2021 JAMB Mathematics Past Questions and Answers

Questions

1. 1.

   Factorize completely 81a$^4$ - 16b$^4$

   1. A.(3a + 2b)(2a - 3b)(9a2 + 4b2)
   2. B.(3a - 2b)(2a - 3b)(4a2 - 9b2)
   3. C.(3a - 2b)(3a + 2b)(9a2 + 4b2)
   4. D.(6a - 2b)(8a - 3b)(4a3 - 9b2)

   Show answer

   Answer: C. (3a - 2b)(3a + 2b)(9a2 + 4b2)

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2. 2.

   determine the maximum value of y=3x$^2$ + 5x - 3

   1. A.6
   2. B.0
   3. C.2
   4. D.No correct option

   Show answer

   Answer: D. No correct option

   AI explanation

   1. Analyze the Question

   You are asked to find the maximum value of the quadratic function $y = 3x^2 + 5x - 3$.

   2. Explain the Concept

   A quadratic function is generally written in the form $y = ax^2 + bx + c$.

   • If $a > 0$ (positive), the parabola opens upwards, meaning the function has a minimum value at its vertex.
   • If $a < 0$ (negative), the parabola opens downwards, meaning the function has a maximum value at its vertex.

   In your equation, $a = 3$. Since $3$ is greater than $0$, this parabola opens upwards. Therefore, the function has a minimum value, not a maximum value.

   3. Show the Solution

   Let's verify the nature of the vertex using the formula for the $x$-coordinate of the vertex: $x = -\frac{b}{2a}$

   Given $a = 3$ and $b = 5$: $x = -\frac{5}{2(3)} = -\frac{5}{6}$

   To find the corresponding $y$-value: $y = 3(-\frac{5}{6})^2 + 5(-\frac{5}{6}) - 3$ $y = 3(\frac{25}{36}) - \frac{25}{6} - 3$ $y = \frac{25}{12} - \frac{50}{12} - \frac{36}{12}$ $y = -\frac{61}{12} \approx -5.08$

   Because the coefficient of $x^2$ is positive, the graph extends infinitely upwards as $x$ increases or decreases. Thus, there is no "maximum" value for this function.

   4. Identify the Answer

   Since the function opens upwards, it has a minimum value but no maximum value. Looking at the provided options, none of them represent a valid maximum value for this function. Therefore, the correct choice is that there is no correct option provided.

   5. Exam Tip

   Always check the sign of the coefficient of $x^2$ before calculating. If you see a positive $x^2$ term, the question asks for a "maximum" to test if you know the difference between a vertex that is a minimum and one that is a maximum. Don't fall for the trap of calculating a value when the function doesn't have one!

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3. 3.

   A trapezium has two parallel sides of lengths 5cm and 9cm. If the area is 91cm$^2$, find the distance between the parallel sides

   1. A.13 cm
   2. B.4 cm
   3. C.6 cm
   4. D.7 cm

   Show answer

   Answer: A. 13 cm

   AI explanation

   1. Analyze the question

   We are given a trapezium with:

   • Length of the first parallel side ($a$) = 5 cm
   • Length of the second parallel side ($b$) = 9 cm
   • Area of the trapezium ($A$) = 91 cm$^2$
   • We need to find the distance between the parallel sides, which is the height ($h$).

   2. Explain the concept

   The area of a trapezium is calculated using the formula:

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

   Where:

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

   3. Show the solution

   Step 1: Substitute the known values into the formula.

   $91 = \frac{1}{2} \times (5 + 9) \times h$

   Step 2: Simplify the expression inside the parentheses.

   $91 = \frac{1}{2} \times (14) \times h$

   Step 3: Multiply $\frac{1}{2}$ by 14.

   $91 = 7 \times h$

   Step 4: Solve for $h$ by dividing both sides by 7.

   $h = \frac{91}{7}$

   $h = 13 \text{ cm}$

   4. Identify the answer

   The distance between the parallel sides is 13 cm. This corresponds to option A.

   5. Tip

   When dealing with area problems for shapes like trapeziums, always remember the "average of the parallel sides multiplied by the height" rule. It makes the formula easier to remember: $Area = (\text{Average of parallel sides}) \times \text{height}$.

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4. 4.

   Simplify $\sqrt{27}$ + $\frac{3}{\sqrt{3}}$

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

   Show answer

   Answer: A. 4$\sqrt{3}$

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5. 5.

   Find the length of a side of a rhombus whose diagonals are 6cm and 8cm

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

   Show answer

   Answer: B. 5cm

   AI explanation

   Analysis

   To find the side length of a rhombus when the diagonals are given, we must use the geometric properties of a rhombus. A rhombus has diagonals that bisect each other at right angles ($90^\circ$).

   Concept

   When the diagonals of a rhombus intersect, they divide the rhombus into four congruent right-angled triangles.

   • The diagonals are $d_1 = 6\text{ cm}$ and $d_2 = 8\text{ cm}$.
   • Because they bisect each other, the legs of each right-angled triangle are half the lengths of the diagonals: • Leg 1 = $\frac{d_1}{2} = \frac{6}{2} = 3\text{ cm}$ • Leg 2 = $\frac{d_2}{2} = \frac{8}{2} = 4\text{ cm}$
   • The side of the rhombus ($s$) acts as the hypotenuse of these right-angled triangles. We can use the Pythagorean theorem: $a^2 + b^2 = c^2$.

   Solution

   1. Identify the legs of the triangle: • $a = 3\text{ cm}$ • $b = 4\text{ cm}$

   2. Apply the Pythagorean theorem: • $s^2 = 3^2 + 4^2$ • $s^2 = 9 + 16$ • $s^2 = 25$

   3. Solve for $s$: • $s = \sqrt{25}$ • $s = 5\text{ cm}$

   The length of the side of the rhombus is 5cm.

   Exam Tip

   Whenever you see a rhombus problem involving diagonals of 6 and 8, remember the 3-4-5 Pythagorean triple. It is a very common pattern in JAMB Mathematics questions, and recognizing it can save you precious time during the exam!

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6. 6.

   The mean of ten positive numbers is 16. When another number is added, the mean becomes 18. Find the eleventh number

   1. A.3
   2. B.16
   3. C.38
   4. D.30

   Show answer

   Answer: C. 38

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7. 7.

   Find the probability that a number selected at random from 41 to 56 is a multiple of 9

   1. A.$\frac{1}{8}$
   2. B.$\frac{2}{15}$
   3. C.$\frac{3}{16}$
   4. D.$\frac{7}{8}$

   Show answer

   Answer: A. $\frac{1}{8}$

   AI explanation

   1. Analyze the Question

   We need to find the probability of picking a multiple of 9 from the set of integers ranging from 41 to 56 (inclusive).

   2. Explain the Concept

   Probability is defined as:

   $\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

   • Total outcomes: The count of numbers from 41 to 56.
   • Favorable outcomes: The count of numbers in that range that are divisible by 9.

   3. Show the Solution

   Step 1: Find the total number of outcomes. To find the number of integers in an inclusive range $[a, b]$, use the formula: $(b - a) + 1$.

   $\text{Total} = (56 - 41) + 1$

   $\text{Total} = 15 + 1 = 16$

   Step 2: Identify the multiples of 9 within the range 41 to 56. Let's list the multiples of 9:

   • $9 \times 4 = 36$ (Too small)
   • $9 \times 5 = 45$ (Within range)
   • $9 \times 6 = 54$ (Within range)
   • $9 \times 7 = 63$ (Too large)

   The multiples of 9 in the range are 45 and 54. There are 2 favorable outcomes.

   Step 3: Calculate the probability.

