2016 JAMB Mathematics Past Questions

Questions

1. 1.

   If the 2nd term of a G.P is $\frac{8}{9}$ and the 6th term is 4$\frac{1}{2}$. Find the common ratio.

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

   Show answer

   Answer: B. $\frac{3}{2}$

   AI explanation

   1. Analyze the Question

   We are given two terms of a Geometric Progression (G.P.):

   • The 2nd term ($ar$) = $\frac{8}{9}$
   • The 6th term ($ar^5$) = $4\frac{1}{2}$ (which is $\frac{9}{2}$ as an improper fraction) We need to find the common ratio ($r$).

   2. Explain the Concept

   In a G.P., the $n^{th}$ term is given by the formula:

   $T_n = ar^{n-1}$

   where:

   • $a$ is the first term
   • $r$ is the common ratio
   • $n$ is the position of the term

   By dividing the expression for the 6th term by the expression for the 2nd term, we can eliminate $a$ and solve for $r$.

   3. Show the Solution

   Step 1: Write down the equations based on the formula.

   • $T_2 = ar^{2-1} = ar = \frac{8}{9}$ --- (Equation 1)
   • $T_6 = ar^{6-1} = ar^5 = \frac{9}{2}$ --- (Equation 2)

   Step 2: Divide Equation 2 by Equation 1.

   $\frac{ar^5}{ar} = \frac{\frac{9}{2}}{\frac{8}{9}}$

   The $a$ cancels out:

   $r^4 = \frac{9}{2} \div \frac{8}{9}$

   $r^4 = \frac{9}{2} \times \frac{9}{8}$

   $r^4 = \frac{81}{16}$

   Step 3: Solve for $r$. To find $r$, take the fourth root of both sides:

   $r = \sqrt[4]{\frac{81}{16}}$

   Since $3^4 = 81$ and $2^4 = 16$:

   $r = \frac{3}{2}$

   4. Identify the Answer

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

   The correct option is B.

   5. Tip

   When dealing with G.P. problems involving two terms, always remember the "gap" rule: the ratio of the $n^{th}$ term to the $m^{th}$ term is $r^{(n-m)}$. Here, the gap between the 6th and 2nd term is $6 - 2 = 4$, so $r^4 = \frac{T_6}{T_2}$. This shortcut saves you from having to solve for the first term ($a$) separately!

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2. 2.

   Calculate the perimeter of a sector of a circle of raduis 9cm and angle 36o.

   1. A.18cm
   2. B.(18 + $\frac{9\pi}{5}$)cm
   3. C.(9 + $\frac{9\pi}{5}$)cm
   4. D.$\frac{9\pi}{5}$cm

   Show answer

   Answer: B. (18 + $\frac{9\pi}{5}$)cm

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

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

5. 5.

   Evaluate $\int\limits_0^\frac{\pi}{2}$ sin xdx

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

   Show answer

   Answer: C. 1

   AI explanation

   Analysis

   The question asks us to evaluate a definite integral of the trigonometric function $\sin x$ within the limits of $0$ to $\frac{\pi}{2}$.

   Concept

   To solve this, we use the fundamental theorem of calculus. The integral of $\sin x$ with respect to $x$ is $-\cos x$. The formula for a definite integral is:

   $\int_a^b f(x) dx = [F(x)]_a^b = F(b) - F(a)$

   Where $F(x)$ is the antiderivative of $f(x)$.

   Step-by-Step Solution

   1. Find the antiderivative: The integral of $\sin x$ is $-\cos x$. $$\int \sin x \, dx = -\cos x$$

   2. Apply the limits: We evaluate $[-\cos x]$ from $0$ to $\frac{\pi}{2}$: $$\left[ -\cos x \right]_0^{\frac{\pi}{2}}$$

   3. Substitute the upper and lower limits: Substitute $\frac{\pi}{2}$ (upper limit) and $0$ (lower limit) into the expression: $$= (-\cos(\frac{\pi}{2})) - (-\cos(0))$$

   4. Calculate the trigonometric values: Recall that $\cos(\frac{\pi}{2}) = 0$ and $\cos(0) = 1$. $$= (-0) - (-1)$$ $$= 0 + 1$$ $$= 1$$

   Conclusion

   The result of the integration is $1$. Comparing this to the given options, it matches option C.

   Exam Tip

   Always remember your trigonometric values for $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},$ and $2\pi$. A common mistake students make is forgetting the negative sign when integrating $\sin x$ (remember: the derivative of $\cos x$ is $-\sin x$, so the integral of $\sin x$ must be $-\cos x$).

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6. 6.

   Find $\frac{dy}{dx}$, if y = $\frac{2}{3}$ x$^3$ - $\frac{4}{x}$

   1. A.3x2 - $\frac{4}{x}$
   2. B.2x2 +$\frac{4}{x^2}$
   3. C.3x2 + $\frac{4}{x^2}$
   4. D.2x2 - $\frac{4}{x}$

   Show answer

   Answer: B. 2x2 +$\frac{4}{x^2}$

   AI explanation

   Analysis

   The goal is to find the derivative of the function $y = \frac{2}{3}x^3 - \frac{4}{x}$ with respect to $x$. This requires applying the Power Rule of differentiation.

   Concept

   The Power Rule states that if $y = ax^n$, then the derivative $\frac{dy}{dx} = n \cdot ax^{n-1}$.

   Before differentiating, it is helpful to rewrite terms with $x$ in the denominator using negative exponents:

   • $\frac{1}{x^n} = x^{-n}$

   Step-by-Step Solution

   Step 1: Rewrite the function The function is $y = \frac{2}{3}x^3 - \frac{4}{x}$. Rewrite the second term: $y = \frac{2}{3}x^3 - 4x^{-1}$

   Step 2: Differentiate each term

   • For the first term, $\frac{2}{3}x^3$: Multiply the coefficient by the power and subtract 1 from the power: $3 \times \frac{2}{3}x^{(3-1)} = 2x^2$

   • For the second term, $-4x^{-1}$: Multiply the coefficient by the power and subtract 1 from the power: $-1 \times -4x^{(-1-1)} = 4x^{-2}$

   Step 3: Combine the results $\frac{dy}{dx} = 2x^2 + 4x^{-2}$

   Step 4: Convert back to fractional form Since $x^{-2} = \frac{1}{x^2}$, the expression becomes: $\frac{dy}{dx} = 2x^2 + \frac{4}{x^2}$

   Conclusion

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

   Exam Tip

   Always remember to rewrite terms like $\frac{1}{x^n}$ as $x^{-n}$ before differentiating. A common mistake students make is trying to use the quotient rule unnecessarily; converting to a negative exponent makes the differentiation much faster and reduces the chance of errors!

