A.1$\frac{1}{4}$
    2. B.2$\frac{1}{4}$
    3. C.6
    4. D.10

    Show answer

    Answer: A. 1$\frac{1}{4}$

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12. 12.

    In a class of 40 students, 32 offer Mathematics, 24 offer Physics and 4 offer neither Mathematics nor Physics. How many offer both Mathematics and Physics?

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

    Show answer

    Answer: D. 20

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13. 13.

    Find the values of x for which $\frac {x+2}{4}$ - $\frac{2x - 3}{3}$ < 4

    1. A.x < 8
    2. B.x > -6
    3. C.x < 4
    4. D.x > -3

    Show answer

    Answer: B. x > -6

    AI explanation

    To solve this inequality, we need to isolate $x$. Here is the step-by-step breakdown:

    1. Analyze the inequality

    We are given:

    $\frac{x+2}{4} - \frac{2x-3}{3} < 4$

    2. Clear the fractions

    To make the calculation easier, find the Least Common Multiple (LCM) of the denominators (4 and 3), which is 12. Multiply every term in the inequality by 12:

    $12 \left( \frac{x+2}{4} \right) - 12 \left( \frac{2x-3}{3} \right) < 12(4)$

    Simplify the fractions:

    $3(x + 2) - 4(2x - 3) < 48$

    3. Expand the brackets

    Distribute the numbers into the parentheses:

    $(3x + 6) - (8x - 12) < 48$

    Be careful with the negative sign when expanding the second bracket:

    $3x + 6 - 8x + 12 < 48$

    4. Simplify and solve for $x$

    Combine the like terms:

    $(3x - 8x) + (6 + 12) < 48$

    $-5x + 18 < 48$

    Subtract 18 from both sides:

    $-5x < 48 - 18$

    $-5x < 30$

    Now, divide both sides by -5. Crucial Rule: When you divide or multiply an inequality by a negative number, you must reverse the inequality sign.

    $x > \frac{30}{-5}$

    $x > -6$

    Conclusion

    Comparing our result to the given options, we find that $x > -6$ matches option B.

    Exam Tip: Always remember the "Negative Flip" rule! Whenever you divide or multiply both sides of an inequality by a negative number, immediately flip the sign (e.g., $<$ becomes $>$, and vice versa). This is a common trap in JAMB Mathematics questions.

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14. 14.

    A machine valued at N20,000 depreciates by 10% every year. What will be the value of the machine at the end of two years?

    1. A.N16,200
    2. B.N14,200
    3. C.N12,000
    4. D.N8,000

    Show answer

    Answer: A. N16,200

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

16. 16.

    Calculate 243$_{six}$ – 243$_{five}$ expressing your answer in base 10

    1. A.0
    2. B.1
    3. C.26
    4. D.46

    Show answer

    Answer: C. 26

    AI explanation

    To solve this problem, we need to convert both numbers from their respective bases to base 10 and then find the difference.

    1. Analyze the question

    We are given two numbers in different bases:

    • $243_{six}$
    • $243_{five}$ We need to convert both to base 10 and subtract the second from the first.

    2. Explain the concept

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

    The formula is: $d_n \times n^n + ... + d_1 \times n^1 + d_0 \times n^0$

    3. Show the solution

    Step 1: Convert $243_{six}$ to base 10 $243_{six} = (2 \times 6^2) + (4 \times 6^1) + (3 \times 6^0)$ $= (2 \times 36) + (4 \times 6) + (3 \times 1)$ $= 72 + 24 + 3$ $= 99_{ten}$

    Step 2: Convert $243_{five}$ to base 10 $243_{five} = (2 \times 5^2) + (4 \times 5^1) + (3 \times 5^0)$ $= (2 \times 25) + (4 \times 5) + (3 \times 1)$ $= 50 + 20 + 3$ $= 73_{ten}$

    Step 3: Subtract the results $99 - 73 = 26$

    4. Identify the answer

    Comparing our result ($26$) with the given options: A. 0 B. 1 C. 26 D. 46

    The correct option is C.

    5. Tip

    When converting from any base to base 10, always remember to start your powers from the rightmost digit (the units place) starting at 0. A common mistake is starting from 1; always remember that $x^0 = 1$, not 0!

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

18. 18.

    The probabilities that John and James pass an examination are $\frac {3}{4}$and $\frac {3}{5}$ respectively. Find the probability of both boys failing the examination.

    1. A.$\frac {1}{10}$
    2. B.$\frac {2}{10}$
    3. C.$\frac {9}{20}$
    4. D.$\frac {11}{20}$

    Show answer

    Answer: A. $\frac {1}{10}$

    AI explanation

    Analysis

    To solve this problem, we need to understand the concept of independent events in probability. Since John passing or failing does not affect James passing or failing, their probabilities are independent.

    Concept

    1. Complementary Events: The probability of an event happening ($P(A)$) and the probability of it not happening ($P(A')$) must sum to 1. • $P(\text{Fail}) = 1 - P(\text{Pass})$
    2. Independent Events: For two independent events $A$ and $B$, the probability of both occurring is $P(A \cap B) = P(A) \times P(B)$.

    Step-by-Step Solution

    Step 1: Find the probability of each boy failing.

    • Probability that John passes, $P(J_p) = \frac{3}{4}$
    • Probability that John fails, $P(J_f) = 1 - \frac{3}{4} = \frac{1}{4}$

    • Probability that James passes, $P(M_p) = \frac{3}{5}$
    • Probability that James fails, $P(M_f) = 1 - \frac{3}{5} = \frac{2}{5}$

    Step 2: Find the probability of both failing. Since these are independent events, we multiply the probability of John failing by the probability of James failing:

    • $P(\text{Both fail}) = P(J_f) \times P(M_f)$
    • $P(\text{Both fail}) = \frac{1}{4} \times \frac{2}{5}$

    Step 3: Simplify the fraction.

    • $P(\text{Both fail}) = \frac{1 \times 2}{4 \times 5} = \frac{2}{20}$
    • Reducing the fraction: $\frac{2}{20} = \frac{1}{10}$

    Conclusion

    The probability that both boys fail the examination is $\frac{1}{10}$.

    Exam Tip: Always remember that "failing" is the complement of "passing." If you are ever asked for the probability of "at least one passing," it is often easier to calculate $1 - P(\text{both failing})$ rather than adding up the individual probabilities of different success scenarios!

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19. 19.

    If $N = \begin{pmatrix} 3 & 5 & -4 \\ 6 & -3 & -5 \\ -2 & 2 & 1 \end{pmatrix}$, find $|N|$.

    1. A.65
    2. B.23
    3. C.17
    4. D.91

    Show answer

    Answer: C. 17

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20. 20.

    Use the quadratic equation curve to answer this questions What is the minimum value of the graph?

    1. A.-5.3
    2. B.0.5
    3. C.3
    4. D.8

    Show answer

    Answer: A. -5.3

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21. 21.

    A man's initial salary is N540.00 a month and increases after each period of six months by N36.00. Find his salary in the eight month of the third year.

    1. A.N828.00
    2. B.N756.00
    3. C.N720.00
    4. D.N684.00

    Show answer

    Answer: C. N720.00

    AI explanation

    Analysis

    To solve this problem, we need to track how many times the salary increase has occurred by the time the man reaches the eighth month of his third year.

    Concept

    This is an arithmetic progression problem where:

    • Initial Salary ($a$): N540.00
    • Increase ($d$): N36.00
    • Interval: Every 6 months.