   $\text{Probability} = \frac{2}{16}$

   Simplify the fraction by dividing both the numerator and the denominator by 2:

   $\text{Probability} = \frac{1}{8}$

   4. Identify the Answer

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

   5. Exam Tip

   When counting numbers in an inclusive range (e.g., "from X to Y"), students often forget to add the "+1". Always remember: if you are counting from 1 to 3, there are 3 numbers (1, 2, 3). If you just did $3 - 1$, you would get 2, which is incorrect. Adding the 1 ensures you include both endpoints!

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8. 8.

   What is the n-th term of the sequence 2, 6, 12, 20...?

   1. A.4n - 2
   2. B.2(3n - 1)
   3. C.n2 + n
   4. D.n2 + 3n + 2

   Show answer

   Answer: C. n2 + n

   AI explanation

   Analysis

   To find the $n$-th term of the sequence $2, 6, 12, 20, \dots$, we need to identify the pattern or the rule that generates each term based on its position ($n$).

   Concept

   A sequence can often be solved by looking at the relationship between the position $n$ (where $n = 1, 2, 3, 4, \dots$) and the value of the term.

   • For $n = 1$, the term is $2$.
   • For $n = 2$, the term is $6$.
   • For $n = 3$, the term is $12$.
   • For $n = 4$, the term is $20$.

   Step-by-Step Solution

   Step 1: Test the options by substituting $n = 1, 2, 3$

   • Option A ($4n - 2$): • $n=1: 4(1) - 2 = 2$ (Correct) • $n=2: 4(2) - 2 = 6$ (Correct) • $n=3: 4(3) - 2 = 10$ (Incorrect, the sequence has 12)

   • Option B ($2(3n - 1)$): • $n=1: 2(3(1) - 1) = 2(2) = 4$ (Incorrect, the sequence starts with 2)

   • Option C ($n^2 + n$): • $n=1: (1)^2 + 1 = 1 + 1 = 2$ (Correct) • $n=2: (2)^2 + 2 = 4 + 2 = 6$ (Correct) • $n=3: (3)^2 + 3 = 9 + 3 = 12$ (Correct) • $n=4: (4)^2 + 4 = 16 + 4 = 20$ (Correct)

   • Option D ($n^2 + 3n + 2$): • $n=1: (1)^2 + 3(1) + 2 = 1 + 3 + 2 = 6$ (Incorrect, the sequence starts with 2)

   Step 2: Verification The formula $n^2 + n$ (which can also be written as $n(n+1)$) perfectly generates every term in the given sequence.

   Conclusion

   The formula that satisfies the sequence $2, 6, 12, 20$ is $n^2 + n$.

   Exam Tip

   When you see a sequence where the differences between terms are increasing (in this case: $6-2=4$, $12-6=6$, $20-12=8$), it is usually a quadratic sequence. If the differences are $4, 6, 8 \dots$, the second difference is constant ($2$), which confirms the formula will involve an $n^2$ term. Always test your options starting with $n=1$ to save time!

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9. 9.

   If (x + 2) and (x - 1) are factors of the expression $Lx + 2kx^{2} + 24$, find the values of L and k.

   1. A.l = -12, k = -6
   2. B.l = -2 , k = 1
   3. C.l = -2 , k = -1
   4. D.l = 0, k = 1

   Show answer

   Answer: A. l = -12, k = -6

   AI explanation

   1. Analyze the Question

   We are given a polynomial $P(x) = 2kx^2 + Lx + 24$ (rearranging the terms for clarity). We are told that $(x + 2)$ and $(x - 1)$ are factors of this expression. According to the Factor Theorem, if $(x - a)$ is a factor of $P(x)$, then $P(a) = 0$.

   2. Explain the Concept

   • Factor Theorem: If $(x + 2)$ is a factor, then $P(-2) = 0$.
   • Factor Theorem: If $(x - 1)$ is a factor, then $P(1) = 0$. We will use these two conditions to create a system of simultaneous equations to solve for $L$ and $k$.

   3. Show the Solution

   Given $P(x) = 2kx^2 + Lx + 24$:

   Step 1: Use the factor $(x + 2)$ Set $x = -2$: $P(-2) = 2k(-2)^2 + L(-2) + 24 = 0$ $2k(4) - 2L + 24 = 0$ $8k - 2L + 24 = 0$ Divide by 2 to simplify: $4k - L + 12 = 0$ $4k - L = -12$ --- (Equation 1)

   Step 2: Use the factor $(x - 1)$ Set $x = 1$: $P(1) = 2k(1)^2 + L(1) + 24 = 0$ $2k + L + 24 = 0$ $2k + L = -24$ --- (Equation 2)

   Step 3: Solve the simultaneous equations Add Equation 1 and Equation 2: $(4k - L) + (2k + L) = -12 + (-24)$ $6k = -36$ $k = -36 / 6$ $k = -6$

   Substitute $k = -6$ into Equation 2: $2(-6) + L = -24$ $-12 + L = -24$ $L = -24 + 12$ $L = -12$

   4. Identify the Answer

   Comparing our results ($L = -12, k = -6$) with the given options, we find they match Option A.

   5. Tip

   When you see "factors" in a polynomial question, always think "Factor Theorem." Immediately set the expression to zero using the root of the factor (e.g., for $x+2$, use $-2$). If you have two variables to find, you will always need two factors to create two equations.

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10. 10.

    If the binary operation $\ast$ is defined by m $\ast$ n = mn + m + n for any real number m and n, find the identity of the elements under this operation

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

    Show answer

    Answer: D. e = 0

    AI explanation

    Analysis

    To find the identity element ($e$) of a binary operation, we use the fundamental property: $m \ast e = m$ (or $e \ast m = m$) This means that when you perform the operation between any element $m$ and the identity $e$, the result must be the element $m$ itself.

    Concept

    The identity element is the "neutral" element of an operation. Just as $0$ is the identity for addition ($m + 0 = m$) and $1$ is the identity for multiplication ($m \times 1 = m$), we need to find the value of $e$ that satisfies the given rule: $m \ast n = mn + m + n$

    Step-by-Step Solution

    1. Set up the equation: Using the definition $m \ast e = m$, substitute $e$ for $n$ in the given formula: $m \ast e = me + m + e$

    2. Equate to $m$: Since $m \ast e$ must equal $m$: $me + m + e = m$

    3. Solve for $e$: Subtract $m$ from both sides of the equation: $me + e = 0$

    4. Factor out $e$: $e(m + 1) = 0$

    5. Determine the value of $e$: For this equation to hold true for any real number $m$, the term $(m + 1)$ cannot be zero (unless $m = -1$). Therefore, the only way for the product to be zero for all $m$ is if $e = 0$.

    Verification

    Let's check if $e = 0$ works: $m \ast 0 = (m \times 0) + m + 0$ $m \ast 0 = 0 + m + 0$ $m \ast 0 = m$ Since the result is $m$, $e = 0$ is indeed the identity element.

    Conclusion

    The correct option is D.

    Exam Tip: Whenever you are asked to find the identity element, always remember the rule $a \ast e = a$. If you are asked for the inverse element later, remember that $a \ast a^{-1} = e$ (where $e$ is the identity you just found).

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11. 11.

    At what rate would a sum of N100.00 deposited for 5 years raise an interest of N7.50?

    1. A.$\frac{1}{2}$%
    2. B.2$\frac{1}{2}$%
    3. C.1.5%
    4. D.25%

    Show answer

    Answer: C. 1.5%

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12. 12.

    Three brothers in a business deal share the profit at the end of a contract. The first received $\frac{1}{3}$ of the profit and the second $\frac{2}{3}$ of the remainder. If the third received the remaining N12000.00 how much profit did they share?