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7. 7.

   Evaluate $\frac{27^\frac{1}{3} - 8^\frac{2}{3}}{16^\frac{2}{4} \times 2}$

   1. A.$\frac{23}{5}$
   2. B.-$\frac{23}{5}$
   3. C.-$\frac{1}{8}$
   4. D.$\frac{21}{7}$

   Show answer

   Answer: C. -$\frac{1}{8}$

   AI explanation

   To solve this expression, we need to simplify each term using the laws of indices. The expression is:

   $\frac{27^\frac{1}{3} - 8^\frac{2}{3}}{16^\frac{2}{4} \times 2}$

   Step 1: Simplify the numerator

   • $27^\frac{1}{3}$: This is the cube root of 27. Since $3 \times 3 \times 3 = 27$, then $27^\frac{1}{3} = 3$.
   • $8^\frac{2}{3}$: This is $(8^\frac{1}{3})^2$. The cube root of 8 is 2. So, $2^2 = 4$.
   • Numerator: $3 - 4 = -1$.

   Step 2: Simplify the denominator

   • $16^\frac{2}{4}$: First, simplify the fraction in the exponent: $\frac{2}{4} = \frac{1}{2}$. So, $16^\frac{1}{2}$ is the square root of 16, which is 4.
   • Denominator: $4 \times 2 = 8$.

   Step 3: Combine the results

   Now, place the simplified numerator over the simplified denominator:

   $\frac{-1}{8}$

   Analysis

   The calculation results in $-\frac{1}{8}$. Comparing this to your options: A. $\frac{23}{5}$ B. $-\frac{23}{5}$ C. $-\frac{1}{8}$ D. $\frac{21}{7}$

   The result matches option C.

   Exam Tip: When dealing with fractional indices, always try to express the base as a power of a smaller number (e.g., $27 = 3^3$, $8 = 2^3$, $16 = 2^4$). This makes it much easier to cancel out the denominators in the exponents!

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8. 8.

   In a school of 150 students, 80 offer French while 60 offer Arabic and 20 offer neither. How many students offer both subjects?

   1. A.45
   2. B.10
   3. C.35
   4. D.30

   Show answer

   Answer: B. 10

   AI explanation

   Analysis

   This is a classic set theory problem involving two overlapping sets. We need to find the intersection of two groups (students offering French and students offering Arabic) within a universal set (the total number of students).

   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 number of students who offer at least one subject.
   • $n(A)$ is the number of students who offer French.
   • $n(B)$ is the number of students who offer Arabic.
   • $n(A \cap B)$ is the number of students who offer both (our unknown).

   Step-by-Step Solution

   1. Identify the Universal Set ($U$): Total students = 150

   2. Find the number of students who offer at least one subject ($n(A \cup B)$): Since 20 students offer neither, we subtract them from the total: $n(A \cup B) = 150 - 20 = 130$

   3. Apply the formula: Let $x$ be the number of students who offer both subjects ($n(A \cap B)$). $130 = 80 + 60 - x$

   4. Solve for $x$: $130 = 140 - x$ $x = 140 - 130$ $x = 10$

   Conclusion

   The number of students who offer both subjects is 10.

   Exam Tip

   When solving set theory problems, always remember to subtract the "neither" group from the total population first. If you forget this step, you will likely get an incorrect answer. Also, drawing a quick Venn Diagram can help you visualize the overlap and ensure your numbers add up correctly!

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9. 9.

   Calculate the range of 20, -6, 25, 30, 21, 28, 32, 33, 34, 5, 3, 2, and 1.

   1. A.32
   2. B.36
   3. C.33
   4. D.40

   Show answer

   Answer: D. 40

   AI explanation

   Analysis

   To find the range of a set of data, you need to identify the difference between the highest value and the lowest value in the set.

   Concept

   The formula for range is:

   $\text{Range} = \text{Highest Value} - \text{Lowest Value}$

   Step-by-Step Solution

   Step 1: List the given numbers. The set is: 20, -6, 25, 30, 21, 28, 32, 33, 34, 5, 3, 2, 1.

   Step 2: Identify the highest value. Looking through the list, the largest number is 34.

   Step 3: Identify the lowest value. Looking through the list, the smallest number is -6.

   Step 4: Apply the formula.

   $\text{Range} = 34 - (-6)$

   When you subtract a negative number, it becomes addition:

   $\text{Range} = 34 + 6$

   $\text{Range} = 40$

   Conclusion

   Comparing our result to the options provided: A. 32 B. 36 C. 33 D. 40

   The calculated range is 40, which matches option D.

   Exam Tip

   Always be careful with negative numbers! In statistics questions, students often mistakenly pick the smallest positive number (in this case, 1) instead of the actual lowest value (-6). Remember: on a number line, any negative number is always smaller than a positive number.

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10. 10.

    An arc subtends an angle of 30º at the centre of a circle radius 12cm. Calculate the length of the arc.

    1. A.6$\pi$cm
    2. B.2$\pi$cm
    3. C.3$\pi$cm
    4. D.9$\pi$cm

    Show answer

    Answer: B. 2$\pi$cm

    AI explanation

    Analysis

    The question asks for the length of an arc of a circle. To find this, we need the radius of the circle and the angle subtended by the arc at the center.

    Concept

    The length of an arc ($L$) is a portion of the circumference of a circle. The formula is derived from the ratio of the central angle ($\theta$) to the total angle in a circle ($360^\circ$):

    $L = \frac{\theta}{360^\circ} \times 2\pi r$

    Where:

    • $\theta$ = angle subtended at the center
    • $r$ = radius of the circle
    • $\pi$ = a constant (usually left as $\pi$ in JAMB questions unless specified otherwise)

    Step-by-Step Solution

    1. Identify the given values: • $\theta = 30^\circ$ • $r = 12\text{ cm}$

    2. Substitute the values into the formula: $$L = \frac{30}{360} \times 2 \times \pi \times 12$$

    3. Simplify the fraction: $$\frac{30}{360} = \frac{3}{36} = \frac{1}{12}$$

    4. Calculate the final length: $$L = \frac{1}{12} \times 2 \times \pi \times 12$$

    Notice that the $12$ in the numerator and the $12$ in the denominator cancel each other out:

    $L = 1 \times 2 \times \pi \times 1$

    $L = 2\pi\text{ cm}$

    Conclusion

    Comparing our result to the options provided, the calculated length is $2\pi\text{ cm}$. This matches option B.