    We need to determine how many "six-month periods" have passed before the 8th month of the 3rd year.

    Step-by-Step Solution

    1. Calculate the total number of months elapsed: • Year 1: 12 months • Year 2: 12 months • Year 3 (up to the 8th month): 8 months • Total months: $12 + 12 + 8 = 32$ months.

    2. Determine the number of increases: The salary increases every 6 months. We divide the total months by 6 to see how many increases have been applied: • $32 \div 6 = 5$ remainder $2$. • This means the salary has increased 5 times by the time he reaches the 8th month of the 3rd year. (The 6th increase would only happen at the end of the 36th month).

    3. Calculate the new salary: • Total increase = $5 \times \text{N}36.00 = \text{N}180.00$ • New Salary = Initial Salary + Total Increase • New Salary = $\text{N}540.00 + \text{N}180.00 = \text{N}720.00$

    Conclusion

    The man's salary in the eighth month of the third year is N720.00.

    Exam Tip

    When dealing with "increase after every X period" questions, always calculate the total time elapsed first, then use integer division (ignoring the remainder) to find the number of times the increase has been applied. Don't be tempted to add an extra increase just because you are in a new year!

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22. 22.

    if y = 23$_{five}$ + 101$_{three}$ find y leaving your answer in base two

    1. A.1110
    2. B.10111
    3. C.11101
    4. D.111100

    Show answer

    Answer: B. 10111

    AI explanation

    To solve this problem, we need to convert each number from its respective base into base ten, add them together, and then convert the result into base two.

    Step 1: Convert $23_{five}$ to base ten

    To convert from base five to base ten, we multiply each digit by 5 raised to the power of its position (starting from 0 on the right): $23_{five} = (2 \times 5^1) + (3 \times 5^0)$ $23_{five} = (2 \times 5) + (3 \times 1)$ $23_{five} = 10 + 3 = 13_{ten}$

    Step 2: Convert $101_{three}$ to base ten

    Similarly, for base three, we multiply each digit by 3 raised to the power of its position: $101_{three} = (1 \times 3^2) + (0 \times 3^1) + (1 \times 3^0)$ $101_{three} = (1 \times 9) + (0 \times 3) + (1 \times 1)$ $101_{three} = 9 + 0 + 1 = 10_{ten}$

    Step 3: Add the base ten values

    $y = 13_{ten} + 10_{ten} = 23_{ten}$

    Step 4: Convert $23_{ten}$ to base two

    To convert to base two, we perform repeated division by 2 and record the remainders:

    • $23 \div 2 = 11$ remainder 1
    • $11 \div 2 = 5$ remainder 1
    • $5 \div 2 = 2$ remainder 1
    • $2 \div 2 = 1$ remainder 0
    • $1 \div 2 = 0$ remainder 1

    Reading the remainders from bottom to top, we get 10111.

    Conclusion

    Comparing our result to the given options, we find that $10111_2$ matches option B.

    Exam Tip: When converting from base ten to any other base, always remember to read your remainders from the last one calculated to the first one (bottom to top). A common mistake is reading them top to bottom, which will give you the wrong answer!

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23. 23.

    The pie chart shows the monthly expenditure of a public servant. The monthly expenditure on housing is twice that of school fees. How much does the worker spend on housing if his monthly income is N7200?

    1. A.1000
    2. B.2000
    3. C.3000
    4. D.4000

    Show answer

    Answer: C. 3000

    Get a step-by-step AI explanation in the appGet a step-by-step AI explanation in the app

24. 24.

    Find the equation of the line through (5,7) parallel to the line 7x + 5y = 12

    1. A.5x + 7y = 20
    2. B.7x + 5y = 70
    3. C.xy = 7
    4. D.15x + 17y = 90

    Show answer

    Answer: B. 7x + 5y = 70

    AI explanation

    Analysis

    To find the equation of a line parallel to a given line, we must understand that parallel lines have the same slope (gradient). We are given a point $(5, 7)$ and the equation of a reference line $7x + 5y = 12$.

    Concept

    1. Slope-Intercept Form: The equation of a line is often written as $y = mx + c$, where $m$ is the slope.
    2. Parallel Lines: If two lines are parallel, their slopes are equal ($m_1 = m_2$).
    3. Point-Slope Form: Once we have the slope and a point $(x_1, y_1)$, we can use the formula: $y - y_1 = m(x - x_1)$

    Step-by-Step Solution

    Step 1: Find the slope of the given line. The given equation is $7x + 5y = 12$. Let's rewrite this in the form $y = mx + c$: $5y = -7x + 12$ $y = -\frac{7}{5}x + \frac{12}{5}$ The slope ($m$) of this line is $-\frac{7}{5}$. Since our new line is parallel, its slope must also be $-\frac{7}{5}$.

    Step 2: Use the point-slope formula. Using the point $(5, 7)$ and the slope $m = -\frac{7}{5}$: $y - 7 = -\frac{7}{5}(x - 5)$

    Step 3: Simplify the equation. Multiply both sides by 5 to clear the fraction: $5(y - 7) = -7(x - 5)$ $5y - 35 = -7x + 35$

    Step 4: Rearrange into the standard form ($Ax + By = C$). Add $7x$ to both sides and add $35$ to both sides: $7x + 5y = 35 + 35$ $7x + 5y = 70$

    Conclusion

    Comparing our result $7x + 5y = 70$ with the given options, it matches option B.

    Exam Tip: Whenever you see "parallel to," immediately look at the coefficients of $x$ and $y$ in the original equation. A parallel line will always have the same $Ax + By$ part; you only need to find the new constant ($C$) by plugging in your given coordinates!

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25. 25.

    A boy walks 800m in 20 minutes. Calculate his average speed in Km/H

    1. A.2.4
    2. B.4
    3. C.24
    4. D.6

    Show answer

    Answer: A. 2.4

    AI explanation

    1. Analyze the question

    We are given the distance in meters (m) and the time in minutes (min). We need to calculate the average speed, but the final answer must be in kilometers per hour (km/h).

    2. Explain the concept

    The formula for speed is:

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

    To get the answer in km/h, we must ensure:

    • Distance is in kilometers (km): Since $1000\text{m} = 1\text{km}$, we divide the distance in meters by 1000.
    • Time is in hours (h): Since $60\text{min} = 1\text{hour}$, we divide the time in minutes by 60.

    3. Show the solution

    Step 1: Convert distance to kilometers

    $\text{Distance} = 800\text{m} = \frac{800}{1000}\text{km} = 0.8\text{km}$

    Step 2: Convert time to hours

    $\text{Time} = 20\text{minutes} = \frac{20}{60}\text{hours} = \frac{1}{3}\text{hour}$

    Step 3: Calculate speed

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

    $\text{Speed} = \frac{0.8}{\frac{1}{3}}$

    $\text{Speed} = 0.8 \times 3$

    $\text{Speed} = 2.4\text{km/h}$

    4. Identify the answer

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

    5. Tip

    When converting units, always check the required units first. If you are converting from m/min to km/h, you can use the conversion factor: multiply by 60 and divide by 1000 (or simply multiply by 0.06). $800 \times 0.06 = 48$; then $48 / 20 = 2.4$. This is a quick way to verify your work during the exam!

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26. 26.

    Find the equation of the tangent at the point (2, 0) to the curve y = x$^2$ - 2x

    1. A.y = 2x - 4
    2. B.y = 2x + 4
    3. C.y = 2x - 2
    4. D.y = 2x + 2

    Show answer

    Answer: A. y = 2x - 4

    AI explanation

    Analysis

    To find the equation of a tangent to a curve at a specific point, we need two things:

    1. The slope (gradient) of the tangent at that point.
    2. The equation of a line using the point-slope form: $y - y_1 = m(x - x_1)$.

