2022 JAMB Mathematics Past Questions and Answers

Questions

1. 1.

   Three times a certain number (x), minus 2 is less than the number minus 6.Find the possible values of x.

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

   Show answer

   Answer: A. x <-2

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2. 2.

   A poultry farmer realized 200 eggs from his birds and sold 5 $\frac{1}{2}$ crates of eggs. What percentages of eggs are left?

   1. A.82.5%
   2. B.15%
   3. C.17.5%
   4. D.85%

   Show answer

   Answer: C. 17.5%

   AI explanation

   1. Analyze the question

   • Total eggs: 200
   • Eggs sold: 5 $\frac{1}{2}$ crates.
   • Standard crate size: In the Nigerian poultry industry, a standard crate holds 30 eggs.
   • Goal: Find the percentage of eggs remaining.

   2. Explain the concept

   To solve this, we need to:

   1. Calculate the total number of eggs sold by multiplying the number of crates by the number of eggs per crate (30).
   2. Subtract the sold eggs from the total (200) to find the number of eggs left.
   3. Calculate the percentage left using the formula: $$\text{Percentage left} = \left( \frac{\text{Number left}}{\text{Total number}} \right) \times 100$$

   3. Show the solution

   Step 1: Calculate eggs sold

   • Crates sold = $5 \frac{1}{2} = 5.5$
   • Eggs per crate = 30
   • Total eggs sold = $5.5 \times 30 = 165$ eggs.

   Step 2: Calculate eggs left

   • Total eggs = 200
   • Eggs left = $200 - 165 = 35$ eggs.

   Step 3: Calculate the percentage left

   • Percentage left = $\left( \frac{35}{200} \right) \times 100$
   • Percentage left = $\frac{35}{2} = 17.5\%$

   4. Identify the answer

   Comparing our result to the options provided: A. 82.5% B. 15% C. 17.5% D. 85%

   The correct option is C.

   5. Tip

   In JAMB questions involving "crates of eggs," always remember that the standard unit is 30 eggs per crate. If the question doesn't explicitly state the crate size, assume 30, as this is the standard practice in Nigerian poultry farming and mathematics examinations.

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3. 3.

   The number line represented by the inequality

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

   Show answer

   Answer: C. x≥2

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4. 4.

   Given that S = $3t^{2}$ + 5t - 10 is displacement of a particle in metres, calculate it initial velocity.

   1. A.10m/s
   2. B.2m/s
   3. C.5m/s
   4. D.6m/s

   Show answer

   Answer: C. 5m/s

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

6. 6.

   A rectangular pyramid has an area 24cm$^2$ and height 7.5cm. Find its volume?

   1. A.65.0cm$^3$
   2. B.70.5cm$^3$
   3. C.56.5cm$^3$
   4. D.60.0cm$^3$

   Show answer

   Answer: D. 60.0cm$^3$

   AI explanation

   Analysis

   To find the volume of a pyramid, we need to use the standard geometric formula that relates the base area and the perpendicular height of the pyramid.

   Concept

   The volume ($V$) of any pyramid (whether rectangular, square, or triangular) is given by the formula:

   $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$

   This formula applies because a pyramid occupies exactly one-third of the volume of a prism with the same base and height.

   Solution

   Step 1: Identify the given values.

   • Base Area ($A$) = $24 \text{ cm}^2$
   • Height ($h$) = $7.5 \text{ cm}$

   Step 2: Substitute the values into the formula.

   $V = \frac{1}{3} \times 24 \times 7.5$

   Step 3: Perform the calculation. First, simplify the fraction:

   $V = (\frac{24}{3}) \times 7.5$

   $V = 8 \times 7.5$

   Now, multiply $8$ by $7.5$:

   $8 \times 7 = 56$

   $8 \times 0.5 = 4$

   $56 + 4 = 60$

   So, $V = 60.0 \text{ cm}^3$.

   Conclusion

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

   Exam Tip

   When dealing with volume problems for pyramids and cones, always remember the "one-third" rule. Students often forget to divide by 3 and end up calculating the volume of a prism instead. If you see "pyramid" or "cone," immediately write down $\frac{1}{3}$ to remind yourself!

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7. 7.

   If sin θ = -  $\frac{3}{5}$ and θ lies in the third quadrant, find cos θ

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

   Show answer

   Answer: D. - $\frac{4}{5}$

   AI explanation

   Analysis

   We are given the value of $\sin \theta$ and the specific quadrant where $\theta$ lies. We need to determine the value of $\cos \theta$.

   Concept: Trigonometric Identities and Quadrants

   1. Pythagorean Identity: The fundamental relationship between sine and cosine is: $$\sin^2 \theta + \cos^2 \theta = 1$$
   2. Quadrant Rules (CAST Rule): • 1st Quadrant (All): All trigonometric ratios are positive. • 2nd Quadrant (Sine): Only $\sin \theta$ is positive. • 3rd Quadrant (Tangent): Only $\tan \theta$ is positive ($\sin \theta$ and $\cos \theta$ are negative). • 4th Quadrant (Cosine): Only $\cos \theta$ is positive.

   Step-by-Step Solution

   Step 1: Use the Pythagorean identity. Given $\sin \theta = -\frac{3}{5}$, substitute this into the identity:

   $(-\frac{3}{5})^2 + \cos^2 \theta = 1$

   $\frac{9}{25} + \cos^2 \theta = 1$

   Step 2: Solve for $\cos^2 \theta$.

   $\cos^2 \theta = 1 - \frac{9}{25}$

   $\cos^2 \theta = \frac{25 - 9}{25}$

   $\cos^2 \theta = \frac{16}{25}$

   Step 3: Find $\cos \theta$ by taking the square root.

   $\cos \theta = \pm \sqrt{\frac{16}{25}}$

   $\cos \theta = \pm \frac{4}{5}$

   Step 4: Determine the sign based on the quadrant. The question states that $\theta$ lies in the third quadrant. According to the CAST rule, only tangent is positive in the third quadrant, meaning both sine and cosine must be negative. Therefore, we choose the negative value:

   $\cos \theta = -\frac{4}{5}$

   Conclusion

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

   Exam Tip: Always remember the CAST mnemonic (starting from the 4th quadrant or 1st quadrant counter-clockwise): Cosine (4th), All (1st), Sine (2nd), Tangent (3rd). It is a lifesaver for questions involving quadrants!