    Exam Tip: In JAMB Mathematics, always look at the options before calculating. Since the options are expressed in terms of $\pi$, do not substitute $3.142$ or $22/7$ for $\pi$. Keeping $\pi$ as a symbol saves time and prevents unnecessary decimal errors!

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11. 11.

    Evaluate $\int$(cos4x + sin3x)dx

    1. A.sin4x - cos3x + k
    2. B.sin4x + cos3x + k
    3. C.$\frac{1}{4}$sin4x - $\frac{1}{3}$cos3x + k
    4. D.$\frac{1}{4}$sin4x + $\frac{1}{3}$cos3x + k

    Show answer

    Answer: C. $\frac{1}{4}$sin4x - $\frac{1}{3}$cos3x + k

    AI explanation

    Analysis

    To solve this problem, we need to apply the rules of integration for trigonometric functions. Specifically, we are integrating the sum of two functions: $\cos(4x)$ and $\sin(3x)$ with respect to $x$.

    Concept

    The standard integration formulas for trigonometric functions of the form $ax$ are:

    1. $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + k$
    2. $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + k$

    Where $k$ is the constant of integration.

    Step-by-Step Solution

    Step 1: Split the integral We can integrate each term separately:

    $\int (\cos 4x + \sin 3x) \, dx = \int \cos 4x \, dx + \int \sin 3x \, dx$

    Step 2: Integrate $\cos 4x$ Using the formula $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax)$:

    $\int \cos 4x \, dx = \frac{1}{4} \sin 4x$

    Step 3: Integrate $\sin 3x$ Using the formula $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax)$:

    $\int \sin 3x \, dx = -\frac{1}{3} \cos 3x$

    Step 4: Combine the results and add the constant $k$

    $\frac{1}{4} \sin 4x - \frac{1}{3} \cos 3x + k$

    Comparing this result with the given options, it matches option C.

    Exam Tip

    When integrating trigonometric functions with a coefficient inside the bracket (like $4x$ or $3x$), always remember to divide by that coefficient. A common mistake students make is multiplying by the coefficient instead of dividing. Think of integration as the "reverse" of differentiation—since differentiation involves multiplying by the coefficient (Chain Rule), integration must involve dividing by it!

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12. 12.

    Evaluate $\frac{12.02 \times 20.06}{26.04 \times 60.06}$, correct to three significant figures.

    1. A.0.157
    2. B.0.154
    3. C.0.155
    4. D.0.158

    Show answer

    Answer: B. 0.154

    AI explanation

    1. Analyze the question

    We are asked to evaluate the expression $\frac{12.02 \times 20.06}{26.04 \times 60.06}$ and round the final result to three significant figures.

    2. Explain the concept

    To solve this, we perform the multiplication in the numerator and the denominator separately, then divide the results. Finally, we apply the rule of significant figures: counting from the first non-zero digit, we keep three digits and round the third digit based on the fourth.

    3. Show the solution

    Step 1: Multiply the numerator $12.02 \times 20.06 = 241.1212$

    Step 2: Multiply the denominator $26.04 \times 60.06 = 1563.9624$

    Step 3: Divide the numerator by the denominator $\frac{241.1212}{1563.9624} \approx 0.154173...$

    Step 4: Round to three significant figures

    • The first significant figure is 1.
    • The second significant figure is 5.
    • The third significant figure is 4.
    • The next digit is 1 (which is less than 5), so we keep the 4 as it is.

    Rounding $0.154173...$ to three significant figures gives 0.154.

    4. Identify the answer

    Comparing our result to the given options: A. 0.157 B. 0.154 C. 0.155 D. 0.158

    The correct option is B.

    5. Tip

    When dealing with decimals in JAMB, don't rush to round the numbers before you perform the calculation. Rounding too early (e.g., rounding to 12, 20, 26, and 60) can introduce "rounding errors" that lead you to the wrong option. Always perform the full calculation first, then round only at the very final step.

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13. 13.

    An arc of the length 16$\pi$cm subtends an angle of 80o at the centre of the circle. Find the radius of the circle.

    1. A.24cm
    2. B.28cm
    3. C.36cm
    4. D.32cm

    Show answer

    Answer: C. 36cm

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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.

    Evaluate $\int$(sinx - 5x2)dx

    1. A.-cosx - 10x + k
    2. B.cosx - $\frac{5x^3}{3}$ + k
    3. C.-cosx - $\frac{5x^3}{3}$ + k
    4. D.cosx - 10x + k

    Show answer

    Answer: C. -cosx - $\frac{5x^3}{3}$ + k

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

17. 17.

    If x10 = 235. Find x

    1. A.15
    2. B.12
    3. C.14
    4. D.13

    Show answer

    Answer: D. 13

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18. 18.

    Determine the mean score of the student that took the mathematics test?

    1. A.4.5
    2. B.4.3
    3. C.4.2
    4. D.4.6

    Show answer

    Answer: D. 4.6

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19. 19.

    The mean of 2-t, 4+t, 3-2t, 2+t and t-1 is

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

    Show answer

    Answer: A. 2

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20. 20.

    If line p = 5x + 3 is parallel to line p = wx + 5. Find the value of w.

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

    Show answer

    Answer: D. 5

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21. 21.

    The venn diagram shows a class of 50 students with the games they play. How many students play only two games?

    1. A.15
    2. B.16
    3. C.20
    4. D.18

    Show answer

    Answer: C. 20

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22. 22.

    If Q is a factor of 18 and T is prime numbers between 2 and 18. What is Q$\cap$T?

    1. A.(2,3)
    2. B.(2,3,18)
    3. C.(2,3,9)
    4. D.(2,3,6)

    Show answer

    Answer: A. (2,3)

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23. 23.

    OGIVE is constructed using

    1. A.Third quartile range
    2. B.Semi-quartile range
    3. C.Cummulative frequency table
    4. D.Inter-quartile table

    Show answer

    Answer: C. Cummulative frequency table

    AI explanation

    Analysis

    The question asks for the statistical tool or data representation used to construct an Ogive. An Ogive is a graphical representation of data that shows the running total of frequencies.

    Concept Explanation

    In statistics, when we deal with grouped data, we often need to visualize how the data accumulates as we move from the lowest class interval to the highest.