    1. A.N60 000.00
    2. B.N54 000.00
    3. C.N48 000.00
    4. D.N42 000.00

    Show answer

    Answer: B. N54 000.00

    AI explanation

    Analysis

    To solve this problem, we need to determine the total profit by tracking the fractions given to each brother and equating the final remaining amount to the total.

    Concept

    We represent the total profit as $x$. We will subtract the portions taken by the first and second brothers from the whole ($1$) to find the fraction that represents the third brother's share.

    Step-by-Step Solution

    1. Define the total profit: Let the total profit be $x$.

    2. Calculate the first brother's share: The first brother receives $\frac{1}{3}x$. The remainder after the first brother is: $1 - \frac{1}{3} = \frac{2}{3}x$

    3. Calculate the second brother's share: The second brother receives $\frac{2}{3}$ of the remainder. Share = $\frac{2}{3} \times (\frac{2}{3}x) = \frac{4}{9}x$

    4. Calculate the third brother's share: The third brother receives what is left after the first and second brothers have taken their shares. Total taken by first and second = $\frac{1}{3}x + \frac{4}{9}x$ To add these, find a common denominator (9): $\frac{3}{9}x + \frac{4}{9}x = \frac{7}{9}x$

    The third brother's share is the remainder: $1 - \frac{7}{9} = \frac{2}{9}x$

    5. Solve for $x$: We are told the third brother's share is N12,000. $\frac{2}{9}x = 12,000$ Multiply both sides by 9: $2x = 108,000$ Divide by 2: $x = 54,000$

    Conclusion

    The total profit shared by the three brothers is N54,000.00.

    Exam Tip: When a question mentions "of the remainder," always calculate the remaining fraction first before multiplying by the next fraction. A common mistake is to multiply the second fraction by the original total instead of the remainder.

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13. 13.

    The locus of a point which moves so that it is equidistant from two intersecting straight lines is the?

    1. A.perpendicular bisector of the two lines
    2. B.angle bisector of the two lines
    3. C.bisector of the two lines
    4. D.line parallel to the two lines

    Show answer

    Answer: B. angle bisector of the two lines

    AI explanation

    Analysis

    The question asks for the geometric path (locus) traced by a point that maintains an equal distance from two lines that intersect at a point.

    Concept: Locus of Points

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

    1. When you have two intersecting lines, they form angles at the point of intersection.
    2. Any point that is equidistant from two intersecting lines must lie on the line that divides the angle between those two lines into two equal parts.
    3. This line is mathematically referred to as the angle bisector.

    Step-by-Step Solution

    1. Visualize the lines: Imagine two lines crossing each other like an "X".
    2. Define the condition: We are looking for a point $P(x, y)$ such that the perpendicular distance from $P$ to Line 1 is equal to the perpendicular distance from $P$ to Line 2.
    3. Apply the property: By the Angle Bisector Theorem, the set of all points equidistant from two intersecting lines forms the bisector of the angles formed by those lines.
    4. Evaluate the options: • A perpendicular bisector is used for a line segment, not two intersecting lines. • A line parallel to the two lines would not be equidistant to both simultaneously. • The "angle bisector" is the precise geometric term for this locus.

    Conclusion

    The locus of a point equidistant from two intersecting lines is the angle bisector of the two lines. This ensures that for any point chosen on this bisector, the perpendicular distance to both original lines remains identical.

    Exam Tip

    Whenever you see the word "equidistant" in a locus question:

    • If it is from two fixed points, the locus is the perpendicular bisector.
    • If it is from two intersecting lines, the locus is the angle bisector.

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14. 14.

    A car travels from calabar to Enugu, a distance of P km with an average speed of U km per hour and continues to benin, a distance of Q km, with an average speed of Wkm per hour. Find its average speed from Calabar to Benin

    1. A.$\frac{(p + q)}{pw + qu}$
    2. B.$\frac{uw(p + q)}{pw + qu}$
    3. C.$\frac{uw(p + q)}{pw}$
    4. D.$\frac{uw}{pw + qu}$

    Show answer

    Answer: B. $\frac{uw(p + q)}{pw + qu}$

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15. 15.

    Correct 241.34(3 x 10$^{-3}$)$^2$ to 4 significant figures

    1. A.0.0014
    2. B.0.001448
    3. C.0.0022
    4. D.0.002172

    Show answer

    Answer: D. 0.002172

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16. 16.

    Find the mean deviation of 1, 2, 3 and 4

    1. A.1.0
    2. B.1.5
    3. C.2.0
    4. D.2.5

    Show answer

    Answer: A. 1.0

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17. 17.

    X is due east point of y on a coast. Z is another point on the coast but 6.0km due south of Y. If the distance ZX is 12km, calculate the bearing of Z from X

    1. A.240°
    2. B.150°
    3. C.60°
    4. D.270°

    Show answer

    Answer: A. 240°

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18. 18.

    If w varies inversely as $\frac{uv}{u + v}$ and w = 8 when u = 2 and v = 6, find a relationship between u, v, w.

    1. A.uvw = 16(u + v)
    2. B.16uv = 3w(u + v)
    3. C.uvw = 12(u + v)
    4. D.12uvw = u + v

    Show answer

    Answer: C. uvw = 12(u + v)

    AI explanation

    1. Analyze the question

    We are given that $w$ varies inversely as $\frac{uv}{u + v}$. We need to find the constant of proportionality ($k$) using the provided values ($w=8, u=2, v=6$) and then express the relationship between $u, v,$ and $w$.

    2. Explain the concept

    In mathematics, when a variable $w$ varies inversely as a quantity $X$, it means: $w = \frac{k}{X}$ or $w \cdot X = k$ where $k$ is the constant of proportionality.

    In this problem, $X = \frac{uv}{u + v}$. Therefore: $w = \frac{k}{\left( \frac{uv}{u + v} \right)}$

    This simplifies to: $w = \frac{k(u + v)}{uv}$

    3. Show the solution

    Step 1: Find the constant $k$ Substitute the given values $w = 8, u = 2, v = 6$ into the equation: $8 = \frac{k(2 + 6)}{2 \times 6}$ $8 = \frac{k(8)}{12}$

    Multiply both sides by 12: $8 \times 12 = 8k$ $96 = 8k$ $k = \frac{96}{8}$ $k = 12$

    Step 2: Establish the relationship Now, substitute $k = 12$ back into the general equation: $w = \frac{12(u + v)}{uv}$

    To match the format of the options, multiply both sides by $uv$: $w \cdot uv = 12(u + v)$ $uvw = 12(u + v)$

    4. Identify the answer

    Comparing our result ($uvw = 12(u + v)$) with the given options, it matches option C.

    5. Tip

    When dealing with "inverse variation," always remember that the product of the variables is constant ($xy = k$). If the variation is "direct," the quotient is constant ($\frac{x}{y} = k$). Don't let the fraction $\frac{uv}{u+v}$ intimidate you; treat the entire expression as a single block!

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19. 19.

    List all integers satisfying the inequality in -2 < 2x-6 < 4

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

    Show answer

    Answer: D. 3,4

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20. 20.

    4, 16, 30, 20, 10, 14 and 26 are represented on a pie chart. Find the sum of the angles of the bisectors representing all numbers equals to or greater than 16

    1. A.48o
    2. B.84o
    3. C.92o
    4. D.276o

    Show answer

    Answer: D. 276o

    AI explanation

    1. Analyze the question

    We are given a set of numbers: 4, 16, 30, 20, 10, 14, and 26. We need to find the sum of the angles on a pie chart that represent the numbers that are 16 or greater.

