2025 JAMB Mathematics Past Questions and Answers

Questions

1. 1.

   P is partly constant and varies partly as Q. If P = 32 when Q = 16 and P = 20 when Q = 12, find P when Q = 28

   1. A.68
   2. B.64
   3. C.66
   4. D.62

   Show answer

   Answer: A. 68

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2. 2.

   A bird flies from a tree P on a bearing of N60º E to a building, Q, a distance of  200 km. It then changes course and flies to another tree R on a bearing of S30ºE. Tree R is directly east of tree P. Calculate the distance of the building to tree R.

   1. A.200$\sqrt{3}$
   2. B.100$\sqrt{3}$
   3. C.100
   4. D.$\frac{200}{\sqrt{3}}$

   Show answer

   Answer: D. $\frac{200}{\sqrt{3}}$

   AI explanation

   1. Analyze the Question

   We are given the movement of a bird starting from point P (tree) to Q (building), and then to R (another tree).

   • P to Q: Distance = 200 km, Bearing = N60°E.
   • Q to R: Bearing = S30°E.
   • R is directly East of P: This means the line PR is a horizontal line (a line of latitude).
   • Goal: Find the distance QR.

   2. Explain the Concept

   This is a problem involving bearings and trigonometry.

   • Bearings: Always measured clockwise from the North.
   • Geometry: Since R is directly East of P, the line PR is horizontal. We can form a triangle PQR.
   • Angles: • At point P, the bearing N60°E means the angle between the North line and PQ is 60°. Since the East line is 90° from North, the angle inside the triangle at P (angle QPR) is $90° - 60° = 30°$. • At point Q, the bearing S30°E means the angle between the South line and QR is 30°. Since the North-South line is vertical, the angle between the vertical line at Q and the line QR is 30°. • Using alternate angles, if we draw a vertical line through Q, the angle between PQ and the vertical is 60° (alternate to the angle at P). Therefore, the interior angle at Q (angle PQR) is $60° + 30° = 90°$.

   3. Show the Solution

   We now have a right-angled triangle PQR where:

   • $\angle PQR = 90°$
   • $\angle QPR = 30°$
   • Hypotenuse $PR$ is the side opposite the 90° angle.
   • Side $PQ = 200$ km.
   • We need to find side $QR$.

   In a right-angled triangle, we use the trigonometric ratio:

   $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

   For angle $P$ (30°):

   $\tan(30°) = \frac{QR}{PQ}$

   Substitute the known values:

   $\frac{1}{\sqrt{3}} = \frac{QR}{200}$

   Solve for $QR$:

   $QR = \frac{200}{\sqrt{3}}$

   4. Identify the Answer

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

   5. Tip

   When solving bearing problems, always draw a neat sketch. Mark your North lines at every point where the direction changes. Remember that "Directly East" or "Directly North" implies a 90-degree relationship with the cardinal axes, which often helps you identify right-angled triangles quickly!

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3. 3.

   If I is a 2 × 2 identity matrix, find the determinant of the matrix.

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

   Show answer

   Answer: A. 1

   AI explanation

   Analysis

   The question asks for the determinant of a $2 \times 2$ identity matrix, denoted as $I$.

   Concept

   An identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. For a $2 \times 2$ matrix, the identity matrix $I$ is defined as:

   $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

   The determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is calculated using the formula:

   $\text{det}(A) = (ad) - (bc)$

   Solution

   1. Identify the elements of the matrix: For the $2 \times 2$ identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$: • $a = 1$ • $b = 0$ • $c = 0$ • $d = 1$

   2. Apply the determinant formula: $$\text{det}(I) = (1 \times 1) - (0 \times 0)$$ $$\text{det}(I) = 1 - 0$$ $$\text{det}(I) = 1$$

   Conclusion

   The determinant of the $2 \times 2$ identity matrix is 1.

   Exam Tip

   Always remember that the determinant of any identity matrix (whether $2 \times 2$, $3 \times 3$, or $n \times n$) is always 1. This is a useful shortcut that can save you time during your JAMB exam!

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4. 4.

   Find the value of t for which ($\frac{1}{2}$)$^{t - 1}$ = 64

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

   Show answer

   Answer: B. -5

   AI explanation

   Analysis

   To solve this equation, we need to express both sides of the equation with the same base. Once the bases are identical, we can equate the exponents to solve for $t$.

   Concept

   This is an exponential equation. The key principle is: If $a^x = a^y$, then $x = y$.

   We know that:

   • $\frac{1}{2}$ can be written as $2^{-1}$.
   • $64$ is a power of $2$ ($2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$, so $64 = 2^6$).

   Step-by-Step Solution

   Step 1: Rewrite the equation using base 2. The original equation is: $(\frac{1}{2})^{t - 1} = 64$

   Substitute $\frac{1}{2}$ with $2^{-1}$: $(2^{-1})^{t - 1} = 2^6$

   Step 2: Apply the law of indices $(a^m)^n = a^{m \times n}$. Multiply the exponents on the left side: $2^{-1(t - 1)} = 2^6$ $2^{-t + 1} = 2^6$

   Step 3: Equate the exponents. Since the bases are now the same (both are 2), we can set the exponents equal to each other: $-t + 1 = 6$

   Step 4: Solve for $t$. Subtract 1 from both sides: $-t = 6 - 1$ $-t = 5$

   Multiply both sides by -1: $t = -5$

   Conclusion

   The value of $t$ that satisfies the equation is $-5$.