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8. 8.

   Simplify $\frac{1}{3-√2}$ in the form of p + q√2

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

   Show answer

   Answer: D. $\frac{3}{7}$ + $\frac{√2}{7}$

   AI explanation

   Analysis

   The goal is to simplify the expression $\frac{1}{3-\sqrt{2}}$ by removing the square root from the denominator. This process is known as rationalizing the denominator.

   Concept

   To rationalize a denominator in the form $(a - \sqrt{b})$, you multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(3 - \sqrt{2})$ is $(3 + \sqrt{2})$.

   Multiplying by the conjugate works because of the difference of two squares identity: $(a - b)(a + b) = a^2 - b^2$

   Step-by-Step Solution

   Step 1: Multiply the numerator and denominator by the conjugate.

   $\frac{1}{3 - \sqrt{2}} \times \frac{3 + \sqrt{2}}{3 + \sqrt{2}}$

   Step 2: Expand the numerator and the denominator. Numerator: $1 \times (3 + \sqrt{2}) = 3 + \sqrt{2}$

   Denominator: $(3 - \sqrt{2})(3 + \sqrt{2})$ Using the identity $(a - b)(a + b) = a^2 - b^2$: $= 3^2 - (\sqrt{2})^2$ $= 9 - 2$ $= 7$

   Step 3: Combine the results.

   $\frac{3 + \sqrt{2}}{7}$

   Step 4: Express in the form $p + q\sqrt{2}$. Split the fraction:

   $\frac{3}{7} + \frac{\sqrt{2}}{7}$

   This matches the form $p + q\sqrt{2}$, where $p = \frac{3}{7}$ and $q = \frac{1}{7}$.

   Conclusion

   Comparing our result $\frac{3}{7} + \frac{\sqrt{2}}{7}$ with the given options, it matches option D.

   Exam Tip: Whenever you see a surd in the denominator, immediately think "conjugate." If the denominator is $(a - \sqrt{b})$, multiply by $(a + \sqrt{b})$. If it is $(a + \sqrt{b})$, multiply by $(a - \sqrt{b})$. This will always clear the root from the bottom!

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9. 9.

   Given that r = $\sqrt \frac{3v}{\pi h}$, make v the subject of the formula

   1. A.v = 3 $πr^2$ h
   2. B.v = $\frac{πrh}{3}$
   3. C.v =  $\frac{πr^2h}{3}$
   4. D.v = 3πrh

   Show answer

   Answer: C. v =  $\frac{πr^2h}{3}$

   AI explanation

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

   1. Analyze the equation

   We are given:

   $r = \sqrt{\frac{3v}{\pi h}}$

   Our goal is to get $v$ by itself.

   2. Explain the concept

   To solve for a variable trapped inside a square root, we must perform the inverse operation, which is squaring both sides. Once the square root is removed, we can use basic algebraic manipulation (multiplication and division) to isolate $v$.

   3. Step-by-step solution

   Step 1: Square both sides to remove the square root.

   $r^2 = \left( \sqrt{\frac{3v}{\pi h}} \right)^2$

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

   Step 2: Clear the fraction by multiplying both sides by $(\pi h)$.

   $r^2 \times (\pi h) = 3v$

   $\pi r^2 h = 3v$

   Step 3: Isolate $v$ by dividing both sides by 3.

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

   Rearranging it to look like the standard form:

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

   4. Identify the answer

   Comparing our result to the given options: A. $v = 3\pi r^2 h$ B. $v = \frac{\pi rh}{3}$ C. $v = \frac{\pi r^2 h}{3}$ D. $v = 3\pi rh$

   The correct option is C.

   5. Exam Tip

   When dealing with "change of subject" questions involving square roots, always square both sides first to "free" the variables inside. If you have a fraction, multiply to clear the denominator immediately after squaring. This prevents common errors like forgetting to square the other side of the equation!

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10. 10.

    If A = $\begin{pmatrix} 2 & 1 \\ 2 & 3 \\ 1 & 2 \end{pmatrix}$ and B = $\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}$. Find AB

    1. A.$\begin{pmatrix} 18 & 6 \\ 12 & 10 \\ 10 & 6 \end{pmatrix}$
    2. B.$\begin{pmatrix} 10 & 6 \\ 13 & 10 \\ 12 & 6 \end{pmatrix}$
    3. C.$\begin{pmatrix} 10 & 6 \\ 12 & 10 \\ 11 & 6 \end{pmatrix}$
    4. D.$\begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}$

    Show answer

    Answer: D. $\begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}$

    AI explanation

    Analysis

    To find the product of two matrices $AB$, we must multiply the rows of matrix $A$ by the columns of matrix $B$.

    • Matrix $A$ is a $3 \times 2$ matrix.
    • Matrix $B$ is a $2 \times 2$ matrix.
    • The resulting matrix $AB$ will be a $3 \times 2$ matrix.

    Concept

    For matrix multiplication, we multiply the elements of the row of the first matrix by the corresponding elements of the column of the second matrix and sum them up. If $C = AB$, then $C_{ij} = (\text{Row } i \text{ of } A) \cdot (\text{Column } j \text{ of } B)$.

    Solution

    Given: $A = \begin{pmatrix} 2 & 1 \\ 2 & 3 \\ 1 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}$

    Step 1: Calculate the first row of the result

    • Row 1, Col 1: $(2 \times 3) + (1 \times 4) = 6 + 4 = 10$
    • Row 1, Col 2: $(2 \times 2) + (1 \times 2) = 4 + 2 = 6$

    Step 2: Calculate the second row of the result

    • Row 2, Col 1: $(2 \times 3) + (3 \times 4) = 6 + 12 = 18$
    • Row 2, Col 2: $(2 \times 2) + (3 \times 2) = 4 + 6 = 10$

    Step 3: Calculate the third row of the result

    • Row 3, Col 1: $(1 \times 3) + (2 \times 4) = 3 + 8 = 11$
    • Row 3, Col 2: $(1 \times 2) + (2 \times 2) = 2 + 4 = 6$

    Putting it all together: $AB = \begin{pmatrix} 10 & 6 \\ 18 & 10 \\ 11 & 6 \end{pmatrix}$

    Conclusion

    Comparing our result with the given options, we see that it matches option D.