    Concept

    • Slope ($m$): The slope of the tangent to a curve $y = f(x)$ at any point is given by the derivative of the function, $\frac{dy}{dx}$, evaluated at that point.
    • Point-Slope Form: Once we have the slope $m$ and the point $(x_1, y_1)$, we substitute them into the formula $y - y_1 = m(x - x_1)$.

    Step-by-Step Solution

    Step 1: Find the derivative of the curve. Given the curve: $y = x^2 - 2x$ Differentiate with respect to $x$: $\frac{dy}{dx} = 2x - 2$

    Step 2: Calculate the slope ($m$) at the point (2, 0). Substitute $x = 2$ into the derivative: $m = 2(2) - 2$ $m = 4 - 2$ $m = 2$

    Step 3: Use the point-slope form to find the equation. We have $m = 2$ and the point $(x_1, y_1) = (2, 0)$. $y - 0 = 2(x - 2)$ $y = 2x - 4$

    Conclusion

    The equation of the tangent is $y = 2x - 4$. This matches option A.

    Exam Tip

    Always remember that the derivative $\frac{dy}{dx}$ represents the slope of the tangent at any point on the curve. If the question asks for the normal instead of the tangent, remember that the slope of the normal is the negative reciprocal of the tangent slope ($m_{normal} = -\frac{1}{m_{tangent}}$).

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27. 27.

    Divide the L.C.M of 48, 64 and 80 by their H.C.F

    1. A.20
    2. B.30
    3. C.48
    4. D.60

    Show answer

    Answer: D. 60

    AI explanation

    To solve this problem, we need to find the Least Common Multiple (L.C.M) and the Highest Common Factor (H.C.F) of the numbers 48, 64, and 80.

    Step 1: Find the Prime Factors of each number

    • 48: $2 \times 24 = 2^2 \times 12 = 2^3 \times 6 = 2^4 \times 3$
    • 64: $2 \times 32 = 2^2 \times 16 = 2^3 \times 8 = 2^4 \times 4 = 2^5 \times 2 = 2^6$
    • 80: $2 \times 40 = 2^2 \times 20 = 2^3 \times 10 = 2^4 \times 5$

    Step 2: Calculate the H.C.F

    The H.C.F is the product of the lowest power of each common prime factor.

    • The only common prime factor is 2.
    • The lowest power of 2 among $2^4, 2^6,$ and $2^4$ is $2^4$.
    • H.C.F = $2^4 = 16$.

    Step 3: Calculate the L.C.M

    The L.C.M is the product of the highest power of every prime factor present in the numbers.

    • The prime factors involved are 2, 3, and 5.
    • Highest power of 2 is $2^6$.
    • Highest power of 3 is $3^1$.
    • Highest power of 5 is $5^1$.
    • L.C.M = $2^6 \times 3 \times 5 = 64 \times 15 = 960$.

    Step 4: Divide the L.C.M by the H.C.F

    Now, we perform the final calculation as requested:

    $\frac{\text{L.C.M}}{\text{H.C.F}} = \frac{960}{16}$

    $960 \div 16 = 60$

    Comparing this result to the given options, we find that it matches option D.

    Exam Tip: When finding the L.C.M and H.C.F of large numbers, prime factorization is much safer than listing multiples or factors, as it prevents you from missing a number and saves time during the JAMB exam.

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28. 28.

    A school girl spends $\frac{1}{4}$ of her pocket money on books and $\frac{1}{3}$ on dress. What fraction remains?

    1. A.$\frac{5}{6}$
    2. B.$\frac{7}{12}$
    3. C.$\frac{5}{12}$
    4. D.$\frac{1}{6}$

    Show answer

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

    AI explanation

    Analysis

    To find the remaining fraction of the pocket money, we need to calculate the total fraction spent and subtract that value from the whole (which is represented as 1).

    Concept

    When dealing with fractions of a whole, the total amount is always represented by $1$. If you spend parts of that whole, you subtract the sum of the spent parts from $1$ to find what is left.

    Step-by-Step Solution

    Step 1: Calculate the total fraction spent. The girl spent $\frac{1}{4}$ on books and $\frac{1}{3}$ on dress. To add these, we find the Least Common Multiple (LCM) of the denominators (4 and 3), which is 12.

    $\text{Total spent} = \frac{1}{4} + \frac{1}{3}$

    Convert the fractions to have a denominator of 12:

    $\frac{1 \times 3}{4 \times 3} + \frac{1 \times 4}{3 \times 4} = \frac{3}{12} + \frac{4}{12}$

    $\text{Total spent} = \frac{3 + 4}{12} = \frac{7}{12}$

    Step 2: Calculate the remaining fraction. Subtract the total spent from the whole (1):

    $\text{Remaining} = 1 - \frac{7}{12}$

    Represent 1 as a fraction with the same denominator:

    $\text{Remaining} = \frac{12}{12} - \frac{7}{12}$

    $\text{Remaining} = \frac{12 - 7}{12} = \frac{5}{12}$

    Conclusion

    The fraction of the pocket money that remains is $\frac{5}{12}$. This matches option C.

    Exam Tip

    When adding or subtracting fractions, always look for the LCM of the denominators first. If you are ever unsure about your fraction arithmetic, imagine the pocket money as a specific amount (like ₦120).

    • $\frac{1}{4}$ of 120 = ₦30
    • $\frac{1}{3}$ of 120 = ₦40
    • Total spent = ₦70
    • Remaining = 120 - 70 = ₦50
    • Fraction remaining = $\frac{50}{120} = \frac{5}{12}$. Using a "dummy" number like this is a great way to double-check your work during the exam!

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29. 29.

    If $\frac{x}{a + 1}$ + $\frac{y}{b}$ = 1. Make y the subject of the relation.

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

    Show answer

    Answer: A. $\frac{b(a + 1 - x)}{a + 1}$

    AI explanation

    To make $y$ the subject of the formula, we need to isolate $y$ on one side of the equation. Let’s break it down step-by-step.

    1. Analyze the equation

    The given equation is:

    $\frac{x}{a + 1} + \frac{y}{b} = 1$

    2. Explain the concept

    To isolate $y$, we follow the order of operations in reverse:

    • First, move the term not containing $y$ to the other side of the equation.
    • Second, multiply by the denominator of the $y$ term to clear the fraction.

    3. Step-by-step solution

    Step 1: Subtract $\frac{x}{a + 1}$ from both sides

    $\frac{y}{b} = 1 - \frac{x}{a + 1}$

    Step 2: Simplify the right side by finding a common denominator The common denominator is $(a + 1)$:

    $\frac{y}{b} = \frac{a + 1}{a + 1} - \frac{x}{a + 1}$

    $\frac{y}{b} = \frac{a + 1 - x}{a + 1}$

    Step 3: Solve for $y$ by multiplying both sides by $b$

    $y = b \times \left( \frac{a + 1 - x}{a + 1} \right)$

    $y = \frac{b(a + 1 - x)}{a + 1}$

    4. Identify the answer

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

    5. Exam Tip

    When dealing with algebraic fractions in JAMB, always look for the "common denominator" first. If you have a constant (like $1$) and a fraction, you can always write the constant as a fraction with the same denominator (e.g., $1 = \frac{a+1}{a+1}$) to make subtraction much easier and avoid errors with signs.