   Exam Tip: Whenever you see a fraction like $\frac{1}{2}, \frac{1}{3},$ or $\frac{1}{4}$ in an exponential equation, remember that they are just negative powers of the denominator (e.g., $\frac{1}{2} = 2^{-1}$, $\frac{1}{4} = 4^{-1}$ or $2^{-2}$). Converting to negative powers early makes the calculation much easier!

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5. 5.

   Given that $P = \begin{pmatrix} 1 & 3 \\ 2 & -5 \end{pmatrix}$ and Q = $\begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix}$ . Find P + 2Q

   1. A.$\begin{pmatrix} 7 & -11 \\ 4 & -1 \end{pmatrix}$
   2. B.$\begin{pmatrix} 7 & -11 \\ 4 & -9 \end{pmatrix}$
   3. C.$\begin{pmatrix} -7 & 11 \\ -4 & -1 \end{pmatrix}$
   4. D.$\begin{pmatrix}11 & -7 \\ 4 & -1 \end{pmatrix}$

   Show answer

   Answer: A. $\begin{pmatrix} 7 & -11 \\ 4 & -1 \end{pmatrix}$

   AI explanation

   Analysis

   To solve this problem, we need to perform two matrix operations:

   1. Scalar Multiplication: Multiply every element in matrix $Q$ by 2.
   2. Matrix Addition: Add the resulting matrix ($2Q$) to matrix $P$ by summing the corresponding elements.

   Concept

   If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$, then $A + B = \begin{pmatrix} a+e & b+f \\ c+g & d+h \end{pmatrix}$. Scalar multiplication $kQ$ means multiplying every entry in $Q$ by $k$.

   Step-by-Step Solution

   Step 1: Calculate $2Q$ Given $Q = \begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix}$, multiply each element by 2: $2Q = 2 \times \begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 2(3) & 2(-7) \\ 2(1) & 2(2) \end{pmatrix} = \begin{pmatrix} 6 & -14 \\ 2 & 4 \end{pmatrix}$

   Step 2: Calculate $P + 2Q$ Now, add matrix $P$ to the result of $2Q$: $P + 2Q = \begin{pmatrix} 1 & 3 \\ 2 & -5 \end{pmatrix} + \begin{pmatrix} 6 & -14 \\ 2 & 4 \end{pmatrix}$

   Perform the addition element by element:

   • Top-left: $1 + 6 = 7$
   • Top-right: $3 + (-14) = 3 - 14 = -11$
   • Bottom-left: $2 + 2 = 4$
   • Bottom-right: $-5 + 4 = -1$

   Resulting matrix: $\begin{pmatrix} 7 & -11 \\ 4 & -1 \end{pmatrix}$

   Conclusion

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

   Exam Tip

   When dealing with matrix addition or subtraction, always double-check your signs, especially when dealing with negative numbers (like $-5 + 4$). A common mistake is to accidentally add the absolute values instead of performing the algebraic sum. Take your time with the arithmetic!

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6. 6.

   Simplify 125$^{−\frac{1}{3}}$ × 49$^{−\frac{1}{2}}$ × 10$^0$

   1. A.$\frac{1}{35}$
   2. B.$\frac{1}{350}$
   3. C.350
   4. D.35

   Show answer

   Answer: A. $\frac{1}{35}$

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7. 7.

   Find the limit of y = $\frac{(x^3 - 2x^2 + 6x - 12)}{(x - 2)}$ as x goes to 2.

   1. A.Infinity
   2. B.10
   3. C.0
   4. D.12

   Show answer

   Answer: B. 10

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8. 8.

   An amount of # 600,000.00 was realized when a principal y was saved for 5% simple interest for 4 years, find the value of y

   1. A.# 570,000
   2. B.# 500,000
   3. C.# 300,000
   4. D.# 400,000

   Show answer

   Answer: B. # 500,000

   AI explanation

   1. Analyze the question

   We are given:

   • Amount ($A$): # 600,000.00 (This is the total sum of the principal and the interest).
   • Rate ($R$): 5% per annum.
   • Time ($T$): 4 years.
   • Principal ($P$): $y$ (This is what we need to find).

   2. Explain the concept

   In simple interest, the total amount ($A$) is the sum of the principal ($P$) and the simple interest ($I$). The formulas are:

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

   By substituting the interest formula into the amount formula, we get: $A = P + \frac{P \times R \times T}{100}$ $A = P(1 + \frac{RT}{100})$

   3. Show the solution

   Given: $A = 600,000$ $R = 5$ $T = 4$ $P = y$

   Substitute these values into the formula: $600,000 = y(1 + \frac{5 \times 4}{100})$

   Simplify the expression inside the bracket: $600,000 = y(1 + \frac{20}{100})$ $600,000 = y(1 + 0.2)$ $600,000 = y(1.2)$

   Now, solve for $y$: $y = \frac{600,000}{1.2}$

   To make the division easier, multiply both numerator and denominator by 10: $y = \frac{6,000,000}{12}$ $y = 500,000$

   4. Identify the answer

   The value of $y$ is # 500,000. This corresponds to option B.