    1. Frequency Table: Lists the frequency of each individual class.
    2. Cumulative Frequency Table: This is created by adding the frequency of a current class to the sum of the frequencies of all previous classes.
    3. Ogive (Cumulative Frequency Curve): This is a graph plotted by using the upper class boundaries on the horizontal axis (x-axis) and the cumulative frequencies on the vertical axis (y-axis). Because it relies entirely on the cumulative frequency values to show the "build-up" of data, the cumulative frequency table is the essential foundation for its construction.

    Step-by-Step Solution

    • Step 1: To draw an Ogive, you must first calculate the cumulative frequency for each class interval.
    • Step 2: You plot the points $(x, y)$ where $x$ is the upper class boundary and $y$ is the corresponding cumulative frequency.
    • Step 3: Connecting these points with a smooth curve or straight lines results in the Ogive.
    • Step 4: Comparing this process to the options provided, it is clear that the Cumulative frequency table is the required data set.

    Conclusion

    Options A, B, and D refer to measures of dispersion or specific statistical ranges, which are calculated from data but are not the tools used to construct the graph itself. The Ogive is specifically the graphical representation of a cumulative frequency distribution.

    Exam Tip

    Whenever you see the word "Ogive" in a JAMB or WASSCE question, immediately associate it with "Cumulative Frequency." If the question asks what an Ogive is used to find, remember it is the best tool for estimating the Median, Quartiles, and Percentiles of a distribution!

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24. 24.

    A number of pencils were shared out among Bisi, Sola and Tunde in the ratio of 2:3:5 respectively. If Bisi got 5, how many were shared out?

    1. A.15
    2. B.25
    3. C.30
    4. D.50

    Show answer

    Answer: B. 25

    AI explanation

    1. Analyze the question

    We are given a total number of pencils shared among three people (Bisi, Sola, and Tunde) in a specific ratio (2:3:5). We know the exact amount Bisi received, and we need to find the total number of pencils shared.

    2. Explain the concept

    In ratio problems, the ratio represents parts of a whole.

    • Bisi's share = 2 parts
    • Sola's share = 3 parts
    • Tunde's share = 5 parts
    • Total parts = 2 + 3 + 5 = 10 parts

    If we know the value of one "part," we can find the total by multiplying that value by the total number of parts.

    3. Show the solution

    Step 1: Find the value of one part. We are told Bisi received 5 pencils, and her ratio share is 2. Let $x$ be the value of one part. $2x = 5$ $x = \frac{5}{2} = 2.5$

    Step 2: Calculate the total number of pencils. The total ratio is $2 + 3 + 5 = 10$ parts. Total pencils = (Total parts) × (Value of one part) Total pencils = $10 \times 2.5$ Total pencils = $25$

    4. Identify the answer

    The total number of pencils shared is 25. This corresponds to option B.

    )*

    5. Exam Tip

    When dealing with ratios, always sum the ratio parts first to get the "total parts." If you are given the value of one person's share, divide their actual amount by their ratio share to find the "unit value." Once you have the unit value, you can find any individual's share or the grand total easily!

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25. 25.

    If N = $\frac{p}{2}$($\frac{T_1 - T_2}{T_1}$). Find P when N = 12, T1 = 27 and T2 = 24.

    1. A.48
    2. B.108
    3. C.54
    4. D.216

    Show answer

    Answer: D. 216

    AI explanation

    1. Analyze the question

    We are given the formula $N = \frac{p}{2} \left( \frac{T_1 - T_2}{T_1} \right)$ and the values $N = 12$, $T_1 = 27$, and $T_2 = 24$. We need to isolate $p$ and calculate its value.

    2. Explain the concept

    This is a problem of substitution and algebraic manipulation. To solve for $p$, we need to rearrange the formula to make $p$ the subject of the equation. Once rearranged, we substitute the known values into the equation and simplify.

    3. Show the solution

    Step 1: Write down the formula

    $N = \frac{p}{2} \left( \frac{T_1 - T_2}{T_1} \right)$

    Step 2: Substitute the given values

    $12 = \frac{p}{2} \left( \frac{27 - 24}{27} \right)$

    Step 3: Simplify the fraction inside the bracket

    $12 = \frac{p}{2} \left( \frac{3}{27} \right)$

    Since $\frac{3}{27}$ simplifies to $\frac{1}{9}$:

    $12 = \frac{p}{2} \times \frac{1}{9}$

    $12 = \frac{p}{18}$

    Step 4: Solve for $p$ Multiply both sides by 18:

    $p = 12 \times 18$

    $p = 216$

    4. Identify the answer

    Comparing our result to the given options: A. 48 B. 108 C. 54 D. 216

    The calculated value matches option D.

    5. Tip

    When dealing with algebraic fractions in JAMB exams, always simplify the numbers inside the brackets first before performing cross-multiplication. This reduces the size of the numbers you are working with and minimizes the risk of calculation errors.

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26. 26.

    Find $\frac{dy}{dx}$. If y = 3x3 + 2x2 + 3x + 1

    1. A.9x2 + 4x + 3
    2. B.9x2 - 4x - 3
    3. C.9x2 + 4x - 3
    4. D.9x2 - 4x + 3

    Show answer

    Answer: A. 9x2 + 4x + 3

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27. 27.

    Find the mean of 10, 8, 5, 11, 12, 9, 6, 3, 15, and 23.

    1. A.11.2
    2. B.10
    3. C.10.2
    4. D.11.3

    Show answer

    Answer: C. 10.2

    AI explanation

    1. Analyze the question

    You are asked to calculate the mean (also known as the arithmetic average) of a given set of ten numbers: 10, 8, 5, 11, 12, 9, 6, 3, 15, and 23.

    2. Explain the concept

    The mean of a set of numbers is calculated by finding the sum of all the values in the set and dividing that sum by the total number of values (the count).

    Formula:

    $\text{Mean} = \frac{\sum x}{n}$

    Where:

    • $\sum x$ = Sum of all the numbers
    • $n$ = Total count of the numbers

    3. Show the solution

    Step 1: Find the sum of the numbers ($\sum x$) Sum = 10 + 8 + 5 + 11 + 12 + 9 + 6 + 3 + 15 + 23

    Let's add them systematically:

    • 10 + 8 = 18
    • 18 + 5 = 23
    • 23 + 11 = 34
    • 34 + 12 = 46
    • 46 + 9 = 55
    • 55 + 6 = 61
    • 61 + 3 = 64
    • 64 + 15 = 79
    • 79 + 23 = 102

    Total Sum = 102

    Step 2: Count the number of values ($n$) Counting the list (10, 8, 5, 11, 12, 9, 6, 3, 15, 23), we see there are exactly 10 numbers. $n = 10$

    Step 3: Calculate the mean

    $\text{Mean} = \frac{102}{10}$

    $\text{Mean} = 10.2$

    4. Identify the answer

    Comparing our result to the given options: A. 11.2 B. 10 C. 10.2 D. 11.3

    The correct option is C.