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30. 30.

    Calculate the total surface area of a cupboard which measures 12cm by 10cm by 8cm

    1. A.1920cm$^2$
    2. B.592cm$^2$
    3. C.296cm$^2$
    4. D.148cm$^2$

    Show answer

    Answer: B. 592cm$^2$

    AI explanation

    Analysis

    The question asks for the total surface area of a cupboard. Since the cupboard has three different dimensions (length, width, and height), it is a rectangular prism (or cuboid).

    Concept

    The total surface area ($A$) of a cuboid is the sum of the areas of all its six faces. A cuboid has three pairs of identical faces. The formula is:

    $A = 2(lb + lh + bh)$

    Where:

    • $l$ = length
    • $b$ = breadth (width)
    • $h$ = height

    Solution

    Given the dimensions:

    • $l = 12\text{ cm}$
    • $b = 10\text{ cm}$
    • $h = 8\text{ cm}$

    Step 1: Calculate the area of each pair of faces.

    • $l \times b = 12 \times 10 = 120\text{ cm}^2$
    • $l \times h = 12 \times 8 = 96\text{ cm}^2$
    • $b \times h = 10 \times 8 = 80\text{ cm}^2$

    Step 2: Sum these areas and multiply by 2.

    • Sum $= 120 + 96 + 80 = 296\text{ cm}^2$
    • Total Surface Area $= 2 \times 296 = 592\text{ cm}^2$

    Conclusion

    Comparing our result to the given options, the correct value is $592\text{ cm}^2$. This matches option B.

    Exam Tip

    When calculating the surface area of a cuboid, a common mistake is to forget to multiply by 2 or to miss one of the three pairs of sides. Always remember that a box has a top/bottom, front/back, and left/right side—that is why we sum three products and multiply the result by two!

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31. 31.

    $\frac{0.00256 \times 0.0064}{0.025 \times 0.08}$

    1. A.8.8 x 10$^{-1}$
    2. B.8.8 x 10$^{-2}$
    3. C.8.2 x 10$^{-3}$
    4. D.8.8 x 10$^{3}$

    Show answer

    Answer: C. 8.2 x 10$^{-3}$

    AI explanation

    To solve this problem, we will convert the decimal numbers into standard form (scientific notation) to make the calculation easier and less prone to errors.

    1. Analyze the expression

    The expression is:

    $\frac{0.00256 \times 0.0064}{0.025 \times 0.08}$

    2. Convert to standard form

    • $0.00256 = 256 \times 10^{-5} = 2.56 \times 10^{-3}$
    • $0.0064 = 64 \times 10^{-4} = 6.4 \times 10^{-3}$
    • $0.025 = 25 \times 10^{-3} = 2.5 \times 10^{-2}$
    • $0.08 = 8 \times 10^{-2}$

    Let's rewrite the expression using powers of 10 to simplify:

    $\frac{(256 \times 10^{-5}) \times (64 \times 10^{-4})}{(25 \times 10^{-3}) \times (8 \times 10^{-2})}$

    3. Simplify the numerator and denominator

    Numerator: $(256 \times 64) \times (10^{-5} \times 10^{-4}) = 16384 \times 10^{-9}$

    Denominator: $(25 \times 8) \times (10^{-3} \times 10^{-2}) = 200 \times 10^{-5}$

    4. Perform the division

    Now divide the two results:

    $\frac{16384 \times 10^{-9}}{200 \times 10^{-5}}$

    Divide the coefficients: $16384 \div 200 = 81.92$

    Divide the powers of 10: $10^{-9} \div 10^{-5} = 10^{-9 - (-5)} = 10^{-9 + 5} = 10^{-4}$

    Combine them: $81.92 \times 10^{-4}$

    Convert to standard form: $8.192 \times 10^1 \times 10^{-4} = 8.192 \times 10^{-3}$

    Rounding to two significant figures (as seen in the options), we get $8.2 \times 10^{-3}$.

    5. Conclusion

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

    Note: While the calculation yields 8.192, in JAMB examinations, you should look for the option that is the closest approximation if the exact value isn't listed.

    Exam Tip: When dealing with decimals in JAMB, always convert them to whole numbers multiplied by powers of 10. It prevents "decimal point errors" which are the most common cause of lost marks in these types of questions.

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32. 32.

    Add 54 $_{eight}$ and 67$_{eight}$ giving your answers in base eight

    1. A.111
    2. B.121
    3. C.123
    4. D.143

    Show answer

    Answer: D. 143

    AI explanation

    Analysis

    To solve this problem, we need to perform addition in base eight (octal). In base eight, the digits allowed are 0, 1, 2, 3, 4, 5, 6, and 7. Whenever the sum of two digits is 8 or more, we must "carry over" to the next column, just as we do in base ten, but using 8 as the threshold.

    Concept: Addition in Base Eight

    When adding in base eight:

    1. Add the digits in each column starting from the right.
    2. If the sum is less than 8, write it down.
    3. If the sum is 8 or greater, divide the sum by 8. The remainder is written down, and the quotient is carried over to the next column.

    Step-by-Step Solution

    Step 1: Add the rightmost column (units column) $4 + 7 = 11$ (in base ten). Since 11 is greater than 8, we divide by 8: $11 \div 8 = 1$ remainder $3$. Write down 3 and carry over 1.

    Step 2: Add the next column (eights column) $5 + 6 = 11$. Now, add the carry-over from the previous step: $11 + 1 = 12$ (in base ten). Since 12 is greater than 8, we divide by 8: $12 \div 8 = 1$ remainder $4$. Write down 4 and carry over 1.

    Step 3: Add the final column There are no more digits to add, so we bring down the carried-over 1.

    Result: Combining these, we get 143.

    Conclusion

    The sum of $54_8$ and $67_8$ is $143_8$. This matches option D.

    Exam Tip

    Always remember that in any base $n$, the maximum digit you can have is $n-1$. If you are adding in base eight and your sum is 8, you write 0 and carry 1. If your sum is 9, you write 1 and carry 1. Never write a digit 8 or higher in your final answer for base eight!

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33. 33.

    From a point R, 300m north of P, a man walks eastwards to a place; Q which is 600m from P. Find the bearing of P from Q correct to the nearest degree

    1. A.026$^o$
    2. B.045$^o$
    3. C.210$^o$
    4. D.240$^o$

    Show answer

    Answer: D. 240$^o$

    AI explanation

    1. Analyze the question

    We are given two points, P and R. Point R is 300m North of P. A man walks East from R to a point Q, such that the distance from P to Q is 600m. We need to find the bearing of P from Q.

    2. Explain the concept

    • Geometry: We have a right-angled triangle $\triangle PRQ$, where $\angle PRQ = 90^\circ$ (because R is North of P and Q is East of R).
    • Trigonometry: In a right-angled triangle, we can use SOH-CAH-TOA.
    • Bearing: The bearing of P from Q is the angle measured clockwise from the North line at Q to the line QP.

    3. Show the solution

    Step 1: Visualize the triangle

    • $PR = 300\text{m}$ (Vertical side)
    • $PQ = 600\text{m}$ (Hypotenuse)
    • $\angle RPQ$ is the angle at P inside the triangle.