    Exam Tip

    When multiplying matrices, always remember the phrase "Row by Column". Use your left index finger to trace the row of the first matrix and your right index finger to trace the column of the second matrix simultaneously. This prevents you from skipping elements or getting confused during the calculation.

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11. 11.

    The equivalent of (10110.011)$_2$ in base 10 is?

    1. A.26.325
    2. B.24.372
    3. C.42.443
    4. D.22.375

    Show answer

    Answer: D. 22.375

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12. 12.

    If a fair coin is tossed twice, what is the probability of obtaining at least one head?

    1. A.0.25
    2. B.0.75
    3. C.0.5
    4. D.0.33

    Show answer

    Answer: B. 0.75

    AI explanation

    1. Analyze the question

    We are tossing a fair coin twice. We need to find the probability of getting at least one head. "At least one" means we are looking for the outcomes that have either one head or two heads.

    2. Explain the concept

    When tossing a coin, there are two possible outcomes for each toss: Head (H) or Tail (T). When tossing a coin twice, the total number of possible outcomes is calculated as $2^n$, where $n$ is the number of tosses. Here, $2^2 = 4$.

    The sample space (all possible outcomes) is:

    • (H, H) — Two heads
    • (H, T) — Head then Tail
    • (T, H) — Tail then Head
    • (T, T) — Two tails

    3. Show the solution

    • Total outcomes: 4
    • Favorable outcomes (at least one head): • (H, H) - Yes • (H, T) - Yes • (T, H) - Yes • (T, T) - No (This has zero heads)

    There are 3 favorable outcomes.

    Probability formula:

    $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

    $P(\text{at least one head}) = \frac{3}{4}$

    Converting the fraction to a decimal:

    $3 \div 4 = 0.75$

    4. Identify the answer

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

    5. Exam Tip

    Whenever you see the phrase "at least one," it is often easier to use the complement rule:

    $P(\text{at least one}) = 1 - P(\text{none})$

    The only outcome with "no heads" is (T, T), which has a probability of $\frac{1}{4}$ (or 0.25). So, $1 - 0.25 = 0.75$. This trick saves time in exams!

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

14. 14.

    Factorize 4a$^2$ - 9b$^2$

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

    Show answer

    Answer: A. (2a-3b) (2a+3b)

    AI explanation

    Analysis

    The expression given is $4a^2 - 9b^2$. This is a classic algebraic expression that fits the Difference of Two Squares pattern.

    Concept: Difference of Two Squares

    The difference of two squares is a special algebraic identity that states:

    $x^2 - y^2 = (x - y)(x + y)$

    To use this, you must be able to express both terms in the given problem as perfect squares.

    Step-by-Step Solution

    1. Identify the squares: • The first term is $4a^2$. Since $4 = 2^2$, we can write this as $(2a)^2$. • The second term is $9b^2$. Since $9 = 3^2$, we can write this as $(3b)^2$.

    2. Rewrite the expression: Substitute these back into the original expression: $$(2a)^2 - (3b)^2$$

    3. Apply the identity: Using the formula $x^2 - y^2 = (x - y)(x + y)$, where $x = 2a$ and $y = 3b$: $$(2a)^2 - (3b)^2 = (2a - 3b)(2a + 3b)$$

    Conclusion

    Comparing our result $(2a - 3b)(2a + 3b)$ with the given options, it matches option A perfectly.

    Exam Tip

    Whenever you see a binomial (two terms) with a minus sign in the middle, always check if both terms are perfect squares (like 1, 4, 9, 16, 25, $a^2$, $x^4$, etc.). If they are, you can immediately apply the $(x-y)(x+y)$ rule to save time during your JAMB exam!

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15. 15.

    Integrate (2x+1)$^3$

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

    Show answer

    Answer: B. $\frac{{2x+1}^4}{8}$ + C

    AI explanation

    To solve the integral of $(2x + 1)^3$ with respect to $x$, we use the method of Integration by Substitution (or the reverse chain rule).

    1. Analyze the question

    We are asked to find the indefinite integral:

    $\int (2x + 1)^3 \, dx$

    2. Explain the concept

    When integrating a function of the form $(ax + b)^n$, where $a$ and $b$ are constants, we use the rule:

    $\int (ax + b)^n \, dx = \frac{(ax + b)^{n+1}}{a(n+1)} + C$

    This rule works because we increase the power by 1 and then divide by the new power, while also dividing by the derivative of the inner function $(2x + 1)$, which is $2$.

    3. Show the solution

    • Step 1: Identify the components. Here, $n = 3$ and the derivative of the inner function $(2x + 1)$ is $2$.
    • Step 2: Apply the power rule for integration. Increase the exponent by 1: $$(2x + 1)^{3+1} = (2x + 1)^4$$
    • Step 3: Divide by the new exponent ($4$) and multiply by the derivative of the inner function ($2$): $$\text{Result} = \frac{(2x + 1)^4}{4 \times 2} + C$$
    • Step 4: Simplify the denominator: $$\text{Result} = \frac{(2x + 1)^4}{8} + C$$

    4. Identify the answer

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

    5. Exam Tip

    Whenever you see a linear expression inside a power (like $ax+b$), don't forget to divide by the coefficient of $x$ (the $a$ value). Many students forget this step and choose the wrong answer. You can always check your answer by differentiating it—if you differentiate $\frac{(2x+1)^4}{8} + C$, you will get back to $(2x+1)^3$ using the chain rule!

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16. 16.

    If the mean of 2, 5, (x+1), (x+2), 7 and 9 is 6. Find the median

    1. A.5.5
    2. B.5
    3. C.6.5
    4. D.6

    Show answer

    Answer: C. 6.5

    AI explanation

    1. Analyze the question

    We are given a set of six numbers: 2, 5, (x+1), (x+2), 7, and 9. We are told their mean is 6. We need to find the value of $x$ first, then arrange the numbers in order to determine the median.

    2. Explain the concept

    • Mean: The sum of all values divided by the total number of values. $$\text{Mean} = \frac{\sum x}{n}$$
    • Median: The middle value of a data set when arranged in ascending or descending order. If there is an even number of values (like 6 here), the median is the average of the two middle terms.

    3. Show the solution

    Step 1: Find the value of $x$ The mean is 6, and there are 6 numbers.