    2. Explain the concept

    To represent data on a pie chart, we follow these steps:

    • Find the total sum of all the numbers in the data set.
    • Identify the specific values that meet the condition (in this case, $\geq 16$).
    • Calculate the sum of these specific values.
    • Convert to degrees: Since a full circle is $360^\circ$, the angle for a specific group is calculated as: $$\text{Angle} = \left( \frac{\text{Sum of specific values}}{\text{Total sum}} \right) \times 360^\circ$$

    3. Show the solution

    Step 1: Find the total sum of the numbers. Total = $4 + 16 + 30 + 20 + 10 + 14 + 26$ Total = $120$

    Step 2: Identify numbers $\geq 16$. The numbers in the set that are 16 or greater are: 16, 30, 20, and 26.

    Step 3: Find the sum of these specific numbers. Sum of values $\geq 16 = 16 + 30 + 20 + 26$ Sum = $92$

    Step 4: Calculate the corresponding angle.

    $\text{Angle} = \left( \frac{92}{120} \right) \times 360^\circ$

    Simplify the fraction: $\frac{92}{120} = \frac{92}{12} \times \frac{360}{10}$ (or simply divide 360 by 120 first) $360 \div 120 = 3$ $\text{Angle} = 92 \times 3$ $\text{Angle} = 276^\circ$

    4. Identify the answer

    Comparing our result to the given options, the correct value is $276^\circ$.

    5. Tip

    When dealing with pie chart questions, always check if the total sum is a factor of 360. If it is (like 120 is here), the calculation becomes much faster because $360/120 = 3$. You simply multiply your target sum by that constant!

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21. 21.

    Simplify 2log $\frac{2}{5}$ - log$\frac{72}{125}$ + log 9

    1. A.1 - 4 log3
    2. B.-1 + 2 log 3
    3. C.-1 + 5 log2
    4. D.1 - 2log 2

    Show answer

    Answer: D. 1 - 2log 2

    AI explanation

    To simplify the expression $2\log(\frac{2}{5}) - \log(\frac{72}{125}) + \log(9)$, we will use the fundamental laws of logarithms.

    1. Analyze the expression

    The expression is: $2\log(\frac{2}{5}) - \log(\frac{72}{125}) + \log(9)$

    2. Explain the concepts (Logarithm Laws)

    • Power Rule: $n\log(x) = \log(x^n)$
    • Subtraction Rule: $\log(a) - \log(b) = \log(\frac{a}{b})$
    • Addition Rule: $\log(a) + \log(b) = \log(a \times b)$

    3. Step-by-step Solution

    Step 1: Apply the Power Rule to the first term $2\log(\frac{2}{5}) = \log((\frac{2}{5})^2) = \log(\frac{4}{25})$

    Now the expression is: $\log(\frac{4}{25}) - \log(\frac{72}{125}) + \log(9)$

    Step 2: Combine the first two terms using the Subtraction Rule $\log(\frac{4}{25}) - \log(\frac{72}{125}) = \log(\frac{4/25}{72/125})$ $= \log(\frac{4}{25} \times \frac{125}{72})$

    Simplify the fraction: $= \log(\frac{1}{1} \times \frac{5}{18}) = \log(\frac{5}{18})$

    Step 3: Add the final term using the Addition Rule $\log(\frac{5}{18}) + \log(9) = \log(\frac{5}{18} \times 9)$ $= \log(\frac{5}{2}) = \log(5) - \log(2)$

    Wait! Let's re-examine the options. Since the options involve $\log(2)$ and $\log(3)$, let's express the numbers in terms of their prime factors (2, 3, and 5).

    $\log(\frac{4}{25}) - \log(\frac{72}{125}) + \log(9)$ $= (\log 4 - \log 25) - (\log 72 - \log 125) + \log 9$ $= (2\log 2 - 2\log 5) - (3\log 2 + 2\log 3 - 3\log 5) + 2\log 3$ $= 2\log 2 - 2\log 5 - 3\log 2 - 2\log 3 + 3\log 5 + 2\log 3$

    Step 4: Group like terms $= (2\log 2 - 3\log 2) + (-2\log 5 + 3\log 5) + (-2\log 3 + 2\log 3)$ $= -\log 2 + \log 5$

    Since $\log 5 = \log(\frac{10}{2}) = \log 10 - \log 2 = 1 - \log 2$: $= -\log 2 + (1 - \log 2)$ $= 1 - 2\log 2$

    4. Identify the answer

    Comparing our result ($1 - 2\log 2$) to the given options, it matches option D.

    5. Exam Tip

    When you see logarithms with different bases or fractions, always try to break the numbers down into their prime factors (2, 3, 5). This makes it much easier to cancel out terms and reach the simplified form required by the options!

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22. 22.

    P(-6, 1) and Q(6, 6) are the two ends of the diameter of a given circle. Calculate the radius.

    1. A.6.5 units
    2. B.13.0 units
    3. C.3.5 units
    4. D.7.0 units

    Show answer

    Answer: A. 6.5 units

    AI explanation

    Analysis

    To find the radius of a circle when the coordinates of the endpoints of its diameter are given, we need to:

    1. Calculate the total length of the diameter using the distance formula.
    2. Divide the diameter by 2, since the radius is half the length of the diameter.

    Concept: The Distance Formula

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

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

    Step-by-Step Solution

    Step 1: Identify the coordinates

    • $P(x_1, y_1) = (-6, 1)$
    • $Q(x_2, y_2) = (6, 6)$

    Step 2: Calculate the length of the diameter ($PQ$) Substitute the values into the distance formula:

    $PQ = \sqrt{(6 - (-6))^2 + (6 - 1)^2}$

    $PQ = \sqrt{(6 + 6)^2 + (5)^2}$

    $PQ = \sqrt{(12)^2 + (5)^2}$

    $PQ = \sqrt{144 + 25}$

    $PQ = \sqrt{169}$

    $PQ = 13 \text{ units}$

    Step 3: Calculate the radius The radius ($r$) is half of the diameter:

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

    $r = \frac{13}{2}$

    $r = 6.5 \text{ units}$

    Conclusion

    Comparing our result to the options provided, the correct value is 6.5 units.

    Exam Tip

    When dealing with coordinate geometry, always be careful with signs when subtracting negative numbers (e.g., $6 - (-6)$ becomes $6 + 6$). Also, remember the common Pythagorean triple $(5, 12, 13)$; recognizing this can save you time during the exam as you won't need to manually calculate the square root of 169!

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23. 23.

    Musa borrows N10.00 at 2% per month simple interest and repays N8.00 after 4 months. How much does he still owe?

    1. A.N10.80
    2. B.N10.67
    3. C.N2.80
    4. D.N2.67

    Show answer

    Answer: C. N2.80

    AI explanation

    1. Analyze the Question

    We need to calculate the total amount Musa owes after 4 months, including the interest accrued on the principal, and then subtract the amount he has already repaid.

    2. Explain the Concept

    To solve this, we use the Simple Interest formula: Simple Interest (I) = (P × R × T) / 100 Where:

    • P = Principal (the amount borrowed) = N10.00
    • R = Rate per month = 2%
    • T = Time in months = 4 months

    The Total Amount (A) to be repaid is the Principal plus the Interest: A = P + I

    Finally, the Balance Owed is: Balance = Total Amount (A) - Repayment

    3. Show the Solution

    Step 1: Calculate the Interest (I) $I = \frac{P \times R \times T}{100}$ $I = \frac{10 \times 2 \times 4}{100}$ $I = \frac{80}{100}$ $I = N0.80$

    Step 2: Calculate the Total Amount (A) Musa owes after 4 months $A = P + I$ $A = 10.00 + 0.80$ $A = N10.80$

    Step 3: Calculate the remaining balance after the repayment Musa repaid N8.00. $Balance = 10.80 - 8.00$ $Balance = N2.80$

    4. Identify the Answer

    Comparing our result to the options provided, we see that N2.80 corresponds to option C.