    Using trigonometry in $\triangle PRQ$: $\cos(\angle RPQ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{PR}{PQ}$ $\cos(\angle RPQ) = \frac{300}{600} = 0.5$ $\angle RPQ = \cos^{-1}(0.5) = 60^\circ$

    Step 2: Determine the bearing

    • At point Q, draw a North line. Since Q is East of R, the line RQ is horizontal (West-East).
    • The angle of the line QP relative to the South line at Q is equal to the alternate angle $\angle RPQ = 60^\circ$.
    • The bearing is measured clockwise from the North line at Q.
    • The angle from North to South is $180^\circ$.
    • The bearing of P from Q is $180^\circ + (\text{angle between South and QP})$.
    • Since the angle between the vertical (South) and QP is $60^\circ$, the bearing is $180^\circ + 60^\circ = 240^\circ$.

    4. Identify the answer

    The calculated bearing is $240^\circ$. Looking at the options provided: A. $026^\circ$ B. $045^\circ$ C. $210^\circ$ D. $240^\circ$

    The correct option is D.

    5. Tip

    When dealing with bearing questions, always draw a neat sketch. Remember that "North" is always $000^\circ/360^\circ$, "East" is $090^\circ$, "South" is $180^\circ$, and "West" is $270^\circ$. Using alternate angles (the "Z" rule) is the fastest way to transfer angles between your North lines!

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34. 34.

    Convert 0.04945 to two significant figures

    1. A.0.040
    2. B.0.049
    3. C.0.050
    4. D.0.49

    Show answer

    Answer: B. 0.049

    AI explanation

    1. Analyze the question

    You are asked to round the decimal number 0.04945 to two significant figures.

    2. Explain the concept

    Significant figures are the digits in a number that carry meaning contributing to its precision. When counting significant figures in a decimal number:

    • Leading zeros (zeros to the left of the first non-zero digit) are not significant.
    • In 0.04945, the first two zeros are placeholders and are not significant.
    • The first significant figure is 4.
    • The second significant figure is 9.
    • To round to two significant figures, we look at the digit immediately to the right of the second significant figure (the third digit). If it is 5 or greater, we round up; if it is less than 5, we keep the second digit as it is.

    3. Show the solution

    • Step 1: Identify the first two significant figures in 0.04945. • The first non-zero digit is 4 (this is the 1st significant figure). • The next digit is 9 (this is the 2nd significant figure).
    • Step 2: Look at the digit immediately to the right of the 2nd significant figure (which is the digit 4). • The number is 0.04945.
    • Step 3: Apply the rounding rule. • Since the digit to the right (4) is less than 5, we do not round up. We simply drop the remaining digits. • The number becomes 0.049.

    4. Identify the answer

    Comparing our result to the options provided: A. 0.040 B. 0.049 C. 0.050 D. 0.49

    The correct option is B.

    5. Tip

    Always remember: Leading zeros never count. If you are ever confused, write the number in standard form (scientific notation). 0.04945 is $4.945 \times 10^{-2}$. Looking at the digits in the decimal part ($4.945$), it is much easier to see that rounding to two figures gives $4.9 \times 10^{-2}$, which is $0.049$.

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35. 35.

    Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

    1. A.210
    2. B.1050
    3. C.21400
    4. D.25200

    Show answer

    Answer: D. 25200

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

37. 37.

    Given that Sin (5$_x$ − 28)$^o$ = Cos(3$_x$ − 50)$^o$, o < x < 90$^o$ Find the value of x

    1. A.14$^o$
    2. B.21$^o$
    3. C.32$^o$
    4. D.39$^o$

    Show answer

    Answer: B. 21$^o$

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38. 38.

    Simplify $\frac {25^{\frac{2}{3}} \div  25^{\frac{1}{6}}} {(\frac{1}{5})^{\frac{7}{6}} \div (\frac{1}{5})^{\frac{1}{6}}}$

    1. A.25
    2. B.1
    3. C.$\frac{1}{5}$
    4. D.$\frac{1}{25}$

    Show answer

    Answer: A. 25

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39. 39.

    Solve for t in the equation $\frac{3}{4}$t + $\frac{1}{3}$(21 - t) = 11

    1. A.$\frac{9}{13}$
    2. B.$\frac{7}{13}$
    3. C.5
    4. D.9$\frac{3}{5}$

    Show answer

    Answer: D. 9$\frac{3}{5}$

    AI explanation

    Analysis

    To solve for $t$ in the equation $\frac{3}{4}t + \frac{1}{3}(21 - t) = 11$, we need to clear the fractions, expand the brackets, and isolate the variable $t$.

    Concept

    When dealing with equations involving fractions, the easiest way to simplify them is to find the Least Common Multiple (LCM) of the denominators and multiply every term in the equation by that value. This eliminates the fractions and makes the equation much easier to solve.

    Step-by-Step Solution

    Step 1: Identify the LCM The denominators are 4 and 3. The LCM of 4 and 3 is 12.

    Step 2: Multiply every term by 12 $12 \times (\frac{3}{4}t) + 12 \times (\frac{1}{3}(21 - t)) = 12 \times 11$

    Step 3: Simplify the terms

    • $12 \div 4 = 3$, so $3 \times 3t = 9t$
    • $12 \div 3 = 4$, so $4 \times (21 - t) = 84 - 4t$
    • $12 \times 11 = 132$

    Now the equation looks like this: $9t + 84 - 4t = 132$

    Step 4: Combine like terms $9t - 4t + 84 = 132$ $5t + 84 = 132$

    Step 5: Isolate $t$ Subtract 84 from both sides: $5t = 132 - 84$ $5t = 48$

    Divide by 5: $t = \frac{48}{5}$

    Step 6: Convert to a mixed fraction $48 \div 5 = 9$ remainder $3$. So, $t = 9\frac{3}{5}$

    Conclusion

    Comparing our result to the given options, we find that $9\frac{3}{5}$ matches option D.

    Exam Tip

    When you have an equation with fractions, don't be tempted to work with decimals unless necessary. Clearing the fraction by multiplying by the LCM is a "golden rule" in JAMB Mathematics—it prevents errors that often occur when adding or subtracting fractions with different denominators.

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40. 40.

    A car uses one litre of petrol for every 14km. If one litre of petrol cost N63.00, how far can the car go with N900.00 worth of petrol?

    1. A.420Km
    2. B.405Km
    3. C.210Km
    4. D.200Km

    Show answer

    Answer: D. 200Km

    AI explanation

    Analysis

    To solve this problem, we need to determine how many litres of petrol can be purchased with N900.00 and then calculate the total distance the car can cover based on its fuel efficiency (14 km per litre).

    Concept

    This is a unitary method problem. We follow two main steps:

    1. Find the total volume of fuel: Divide the total money available by the cost per litre.
    2. Find the total distance: Multiply the total volume of fuel by the distance the car travels per litre.

    Step-by-Step Solution

    Step 1: Calculate the total litres of petrol bought. Total money = N900.00 Cost per litre = N63.00

    $\text{Total litres} = \frac{\text{Total money}}{\text{Cost per litre}}$

    $\text{Total litres} = \frac{900}{63}$

    To simplify this fraction, we can divide both numbers by 9:

    $\frac{900 \div 9}{63 \div 9} = \frac{100}{7} \text{ litres}$

    Step 2: Calculate the total distance. The car travels 14 km for every 1 litre.

    $\text{Total distance} = \text{Total litres} \times \text{Distance per litre}$

    $\text{Total distance} = \left( \frac{100}{7} \right) \times 14$

    Now, simplify the calculation:

    $\text{Total distance} = 100 \times \left( \frac{14}{7} \right)$

    $\text{Total distance} = 100 \times 2$

    $\text{Total distance} = 200 \text{ km}$

    Conclusion

    Based on the calculation, the car can travel 200 km.