    $\frac{2 + 5 + (x + 1) + (x + 2) + 7 + 9}{6} = 6$

    Combine the constants:

    $2 + 5 + 1 + 2 + 7 + 9 = 26$

    So, the equation becomes:

    $\frac{26 + 2x}{6} = 6$

    Multiply both sides by 6:

    $26 + 2x = 36$

    $2x = 36 - 26$

    $2x = 10$

    $x = 5$

    Step 2: List the numbers Substitute $x = 5$ into the expressions:

    • $x + 1 = 5 + 1 = 6$
    • $x + 2 = 5 + 2 = 7$

    The set of numbers is: 2, 5, 6, 7, 7, 9

    Step 3: Find the median Since there are 6 numbers (an even number), the median is the average of the 3rd and 4th terms:

    • 3rd term = 6
    • 4th term = 7

    $\text{Median} = \frac{6 + 7}{2} = \frac{13}{2} = 6.5$

    4. Identify the answer

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

    5. Tip

    When finding the median of an even-numbered set, don't just pick the middle number—always remember to add the two central values and divide by 2. Also, always re-arrange your data in ascending order before looking for the median; if you had used the original order, you might have gotten the wrong answer!

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17. 17.

    In how many ways can the letter of ZOOLOGY be arranged?

    1. A.720
    2. B.360
    3. C.840
    4. D.120

    Show answer

    Answer: C. 840

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18. 18.

    Calculate the median of 14, 17, 10, 13, 18 and 10.

    1. A.12.5
    2. B.13.5
    3. C.13.2
    4. D.14.5

    Show answer

    Answer: B. 13.5

    AI explanation

    Analysis

    To find the median of a set of numbers, you must first arrange them in numerical order (either ascending or descending). The median is the middle value of the data set. If the set has an even number of values, the median is the average (mean) of the two middle numbers.

    Concept

    1. Arrange the data: Sort the numbers from smallest to largest.
    2. Identify the middle: Since there are 6 numbers (an even count), there is no single middle number. We must take the two middle values.
    3. Calculate the average: Add the two middle numbers together and divide by 2.

    Step-by-Step Solution

    Step 1: Arrange the numbers in ascending order. The given numbers are: 14, 17, 10, 13, 18, 10. Sorted: 10, 10, 13, 14, 17, 18.

    Step 2: Identify the two middle numbers. Since there are 6 numbers, the middle positions are the 3rd and 4th values.

    • 1st: 10
    • 2nd: 10
    • 3rd: 13
    • 4th: 14
    • 5th: 17
    • 6th: 18

    The two middle numbers are 13 and 14.

    Step 3: Calculate the average of the two middle numbers.

    $\text{Median} = \frac{13 + 14}{2}$

    $\text{Median} = \frac{27}{2}$

    $\text{Median} = 13.5$

    Conclusion

    Comparing our result to the given options, 13.5 matches option B.

    Exam Tip

    Always count the number of items in your data set first. If the count is odd, the median is just the middle number. If the count is even, you must average the two middle numbers. Don't forget to re-count your list after sorting to ensure you haven't accidentally left out a number!

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19. 19.

    Find the equation of a straight line parallel to the line 2x - y = 5 and having intercept of 5

    1. A.2x + y = 5
    2. B.2x + y = -5
    3. C.2x - y = -5
    4. D.2x - y = 5

    Show answer

    Answer: C. 2x - y = -5

    AI explanation

    1. Analyze the question

    We are asked to find the equation of a line that satisfies two conditions:

    • It is parallel to the line $2x - y = 5$.
    • It has a y-intercept of $5$.

    2. Explain the concept

    • Parallel Lines: Two lines are parallel if and only if they have the same gradient (slope). To find the gradient, we rewrite the equation in the slope-intercept form: $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.
    • Y-intercept: The y-intercept is the value of $y$ when $x = 0$. In the form $y = mx + c$, the constant $c$ represents the y-intercept.

    3. Show the solution

    Step 1: Find the gradient of the given line. The given equation is $2x - y = 5$. Rearrange it to make $y$ the subject of the formula: $-y = -2x + 5$ Multiply through by $-1$: $y = 2x - 5$ Comparing this to $y = mx + c$, the gradient ($m$) is 2.

    Step 2: Form the new equation. Since the new line is parallel to the given line, it must also have a gradient ($m$) of 2. We are given that the y-intercept ($c$) is 5. Using the slope-intercept form $y = mx + c$: $y = 2x + 5$

    Step 3: Rearrange to match the options. Looking at the options provided, they are written in the form $Ax + By = C$. Let's rearrange $y = 2x + 5$: $-2x + y = 5$ Or, multiplying by $-1$: $2x - y = -5$

    4. Identify the answer

    Comparing our result ($2x - y = -5$) with the given options: A. $2x + y = 5$ B. $2x + y = -5$ C. $2x - y = -5$ D. $2x - y = 5$

    Our calculated equation matches option C.

    )*

    5. Tip

    When dealing with parallel lines in the form $Ax + By = C$, the coefficients of $x$ and $y$ (the $A$ and $B$ parts) remain the same. Only the constant term ($C$) changes. So, for $2x - y = 5$, any parallel line will look like $2x - y = k$. Since the intercept is 5, we plug in $(0, 5)$ into $2x - y = k$: $2(0) - 5 = k \implies k = -5$. Thus, $2x - y = -5$.

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20. 20.

    find the limit of y = $\frac{x^3 + 6x - 7}{x-1}$ as x tends to 1

    1. A.9
    2. B.8
    3. C.0
    4. D.7

    Show answer

    Answer: A. 9

    AI explanation

    1. Analyze the question

    We are asked to find the limit of the function $f(x) = \frac{x^3 + 6x - 7}{x-1}$ as $x \to 1$.

    If we substitute $x = 1$ directly into the expression, we get: $\frac{1^3 + 6(1) - 7}{1 - 1} = \frac{1 + 6 - 7}{0} = \frac{0}{0}$

    This is an indeterminate form. In calculus, when you encounter $\frac{0}{0}$, it means there is a common factor in the numerator and denominator that can be cancelled out.

    2. Explain the concept

    To solve limits that result in $\frac{0}{0}$, we can use two main methods:

    • Factorization: Simplify the expression by factoring the numerator.
    • L'Hôpital's Rule: Differentiate the numerator and the denominator separately with respect to $x$ and then substitute the limit value.