   5. Tip

   When dealing with simple interest problems, always remember that the "Amount" is the Principal + Interest. A common mistake students make is calculating the interest and forgetting to add it back to the principal, or confusing "Amount" with "Interest." Always check if the question asks for the interest earned or the total amount.

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9. 9.

   Obtain the equation of a straight line passing through (3, 15) whose slope = 3$\frac{1}{5}$.

   1. A.5y + 16x + 27 = 0
   2. B.5y - 16x + 27 = 0
   3. C.5y - 16x - 27 = 0
   4. D.5y + 16x - 27 = 0

   Show answer

   Answer: C. 5y - 16x - 27 = 0

   AI explanation

   Analysis

   To find the equation of a straight line when you are given a point $(x_1, y_1)$ and the slope ($m$), you use the Point-Slope Form of a linear equation.

   Concept

   The point-slope formula is:

   $y - y_1 = m(x - x_1)$

   Where:

   • $(x_1, y_1) = (3, 15)$
   • $m = 3\frac{1}{5} = \frac{16}{5}$

   Solution

   Step 1: Substitute the values into the formula.

   $y - 15 = \frac{16}{5}(x - 3)$

   Step 2: Eliminate the fraction by multiplying both sides by 5.

   $5(y - 15) = 16(x - 3)$

   Step 3: Expand both sides.

   $5y - 75 = 16x - 48$

   Step 4: Rearrange the equation to match the format of the options (moving all terms to one side). Subtract $16x$ and add $48$ to both sides to set the equation to zero:

   $5y - 16x - 75 + 48 = 0$

   $5y - 16x - 27 = 0$

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

   Tip

   When dealing with fractional slopes in exam questions, always convert the mixed fraction to an improper fraction immediately. Multiplying the entire equation by the denominator is the fastest way to clear the fraction and avoid calculation errors with decimals!

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10. 10.

    Find the probability of getting an even number in a single throw of a six-sided die.

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

    Show answer

    Answer: C. $\frac{1}{2}$

    AI explanation

    Analysis

    To solve this problem, we need to determine the probability of an event occurring. Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in a sample space.

    Concept

    1. Sample Space ($S$): A standard six-sided die has six faces numbered 1, 2, 3, 4, 5, and 6. Therefore, the total number of possible outcomes $n(S) = 6$.
    2. Favorable Outcomes ($E$): We are looking for an "even number." From the set $\{1, 2, 3, 4, 5, 6\}$, the even numbers are 2, 4, and 6.
    3. Count of Favorable Outcomes: There are 3 even numbers, so $n(E) = 3$.
    4. Probability Formula: $P(E) = \frac{n(E)}{n(S)}$

    Solution

    • Step 1: Identify the total outcomes: $\{1, 2, 3, 4, 5, 6\}$, so $n(S) = 6$.
    • Step 2: Identify the favorable outcomes (even numbers): $\{2, 4, 6\}$, so $n(E) = 3$.
    • Step 3: Apply the formula: $$P(\text{even}) = \frac{3}{6}$$
    • Step 4: Simplify the fraction: $$\frac{3}{6} = \frac{1}{2}$$

    Exam Tip

    When dealing with dice problems, always list out your sample space if you are unsure. For a single die, the sample space is always $\{1, 2, 3, 4, 5, 6\}$. If the question asked for a prime number, remember that 1 is not a prime number, so your favorable outcomes would be $\{2, 3, 5\}$.

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11. 11.

    If A = $\frac{\theta}{360}$$\pi r^2$, make $\theta$ the subject of the formula

    1. A.$\theta = \frac{360}{\pi A}$
    2. B.$\theta = \frac{360A}{\pi \theta}$
    3. C.$\theta = \frac{360A}{\pi r^2}$
    4. D.$\theta = \frac{360 r^2}{\pi A}$

    Show answer

    Answer: C. $\theta = \frac{360A}{\pi r^2}$

    AI explanation

    To make $\theta$ the subject of the formula, we need to isolate it on one side of the equation. Here is the step-by-step breakdown:

    1. Analyze the equation

    The given equation is:

    $A = \frac{\theta}{360} \pi r^2$

    Our goal is to get $\theta$ by itself. Currently, $\theta$ is being multiplied by $\pi r^2$ and divided by $360$.

    2. Explain the concept

    To isolate a variable, we perform inverse operations. If a term is dividing, we multiply it to the other side. If a term is multiplying, we divide it to the other side.

    3. Show the solution

    Step 1: Multiply both sides by 360 to remove the fraction.

    $360 \times A = \theta \pi r^2$

    $360A = \theta \pi r^2$

    Step 2: Divide both sides by $(\pi r^2)$ to isolate $\theta$.

    $\frac{360A}{\pi r^2} = \theta$

    Rearranging this to put $\theta$ on the left side:

    $\theta = \frac{360A}{\pi r^2}$

    4. Identify the answer

    Comparing our result to the given options: A. $\theta = \frac{360}{\pi A}$ B. $\theta = \frac{360A}{\pi \theta}$ C. $\theta = \frac{360A}{\pi r^2}$ D. $\theta = \frac{360 r^2}{\pi A}$

    The correct option is C.

    5. Exam Tip

    When asked to "make a variable the subject," treat the equation like a balance scale. Whatever operation you perform to move terms away from your target variable, you must apply it to the entire opposite side. Also, always check your final expression to ensure the target variable ($\theta$) does not appear on the right-hand side of the final answer!