    5. Tip

    When calculating the mean of a long list of numbers, group them into pairs or sets of tens to make the addition easier and reduce the risk of a simple arithmetic error. Always double-check your count ($n$) before dividing, as miscounting the number of items is the most common mistake students make in this topic!

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28. 28.

    Simplify $\frac{0.026 \times 0.36}{0.69}$. Leave your answer in standard form

    1. A.1.36 x 10-4
    2. B.1.36 x 10-3
    3. C.1.36 x 10-1
    4. D.1.36 x 10-2

    Show answer

    Answer: D. 1.36 x 10-2

    AI explanation

    To solve this problem, we will break it down into simple arithmetic steps and then convert the final result into standard form.

    1. Analyze the question

    We are asked to simplify the expression $\frac{0.026 \times 0.36}{0.69}$ and express the result in standard form (scientific notation).

    2. Explain the concept

    • Standard Form: A number in standard form is written as $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.
    • Decimal Multiplication/Division: It is often easier to convert decimals into fractions or whole numbers by multiplying by powers of 10 to remove the decimal points.

    3. Show the solution

    Step 1: Simplify the expression by removing decimals. To make the calculation easier, let's multiply the numerator and denominator by a power of 10 that clears the decimals:

    $\frac{0.026 \times 0.36}{0.69} = \frac{26 \times 10^{-3} \times 36 \times 10^{-2}}{69 \times 10^{-2}}$

    Step 2: Simplify the powers of 10. The $10^{-2}$ in the numerator and denominator cancel each other out:

    $\frac{26 \times 36 \times 10^{-3}}{69}$

    Step 3: Perform the multiplication and division. First, multiply $26 \times 36$: $26 \times 36 = 936$

    Now, divide by $69$: $936 \div 69 \approx 13.565...$

    Wait, let's re-examine the calculation. If we calculate $0.026 \times 0.36 = 0.00936$. Now, divide $0.00936$ by $0.69$: $0.00936 \div 0.69 \approx 0.013565...$

    Rounding this to two decimal places for the coefficient (as suggested by the options): $0.013565 \approx 0.0136$

    Step 4: Convert to standard form. To write $0.0136$ in standard form, we move the decimal point two places to the right: $0.0136 = 1.36 \times 10^{-2}$

    4. Identify the answer

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

    5. Tip

    When dealing with decimals in JAMB, if the numbers look messy, convert them to whole numbers using powers of 10 (e.g., $0.026 = 26 \times 10^{-3}$). This prevents errors caused by misplacing the decimal point during long division.

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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.

    From the cyclic quadrilateral above, find < TSV

    1. A.60o
    2. B.80o
    3. C.70o
    4. D.50o

    Show answer

    Answer: C. 70o

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31. 31.

    Simplify 1 - ($\frac{1}{7}$ x 3 $\frac{1}{2}$) $\div$ $\frac{3}{4}$

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

    Show answer

    Answer: B. $\frac{1}{3}$

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32. 32.

    If the mean of 4, y, 8 and 10 is 7. Find Y?

    1. A.6
    2. B.10
    3. C.7
    4. D.9

    Show answer

    Answer: A. 6

    AI explanation

    Analysis

    To find the value of an unknown variable in a set of numbers when the mean is given, we use the definition of the arithmetic mean. The mean is the sum of all values divided by the total number of values.

    Concept

    The formula for the mean ($\bar{x}$) is:

    $\text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}}$

    In this problem:

    • The values are: 4, $y$, 8, and 10.
    • The number of values ($n$) is 4.
    • The mean is 7.

    Step-by-Step Solution

    1. Set up the equation: $$\frac{4 + y + 8 + 10}{4} = 7$$

    2. Simplify the numerator: Add the known numbers: $4 + 8 + 10 = 22$. So, the equation becomes: $$\frac{22 + y}{4} = 7$$

    3. Clear the fraction: Multiply both sides by 4 to isolate the numerator: $$22 + y = 7 \times 4$$ $$22 + y = 28$$

    4. Solve for $y$: Subtract 22 from both sides: $$y = 28 - 22$$ $$y = 6$$

    Conclusion

    The value of $y$ is 6.

    Exam Tip

    When dealing with mean questions, a quick way to check your work is to multiply the mean by the number of items to get the "Total Sum." Here, $7 \times 4 = 28$. Then, subtract the sum of the known numbers ($4 + 8 + 10 = 22$) from the total sum ($28 - 22 = 6$). This is a faster way to solve it during the actual JAMB exam!

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33. 33.

    The nth term of the sequence 3, 9, 27, 81.....is

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

    Show answer

    Answer: B. 3 x 3n-1

    AI explanation

    Analysis

    The sequence provided is 3, 9, 27, 81, ... We need to find the general formula (the $n^{th}$ term) that generates these numbers.

    Concept: Geometric Progression (GP)

    A sequence is a geometric progression if the ratio between consecutive terms is constant. This constant is called the common ratio ($r$).

    • The first term ($a$) = 3
    • The second term = 9
    • The third term = 27

    To find the common ratio ($r$): $r = \frac{\text{second term}}{\text{first term}} = \frac{9}{3} = 3$

    The formula for the $n^{th}$ term ($T_n$) of a geometric progression is: $T_n = a \cdot r^{(n-1)}$

    Step-by-Step Solution

    1. Identify the variables: • $a = 3$ • $r = 3$

    2. Substitute these values into the formula: $T_n = 3 \cdot 3^{(n-1)}$

    3. Verify the formula: • For $n = 1$: $T_1 = 3 \cdot 3^{(1-1)} = 3 \cdot 3^0 = 3 \cdot 1 = 3$ (Correct) • For $n = 2$: $T_2 = 3 \cdot 3^{(2-1)} = 3 \cdot 3^1 = 3 \cdot 3 = 9$ (Correct) • For $n = 3$: $T_3 = 3 \cdot 3^{(3-1)} = 3 \cdot 3^2 = 3 \cdot 9 = 27$ (Correct)

    Comparing our derived formula $3 \cdot 3^{n-1}$ with the given options, it matches option B.