    5. Exam Tip

    When dealing with "Simple Interest" questions in JAMB, always pay close attention to the unit of time. If the rate is given "per month," ensure your time (T) is also in months. If the rate was "per annum" (per year) but the time was in months, you would have to convert the time to years (e.g., 4 months = 4/12 years).

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24. 24.

    Solve the following equation: $\frac{2}{(2r - 1)}$ - $\frac{5}{3}$ =  $\frac{1}{(r + 2)}$

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

    Show answer

    Answer: B. ( 1, - $\frac{5}{2}$ )

    AI explanation

    1. Analyze the question

    We are given a rational equation:

    $\frac{2}{2r - 1} - \frac{5}{3} = \frac{1}{r + 2}$

    Our goal is to find the values of $r$ that satisfy this equation.

    2. Explain the concept

    To solve an equation involving fractions, we need to eliminate the denominators by multiplying every term by the Least Common Multiple (LCM) of the denominators. Once the fractions are cleared, we will have a quadratic equation, which we can solve by factorization or the quadratic formula.

    3. Show the solution

    Step 1: Identify the LCM The denominators are $(2r - 1)$, $3$, and $(r + 2)$. The LCM is $3(2r - 1)(r + 2)$.

    Step 2: Multiply every term by the LCM

    $3(2r - 1)(r + 2) \left[ \frac{2}{2r - 1} \right] - 3(2r - 1)(r + 2) \left[ \frac{5}{3} \right] = 3(2r - 1)(r + 2) \left[ \frac{1}{r + 2} \right]$

    Step 3: Simplify

    $6(r + 2) - 5(2r - 1)(r + 2) = 3(2r - 1)$

    Expand the brackets:

    $6r + 12 - 5(2r^2 + 4r - r - 2) = 6r - 3$

    $6r + 12 - 5(2r^2 + 3r - 2) = 6r - 3$

    $6r + 12 - 10r^2 - 15r + 10 = 6r - 3$

    Step 4: Rearrange into standard quadratic form ($ar^2 + br + c = 0$)

    $-10r^2 - 9r + 22 = 6r - 3$

    $-10r^2 - 15r + 25 = 0$

    Divide the entire equation by $-5$ to simplify:

    $2r^2 + 3r - 5 = 0$

    Step 5: Factorize We need two numbers that multiply to $(2 \times -5) = -10$ and add to $3$. These are $5$ and $-2$.

    $2r^2 - 2r + 5r - 5 = 0$

    $2r(r - 1) + 5(r - 1) = 0$

    $(2r + 5)(r - 1) = 0$

    Setting each factor to zero: $2r + 5 = 0 \implies r = -\frac{5}{2}$ $r - 1 = 0 \implies r = 1$

    The solutions are $1$ and $-\frac{5}{2}$.

    4. Identify the answer

    Comparing our results ($1, -\frac{5}{2}$) with the given options, we see that it matches option B.

    5. Tip

    When dealing with rational equations, always check your final answers against the original equation. If any value of $r$ makes a denominator zero (e.g., $r = 0.5$ or $r = -2$ in this case), that value is "undefined" and must be rejected. Here, both our answers are valid!

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25. 25.

    If g(x) = x$^2$ + 3x  find g(x + 1) - g(x)

    1. A.(x + 2)
    2. B.2(x + 2)
    3. C.(2x + 1)
    4. D.(x2 + 4)

    Show answer

    Answer: B. 2(x + 2)

    AI explanation

    Analysis

    To solve this problem, we need to evaluate the function $g(x)$ at two different points: $x+1$ and $x$, and then find the difference between the two results.

    Concept

    The function is given as $g(x) = x^2 + 3x$. To find $g(x+1)$, we substitute $(x+1)$ wherever we see $x$ in the original function. After finding both expressions, we subtract $g(x)$ from $g(x+1)$.

    Step-by-Step Solution

    Step 1: Find $g(x+1)$ Substitute $(x+1)$ into the function $g(x) = x^2 + 3x$: $g(x+1) = (x+1)^2 + 3(x+1)$

    Expand the terms: $(x+1)^2 = x^2 + 2x + 1$ $3(x+1) = 3x + 3$

    Combine them: $g(x+1) = (x^2 + 2x + 1) + (3x + 3)$ $g(x+1) = x^2 + 5x + 4$

    Step 2: Find $g(x+1) - g(x)$ Now, subtract the original function $g(x) = x^2 + 3x$ from the result in Step 1: $[x^2 + 5x + 4] - [x^2 + 3x]$

    Distribute the negative sign: $x^2 + 5x + 4 - x^2 - 3x$

    Group like terms: $(x^2 - x^2) + (5x - 3x) + 4$ $0 + 2x + 4$ $2x + 4$

    Step 3: Factorize the result Factor out 2 from the expression: $2(x + 2)$

    Conclusion

    Comparing our result $2(x + 2)$ with the given options, it matches option B.

    Exam Tip

    When dealing with functions like $g(x+a)$, always expand the brackets carefully. A common mistake is to write $(x+1)^2$ as $x^2 + 1$ instead of $x^2 + 2x + 1$. Always remember the identity $(a+b)^2 = a^2 + 2ab + b^2$.

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26. 26.

    The ratio of the length of two similar rectangular blocks is 2 : 3. If the volume of the larger block is 351cm$^3$, then the volume of the other block is?

    1. A.234.00 cm3
    2. B.526.50 cm3
    3. C.166.00 cm3
    4. D.687cm3

    Show answer

    Answer: C. 166.00 cm3

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27. 27.

    Two numbers are removed at random from the numbers 1, 2, 3 and 4. What is the probability that the sum of the numbers removed is even?

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

    Show answer

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

    AI explanation

    1. Analyze the question

    We are given a set of four numbers: $\{1, 2, 3, 4\}$. We need to select two numbers at random and find the probability that their sum is an even number.

    2. Explain the concept

    • Total Outcomes: We need to find the number of ways to choose 2 numbers out of 4. This is a combination problem: $^nC_r = \frac{n!}{r!(n-r)!}$.
    • Condition for an Even Sum: The sum of two numbers is even if: • Both numbers are even (Even + Even = Even) • Both numbers are odd (Odd + Odd = Even)
    • Probability Formula: $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

    3. Show the solution

    Step 1: Calculate the total number of ways to choose 2 numbers from 4.

    $_4C_2 = \frac{4 \times 3}{2 \times 1} = 6$

    The possible pairs are: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)

    Step 2: Identify the pairs that result in an even sum.

    • (1, 2) $\rightarrow$ Sum = 3 (Odd)
    • (1, 3) $\rightarrow$ Sum = 4 (Even)
    • (1, 4) $\rightarrow$ Sum = 5 (Odd)
    • (2, 3) $\rightarrow$ Sum = 5 (Odd)
    • (2, 4) $\rightarrow$ Sum = 6 (Even)
    • (3, 4) $\rightarrow$ Sum = 7 (Odd)

    There are 2 favorable outcomes: (1, 3) and (2, 4).

    Step 3: Calculate the probability.

    $P(\text{Even Sum}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{2}{6}$

    Simplifying the fraction:

    $\frac{2}{6} = \frac{1}{3}$

    4. Identify the answer

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

    5. Exam Tip

    When dealing with "sum is even/odd" problems, remember the parity rules:

    • Odd + Odd = Even
    • Even + Even = Even
    • Odd + Even = Odd Instead of listing all pairs, you can quickly count how many odd and even numbers are in the set and use combinations to find the favorable pairs. This saves time in the JAMB exam hall!