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

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

14. 14.

    Convert 137 to base 5

    1. A.$1022_5$
    2. B.$2102_5$.
    3. C.$2210_5$.
    4. D.$2201_5$.

    Show answer

    Answer: A. $1022_5$

    AI explanation

    To convert a number from base 10 to another base, we use the method of successive division. This involves dividing the number by the target base (in this case, 5) repeatedly and recording the remainders until the quotient becomes zero.

    Step-by-Step Solution:

    1. Divide 137 by 5: $137 \div 5 = 27$ remainder 2

    2. Divide 27 by 5: $27 \div 5 = 5$ remainder 2

    3. Divide 5 by 5: $5 \div 5 = 1$ remainder 0

    4. Divide 1 by 5: $1 \div 5 = 0$ remainder 1

    Reading the Result:

    To get the final answer, we read the remainders from the bottom to the top (from the last remainder to the first).

    • Last remainder: 1
    • Third remainder: 0
    • Second remainder: 2
    • First remainder: 2

    Putting these together, we get $1022_5$.

    Verification:

    We can check our work by converting $1022_5$ back to base 10: $(1 \times 5^3) + (0 \times 5^2) + (2 \times 5^1) + (2 \times 5^0)$ $= (1 \times 125) + (0 \times 25) + (2 \times 5) + (2 \times 1)$ $= 125 + 0 + 10 + 2$ $= 137_{10}$

    The calculation is correct.

    Exam Tip: When converting to a new base, always remember to write your remainders clearly to the side. If you ever get a remainder larger than the base you are converting to (e.g., getting a 6 when dividing by 5), you know you've made a calculation error!

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15. 15.

    Find the derivatives of y = sin 4x

    1. A.-4sin x
    2. B.-4cos x
    3. C.4sin4x
    4. D.4cos4x

    Show answer

    Answer: D. 4cos4x

    AI explanation

    Analysis

    To find the derivative of $y = \sin 4x$, we need to apply the Chain Rule of differentiation. The Chain Rule is used when we have a composite function (a function inside another function).

    Concept: The Chain Rule

    The Chain Rule states that if you have a function $y = f(g(x))$, then the derivative is:

    $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

    In this specific problem:

    1. The "outer" function is $\sin(u)$, where $u = 4x$. The derivative of $\sin(u)$ is $\cos(u)$.
    2. The "inner" function is $u = 4x$. The derivative of $4x$ with respect to $x$ is $4$.

    Step-by-Step Solution

    1. Identify the components: Let $u = 4x$. Then $y = \sin(u)$.

    2. Differentiate the outer function: $\frac{dy}{du} = \cos(u)$

    3. Differentiate the inner function: $\frac{du}{dx} = 4$

    4. Multiply them together (Chain Rule): $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$ $\frac{dy}{dx} = \cos(u) \cdot 4$

    5. Substitute $u = 4x$ back into the equation: $\frac{dy}{dx} = 4 \cos(4x)$

    Conclusion

    Comparing our result to the given options, we can see that $4 \cos 4x$ matches option D.

    Exam Tip

    Whenever you differentiate a trigonometric function like $\sin(ax)$ or $\cos(ax)$, always remember to multiply by the derivative of the angle ($a$).

    • Derivative of $\sin(ax) = a \cos(ax)$
    • Derivative of $\cos(ax) = -a \sin(ax)$ This quick mental check will save you time in the exam hall!

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16. 16.

    Find the coordinates of the midpoint of line PQ given P(-3, 4) and Q(5, 6).

    1. A.(1, 4)
    2. B.(4, 1)
    3. C.(5, 1)
    4. D.(1, 5)

    Show answer

    Answer: D. (1, 5)

    AI explanation

    Analysis

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

    Concept

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

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

    Solution

    Given the points $P(-3, 4)$ and $Q(5, 6)$:

    • $x_1 = -3, y_1 = 4$
    • $x_2 = 5, y_2 = 6$

    Step 1: Calculate the x-coordinate of the midpoint

    $x = \frac{x_1 + x_2}{2} = \frac{-3 + 5}{2}$

    $x = \frac{2}{2} = 1$

    Step 2: Calculate the y-coordinate of the midpoint

    $y = \frac{y_1 + y_2}{2} = \frac{4 + 6}{2}$

    $y = \frac{10}{2} = 5$

    Step 3: Combine the coordinates The midpoint is $(1, 5)$.

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

    Exam Tip

    When dealing with midpoint or gradient questions, always label your coordinates $(x_1, y_1)$ and $(x_2, y_2)$ clearly before you start. This simple habit prevents "sign errors" (like mixing up a negative sign), which is the most common reason students lose marks in coordinate geometry.

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17. 17.

    Find the number of permutations of the letters of the word SCHOOL.

    1. A.600
    2. B.360
    3. C.480
    4. D.300

    Show answer

    Answer: B. 360

    AI explanation

    Analysis

    To find the number of permutations of the letters in the word SCHOOL, we need to determine the total number of letters and identify if any letters are repeated.

    Concept

    The number of permutations of $n$ objects, where some objects are identical, is given by the formula:

    $\text{Permutations} = \frac{n!}{p! \cdot q! \cdot \dots}$

    Where:

    • $n$ is the total number of items.
    • $n!$ (n factorial) is the product of all positive integers up to $n$.
    • $p!, q!$ represent the factorials of the counts of each repeated item.