    3. Show the solution

    Method 1: Factorization Since $(x-1)$ is a factor of the denominator, it must also be a factor of the numerator. We can perform polynomial long division or synthetic division to divide $(x^3 + 6x - 7)$ by $(x-1)$.

    Dividing $x^3 + 0x^2 + 6x - 7$ by $(x-1)$:

    • $x^3 \div x = x^2$
    • $x^2(x-1) = x^3 - x^2$
    • Subtracting: $(x^3 + 0x^2) - (x^3 - x^2) = x^2$
    • Bring down $6x$: $x^2 + 6x$
    • $x^2 \div x = x$
    • $x(x-1) = x^2 - x$
    • Subtracting: $(x^2 + 6x) - (x^2 - x) = 7x$
    • Bring down $-7$: $7x - 7$
    • $7x \div x = 7$
    • $7(x-1) = 7x - 7$
    • Remainder = 0

    So, $x^3 + 6x - 7 = (x-1)(x^2 + x + 7)$.

    Now, substitute this back into the limit: $\lim_{x \to 1} \frac{(x-1)(x^2 + x + 7)}{(x-1)}$

    Cancel the $(x-1)$ terms: $\lim_{x \to 1} (x^2 + x + 7)$

    Substitute $x = 1$: $1^2 + 1 + 7 = 1 + 1 + 7 = 9$

    Method 2: L'Hôpital's Rule Differentiate the numerator and denominator: Numerator: $\frac{d}{dx}(x^3 + 6x - 7) = 3x^2 + 6$ Denominator: $\frac{d}{dx}(x - 1) = 1$

    Now find the limit of the new fraction: $\lim_{x \to 1} \frac{3x^2 + 6}{1} = 3(1)^2 + 6 = 3 + 6 = 9$

    4. Identify the answer

    Both methods confirm that the limit is 9. This corresponds to option A.

    5. Tip

    Whenever you see a limit problem where the denominator becomes zero, always check for the $\frac{0}{0}$ case. If you see it, don't panic! Just factorize the numerator or use L'Hôpital's Rule (if you are comfortable with differentiation). It is a guaranteed way to clear the "undefined" part of the fraction.

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21. 21.

    Let (*) be a binary operation on a natural number defined by a * b = a - b + (ab)$^2$, then find 3 * 5

    1. A.223
    2. B.232
    3. C.32
    4. D.-232

    Show answer

    Answer: A. 223

    AI explanation

    Analysis

    The question asks us to evaluate a binary operation defined as $a * b = a - b + (ab)^2$ for the specific values $a = 3$ and $b = 5$.

    Concept

    A binary operation is simply a rule that combines two elements of a set to produce a third element. In this case, the rule is:

    1. Subtract the second number from the first ($a - b$).
    2. Multiply the two numbers together ($ab$).
    3. Square the result of that multiplication ($(ab)^2$).
    4. Add the results of step 1 and step 3 together.

    Step-by-Step Solution

    Step 1: Identify the variables Given $a * b = a - b + (ab)^2$ We are to find $3 * 5$, so: $a = 3$ $b = 5$

    Step 2: Substitute the values into the formula $3 * 5 = 3 - 5 + (3 \times 5)^2$

    Step 3: Solve the expression inside the parentheses $3 \times 5 = 15$

    Step 4: Square the result $15^2 = 15 \times 15 = 225$

    Step 5: Complete the calculation $3 - 5 + 225$ $-2 + 225 = 223$

    Conclusion

    Following the defined operation, $3 * 5$ equals $223$. This matches option A.

    Exam Tip

    When dealing with binary operations in JAMB, always pay close attention to the order of the variables. If the operation were not commutative (meaning $a * b$ is not the same as $b * a$), swapping the numbers would give you a completely different answer. Always substitute carefully!

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22. 22.

    Find x if the mean of 2x, 4x, 2x - 13 and 6x is 4.

    1. A.1.5
    2. B.2.0
    3. C.1.0
    4. D.0.5

    Show answer

    Answer: D. 0.5

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23. 23.

    The coordinates of the mid-point of the line joining the points (-3,5) and (2,10) is given by?

    1. A.( $\frac{1}{2}$, $\frac{15}{2}$ )
    2. B.( $\frac{1}{2}$, $\frac{-15}{2}$ )
    3. C.( $\frac{-1}{2}$, $\frac{-15}{2}$ )
    4. D.( $\frac{-1}{2}$, $\frac{15}{2}$ )

    Show answer

    Answer: D. ( $\frac{-1}{2}$, $\frac{15}{2}$ )

    AI explanation

    Analysis

    To find the midpoint of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the midpoint formula. This formula calculates the average of the x-coordinates and the average of the y-coordinates.

    Concept

    The midpoint $M$ of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

    $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

    Step-by-Step Solution

    1. Identify the coordinates: • Point 1: $(x_1, y_1) = (-3, 5)$ • Point 2: $(x_2, y_2) = (2, 10)$

    2. Apply the midpoint formula for the x-coordinate: $$x_m = \frac{x_1 + x_2}{2} = \frac{-3 + 2}{2} = \frac{-1}{2}$$

    3. Apply the midpoint formula for the y-coordinate: $$y_m = \frac{y_1 + y_2}{2} = \frac{5 + 10}{2} = \frac{15}{2}$$

    4. Combine the coordinates: The midpoint is $\left( \frac{-1}{2}, \frac{15}{2} \right)$.

    Conclusion

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

    Exam Tip: When dealing with negative numbers in coordinate geometry, always be careful with the signs during addition. A common mistake is to subtract the numbers instead of adding them algebraically (e.g., $-3 + 2$ is $-1$, not $-5$). Always write out the signs clearly before calculating!

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24. 24.

    If a dress is sold for #3800.00 at 20% discount. what is its original price?

    1. A.#3,000.00
    2. B.#4,000.00
    3. C.#5,000.00
    4. D.#4,750.00

    Show answer

    Answer: D. #4,750.00

    AI explanation

    Analysis

    To find the original price of the dress, we need to understand that the selling price represents the original price minus the discount. Since the discount is 20%, the selling price is actually 80% of the original price.