    Conclusion

    The $n^{th}$ term of the sequence is $3 \times 3^{n-1}$.

    Exam Tip

    When you see a sequence where each term is a power of the same number (like 3, 9, 27...), always check if it is a geometric progression. If the first term is the same as the common ratio, the formula will always simplify to $a^n$ or $r^n$. In this case, $3^1, 3^2, 3^3, 3^4...$ which is simply $3^n$. Since the options are written in the form $3 \times 3^{n-1}$, remember that $3^1 \times 3^{n-1} = 3^{1+n-1} = 3^n$.

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34. 34.

    Given U = {x: x is a positive integer less than 15} and P = {x: x is even number from 1 to 14}. Find the compliment of P.

    1. A.{1, 3, 5, 7, 9, 11, 13, 15}
    2. B.{2, 3, 5, 7, 9, 11, 13}
    3. C.{1, 3, 5, 7, 9, 11, 13}
    4. D.{2, 3, 5, 7, 11, 15}

    Show answer

    Answer: C. {1, 3, 5, 7, 9, 11, 13}

    AI explanation

    1. Analyze the question

    We are given a Universal set ($U$) containing positive integers less than 15, and a subset ($P$) containing even numbers from 1 to 14. We need to find the complement of $P$ (denoted as $P'$ or $P^c$), which consists of all elements in $U$ that are not in $P$.

    2. Explain the concept

    • Universal Set ($U$): The set of all elements under consideration. Here, "positive integers less than 15" means $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}$.
    • Subset ($P$): The set of even numbers from 1 to 14. These are $\{2, 4, 6, 8, 10, 12, 14\}$.
    • Complement ($P'$): This is the set of elements that are in $U$ but are not in $P$. Essentially, you remove all even numbers from the universal set, which leaves you with the odd numbers.

    3. Show the solution

    • Step 1: List the elements of $U$ $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}$

    • Step 2: List the elements of $P$ $P = \{2, 4, 6, 8, 10, 12, 14\}$

    • Step 3: Find $P'$ (Elements in $U$ not in $P$) We remove $\{2, 4, 6, 8, 10, 12, 14\}$ from $U$: $P' = \{1, 3, 5, 7, 9, 11, 13\}$

    4. Identify the answer

    Comparing our result $\{1, 3, 5, 7, 9, 11, 13\}$ with the given options:

    • Option A includes 15 (which is not in $U$ because $U$ is less than 15).
    • Option B includes 2 (which is in $P$).
    • Option C matches our result exactly.
    • Option D is missing several odd numbers.

    5. Tip

    When dealing with set complements, always double-check the definition of your Universal set. If the question says "less than 15," 15 is excluded. If it says "15 or less," 15 is included. Always list out the elements of the Universal set first to avoid missing numbers!

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

36. 36.

    If y = 2x3 + 6x2 + 6x + 1, Find $\frac{dy}{dx}$

    1. A.6x2 + 12x + 1
    2. B.6x2 + 6x + 1
    3. C.6x2 + 6x + 6
    4. D.6x2 + 12x + 6

    Show answer

    Answer: D. 6x2 + 12x + 6

    AI explanation

    Analysis

    The question asks for the derivative of the polynomial function $y = 2x^3 + 6x^2 + 6x + 1$ with respect to $x$. This is a standard calculus problem involving the power rule for differentiation.

    Concept: The Power Rule

    To differentiate a term of the form $ax^n$, you use the Power Rule:

    $\frac{d}{dx}(ax^n) = n \cdot ax^{n-1}$

    Essentially, you multiply the coefficient by the exponent and then subtract 1 from the exponent.

    • The derivative of a constant (like 1) is always 0.
    • The derivative of $x$ (which is $x^1$) is 1.

    Step-by-Step Solution

    Given the function:

    $y = 2x^3 + 6x^2 + 6x + 1$

    We differentiate each term individually:

    1. For $2x^3$: Multiply 3 by 2 and subtract 1 from the exponent: $3 \times 2x^{(3-1)} = 6x^2$

    2. For $6x^2$: Multiply 2 by 6 and subtract 1 from the exponent: $2 \times 6x^{(2-1)} = 12x^1 = 12x$

    3. For $6x$: The derivative of $6x$ is simply the coefficient: $1 \times 6x^{(1-1)} = 6x^0 = 6(1) = 6$

    4. For $1$: The derivative of any constant is 0.

    Combining these results:

    $\frac{dy}{dx} = 6x^2 + 12x + 6$

    Conclusion

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

    Exam Tip: Always remember to differentiate the constant term! Many students mistakenly leave the constant in the final answer or forget to change it to zero. Also, double-check your arithmetic when multiplying the exponent by the coefficient—it's a common place for "silly" mistakes under exam pressure.

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37. 37.

    Integrate $\frac{2x^3 + 2x}{x}$ with respect to x

    1. A.$\frac{2x^3}{3}$ - 2x + k
    2. B.x3 + 2x + k
    3. C.$\frac{2x^3}{3}$ + 2x + k
    4. D.x3 - 2x + k

    Show answer

    Answer: C. $\frac{2x^3}{3}$ + 2x + k

    AI explanation

    Analysis

    The question asks us to find the indefinite integral of the function $\frac{2x^3 + 2x}{x}$ with respect to $x$. Before we can integrate, we must simplify the algebraic expression to make it easier to handle.

    Concept: Integration of Algebraic Functions

    To integrate a polynomial, we use the Power Rule for Integration:

    $\int x^n \, dx = \frac{x^{n+1}}{n+1} + k$

    where $k$ is the constant of integration.

    Step-by-Step Solution

    Step 1: Simplify the expression Divide each term in the numerator by the denominator ($x$):

    $\frac{2x^3 + 2x}{x} = \frac{2x^3}{x} + \frac{2x}{x}$

    $= 2x^2 + 2$

    Step 2: Set up the integral Now, integrate the simplified expression with respect to $x$:

    $\int (2x^2 + 2) \, dx$

    Step 3: Integrate term by term Apply the power rule to each term:

    • For $2x^2$: $\int 2x^2 \, dx = 2 \left( \frac{x^{2+1}}{2+1} \right) = \frac{2x^3}{3}$
    • For $2$: $\int 2 \, dx = 2x$

    Step 4: Combine and add the constant

    $\frac{2x^3}{3} + 2x + k$

    Conclusion

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

    Exam Tip

    Whenever you see a fraction in an integration problem, always check if you can simplify it by dividing the numerator by the denominator first. It usually turns a complex-looking problem into a very simple one! Also, never forget to add the constant $k$ (or $c$) at the end of every indefinite integral; examiners often penalize for missing it.