    Solution

    1. Count the total letters: The word SCHOOL has 6 letters ($n = 6$).
    2. Identify repetitions: • S: 1 • C: 1 • H: 1 • O: 2 • L: 1 • The letter 'O' appears twice.
    3. Apply the formula: $$\text{Permutations} = \frac{6!}{2!}$$
    4. Calculate the values: • $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$ • $2! = 2 \times 1 = 2$ • $\text{Permutations} = \frac{720}{2} = 360$

    Conclusion

    The total number of distinct arrangements for the letters in the word SCHOOL is 360.

    Exam Tip: Always remember to divide by the factorial of the number of times a letter repeats. If you forget to divide by the repetitions, you will get 720, which is a common distractor in JAMB questions!

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

19. 19.

    Simplify - log$_{10}$ 0.00001.

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

    Show answer

    Answer: C. 5

    AI explanation

    Analysis

    The question asks us to simplify the expression $-\log_{10} 0.00001$. To solve this, we need to express the decimal number $0.00001$ as a power of $10$ and then apply the laws of logarithms.

    Concept

    The logarithm of a number $x$ to the base $a$ is the exponent to which $a$ must be raised to get $x$. That is:

    $\log_a(a^n) = n$

    Additionally, remember that:

    $0.00001 = \frac{1}{100,000} = \frac{1}{10^5} = 10^{-5}$

    Step-by-Step Solution

    1. Rewrite the decimal in index form: $0.00001$ has five decimal places, so it can be written as $10^{-5}$.

    2. Substitute this into the expression: $-\log_{10} (0.00001) = -\log_{10} (10^{-5})$

    3. Apply the logarithm power rule: The power rule states that $\log_b(x^n) = n \cdot \log_b(x)$. Therefore, $\log_{10} (10^{-5}) = -5 \cdot \log_{10} (10)$.

    4. Simplify the base: Since $\log_{10} (10) = 1$, the expression becomes: $-(-5 \cdot 1) = -(-5)$

    5. Final calculation: $-(-5) = 5$

    Conclusion

    The simplified value of the expression is $5$. Comparing this to the provided options, it matches option C.

    Exam Tip: When dealing with decimals in logarithms, always convert them to standard form (scientific notation) first. For example, $0.001 = 10^{-3}$, $0.01 = 10^{-2}$, and $0.1 = 10^{-1}$. This makes the calculation much faster and reduces the risk of errors with decimal points!

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20. 20.

    Given that a * b = $\frac{a + b}{\text{ab}}$ + (a - b). Find the value of 3 * 2

    1. A.-6
    2. B.$\frac{11}{6}$
    3. C.$\frac{6}{11}$
    4. D.-1 $\frac{5}{6}$

    Show answer

    Answer: B. $\frac{11}{6}$

    AI explanation

    1. Analyze the question

    You are given a binary operation defined as $a * b = \frac{a + b}{ab} + (a - b)$. You need to substitute the values $a = 3$ and $b = 2$ into this formula to find the result.

    2. Explain the concept

    A binary operation is simply a rule that tells you how to combine two numbers to get a new result. In this case, the rule has two parts:

    • A fraction part: $\frac{a + b}{ab}$
    • A subtraction part: $(a - b)$

    To solve, we calculate each part separately and then add them together.

    3. Show the solution

    Step 1: Identify the values $a = 3$ $b = 2$

    Step 2: Substitute into the formula $3 * 2 = \frac{3 + 2}{3 \times 2} + (3 - 2)$

    Step 3: Solve the fraction part $\frac{3 + 2}{3 \times 2} = \frac{5}{6}$

    Step 4: Solve the subtraction part $(3 - 2) = 1$

    Step 5: Combine the results $3 * 2 = \frac{5}{6} + 1$

    To add these, convert $1$ to a fraction with a denominator of $6$: $1 = \frac{6}{6}$

    So, $\frac{5}{6} + \frac{6}{6} = \frac{11}{6}$

    4. Identify the answer

    The calculated value is $\frac{11}{6}$, which corresponds to option B.

    5. Tip

    When dealing with binary operations in JAMB, always use brackets to separate the different parts of the equation. This prevents "sign errors" (like accidentally subtracting when you should be adding) and keeps your work organized. If you get a fraction, always check if it can be simplified or if it needs to be converted to a mixed fraction to match the options provided.

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21. 21.

    What is the minimum value of y = 2 - 4x - 2x$^2$

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

    Show answer

    Answer: A. 4

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22. 22.

    The determinant of the matrix  $A = \begin{pmatrix} -2 & 3 & 1 \\ p & 2 & 1 \\ 1 & 4 & 2\end{pmatrix}$ is -5. Find the value of p.

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

    Show answer

    Answer: C. 3

    AI explanation

    Analysis

    To find the value of $p$, we need to calculate the determinant of the $3 \times 3$ matrix $A$ and set it equal to $-5$. The determinant of a matrix $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ is calculated using the expansion formula: $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$

    Concept

    The determinant of a matrix is a scalar value that provides information about the matrix's properties. For a $3 \times 3$ matrix, we expand along the first row. Remember to alternate the signs of the elements: $(+), (-), (+)$.