    Concept

    When an item is sold at a discount, the relationship is: Selling Price = Original Price × (100% - Discount Percentage)

    Let $x$ be the original price. The discount is 20%, so the customer pays 80% of the original price. Mathematically: $0.80 \times x = \text{Selling Price}$

    Step-by-Step Solution

    1. Identify the given values: • Selling Price = #3,800.00 • Discount = 20% • Percentage paid = 100% - 20% = 80% (or 0.8 in decimal form)

    2. Set up the equation: $0.8 \times x = 3800$

    3. Solve for $x$: $x = \frac{3800}{0.8}$

    4. Perform the division: To make it easier, multiply both numerator and denominator by 10: $x = \frac{38000}{8}$ $38000 \div 8 = 4750$

    The original price of the dress is #4,750.00.

    Conclusion

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

    Exam Tip

    Whenever you are given a selling price after a discount, do not just add 20% of the selling price to the selling price. That is a common mistake! Always remember that the discount is calculated based on the original price, not the reduced price. If you are ever unsure, check your answer by calculating 20% of your result and subtracting it—if you get the original selling price, you are correct!

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25. 25.

    Evaluate $(101_{two})^3$

    1. A.$111101_{two}$
    2. B.$11111101_{two}$
    3. C.$1111101_{two}$
    4. D.$11001_{two}$

    Show answer

    Answer: C. $1111101_{two}$

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26. 26.

    If 8, p, q and 26 is an A.P. Find the values of p and q

    1. A.p = 14 and q = 14
    2. B.p = 20 and q = 14
    3. C.p = 20 and q = 20
    4. D.p = 14 and q = 20

    Show answer

    Answer: D. p = 14 and q = 20

    AI explanation

    Analysis

    The question asks for the missing terms ($p$ and $q$) in an Arithmetic Progression (A.P.) sequence: $8, p, q, 26$. In an A.P., the difference between consecutive terms is constant. This constant is called the common difference ($d$).

    Concept

    An Arithmetic Progression follows the formula: $T_n = a + (n - 1)d$ Where:

    • $a$ is the first term.
    • $d$ is the common difference.
    • $n$ is the position of the term.

    Given the sequence $8, p, q, 26$:

    • $T_1 = a = 8$
    • $T_4 = 26$

    Step-by-Step Solution

    Step 1: Find the common difference ($d$) Using the formula for the 4th term ($T_4$): $T_4 = a + (4 - 1)d$ $26 = 8 + 3d$

    Subtract 8 from both sides: $26 - 8 = 3d$ $18 = 3d$

    Divide by 3: $d = \frac{18}{3} = 6$

    Step 2: Find $p$ $p$ is the second term ($T_2$): $p = a + d$ $p = 8 + 6$ $p = 14$

    Step 3: Find $q$ $q$ is the third term ($T_3$): $q = p + d$ $q = 14 + 6$ $q = 20$

    Conclusion

    We have calculated $p = 14$ and $q = 20$. Comparing this to the given options, it matches option D.

    Exam Tip

    In an A.P. with four terms, you can also use the property that the sum of the extremes equals the sum of the means ($T_1 + T_4 = T_2 + T_3$). Here, $8 + 26 = 34$, and $14 + 20 = 34$. This is a quick way to verify your answer during the exam!

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27. 27.

    If n(P) = 20 and n(Q) = 30 and n(PuQ) = 40, find the value n(PnQ)

    1. A.10
    2. B.30
    3. C.15
    4. D.40

    Show answer

    Answer: A. 10

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28. 28.

    If sec$^2$θ + tan$^2$θ = 3, then the angle θ is equal to?

    1. A.90º
    2. B.30º
    3. C.45º
    4. D.60º

    Show answer

    Answer: C. 45º

    AI explanation

    Analysis

    We are given a trigonometric equation involving $\sec^2\theta$ and $\tan^2\theta$. To solve for $\theta$, we need to express the entire equation in terms of a single trigonometric function using standard identities.

    Concept

    The fundamental trigonometric identity relating secant and tangent is:

    $1 + \tan^2\theta = \sec^2\theta$

    By substituting this identity into the given equation, we can reduce the problem to a simple algebraic equation.

    Step-by-Step Solution

    Step 1: Substitute the identity Given: $\sec^2\theta + \tan^2\theta = 3$ Since $\sec^2\theta = 1 + \tan^2\theta$, substitute this into the equation: $(1 + \tan^2\theta) + \tan^2\theta = 3$

    Step 2: Simplify the equation Combine the like terms ($\tan^2\theta$): $1 + 2\tan^2\theta = 3$

    Step 3: Solve for $\tan^2\theta$ Subtract 1 from both sides: $2\tan^2\theta = 3 - 1$ $2\tan^2\theta = 2$

    Divide by 2: $\tan^2\theta = 1$

    Step 4: Find $\theta$ Take the square root of both sides: $\tan\theta = \sqrt{1}$ $\tan\theta = 1$

    Now, find the angle whose tangent is 1: $\theta = \tan^{-1}(1)$ $\theta = 45^\circ$

    Conclusion

    Comparing our result with the provided options, we find that $45^\circ$ corresponds to option C.

    Exam Tip

    Always remember the "Pythagorean Identities" for trigonometry:

    1. $\sin^2\theta + \cos^2\theta = 1$
    2. $1 + \tan^2\theta = \sec^2\theta$
    3. $1 + \cot^2\theta = \csc^2\theta$

    If you see a mix of squared trig functions in an exam, your first step should always be to use these identities to convert them into the same function!

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29. 29.

    The locus of points equidistant from a fixed point.

    1. A.circle
    2. B.perpendicular lines
    3. C.straight line
    4. D.bisector

    Show answer

    Answer: A. circle

    AI explanation

    1. Analyze the question

    The question asks for the geometric shape formed by all points that maintain a constant distance (equidistant) from a single, fixed point in a two-dimensional plane.

    2. Explain the concept

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

    • A fixed point is referred to as the center.
    • The constant distance from that center is referred to as the radius.

    By definition, a circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center). If you imagine a compass used in your technical drawing set, the needle stays at the fixed point while the pencil traces the locus of points equidistant from that needle.

    3. Show the solution

    • Let the fixed point be $O(h, k)$.
    • Let any point on the locus be $P(x, y)$.
    • The condition "equidistant" means the distance $d$ between $P$ and $O$ is constant ($r$).
    • Using the distance formula: $\sqrt{(x - h)^2 + (y - k)^2} = r$
    • Squaring both sides gives: $(x - h)^2 + (y - k)^2 = r^2$
    • This is the standard equation of a circle.