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38. 38.

    Factorize k2 - 2kp + p2.

    1. A.(k +p)2
    2. B.(k - p)2
    3. C.k2 + p2
    4. D.k2 - p2

    Show answer

    Answer: B. (k - p)2

    AI explanation

    Analysis

    The question asks us to factorize the quadratic expression $k^2 - 2kp + p^2$. This is a classic algebraic expression that follows the pattern of a Perfect Square Trinomial.

    Concept

    A perfect square trinomial follows the general form: $a^2 - 2ab + b^2 = (a - b)^2$

    In this expression, we are looking for two terms that, when squared, give us the first and last terms of our expression, and when multiplied together and doubled, give us the middle term.

    Step-by-Step Solution

    1. Identify the terms: • The first term is $k^2$, which is $(k)^2$. • The last term is $p^2$, which is $(p)^2$. • The middle term is $-2kp$.

    2. Check the middle term: • According to the formula $(a - b)^2 = a^2 - 2ab + b^2$, the middle term should be $-2 \times (\text{first term}) \times (\text{last term})$. • Here, $-2 \times k \times p = -2kp$. • Since this matches our middle term, the expression is indeed a perfect square.

    3. Write the factors: • Since the middle term is negative, we use the subtraction form: $(k - p)^2$.

    4. Verification: • $(k - p)(k - p) = k(k - p) - p(k - p)$ • $= k^2 - kp - kp + p^2$ • $= k^2 - 2kp + p^2$ • This confirms our factorization is correct.

    Conclusion

    Comparing our result $(k - p)^2$ with the given options, it matches option B.

    Exam Tip

    Whenever you see a quadratic expression with three terms where the first and last terms are perfect squares (like $x^2$ and $y^2$), always check if the middle term is exactly twice the product of the square roots of the first and last terms. If it is, you can immediately write it as $(a \pm b)^2$ without needing to go through the long process of splitting the middle term!

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39. 39.

    The sum of the interior angles of a polygon is a given as 1080o. Find the number of the sides of the polygon.

    1. A.5
    2. B.7
    3. C.6
    4. D.8

    Show answer

    Answer: D. 8

    AI explanation

    Analysis

    To find the number of sides of a polygon when the sum of its interior angles is known, we use the geometric formula relating the number of sides ($n$) to the sum of interior angles ($S$).

    Concept

    The formula for the sum of the interior angles of any polygon with $n$ sides is:

    $S = (n - 2) \times 180^\circ$

    Where:

    • $S$ is the sum of the interior angles.
    • $n$ is the number of sides of the polygon.

    Solution

    Step 1: Set up the equation. Given that $S = 1080^\circ$, we substitute this into our formula:

    $1080 = (n - 2) \times 180$

    Step 2: Isolate the term with $n$. Divide both sides of the equation by $180$:

    $\frac{1080}{180} = n - 2$

    Step 3: Perform the division.

    $6 = n - 2$

    Step 4: Solve for $n$. Add 2 to both sides:

    $n = 6 + 2$

    $n = 8$

    The polygon has 8 sides (it is an octagon).

    Conclusion

    Comparing our result to the given options, the correct answer is D.

    Exam Tip

    A quick way to remember this is to divide the given sum by 180 first, then simply add 2 to the result. Also, remember that a triangle ($n=3$) has a sum of $180^\circ$, a quadrilateral ($n=4$) has $360^\circ$, and every time you add a side, you add another $180^\circ$ to the total sum!

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

41. 41.

    Solve for x and y respectively 3x - 5y = 9 6x - 4y = 12

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

    Show answer

    Answer: D. $\frac{4}{3}$, -1

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42. 42.

    Rationalize $\frac{\sqrt{6} - \sqrt{4}}{\sqrt{6} + \sqrt{4}}$

    1. A.5 + 2 $\sqrt{6}$
    2. B.5 - 4 $\sqrt{6}$
    3. C.5 + 4 $\sqrt{6}$
    4. D.5 - 2 $\sqrt{6}$

    Show answer

    Answer: D. 5 - 2 $\sqrt{6}$

    AI explanation

    Analysis

    To rationalize a fraction with a radical expression in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(\sqrt{6} + \sqrt{4})$ is $(\sqrt{6} - \sqrt{4})$.

    Concept

    Rationalization is used to remove square roots from the denominator. Recall the algebraic identity: $(a + b)(a - b) = a^2 - b^2$

    Step-by-Step Solution

    Step 1: Set up the multiplication

    $\frac{\sqrt{6} - \sqrt{4}}{\sqrt{6} + \sqrt{4}} \times \frac{\sqrt{6} - \sqrt{4}}{\sqrt{6} - \sqrt{4}}$

    Step 2: Expand the numerator The numerator is $(\sqrt{6} - \sqrt{4})(\sqrt{6} - \sqrt{4}) = (\sqrt{6} - \sqrt{4})^2$. Using the expansion $(a - b)^2 = a^2 - 2ab + b^2$: $= (\sqrt{6})^2 - 2(\sqrt{6})(\sqrt{4}) + (\sqrt{4})^2$ $= 6 - 2\sqrt{24} + 4$ $= 10 - 2\sqrt{4 \times 6}$ $= 10 - 2(2\sqrt{6})$ $= 10 - 4\sqrt{6}$

    Step 3: Expand the denominator Using the difference of two squares $(a + b)(a - b) = a^2 - b^2$: $= (\sqrt{6})^2 - (\sqrt{4})^2$ $= 6 - 4$ $= 2$

    Step 4: Simplify the fraction

    $\frac{10 - 4\sqrt{6}}{2}$

    Divide both terms in the numerator by 2: $= \frac{10}{2} - \frac{4\sqrt{6}}{2}$ $= 5 - 2\sqrt{6}$

    Conclusion

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

    Exam Tip

    When rationalizing, always check if the square roots can be simplified further (like $\sqrt{4} = 2$ or $\sqrt{24} = 2\sqrt{6}$) before or during the calculation to keep your numbers small and manageable. This prevents errors during multiplication!

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

44. 44.

    Find the derivative of y = ( $\frac{1}{3}$X + 6)$^2$

    1. A.2( $\frac{1}{3}$X + 6
    2. B.$\frac{2}{3}$ ( $\frac{1}{3}$X + 6)
    3. C.$\frac{2}{3}$ ( $\frac{1}{3}$X + 6)2
    4. D.$\frac{1}{3}$ ( $\frac{1}{3}$X + 6)2

    Show answer

    Answer: B. $\frac{2}{3}$ ( $\frac{1}{3}$X + 6)

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45. 45.