    Step-by-Step Solution

    Given matrix $A = \begin{pmatrix} -2 & 3 & 1 \\ p & 2 & 1 \\ 1 & 4 & 2 \end{pmatrix}$ and $|A| = -5$.

    1. Set up the expansion along the first row: $|A| = -2 \begin{vmatrix} 2 & 1 \\ 4 & 2 \end{vmatrix} - 3 \begin{vmatrix} p & 1 \\ 1 & 2 \end{vmatrix} + 1 \begin{vmatrix} p & 2 \\ 1 & 4 \end{vmatrix} = -5$

    2. Calculate the $2 \times 2$ determinants:

    • First part: $-2[(2 \times 2) - (1 \times 4)] = -2[4 - 4] = -2(0) = 0$
    • Second part: $-3[(p \times 2) - (1 \times 1)] = -3[2p - 1] = -6p + 3$
    • Third part: $+1[(p \times 4) - (2 \times 1)] = 1[4p - 2] = 4p - 2$

    3. Combine the results: $0 + (-6p + 3) + (4p - 2) = -5$ $-6p + 4p + 3 - 2 = -5$ $-2p + 1 = -5$

    4. Solve for $p$: $-2p = -5 - 1$ $-2p = -6$ $p = \frac{-6}{-2}$ $p = 3$

    Conclusion

    Comparing our result with the provided options, we find that $p = 3$ matches option C.

    Exam Tip

    When calculating determinants, always be extremely careful with the negative signs in the expansion formula (the middle term is always subtracted). A common mistake is forgetting to distribute the negative sign across the entire bracketed expression!

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23. 23.

    Find the sum of the entries in the inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$

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

    Show answer

    Answer: B. -1

    AI explanation

    1. Analyze the Question

    We are given a $2 \times 2$ matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$. We need to find its inverse, $A^{-1}$, and then calculate the sum of all the elements (entries) within that resulting matrix.

    2. Explain the Concept

    For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the inverse is given by the formula:

    $A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

    Where $\det(A) = (ad - bc)$.

    3. Show the Solution

    Step 1: Find the determinant of A.

    $\det(A) = (1 \times 5) - (2 \times 3)$

    $\det(A) = 5 - 6 = -1$

    Step 2: Find the inverse matrix. Using the formula:

    $A^{-1} = \frac{1}{-1} \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix}$

    Multiply each element by $-1$:

    $A^{-1} = \begin{bmatrix} -5 & 2 \\ 3 & -1 \end{bmatrix}$

    Step 3: Find the sum of the entries. Sum $= (-5) + 2 + 3 + (-1)$ Sum $= -5 + 2 + 3 - 1$ Sum $= -3 + 3 - 1$ Sum $= 0 - 1 = -1$

    4. Identify the Answer

    The sum of the entries is $-1$. Comparing this to the given options, it matches option B.

    5. Exam Tip

    When finding the inverse of a $2 \times 2$ matrix, remember the "Swap and Negate" rule: swap the elements on the leading diagonal ($a$ and $d$) and change the signs of the elements on the other diagonal ($b$ and $c$). Always double-check your determinant calculation, as a sign error there will change the entire result!

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24. 24.

    Integrate the function y = 3x$^2$ + 2x - 5 with respect to x.

    1. A.x$^3$ + x$^2$ - 5x + C
    2. B.x$^2$ + 2x$^2$ - 5x + C
    3. C.x$^2$ + 2x - 5 + C
    4. D.x$^3$ + 2x - 5 + C

    Show answer

    Answer: A. x$^3$ + x$^2$ - 5x + C

    AI explanation

    To solve this integration problem, we use the Power Rule for Integration.

    1. Analyze the question

    We are asked to find the indefinite integral of the polynomial function:

    $\int (3x^2 + 2x - 5) \, dx$

    2. Explain the concept

    The Power Rule for integration states that for any term $ax^n$:

    $\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + C$

    (Where $C$ is the constant of integration, and $n \neq -1$).

    Essentially, you add 1 to the exponent and then divide by the new exponent. For a constant term (like $-5$), it is treated as $-5x^0$, so its integral becomes $-5x^1$.

    3. Show the solution

    Let's integrate each term of the function $3x^2 + 2x - 5$ individually:

    • For $3x^2$: Add 1 to the exponent ($2+1=3$) and divide by 3: $$\frac{3x^3}{3} = x^3$$

    • For $2x$ (which is $2x^1$): Add 1 to the exponent ($1+1=2$) and divide by 2: $$\frac{2x^2}{2} = x^2$$

    • For $-5$ (which is $-5x^0$): Add 1 to the exponent ($0+1=1$) and divide by 1: $$\frac{-5x^1}{1} = -5x$$

    • Don't forget the constant: Always add $+ C$ at the end of an indefinite integral.

    Combining these, we get:

    $x^3 + x^2 - 5x + C$

    4. Identify the answer

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

    5. Exam Tip

    When integrating, always perform a quick mental check by differentiating your answer. If you differentiate $x^3 + x^2 - 5x + C$, you get $3x^2 + 2x - 5$, which is the original function. If you get the original function back, you know your integration is 100% correct!

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25. 25.