    4. Identify the answer

    Comparing our definition and the mathematical derivation to the options provided:

    • A. Circle: This matches our definition perfectly.
    • B. Perpendicular lines: These do not maintain a constant distance from a single point.
    • C. Straight line: This represents points equidistant from two fixed points (the perpendicular bisector), not one.
    • D. Bisector: This is a line that divides an angle or segment into two equal parts.

    The correct option is A.

    5. Exam Tip

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

    • Equidistant from one point = Circle.
    • Equidistant from two points = Perpendicular bisector of the line joining them.
    • Equidistant from two intersecting lines = Angle bisector.

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30. 30.

    Evaluate $^{n^2 + 1}$C$_{n+5}$ if n = 3

    1. A.55
    2. B.45
    3. C.35
    4. D.25

    Show answer

    Answer: B. 45

    AI explanation

    1. Analyze the question

    We are asked to evaluate the combination formula $^{n^2 + 1}C_{n+5}$ given that $n = 3$.

    Recall the general formula for combinations:

    $^nC_r = \frac{n!}{(n-r)!r!}$

    2. Explain the concept

    To solve this, we first substitute the value of $n$ into the expression to determine the actual values for the upper index ($n^2 + 1$) and the lower index ($n + 5$). Once we have the specific values, we apply the combination formula.

    3. Show the solution

    Step 1: Substitute $n = 3$ into the expression.

    • Upper index: $n^2 + 1 = (3)^2 + 1 = 9 + 1 = 10$
    • Lower index: $n + 5 = 3 + 5 = 8$

    So, the expression becomes: $^{10}C_8$

    Step 2: Apply the combination formula.

    $^{10}C_8 = \frac{10!}{(10-8)!8!}$

    $^{10}C_8 = \frac{10!}{2! \times 8!}$

    Step 3: Simplify the factorials. Expand $10!$ until it reaches $8!$ to cancel it out:

    $^{10}C_8 = \frac{10 \times 9 \times 8!}{2 \times 1 \times 8!}$

    $^{10}C_8 = \frac{10 \times 9}{2 \times 1}$

    $^{10}C_8 = \frac{90}{2}$

    $^{10}C_8 = 45$

    4. Identify the answer

    Comparing our result to the given options: A. 55 B. 45 C. 35 D. 25

    The correct option is B.

    5. Tip

    The Symmetry Property: Remember that $^nC_r = ^nC_{n-r}$. In this problem, instead of calculating $^{10}C_8$, you could have calculated $^{10}C_{10-8}$, which is $^{10}C_2$.

    $^{10}C_2 = \frac{10 \times 9}{2 \times 1} = 45$

    This trick saves time and reduces the chance of making errors with large factorials!

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

32. 32.

    Evaluate Log$_2$ 8√2

    1. A.3.0
    2. B.4.5
    3. C.3.5
    4. D.2.5

    Show answer

    Answer: C. 3.5

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33. 33.

    If y varies inversely as x and x = 3 when y =4. Find the value of x when y = 12

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

    Show answer

    Answer: B. 1

    AI explanation

    1. Analyze the question

    We are given that $y$ varies inversely as $x$. We are provided with an initial pair of values ($x=3, y=4$) and asked to find the new value of $x$ when $y=12$.

    2. Explain the concept

    In mathematics, when $y$ varies inversely as $x$, it means that their product is always a constant ($k$). The relationship is expressed as:

    $y \propto \frac{1}{x} \quad \text{or} \quad y = \frac{k}{x}$

    This can also be written as:

    $xy = k$

    Where $k$ is the constant of variation.

    3. Show the solution

    Step 1: Find the constant ($k$) using the initial values. Given $x = 3$ and $y = 4$:

    $k = x \times y$

    $k = 3 \times 4$

    $k = 12$

    Step 2: Use the constant to find the new value of $x$. Now we know the relationship is $xy = 12$. We need to find $x$ when $y = 12$:

    $x \times 12 = 12$

    Divide both sides by 12:

    $x = \frac{12}{12}$

    $x = 1$

    4. Identify the answer

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

    The correct option is B.

    5. Tip

    For inverse variation problems, always remember the "Product Rule": $x_1y_1 = x_2y_2$. This allows you to skip finding the constant $k$ entirely and solve the equation directly: $3 \times 4 = x_2 \times 12$ $12 = 12x_2$ $x_2 = 1$

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34. 34.

    The cost C of running a school is directly proportional to the number of students N, if 20 students cost #10,000, How many students can #1,000,000 cover?

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

    Show answer

    Answer: C. 2000

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35. 35.

    What will be the result obtained when the numerator of $\frac{96}{50}$ is decreased by 37.5% and its denominator decreased by 20%.

    1. A.1.5
    2. B.$\frac{5}{2}$
    3. C.$\frac{96}{48}$
    4. D.0.5

    Show answer

    Answer: A. 1.5

    AI explanation

    1. Analyze the question

    We are given the fraction $\frac{96}{50}$. We need to:

    • Decrease the numerator (96) by 37.5%.
    • Decrease the denominator (50) by 20%.
    • Calculate the new fraction and simplify it to find the result.

    2. Explain the concept

    To decrease a value by a percentage, you subtract that percentage of the value from the original value.

    • Formula: New Value = Original Value - (Percentage $\times$ Original Value)
    • Alternatively, you can multiply the original value by $(100\% - \text{percentage})$. For example, a 20% decrease means the new value is 80% of the original.

    3. Show the solution

    Step 1: Calculate the new numerator The original numerator is 96. We decrease it by 37.5%.

    • Decrease amount = $37.5\% \text{ of } 96 = 0.375 \times 96$
    • $0.375 \times 96 = 36$
    • New numerator = $96 - 36 = 60$

    Step 2: Calculate the new denominator The original denominator is 50. We decrease it by 20%.

    • Decrease amount = $20\% \text{ of } 50 = 0.20 \times 50$
    • $0.20 \times 50 = 10$
    • New denominator = $50 - 10 = 40$

    Step 3: Form the new fraction and simplify

    • New fraction = $\frac{60}{40}$
    • Simplify by dividing both by 20: $\frac{60 \div 20}{40 \div 20} = \frac{3}{2}$
    • Convert to decimal: $\frac{3}{2} = 1.5$

    4. Identify the answer

    Comparing our result (1.5) with the given options, it matches option A.