    Note: Upon reviewing the calculation, the result is 200 km, which matches option D.

    Exam Tip

    When dealing with fractions in JAMB, try not to convert them to decimals immediately (like 14.28...). Keep them as fractions until the very last step. You will notice that the numbers often cancel out perfectly, making your work much faster and reducing the risk of calculation errors!

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41. 41.

    Express 495g as a percentage of 16.5kg

    1. A.3%
    2. B.3 $\frac{1}{3}$%
    3. C.15%
    4. D.30%

    Show answer

    Answer: A. 3%

    AI explanation

    1. Analyze the question

    You are asked to find what percentage 495 grams (g) is of 16.5 kilograms (kg). To solve this, both quantities must be in the same unit before calculating the percentage.

    2. Explain the concept

    To express one quantity as a percentage of another, use the formula:

    $\text{Percentage} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100\%$

    Important: Always ensure the units are consistent. Since $1\text{ kg} = 1000\text{ g}$, we must convert the 16.5 kg into grams first.

    3. Show the solution

    Step 1: Convert units Convert 16.5 kg to grams:

    $16.5 \text{ kg} \times 1000 = 16,500 \text{ g}$

    Step 2: Set up the fraction Now, express 495 g as a fraction of 16,500 g:

    $\frac{495}{16,500}$

    Step 3: Calculate the percentage Multiply the fraction by 100%:

    $\text{Percentage} = \left( \frac{495}{16,500} \right) \times 100\%$

    Simplify by canceling the zeros:

    $\text{Percentage} = \frac{495}{165} \%$

    Divide 495 by 165:

    $495 \div 165 = 3$

    $\text{Percentage} = 3\%$

    4. Identify the answer

    The calculated value is 3%. Comparing this to the given options, it matches option A.

    5. Tip

    In JAMB exams, always check your units first! A common trap is forgetting to convert kg to g. If you see mixed units (like kg and g, or m and cm), convert the larger unit to the smaller one immediately to avoid decimal errors.

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42. 42.

    If y = 23$_{five}$ + 101$_{three}$ , find y, leaving your answer in base two

    1. A.1110
    2. B.10111
    3. C.11101
    4. D.111100

    Show answer

    Answer: B. 10111

    AI explanation

    To solve this problem, we need to convert both numbers from their respective bases (base 5 and base 3) into base 10, add them together, and then convert the result into base 2.

    Step 1: Convert $23_{five}$ to base 10

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

    $23_{five} = (2 \times 5^1) + (3 \times 5^0)$ $23_{five} = (2 \times 5) + (3 \times 1)$ $23_{five} = 10 + 3 = 13_{ten}$

    Step 2: Convert $101_{three}$ to base 10

    $101_{three} = (1 \times 3^2) + (0 \times 3^1) + (1 \times 3^0)$ $101_{three} = (1 \times 9) + (0 \times 3) + (1 \times 1)$ $101_{three} = 9 + 0 + 1 = 10_{ten}$

    Step 3: Add the two values in base 10

    $y = 13_{ten} + 10_{ten} = 23_{ten}$

    Step 4: Convert $23_{ten}$ to base 2

    We perform repeated division by 2 and record the remainders:

    • $23 \div 2 = 11$ remainder 1
    • $11 \div 2 = 5$ remainder 1
    • $5 \div 2 = 2$ remainder 1
    • $2 \div 2 = 1$ remainder 0
    • $1 \div 2 = 0$ remainder 1

    Reading the remainders from bottom to top, we get 10111.

    Conclusion

    Comparing our result to the options provided, we see that $10111_2$ matches option B.

    Exam Tip: When converting from base 10 to any other base, always remember to read your remainders from the bottom to the top (the last remainder is the Most Significant Digit). Also, ensure you don't confuse the powers of the base—always start from $0$ on the far right digit.

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43. 43.

    Calculate the area of an equilateral triangle of side 8cm

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

    Show answer

    Answer: D. 16√3

    AI explanation

    1. Analyze the question

    You are asked to find the area of an equilateral triangle where each side ($a$) is 8 cm.

    2. Explain the concept

    An equilateral triangle is a triangle where all three sides are equal in length and all three internal angles are 60°.

    The standard formula for the area of an equilateral triangle is derived from the general area formula ($\frac{1}{2} \times \text{base} \times \text{height}$):

    $\text{Area} = \frac{\sqrt{3}}{4} a^2$

    Where $a$ is the length of the side of the triangle.

    3. Show the solution

    • Step 1: Identify the given value. $a = 8 \text{ cm}$
    • Step 2: Substitute the value into the formula. $$\text{Area} = \frac{\sqrt{3}}{4} \times (8)^2$$
    • Step 3: Calculate the square of the side. $8^2 = 64$
    • Step 4: Simplify the expression. $$\text{Area} = \frac{\sqrt{3}}{4} \times 64$$
    • Step 5: Divide 64 by 4. $64 \div 4 = 16$
    • Step 6: Combine the results. $$\text{Area} = 16\sqrt{3} \text{ cm}^2$$

    4. Identify the answer

    Comparing our result to the options provided: A. 8√3 B. 16 C. 4√3 D. 16√3

    The correct option is D.

    5. Tip

    To remember this formula easily, think of it as a variation of the standard area formula. Since the height of an equilateral triangle is always $\frac{\sqrt{3}}{2}a$, multiplying this by the base ($a$) and dividing by 2 gives you $\frac{\sqrt{3}}{4}a^2$. Always remember to square the side first before multiplying by the fraction!

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

45. 45.

    If X, Y can take values from the set (1, 2, 3 ,4), find the probability that the product of X and Y is not greater than 6

    1. A.$\frac{5}{8}$
    2. B.$\frac{5}{16}$
    3. C.$\frac{1}{2}$
    4. D.$\frac{3}{8}$

    Show answer

    Answer: A. $\frac{5}{8}$

    AI explanation

    1. Analyze the question

    We are given two variables, $X$ and $Y$, which can each take any value from the set $\{1, 2, 3, 4\}$. We need to find the probability that their product ($X \times Y$) is not greater than 6 (i.e., $X \times Y \leq 6$).

    2. Explain the concept

    • Sample Space: Since $X$ has 4 choices and $Y$ has 4 choices, the total number of possible outcomes is $4 \times 4 = 16$.
    • Probability Formula: $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$
    • Condition: We are looking for pairs $(X, Y)$ such that $X \times Y \leq 6$.

    3. Show the solution

    Let's list all possible outcomes $(X, Y)$ and identify those where the product is $\leq 6$:

    | X \ Y | 1 | 2 | 3 | 4 | | :--- | :--- | :--- | :--- | :--- | | 1 | 1 (Yes) | 2 (Yes) | 3 (Yes) | 4 (Yes) | | 2 | 2 (Yes) | 4 (Yes) | 6 (Yes) | 8 (No) | | 3 | 3 (Yes) | 6 (Yes) | 9 (No) | 12 (No) | | 4 | 4 (Yes) | 8 (No) | 12 (No) | 16 (No) |

    Counting the "Yes" outcomes:

    • When $X=1$: Products are 1, 2, 3, 4 (All 4 are $\leq 6$)
    • When $X=2$: Products are 2, 4, 6 (3 are $\leq 6$)
    • When $X=3$: Products are 3, 6 (2 are $\leq 6$)
    • When $X=4$: Product is 4 (1 is $\leq 6$)

    Total favorable outcomes = $4 + 3 + 2 + 1 = 10$.