    Given the triangle XYZ above, calculate the value of Cot $\theta$ and the length XY, respectively

    1. A.$\frac{\sqrt{48}}{13}$, $\sqrt{48}$
    2. B.$\frac{\sqrt{48}}{11}$, $\sqrt{48}$
    3. C.$\frac{\sqrt{13}}{11}$, $\sqrt{13}$
    4. D.$\frac{13}{\sqrt{48}}$, $\sqrt{48}$

    Show answer

    Answer: C. $\frac{\sqrt{13}}{11}$, $\sqrt{13}$

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26. 26.

    Solve x$^2$ + 3x - 4  ≤ 0

    1. A.-4 $\geq$ x $\geq$ 1
    2. B.4 $\leq$ x $\leq$ - 1
    3. C.4 $\geq$ x $\geq$ 1
    4. D.$- 4 \leq x \leq 1$

    Show answer

    Answer: D. $- 4 \leq x \leq 1$

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27. 27.

    The second and fifth terms of a G.P are 1 and $\frac{1}{8}$ respectively. Find the common ratio

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

    Show answer

    Answer: C. $\frac{1}{2}$

    AI explanation

    Analysis

    To solve this problem, we need to use the formula for the $n^{th}$ term of a Geometric Progression (G.P.):

    $T_n = ar^{n-1}$

    Where:

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

    Concept

    In a G.P., each term is obtained by multiplying the previous term by a constant value called the common ratio ($r$). By setting up equations for the second and fifth terms, we can create a system of equations to isolate $r$.

    Step-by-Step Solution

    Step 1: Write the equations for the given terms.

    • For the second term ($n=2$): $$T_2 = ar^{2-1} = ar = 1 \quad \text{--- (Equation 1)}$$
    • For the fifth term ($n=5$): $$T_5 = ar^{5-1} = ar^4 = \frac{1}{8} \quad \text{--- (Equation 2)}$$

    Step 2: Divide Equation 2 by Equation 1 to eliminate $a$.

    $\frac{ar^4}{ar} = \frac{\frac{1}{8}}{1}$

    $r^3 = \frac{1}{8}$

    Step 3: Solve for $r$ by taking the cube root of both sides.

    $r = \sqrt[3]{\frac{1}{8}}$

    Since $2 \times 2 \times 2 = 8$, the cube root of $\frac{1}{8}$ is $\frac{1}{2}$.

    $r = \frac{1}{2}$

    Conclusion

    The common ratio of the G.P. is $\frac{1}{2}$.

    Exam Tip

    When dealing with G.P. problems where you are given two terms, always remember that the ratio of the two terms $T_n$ and $T_m$ is given by $\frac{T_n}{T_m} = r^{n-m}$. In this case:

    $\frac{T_5}{T_2} = r^{5-2} \implies \frac{1/8}{1} = r^3 \implies r^3 = \frac{1}{8} \implies r = \frac{1}{2}$

    This shortcut saves you time during the JAMB exam!

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

29. 29.

    Simplify ($\frac{3}{4} \div 2\frac{1}{4}$) of 1 $\frac{7}{11}$ (3 $\frac{2}{3}$ - $\frac{15}{6}$)

    1. A.$\frac{7}{11}$.
    2. B.$\frac{11}{7}$.
    3. C.$\frac{5}{11}$.
    4. D.$\frac{11}{5}$.

    Show answer

    Answer: A. $\frac{7}{11}$.

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30. 30.

    The mean of the numbers 0, x + 2, 3x + 6, and 4x + 8 is 4, find the value of x.

    1. A.$\frac{2}{5}$
    2. B.$\frac{2}{9}$
    3. C.0
    4. D.4

    Show answer

    Answer: C. 0

    AI explanation

    Analysis

    To find the value of $x$, we need to use the definition of the mean (average) of a set of numbers. The mean is calculated by finding the sum of all the numbers and dividing that sum by the total count of the numbers.

    Concept

    The formula for the mean is:

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

    In this problem:

    • The numbers are: $0, (x + 2), (3x + 6), (4x + 8)$
    • The total count of numbers is: $4$
    • The mean is given as: $4$

    Step-by-Step Solution

    Step 1: Set up the equation.

    $\frac{0 + (x + 2) + (3x + 6) + (4x + 8)}{4} = 4$

    Step 2: Simplify the numerator by combining like terms.

    • Sum of $x$ terms: $x + 3x + 4x = 8x$
    • Sum of constants: $0 + 2 + 6 + 8 = 16$
    • The equation becomes: $\frac{8x + 16}{4} = 4$

    Step 3: Solve for $x$. Multiply both sides by 4 to clear the fraction:

    $8x + 16 = 4 \times 4$

    $8x + 16 = 16$

    Subtract 16 from both sides:

    $8x = 16 - 16$

    $8x = 0$

    Divide by 8:

    $x = \frac{0}{8}$

    $x = 0$

    Conclusion

    By substituting $x = 0$ back into the original numbers, we get $0, 2, 6, 8$. The sum is $0 + 2 + 6 + 8 = 16$, and $16 \div 4 = 4$, which confirms our result.

    Exam Tip

    When dealing with mean problems involving algebraic expressions, always group your "like terms" (all the $x$'s together and all the constants together) before trying to solve the equation. This prevents simple arithmetic errors that often occur during the addition phase.

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

33. 33.

    Given the progression 3, 5, 7, 9,.... . . . find an expression for the (n - 2)$^{th}$ term of the progression.