    5. Exam Tip

    When dealing with percentage decreases, remember that "decreasing by $x\%$" is the same as multiplying by $(1 - \frac{x}{100})$.

    • For the numerator: $96 \times (1 - 0.375) = 96 \times 0.625 = 60$.
    • For the denominator: $50 \times (1 - 0.20) = 50 \times 0.8 = 40$. This method is often faster and less prone to subtraction errors during the exam!

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36. 36.

    Solve for k in the equation $\frac{1}{8}^{k+2}$ = 1

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

    Show answer

    Answer: C. -2

    AI explanation

    Analysis

    The equation given is an exponential equation: $\left(\frac{1}{8}\right)^{k+2} = 1$. To solve for $k$, we need to express both sides of the equation with the same base or use the property of exponents that states any non-zero number raised to the power of zero equals 1 ($x^0 = 1$).

    Concept

    The fundamental rule here is the Zero Exponent Rule: $a^0 = 1$ (where $a \neq 0$).

    This means that for any base (in this case, $\frac{1}{8}$), if the result is 1, the exponent must be equal to 0.

    Step-by-Step Solution

    1. Set up the equation: $$\left(\frac{1}{8}\right)^{k+2} = 1$$

    2. Apply the Zero Exponent Rule: Since any number raised to the power of 0 is 1, we can rewrite the right side of the equation as: $$\left(\frac{1}{8}\right)^{k+2} = \left(\frac{1}{8}\right)^0$$

    3. Equate the exponents: Because the bases are the same ($\frac{1}{8}$), we can equate the exponents: $$k + 2 = 0$$

    4. Solve for $k$: Subtract 2 from both sides: $$k = -2$$

    Conclusion

    By setting the exponent to zero, we find that $k = -2$. This matches option C.

    Exam Tip

    Whenever you see an exponential equation equal to 1, don't waste time trying to find complex logarithms or common bases. Immediately set the entire exponent expression to zero. It is a classic JAMB "shortcut" question designed to test if you know the zero exponent rule!

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37. 37.

    Find the length of a chord 3cm from the centre of a circle of radius 5cm

    1. A.8cm
    2. B.5.6cm
    3. C.7cm
    4. D.6.5cm

    Show answer

    Answer: A. 8cm

    AI explanation

    1. Analyze the question

    We are given a circle with a radius of $5\text{ cm}$. A chord is drawn inside this circle, and its perpendicular distance from the center of the circle is $3\text{ cm}$. We need to find the total length of this chord.

    2. Explain the concept

    In geometry, a line drawn from the center of a circle perpendicular to a chord bisects the chord (divides it into two equal parts). This creates a right-angled triangle where:

    • The hypotenuse is the radius of the circle ($r = 5\text{ cm}$).
    • One leg is the perpendicular distance from the center to the chord ($d = 3\text{ cm}$).
    • The other leg is half the length of the chord ($x$).

    We can solve for $x$ using the Pythagorean theorem: $a^2 + b^2 = c^2$.

    3. Show the solution

    • Step 1: Set up the Pythagorean equation. Let $x$ be half the length of the chord. $x^2 + 3^2 = 5^2$
    • Step 2: Solve for $x$. $x^2 + 9 = 25$ $x^2 = 25 - 9$ $x^2 = 16$ $x = \sqrt{16}$ $x = 4\text{ cm}$
    • Step 3: Find the total length of the chord. Since $x$ is only half the chord, the total length is $2 \times x$. Total length $= 2 \times 4\text{ cm} = 8\text{ cm}$.

    4. Identify the answer

    The calculated length is $8\text{ cm}$, which corresponds to option A.

    5. Tip

    Whenever you see a circle problem involving a chord and a distance from the center, immediately visualize a 3-4-5 right-angled triangle. This is a common Pythagorean triple that examiners love to use to save you time during the JAMB exam. If you recognize the numbers 3 and 5, you can quickly deduce the missing side is 4 without doing long calculations!

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38. 38.

    Find the determinant of the matrix A = $\begin{pmatrix} 2 & 3 \\ 1 & 3 \end{pmatrix}$

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

    Show answer

    Answer: D. 3

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39. 39.

    The 10th term of an AP is 32. If the first term is 3/2, what is the 4th term?

    1. A.35/3
    2. B.64
    3. C.16
    4. D.35/2

    Show answer

    Answer: A. 35/3

    AI explanation

    Analysis

    To solve this problem, we need to use the formula for the $n^{th}$ term of an Arithmetic Progression (AP): $T_n = a + (n - 1)d$

    Where:

    • $T_n$ is the $n^{th}$ term.
    • $a$ is the first term.
    • $n$ is the position of the term.
    • $d$ is the common difference.

    Concept

    We are given:

    • The 10th term ($T_{10}$) = $32$
    • The first term ($a$) = $\frac{3}{2}$
    • We need to find the 4th term ($T_4$).

    First, we must find the common difference ($d$) using the information for the 10th term, then use that $d$ to calculate the 4th term.

    Step-by-Step Solution

    Step 1: Find the common difference ($d$) Using $T_{10} = a + (10 - 1)d$: $32 = \frac{3}{2} + 9d$

    Subtract $\frac{3}{2}$ from both sides: $32 - \frac{3}{2} = 9d$ $\frac{64 - 3}{2} = 9d$ $\frac{61}{2} = 9d$

    Divide both sides by 9: $d = \frac{61}{2 \times 9} = \frac{61}{18}$

    Step 2: Find the 4th term ($T_4$) Using $T_4 = a + (4 - 1)d$: $T_4 = \frac{3}{2} + 3d$

    Substitute $d = \frac{61}{18}$: $T_4 = \frac{3}{2} + 3(\frac{61}{18})$ $T_4 = \frac{3}{2} + \frac{61}{6}$

    Find a common denominator (which is 6): $T_4 = \frac{9}{6} + \frac{61}{6}$ $T_4 = \frac{70}{6}$

    Simplify the fraction by dividing both numerator and denominator by 2: $T_4 = \frac{35}{3}$

    Conclusion

    The calculated 4th term is $\frac{35}{3}$, which matches option A.

    Exam Tip: When dealing with fractions in an AP, always convert the whole numbers to fractions with the same denominator as early as possible. This prevents calculation errors when subtracting or adding terms.

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