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28. 28.

    factorize m$^3$ - m$^2$ + 2m - 2

    1. A.(m2 + 1)(m - 2)
    2. B.(m - 1)(m + 1)(m + 2)
    3. C.(m - 2)(m + 1)(m - 1)
    4. D.(m2 + 2)(m - 1)

    Show answer

    Answer: D. (m2 + 2)(m - 1)

    AI explanation

    1. Analyze the question

    We are asked to factorize the polynomial expression: m³ - m² + 2m - 2. This is a four-term expression, which suggests that factorization by grouping is the most effective method.

    2. Explain the concept

    Factorization by grouping involves splitting the expression into two pairs of terms. We look for a common factor in each pair. Once we factor out the common terms, we look for a common binomial factor that can be pulled out to complete the factorization.

    3. Show the solution

    Step 1: Group the terms into two pairs. (m³ - m²) + (2m - 2)

    Step 2: Factor out the greatest common factor (GCF) from each group.

    • In the first group (m³ - m²), the GCF is m². m²(m - 1)
    • In the second group (2m - 2), the GCF is 2. 2(m - 1)

    Step 3: Rewrite the expression with the factored groups. m²(m - 1) + 2(m - 1)

    Step 4: Factor out the common binomial (m - 1). Since (m - 1) appears in both parts, we can factor it out: (m - 1)(m² + 2)

    4. Identify the answer

    Comparing our result, (m - 1)(m² + 2), to the provided options, we see that it matches option D.

    *

    5. Tip

    When you see a polynomial with four terms in a JAMB exam, always try grouping the first two and the last two terms first. If the terms inside the brackets after the first step are identical, you are on the right track! If they aren't, try rearranging the terms before grouping.

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29. 29.

    Find the derivative of the function y = 2x$^2$(2x - 1) at the point x = -1?

    1. A.18
    2. B.16
    3. C.-4
    4. D.-6

    Show answer

    Answer: B. 16

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30. 30.

    Three children shared a basket of mangoes in such a way that the first child took $\frac{1}{4}$ of the mangoes and the second $\frac{3}{4}$ of the remainder. What fraction of the mangoes did the third child take?

    1. A.$\frac{3}{16}$
    2. B.$\frac{7}{16}$
    3. C.$\frac{9}{16}$
    4. D.$\frac{13}{16}$

    Show answer

    Answer: A. $\frac{3}{16}$

    AI explanation

    Analysis

    To solve this problem, we need to determine the portion of the basket left after each child takes their share. We start with a whole basket, represented by the fraction $1$.

    Concept

    When dealing with "remainders" in fractions, we subtract the portion taken from the total to find what is left. We then apply the next fraction to that specific remainder.

    Step-by-Step Solution

    Step 1: Find the remainder after the first child. The first child took $\frac{1}{4}$ of the mangoes.

    $\text{Remainder}_1 = 1 - \frac{1}{4} = \frac{3}{4}$

    Step 2: Find the share of the second child. The second child took $\frac{3}{4}$ of the remainder.

    $\text{Second child's share} = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}$

    Step 3: Find the share of the third child. The third child gets whatever is left after the first and second children have taken their portions.

    $\text{Total taken by first and second} = \frac{1}{4} + \frac{9}{16}$

    To add these, find a common denominator (which is 16):

    $\frac{4}{16} + \frac{9}{16} = \frac{13}{16}$

    Now, subtract this total from the whole basket ($1$):

    $\text{Third child's share} = 1 - \frac{13}{16}$

    $\text{Third child's share} = \frac{16}{16} - \frac{13}{16} = \frac{3}{16}$

    Conclusion

    The third child took $\frac{3}{16}$ of the mangoes. This matches option A.

    Exam Tip

    Whenever you see the phrase "of the remainder," always calculate the remaining fraction first before multiplying. A common mistake is to multiply the second fraction by the original total instead of the remainder. Always check that your final fractions add up to $1$ ($\frac{4}{16} + \frac{9}{16} + \frac{3}{16} = \frac{16}{16} = 1$).

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31. 31.

    find the value of p if the line which passes through (-1, -p) and (-2,2) is parallel to the line 2y+8x-17=0?

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

    Show answer

    Answer: D. 2

    AI explanation

    Analysis

    To solve this problem, we need to use the property of parallel lines. Two lines are parallel if and only if their slopes ($m$) are equal. We will find the slope of the line passing through the two given points and equate it to the slope of the given equation.

    Concept: Slope of a Line

    1. Slope from two points $(x_1, y_1)$ and $(x_2, y_2)$: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
    2. Slope from a linear equation ($Ax + By + C = 0$): Rewrite the equation in the form $y = mx + c$, where $m$ is the slope. Alternatively, $m = -\frac{A}{B}$.

    Step-by-Step Solution

    Step 1: Find the slope of the given line $2y + 8x - 17 = 0$. Rearrange the equation to make $y$ the subject: $2y = -8x + 17$ Divide by 2: $y = -4x + \frac{17}{2}$ Comparing this to $y = mx + c$, the slope ($m_1$) is -4.

    Step 2: Find the slope of the line passing through $(-1, -p)$ and $(-2, 2)$. Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$: $m_2 = \frac{2 - (-p)}{-2 - (-1)}$ $m_2 = \frac{2 + p}{-2 + 1}$ $m_2 = \frac{2 + p}{-1}$ $m_2 = -(2 + p) = -2 - p$

    Step 3: Equate the slopes. Since the lines are parallel, $m_1 = m_2$: $-4 = -2 - p$

    Step 4: Solve for $p$. $-4 + 2 = -p$ $-2 = -p$ $p = 2$

    Conclusion

    The value of $p$ is 2, which corresponds to option D.

    Exam Tip

    When dealing with parallel lines, always remember: Parallel = Same Slope. If the question had said the lines were perpendicular, you would have used the rule $m_1 \times m_2 = -1$. Also, always be careful with the signs when subtracting negative coordinates—it is the most common place students make mistakes in coordinate geometry!

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32. 32.

    Simplify and express in standard form $\frac{0.00275 \times 0.00640}{0.025 \times 0.08}$

    1. A.8.8 x 10-1
    2. B.8.8 x 10-2
    3. C.8.8 x 10-3
    4. D.8.8 x 103

    Show answer

    Answer: C. 8.8 x 10-3

    AI explanation

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

    1. Analyze the question

    We need to simplify the expression:

    $\frac{0.00275 \times 0.00640}{0.025 \times 0.08}$

    2. Explain the concept

    Standard form is written as $a \times 10^n$, where $1 \le a < 10$.

    • $0.00275 = 2.75 \times 10^{-3}$
    • $0.00640 = 6.4 \times 10^{-3}$
    • $0.025 = 2.5 \times 10^{-2}$
    • $0.08 = 8.0 \times 10^{-2}$

    3. Show the solution

    Substitute these values into the expression:

    $\frac{(2.75 \times 10^{-3}) \times (6.4 \times 10^{-3})}{(2.5 \times 10^{-2}) \times (8.0 \times 10^{-2})}$

    Step A: Multiply the numerators and denominators separately Numerator: $(2.75 \times 6.4) \times (10^{-3} \times 10^{-3}) = 17.6 \times 10^{-6}$ Denominator: $(2.5 \times 8.0) \times (10^{-2} \times 10^{-2}) = 20 \times 10^{-4}$

    Step B: Divide the results

    $\frac{17.6 \times 10^{-6}}{20 \times 10^{-4}}$

    Divide the coefficients: $17.6 \div 20 = 0.88$ Divide the powers of 10: $10^{-6} \div 10^{-4} = 10^{-6 - (-4)} = 10^{-2}$

    So, we have: $0.88 \times 10^{-2}$

    Step C: Convert to standard form To make $0.88$ a number between 1 and 10, we move the decimal point one place to the right: $0.88 = 8.8 \times 10^{-1}$

    Now, multiply this by the $10^{-2}$ we already had: $(8.8 \times 10^{-1}) \times 10^{-2} = 8.8 \times 10^{-1 + (-2)} = 8.8 \times 10^{-3}$

    4. Identify the answer

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

    5. Exam Tip

    When dealing with many decimal places in JAMB, always convert to powers of 10 first. It prevents "decimal point errors" which are the most common cause of mistakes in these types of questions. Also, remember the law of indices: $10^a \div 10^b = 10^{a-b}$. Be very careful with the signs when subtracting negative exponents!