    Calculating the probability: $P(XY \leq 6) = \frac{10}{16}$

    Simplify the fraction by dividing both numerator and denominator by 2: $\frac{10 \div 2}{16 \div 2} = \frac{5}{8}$

    4. Identify the answer

    The calculated probability is $\frac{5}{8}$, which matches option A.

    5. Tip

    When dealing with probability questions involving two variables, a grid table is your best friend! It helps you visualize all possible outcomes and prevents you from missing any combinations. Always double-check if the question says "greater than" or "not greater than" (which includes the number itself) to avoid silly mistakes.

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46. 46.

    Approximate 0.9875 to 1 decimal place.

    1. A.1.1
    2. B.1.0
    3. C.0.9
    4. D.0.10

    Show answer

    Answer: B. 1.0

    AI explanation

    Analysis

    The question asks us to round off the decimal number 0.9875 to 1 decimal place. This means we want only one digit after the decimal point.

    Concept: Rounding Decimals

    To round a number to a specific decimal place:

    1. Identify the digit in the target decimal place (the first digit after the decimal point).
    2. Look at the digit immediately to its right (the second decimal place).
    3. Rule: If the digit to the right is 5 or greater, round up the target digit by adding 1. If it is less than 5, keep the target digit as it is.

    Step-by-Step Solution

    1. Identify the target digit: In 0.9875, the first decimal place is 9.
    2. Look at the next digit: The digit to the right of 9 is 8.
    3. Apply the rule: Since 8 is greater than 5, we must round up the target digit (9).
    4. Perform the addition: Adding 1 to 9 gives 10. We write down the 0 and carry over the 1 to the units place. • $0.9 + 0.1 = 1.0$

    Therefore, 0.9875 rounded to 1 decimal place is 1.0.

    Conclusion

    Comparing our result with the given options: A. 1.1 B. 1.0 C. 0.9 D. 0.10

    The correct option is B.

    Exam Tip

    When rounding, always look at the digit immediately to the right of your target place. Do not be distracted by the other digits further to the right (like the 7 and 5 in this question); they do not affect the rounding process once you have determined the status of the digit in the second decimal place.

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47. 47.

    A box contains two red balls and four blue balls. A ball is drawn at random from the box and then replaced before a second ball is drawn. Find the probability of drawing two red balls.

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

    Show answer

    Answer: D. $\frac{1}{9}$

    AI explanation

    1. Analyze the Question

    We are asked to find the probability of drawing two red balls from a box containing a mix of red and blue balls. Crucially, the first ball is replaced before the second draw, which means the total number of balls remains the same for both draws.

    2. Explain the Concept

    This is a problem involving independent events. Because the first ball is replaced, the outcome of the first draw does not affect the outcome of the second draw.

    The formula for the probability of two independent events ($A$ and $B$) occurring is:

    $P(A \text{ and } B) = P(A) \times P(B)$

    3. Show the Solution

    • Step 1: Determine the total number of balls. Total balls = 2 (red) + 4 (blue) = 6 balls.

    • Step 2: Find the probability of drawing a red ball on the first draw ($P(R_1)$). $$P(R_1) = \frac{\text{Number of red balls}}{\text{Total number of balls}} = \frac{2}{6} = \frac{1}{3}$$

    • Step 3: Find the probability of drawing a red ball on the second draw ($P(R_2)$). Since the ball was replaced, the box still contains 2 red balls and 6 balls in total. $$P(R_2) = \frac{2}{6} = \frac{1}{3}$$

    • Step 4: Calculate the combined probability. Since the events are independent: $$P(R_1 \text{ and } R_2) = P(R_1) \times P(R_2)$$ $$P = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}$$

    4. Identify the Answer

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

    5. Exam Tip

    Always look for the keyword "replaced" or "with replacement" in probability questions. If you see it, it means the denominator (total number of items) stays the same for every draw. If the question says "without replacement," the denominator will decrease by 1 for the second draw!

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48. 48.

    Find the average of the first four prime numbers greater than 10

    1. A.20
    2. B.19
    3. C.17
    4. D.15

    Show answer

    Answer: D. 15

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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.

    What is the place value of 9 in the number 3.0492?

    1. A.$\frac{9}{10000}$
    2. B.$\frac{9}{1000}$
    3. C.$\frac{9}{100}$
    4. D.$\frac{9}{10}$

    Show answer

    Answer: B. $\frac{9}{1000}$

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51. 51.

    Make T the subject of the relation.

    1. A.T = $\frac{R + P3}{15Q}$
    2. B.T = $\frac{R - 15P^3}{Q}$
    3. C.T =R - $\frac{15P^3}{Q}$
    4. D.T = $\frac{15R + Q}{P^3}$

    Show answer

    Answer: C. T =R - $\frac{15P^3}{Q}$

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52. 52.

    An arc of a circle of radius 14cm subtends angle 300° at the centre. Find the perimeter of the sector formed by the arc (take π = $\frac{22}{7}$)

    1. A.14.67cm
    2. B.73.33 cm
    3. C.101.33cm
    4. D.513.33cm

    Show answer

    Answer: C. 101.33cm

    AI explanation

    Analysis

    To find the perimeter of a sector, we need to calculate the sum of the length of the arc and the two radii that enclose the sector.

    Concept

    1. Arc Length Formula: The length of an arc ($L$) is given by: $L = \frac{\theta}{360} \times 2\pi r$ where $\theta$ is the angle subtended at the center and $r$ is the radius.
    2. Perimeter of a Sector: The perimeter ($P$) is the sum of the arc length and the two radii: $P = L + 2r$

    Step-by-Step Solution

    Step 1: Identify the given values

    • Radius ($r$) = 14 cm
    • Angle ($\theta$) = 300°
    • $\pi = \frac{22}{7}$

    Step 2: Calculate the Arc Length ($L$) $L = \frac{300}{360} \times 2 \times \frac{22}{7} \times 14$

    Simplify the fraction $\frac{300}{360}$: $\frac{300}{360} = \frac{30}{36} = \frac{5}{6}$

    Now, calculate $L$: $L = \frac{5}{6} \times 2 \times \frac{22}{7} \times 14$ $L = \frac{5}{6} \times 2 \times 22 \times 2$ (since $14 \div 7 = 2$) $L = \frac{5 \times 88}{6} = \frac{440}{6}$ $L \approx 73.33 \text{ cm}$

    Step 3: Calculate the Perimeter ($P$) The perimeter of the sector is the arc length plus the two radii forming the sector: $P = L + r + r$ $P = 73.33 + 14 + 14$ $P = 73.33 + 28$ $P = 101.33 \text{ cm}$

    Conclusion

    Comparing our result with the given options, we find that 101.33 cm matches option C.

    Exam Tip

    Always remember that a "sector" is like a slice of pizza. The perimeter includes not just the curved crust (the arc), but also the two straight sides (the radii). Students often forget to add the two radii and mistakenly choose the arc length alone!

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53. 53.

    Tossing a coin and rolling a die are two separate events. What is the probability of obtaining a tail on the coin and an even number on the die?

    1. A.$\frac{1}{16}$
    2. B.$\frac{1}{6}$
    3. C.$\frac{1}{4}$
    4. D.$\frac{3}{8}$

    Show answer

    Answer: C. $\frac{1}{4}$

    AI explanation

    Analysis

    This is a problem involving independent events in probability. Two events are independent if the outcome of one does not affect the outcome of the other. Here, tossing a coin has no influence on rolling a die.