    Calculate the perimeter of a sector of a circle of raduis 12cm and angle 60o.

    1. A.(12 + 4$\pi$)cm
    2. B.(24 + 4$\pi$)cm
    3. C.(12 + 6$\pi$)cm
    4. D.(24 + 6$\pi$)cm

    Show answer

    Answer: B. (24 + 4$\pi$)cm

    AI explanation

    1. Analyze the question

    We are asked to find the perimeter of a sector of a circle.

    • Radius ($r$): 12 cm
    • Angle ($\theta$): 60°

    2. Explain the concept

    The perimeter of a sector is not just the length of the curved arc; it is the total distance around the shape. A sector is bounded by two radii and an arc.

    • Formula for Arc Length ($L$): $L = \frac{\theta}{360} \times 2\pi r$
    • Formula for Perimeter of a Sector: $P = \text{Arc Length} + 2r$ (because you must include the two straight radii).

    3. Show the solution

    Step 1: Calculate the Arc Length ($L$) Using the formula $L = \frac{\theta}{360} \times 2\pi r$: $L = \frac{60}{360} \times 2 \times \pi \times 12$ $L = \frac{1}{6} \times 24\pi$ $L = 4\pi \text{ cm}$

    Step 2: Calculate the Perimeter ($P$) The perimeter includes the arc length plus the two radii that form the sides of the sector: $P = L + r + r$ $P = 4\pi + 12 + 12$ $P = 4\pi + 24$ $P = (24 + 4\pi) \text{ cm}$

    4. Identify the answer

    Comparing our result with the given options: A. (12 + 4$\pi$)cm B. (24 + 4$\pi$)cm C. (12 + 6$\pi$)cm D. (24 + 6$\pi$)cm

    The calculated value matches option B.

    5. Tip

    A common mistake in JAMB exams is calculating only the arc length and forgetting to add the two radii. Always visualize the shape: a sector looks like a slice of pizza—to walk around the edge, you must walk along the crust (arc) and then back along both straight sides (radii).

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46. 46.

    If a car travels 120km on 45 litres of petrol, how much petrol is needed for a journey of 600km?

    1. A.720 litres
    2. B.160 litres
    3. C.225 litres
    4. D.960 litres

    Show answer

    Answer: C. 225 litres

    AI explanation

    Analysis

    This is a direct proportion problem. Since the car consumes petrol at a constant rate, the ratio of the distance traveled to the petrol consumed remains the same for any distance.

    Concept

    To solve this, we use the principle of proportionality:

    $\frac{\text{Distance}_1}{\text{Petrol}_1} = \frac{\text{Distance}_2}{\text{Petrol}_2}$

    Where:

    • Distance$_1$ = 120 km
    • Petrol$_1$ = 45 litres
    • Distance$_2$ = 600 km
    • Petrol$_2$ = $x$ (the unknown amount we need to find)

    Step-by-Step Solution

    Step 1: Set up the equation.

    $\frac{120}{45} = \frac{600}{x}$

    Step 2: Cross-multiply to solve for $x$.

    $120 \times x = 600 \times 45$

    Step 3: Simplify the calculation.

    $120x = 27,000$

    Step 4: Divide both sides by 120 to isolate $x$.

    $x = \frac{27,000}{120}$

    $x = \frac{2,700}{12}$

    $x = 225$

    So, the car needs 225 litres of petrol for a 600 km journey.

    Conclusion

    The calculation confirms that 225 litres are required.

    Correct Option: C

    Exam Tip

    When dealing with direct proportion questions in JAMB, you can also use the "Unitary Method":

    1. Find out how much petrol is needed for just 1 km: $45 \div 120 = 0.375$ litres/km.
    2. Multiply that by the new distance: $0.375 \times 600 = 225$ litres. This method is often faster and less prone to errors when you are working under time pressure!

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47. 47.

    In the cyclic quadrilateral above . Find < PRO

    1. A.70o
    2. B.20o
    3. C.50o
    4. D.30o

    Show answer

    Answer: D. 30o

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48. 48.

    Evaluate $\frac{0.8 \times 0.43 \times 0.031}{0.05 \times 0.72 \times 0.021}$. Correct to four significant figures.

    1. A.14.09
    2. B.14.1
    3. C.14.12
    4. D.14.11

    Show answer

    Answer: D. 14.11

    AI explanation

    To solve this expression, we will simplify the decimals by converting them into powers of 10 to make the division easier.

    1. Analyze the expression

    We have:

    $\frac{0.8 \times 0.43 \times 0.031}{0.05 \times 0.72 \times 0.021}$

    2. Convert decimals to standard form

    To remove the decimals, we can express each number as a fraction or multiply both the numerator and denominator by a power of 10 that clears all decimal places.

    • Numerator: $0.8 \times 0.43 \times 0.031 = \frac{8}{10} \times \frac{43}{100} \times \frac{31}{1000} = \frac{8 \times 43 \times 31}{10^6} = \frac{10664}{1,000,000}$
    • Denominator: $0.05 \times 0.72 \times 0.021 = \frac{5}{100} \times \frac{72}{100} \times \frac{21}{1000} = \frac{5 \times 72 \times 21}{10^7} = \frac{7560}{10,000,000}$

    3. Perform the division

    Now, divide the numerator by the denominator:

    $\frac{10664 \times 10^{-6}}{7560 \times 10^{-7}}$

    When dividing powers of 10, subtract the exponents:

    $\frac{10664}{7560} \times 10^{(-6 - (-7))}$

    $\frac{10664}{7560} \times 10^1$

    Now, calculate the division:

    $10664 \div 7560 \approx 1.410582...$

    Multiply by $10^1$:

    $1.410582 \times 10 = 14.10582...$

    4. Round to four significant figures

    Significant figures start from the first non-zero digit.

    1. First: 1
    2. Second: 4
    3. Third: 1
    4. Fourth: 0

    The next digit is 5, so we round the fourth digit up from 0 to 1. Result: 14.11

    5. Conclusion

    Comparing our result to the options provided: A. 14.09 B. 14.1 C. 14.12 D. 14.11

    The calculated value matches option D.

    Exam Tip: When dealing with many decimal places in JAMB, always try to shift the decimal point to the right for both the numerator and denominator by the same number of places. This turns the problem into simple long division of whole numbers, which is much less prone to error!

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