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33. 33.

    Each of the interior angles of a regular polygon is 140°. How many sides has the polygon?

    1. A.9
    2. B.8
    3. C.7
    4. D.5

    Show answer

    Answer: A. 9

    AI explanation

    Analysis

    The question asks for the number of sides ($n$) of a regular polygon, given that each interior angle measures $140^\circ$.

    Concept

    To solve this, we use the relationship between the interior angle and the exterior angle of a polygon.

    1. The sum of an interior angle and its corresponding exterior angle is always $180^\circ$ (angles on a straight line).
    2. The sum of all exterior angles of any convex polygon is always $360^\circ$.
    3. For a regular polygon, each exterior angle is calculated as: $$\text{Exterior Angle} = \frac{360^\circ}{n}$$

    Solution

    Step 1: Find the exterior angle. Since the interior angle is $140^\circ$:

    $\text{Exterior Angle} = 180^\circ - 140^\circ = 40^\circ$

    Step 2: Use the exterior angle formula to find $n$. We know that:

    $40^\circ = \frac{360^\circ}{n}$

    Step 3: Solve for $n$. Multiply both sides by $n$ and divide by $40^\circ$:

    $n = \frac{360^\circ}{40^\circ}$

    $n = 9$

    The polygon has 9 sides (it is a nonagon).

    Conclusion

    Comparing our result with the options provided, the correct answer is A.

    Exam Tip

    Always remember that Exterior Angle = $360^\circ / n$. It is usually much faster to work with the exterior angle than the interior angle formula ($(n-2) \times 180 / n$) because it involves simpler division.

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34. 34.

    Find x if log$_9$x = 1.5

    1. A.27
    2. B.15
    3. C.3.5
    4. D.32

    Show answer

    Answer: A. 27

    AI explanation

    Analysis

    The question asks us to solve for $x$ in the logarithmic equation $\log_9 x = 1.5$. To solve this, we need to convert the logarithmic form into its equivalent exponential form.

    Concept: Logarithms to Exponents

    The fundamental definition of a logarithm states that if $\log_b a = c$, then $b^c = a$. In this equation:

    • The base ($b$) is 9
    • The exponent ($c$) is 1.5
    • The result ($a$) is $x$

    Step-by-Step Solution

    Step 1: Rewrite in exponential form Using the definition above: $x = 9^{1.5}$

    Step 2: Simplify the exponent Recall that $1.5$ can be written as a fraction: $1.5 = \frac{3}{2}$. So, $x = 9^{3/2}$

    Step 3: Evaluate the expression The fractional exponent $\frac{3}{2}$ means we take the square root of the base and then raise it to the power of 3: $x = (\sqrt{9})^3$

    Since $\sqrt{9} = 3$: $x = 3^3$

    Step 4: Calculate the final value $x = 3 \times 3 \times 3$ $x = 27$

    Conclusion

    Comparing our result to the given options, we find that $x = 27$ corresponds to option A.

    Exam Tip

    When you see a decimal exponent like $1.5$ or $0.5$, always convert it to a fraction ($3/2$ or $1/2$) immediately. It makes the calculation much easier to visualize! For example, $x^{0.5}$ is just the square root of $x$.

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35. 35.

    A cylinder pipe, made of metal is 3cm thick.If the internal radius of the pipe is 10cm.Find the volume of metal used in making 3m of the pipe.

    1. A.153$\pi$cm3
    2. B.207$\pi$cm3
    3. C.15 300$\pi$cm3
    4. D.20 700$\pi$cm3

    Show answer

    Answer: D. 20 700$\pi$cm3

    AI explanation

    1. Analyze the question

    We are asked to find the volume of metal used to make a cylindrical pipe.

    • Internal radius ($r$): 10 cm
    • Thickness of the pipe: 3 cm
    • Length (height, $h$) of the pipe: 3 m
    • Goal: Calculate the volume of the metal (the hollow part).

    2. Explain the concept

    A cylindrical pipe is essentially a hollow cylinder. The volume of the metal used is the difference between the outer cylinder and the inner cylinder.

    • Internal radius ($r$): 10 cm
    • External radius ($R$): Internal radius + Thickness = $10\text{ cm} + 3\text{ cm} = 13\text{ cm}$
    • Height ($h$): Since the radii are in cm, we must convert the length to cm. $3\text{ m} = 300\text{ cm}$.
    • Formula for Volume of a hollow cylinder: $V = \pi h(R^2 - r^2)$

    3. Show the solution

    Step 1: Identify the dimensions

    • $R = 13\text{ cm}$
    • $r = 10\text{ cm}$
    • $h = 300\text{ cm}$

    Step 2: Apply the formula $V = \pi \times h \times (R^2 - r^2)$ $V = \pi \times 300 \times (13^2 - 10^2)$

    Step 3: Calculate the squares $13^2 = 169$ $10^2 = 100$

    Step 4: Subtract and multiply $V = \pi \times 300 \times (169 - 100)$ $V = \pi \times 300 \times 69$ $V = 20,700\pi\text{ cm}^3$

    4. Identify the answer

    Comparing our result to the given options: A. $153\pi\text{ cm}^3$ B. $207\pi\text{ cm}^3$ C. $15,300\pi\text{ cm}^3$ D. $20,700\pi\text{ cm}^3$

    The correct option is D.

    5. Tip

    Always check your units before calculating! In JAMB questions, they often mix meters and centimeters to test if you remember to convert them to the same unit. If you had used $h = 3$ instead of $300$, you would have gotten $207\pi$, which is option B—a common "distractor" answer designed to catch students who forget to convert units.

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36. 36.

    In how many ways can 2 students be selected from a group of 5 students in a debating competition?

    1. A.25 ways
    2. B.10 ways
    3. C.15 ways
    4. D.20 ways

    Show answer

    Answer: B. 10 ways

    AI explanation

    Analysis

    The question asks for the number of ways to select 2 students out of a group of 5. In this scenario, the order in which the students are selected does not matter (selecting Student A then Student B is the same as selecting Student B then Student A). Therefore, we use the concept of Combinations.

    Concept

    When selecting $r$ items from a total of $n$ items where order does not matter, we use the combination formula:

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

    Where:

    • $n$ = total number of items (5 students)
    • $r$ = number of items to be selected (2 students)
    • $!$ = factorial (e.g., $3! = 3 \times 2 \times 1 = 6$)

    Solution

    1. Identify the values: $n = 5$ $r = 2$

    2. Apply the formula: $$^5C_2 = \frac{5!}{2!(5 - 2)!}$$

    3. Simplify the expression: $$^5C_2 = \frac{5!}{2! \times 3!}$$

    4. Expand the factorials: $$^5C_2 = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1) \times (3 \times 2 \times 1)}$$

    5. Cancel out common terms: $$^5C_2 = \frac{5 \times 4}{2 \times 1}$$ $$^5C_2 = \frac{20}{2}$$ $$^5C_2 = 10$$

    There are 10 different ways to select the students.