    Concept

    When two events, $A$ and $B$, are independent, the probability of both occurring (the intersection) is the product of their individual probabilities:

    $P(A \text{ and } B) = P(A) \times P(B)$

    Step-by-Step Solution

    Step 1: Find the probability of obtaining a tail on the coin ($P(A)$) A fair coin has two sides: Head (H) and Tail (T).

    • Total outcomes = 2
    • Favorable outcome (Tail) = 1
    • $P(\text{Tail}) = \frac{1}{2}$

    Step 2: Find the probability of obtaining an even number on the die ($P(B)$) A standard six-sided die has the numbers {1, 2, 3, 4, 5, 6}.

    • Total outcomes = 6
    • Even numbers = {2, 4, 6}
    • Number of favorable outcomes = 3
    • $P(\text{Even}) = \frac{3}{6} = \frac{1}{2}$

    Step 3: Calculate the combined probability Since the events are independent, multiply the two probabilities:

    $P(\text{Tail and Even}) = P(\text{Tail}) \times P(\text{Even})$

    $P(\text{Tail and Even}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

    Conclusion

    The probability of obtaining a tail on the coin and an even number on the die is $\frac{1}{4}$.

    Exam Tip: Whenever you see the word "and" in probability questions involving independent events, it is a signal to multiply the probabilities. If you see the word "or", you generally add them.

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54. 54.

    From a point P, Q is 5km due West and R is12km due South of Q. Find the distance between P and R.

    1. A.5km
    2. B.12km
    3. C.13km
    4. D.17km

    Show answer

    Answer: C. 13km

    AI explanation

    1. Analyze the question

    We are given the positions of three points: P, Q, and R.

    • Q is 5km West of P.
    • R is 12km South of Q.
    • We need to find the straight-line distance between P and R.

    2. Explain the concept

    This problem involves bearings and directions, which can be visualized as a right-angled triangle.

    • Moving "West" and "South" creates a 90-degree angle at point Q.
    • Since we have a right-angled triangle where PQ and QR are the two legs, we can use the Pythagoras Theorem to find the hypotenuse (PR).
    • Pythagoras Theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (the side opposite the right angle).

    3. Show the solution

    • Let $PQ = 5\text{ km}$ (base)
    • Let $QR = 12\text{ km}$ (height)
    • Let $PR = d$ (hypotenuse)

    Using the formula: $PR^2 = PQ^2 + QR^2$ $PR^2 = 5^2 + 12^2$ $PR^2 = 25 + 144$ $PR^2 = 169$

    To find $PR$, take the square root of both sides: $PR = \sqrt{169}$ $PR = 13\text{ km}$

    4. Identify the answer

    The distance between P and R is 13km. This corresponds to option C.

    5. Tip

    Whenever you see "North/South" combined with "East/West" in a geometry problem, immediately think of a right-angled triangle. Also, keep an eye out for Pythagorean Triples (like 3, 4, 5 or 5, 12, 13). Recognizing these triples can save you valuable time during your JAMB exam because you won't need to perform the full calculation!

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55. 55.

    What is the modal age?

    1. A.4
    2. B.5
    3. C.6
    4. D.7

    Show answer

    Answer: A. 4

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56. 56.

    Integrate the expression 6x$^2$ - 2x + 1

    1. A.3x$^3$ - 2x$^2$ + x + c
    2. B.2x$^3$ - x$^2$ + x + c
    3. C.2x$^3$ – 3x$^2$ + c
    4. D.x$^3$ + x$^2$ – x + c

    Show answer

    Answer: B. 2x$^3$ - x$^2$ + x + c

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57. 57.

    Which one of the following gives the members of the set A1 n B n C?

    1. A.Φ
    2. B.{s}
    3. C.{t, u}
    4. D.{y, z}

    Show answer

    Answer: C. {t, u}

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58. 58.

    Evaluate ∫$^2_1$ $\frac{5}{x}$ dx

    1. A.1.47
    2. B.2.67
    3. C.3.23
    4. D.3.47

    Show answer

    Answer: D. 3.47

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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.

    Evaluate (2√3 - 4) (2√3 + 4)

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

    Show answer

    Answer: A. -4

    AI explanation

    Analysis

    The expression $(2\sqrt{3} - 4)(2\sqrt{3} + 4)$ is a classic example of the Difference of Two Squares algebraic identity.

    Concept

    The difference of two squares formula states that:

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

    In this problem:

    • $a = 2\sqrt{3}$
    • $b = 4$

    Solution

    Applying the formula $a^2 - b^2$:

    1. Square the first term ($a^2$): $(2\sqrt{3})^2 = 2^2 \times (\sqrt{3})^2$ $= 4 \times 3$ $= 12$

    2. Square the second term ($b^2$): $(4)^2 = 16$

    3. Subtract the second from the first: $12 - 16 = -4$

    Alternatively, you can use the FOIL method (First, Outer, Inner, Last):

    • First: $2\sqrt{3} \times 2\sqrt{3} = 12$
    • Outer: $2\sqrt{3} \times 4 = 8\sqrt{3}$
    • Inner: $-4 \times 2\sqrt{3} = -8\sqrt{3}$
    • Last: $-4 \times 4 = -16$

    Combine them: $12 + 8\sqrt{3} - 8\sqrt{3} - 16$ The middle terms ($8\sqrt{3} - 8\sqrt{3}$) cancel out to zero, leaving: $12 - 16 = -4$

    Conclusion

    The result of the evaluation is -4, which corresponds to option A.

    Exam Tip

    Whenever you see two brackets with the same terms but different signs (e.g., $(x-y)(x+y)$), don't waste time multiplying every single term. Simply square the first term, square the second term, and put a minus sign between them. It saves time and reduces the risk of calculation errors!

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61. 61.

    How many times, correct to the nearest whole number, will a man run round a circular track of diameter 100m to cover a distance of 1000m?

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

    Show answer

    Answer: A. 3

    AI explanation

    1. Analyze the question

    We are given a circular track with a diameter of 100m. We need to find how many complete laps (runs round the track) a man must make to cover a total distance of 1000m.

    2. Explain the concept

    To solve this, we first need to find the distance covered in one single lap. The distance around a circle is its circumference. The formula for the circumference ($C$) of a circle is:

    $C = \pi d$

    where:

    • $\pi \approx 3.142$ (or $\frac{22}{7}$)
    • $d = \text{diameter} = 100\text{m}$

    Once we have the circumference, we divide the total distance (1000m) by the circumference to find the number of laps.

    3. Show the solution

    Step 1: Calculate the circumference of the track.

    $C = \pi \times 100$

    $C = 3.14159 \times 100$

    $C \approx 314.16\text{m}$

    Step 2: Calculate the number of laps.

    $\text{Number of laps} = \frac{\text{Total distance}}{\text{Circumference}}$

    $\text{Number of laps} = \frac{1000}{314.16}$

    $\text{Number of laps} \approx 3.183$

    Step 3: Round to the nearest whole number. Since 3.183 is closer to 3 than to 4, rounding to the nearest whole number gives us 3.

    4. Identify the answer

    Comparing our result to the given options: A. 3 B. 4 C. 5 D. 6

    The correct option is A.

    5. Tip

    When dealing with circular problems in JAMB, always remember that "distance round" is synonymous with "circumference." If you are ever unsure about the value of $\pi$, using 3.142 is usually sufficient for most calculations unless the question specifies otherwise. Keep calm and stay focused—you've got this!

    AI can make mistakes. Report anything that looks wrong.