    Exam Tip

    Whenever you see the word "selected," "chosen," or "committee," think Combination ($^nC_r$). If the question implies a specific order or position (like "President" and "Secretary"), then you should use Permutation ($^nP_r$) instead.

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37. 37.

    A group of market women sell at least one of yam, plantain and maize. 12 of them sell maize, 10 sell yam and 14 sell plantain. 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only while 3 sell all the three items. How many women are in the group?

    1. A.25
    2. B.19
    3. C.18
    4. D.17

    Show answer

    Answer: A. 25

    AI explanation

    Analysis

    To solve this problem, we use a Venn diagram approach for three sets: Y (Yam), P (Plantain), and M (Maize). We are given the total number of people in each set and the intersections.

    Concept

    We need to find the total number of women, which is the union of the three sets: $n(Y \cup P \cup M)$. The formula for the union of three sets is: $n(Y \cup P \cup M) = n(Y) + n(P) + n(M) - [n(Y \cap P) + n(Y \cap M) + n(P \cap M)] + n(Y \cap P \cap M)$

    Step-by-Step Solution

    1. Identify the given values:

    • $n(M) = 12$
    • $n(Y) = 10$
    • $n(P) = 14$
    • $n(P \cap M) = 5$
    • $n(Y \cap M) = 4$
    • $n(Y \cap P \text{ only}) = 2$
    • $n(Y \cap P \cap M) = 3$

    2. Determine the intersections:

    • Since $n(Y \cap P \text{ only}) = 2$ and $n(Y \cap P \cap M) = 3$, the total intersection $n(Y \cap P)$ is: $n(Y \cap P) = n(Y \cap P \text{ only}) + n(Y \cap P \cap M) = 2 + 3 = 5$

    3. Calculate the total using the principle of inclusion-exclusion: We have all the components needed for the formula:

    • Sum of individual sets: $12 + 10 + 14 = 36$
    • Sum of double intersections: $n(Y \cap P) + n(Y \cap M) + n(P \cap M) = 5 + 4 + 5 = 14$
    • Triple intersection: $3$

    Applying the formula: Total $= (12 + 10 + 14) - (5 + 4 + 5) + 3$ Total $= 36 - 14 + 3$ Total $= 22 + 3 = 25$

    Verification

    If we fill the Venn diagram regions:

    • All three: 3
    • Yam and Plantain only: 2
    • Yam and Maize only: $4 - 3 = \mathbf{1}$
    • Plantain and Maize only: $5 - 3 = \mathbf{2}$
    • Yam only: $10 - (3 + 2 + 1) = 10 - 6 = \mathbf{4}$
    • Plantain only: $14 - (3 + 2 + 2) = 14 - 7 = \mathbf{7}$
    • Maize only: $12 - (3 + 1 + 2) = 12 - 6 = \mathbf{6}$

    Summing these up: $3 + 2 + 1 + 2 + 4 + 7 + 6 = 25$.

    Exam Tip

    When solving Venn diagram problems, always start from the "center" (the intersection of all three sets) and work your way outwards. If the question says "only," do not subtract the triple intersection again, as it has already been excluded.

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38. 38.

    The angles of a quadrilateral are 5x-30, 4x+60, 60-x and 3x+61.find the smallest of these angles​

    1. A.5x - 30
    2. B.4x + 60
    3. C.60 - x
    4. D.3x + 61

    Show answer

    Answer: C. 60 - x

    AI explanation

    1. Analyze the question

    We are given the four interior angles of a quadrilateral in terms of $x$. We need to find the value of $x$ first, then substitute it back into each expression to determine which one yields the smallest numerical value.

    2. Explain the concept

    The sum of the interior angles of any quadrilateral is always 360°. Therefore, we can set up an equation by adding all four given expressions and equating them to 360: $(5x - 30) + (4x + 60) + (60 - x) + (3x + 61) = 360$

    3. Show the solution

    Step 1: Solve for $x$ Group the like terms: $(5x + 4x - x + 3x) + (-30 + 60 + 60 + 61) = 360$ $(11x) + (151) = 360$

    Subtract 151 from both sides: $11x = 360 - 151$ $11x = 209$

    Divide by 11: $x = \frac{209}{11}$ $x = 19$

    Step 2: Calculate each angle Now, substitute $x = 19$ into each expression:

    • Angle 1: $5x - 30 = 5(19) - 30 = 95 - 30 = \mathbf{65^\circ}$
    • Angle 2: $4x + 60 = 4(19) + 60 = 76 + 60 = \mathbf{136^\circ}$
    • Angle 3: $60 - x = 60 - 19 = \mathbf{41^\circ}$
    • Angle 4: $3x + 61 = 3(19) + 61 = 57 + 61 = \mathbf{118^\circ}$

    Step 3: Identify the smallest angle Comparing the results ($65^\circ, 136^\circ, 41^\circ, 118^\circ$), the smallest angle is 41°, which corresponds to the expression $60 - x$.

    4. Identify the answer

    The expression $60 - x$ corresponds to option C.

    5. Tip

    When dealing with geometry problems involving "sum of angles," always write down the geometric rule first (e.g., "Sum of angles in a quadrilateral = 360°"). This helps you avoid silly mistakes when setting up your algebraic equation. Also, always double-check your arithmetic when adding the constants!

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39. 39.

    The angle of a sector of a circle, radius 10.5cm, is 48°, Calculate the perimeter of the sector

    1. A.8.8cm
    2. B.25.4cm
    3. C.25.6cm
    4. D.29.8cm

    Show answer

    Answer: D. 29.8cm

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40. 40.

    What is the rate of change of the volume V of a hemisphere with respect to its radius r when r = 2?

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

    Show answer

    Answer: A. 8π

    AI explanation

    1. Analyze the question

    We are asked to find the rate of change of the volume ($V$) of a hemisphere with respect to its radius ($r$) at the specific point where $r = 2$. In calculus, the "rate of change" of a quantity with respect to another is represented by the derivative. Therefore, we need to find $\frac{dV}{dr}$ and evaluate it at $r = 2$.

    2. Explain the concept

    • Volume of a Sphere: The volume of a full sphere is given by $V = \frac{4}{3}\pi r^3$.
    • Volume of a Hemisphere: A hemisphere is exactly half of a sphere. Therefore, the volume is: $$V = \frac{1}{2} \times \left( \frac{4}{3}\pi r^3 \right) = \frac{2}{3}\pi r^3$$
    • Differentiation: To find the rate of change, we differentiate $V$ with respect to $r$ using the power rule ($\frac{d}{dx}x^n = nx^{n-1}$).

    3. Show the solution

    Step 1: Write the formula for the volume of a hemisphere.

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

    Step 2: Differentiate $V$ with respect to $r$.

    $\frac{dV}{dr} = \frac{d}{dr} \left( \frac{2}{3}\pi r^3 \right)$

    Using the power rule, bring down the exponent (3) and multiply:

    $\frac{dV}{dr} = \frac{2}{3} \times 3 \times \pi \times r^{(3-1)}$

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

    Step 3: Evaluate the derivative at $r = 2$. Substitute $r = 2$ into the derivative formula:

    $\frac{dV}{dr} = 2\pi(2)^2$

    $\frac{dV}{dr} = 2\pi(4)$

    $\frac{dV}{dr} = 8\pi$

    4. Identify the answer

    Comparing our result ($8\pi$) to the given options, it matches option A.

    5. Tip

    Whenever you see "rate of change" in a calculus question, immediately think of differentiation. If the question asks for the rate of change of volume with respect to radius, you are looking for $\frac{dV}{dr}$. If it asked for the rate of change with respect to time, you would use the chain rule ($\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}$).

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