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

    Show answer

    Answer: D. $2n - 3$

    AI explanation

    1. Analyze the question

    We are given an arithmetic progression (AP): 3, 5, 7, 9, ... and we need to find an expression for the $(n - 2)^{th}$ term.

    2. Explain the concept

    In an arithmetic progression, the $n^{th}$ term ($T_n$) is calculated using 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.

    For this sequence:

    • $a = 3$
    • $d = 5 - 3 = 2$

    3. Show the solution

    First, let's find the general formula for the $n^{th}$ term of this specific sequence:

    $T_n = 3 + (n - 1)2$

    $T_n = 3 + 2n - 2$

    $T_n = 2n + 1$

    Now, the question asks for the $(n - 2)^{th}$ term. To find this, we substitute $(n - 2)$ into the position variable of our general formula:

    $T_{(n-2)} = 2(n - 2) + 1$

    Expand the brackets:

    $T_{(n-2)} = 2n - 4 + 1$

    Simplify the expression:

    $T_{(n-2)} = 2n - 3$

    4. Identify the answer

    Comparing our result ($2n - 3$) with the given options, it matches option D.

    5. Exam Tip

    When asked for a term like $(n - k)^{th}$, don't panic! Simply find the general $n^{th}$ term formula first, then replace every '$n$' in that formula with the expression $(n - k)$. Always remember to use brackets when substituting to avoid sign errors during expansion.

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34. 34.

    In a basket of fruits, there are 6 grapes, 11 bananas, and 13 oranges. If one fruit is chosen at random, what is the probability that the fruit is either a grape or a banana?

    1. A.$\frac{17}{30}$
    2. B.$\frac{11}{30}$
    3. C.$\frac{6}{30}$
    4. D.$\frac{5}{30}$

    Show answer

    Answer: A. $\frac{17}{30}$

    AI explanation

    1. Analyze the question

    We are asked to find the probability of picking either a grape or a banana from a basket containing a specific number of different fruits.

    2. Explain the concept

    Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes:

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

    When we want to find the probability of one event or another event occurring (and they are mutually exclusive, meaning you can't pick a fruit that is both a grape and a banana at the same time), we add their individual probabilities together.

    3. Show the solution

    Step 1: Find the total number of fruits in the basket.

    $\text{Total} = \text{Grapes} + \text{Bananas} + \text{Oranges}$

    $\text{Total} = 6 + 11 + 13 = 30$

    Step 2: Find the number of favorable outcomes. We want either a grape or a banana.

    $\text{Favorable outcomes} = \text{Number of grapes} + \text{Number of bananas}$

    $\text{Favorable outcomes} = 6 + 11 = 17$

    Step 3: Calculate the probability.

    $\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}$

    $\text{Probability} = \frac{17}{30}$

    4. Identify the answer

    Comparing our result to the options provided: A. $\frac{17}{30}$ B. $\frac{11}{30}$ C. $\frac{6}{30}$ D. $\frac{5}{30}$

    The correct option is A.

    5. Tip

    In probability questions involving "OR," always think of addition. If the question had asked for the probability of picking a fruit that is "neither a grape nor a banana," you would simply subtract the probability of picking a grape or banana from 1 (the total probability).

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35. 35.

    A car dealer bought a used car for ₦270,000 and spent ₦70,000 to refurbish it. He later sold the car for ₦490,000. What was the percentage profit?

    1. A.30%
    2. B.35%
    3. C.25%
    4. D.45%

    Show answer

    Answer: D. 45%

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36. 36.

    The average weight of 15 iron bars is 1000 kg. If the heaviest iron bar is removed, the average weight is reduced by 5 kg. Find the weight in kg of the heaviest iron bar.

    1. A.1170kg
    2. B.1270kg
    3. C.1070kg
    4. D.1370kg

    Show answer

    Answer: C. 1070kg

    AI explanation

    1. Analyze the question

    We are given the average weight of 15 iron bars. When one bar (the heaviest) is removed, the number of bars decreases to 14, and the average weight decreases by 5 kg. We need to find the specific weight of that removed bar.

    2. Explain the concept

    The formula for the average is:

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

    From this, we can derive:

    $\text{Sum of values} = \text{Average} \times \text{Number of values}$

    3. Show the solution

    Step 1: Find the total weight of the 15 iron bars.

    $\text{Total weight} = 15 \times 1000 = 15,000 \text{ kg}$

    Step 2: Find the new average and the new number of bars.

    • New number of bars = $15 - 1 = 14$
    • New average = $1000 - 5 = 995 \text{ kg}$

    Step 3: Find the total weight of the remaining 14 iron bars.

    $\text{New total weight} = 14 \times 995$

    $14 \times 995 = 13,930 \text{ kg}$

    Step 4: Calculate the weight of the removed bar.

    $\text{Weight of heaviest bar} = (\text{Original total}) - (\text{New total})$

    $\text{Weight} = 15,000 - 13,930 = 1,070 \text{ kg}$

    4. Identify the answer

    Comparing our result to the given options: A. 1170kg B. 1270kg C. 1070kg D. 1370kg

    The correct option is C.

    5. Tip

    When dealing with "average" problems, always calculate the Total Sum first. It is much easier to work with the totals than to try to manipulate the averages directly. Also, double-check your subtraction—a small error in the final step is the most common way students lose marks in JAMB Mathematics!

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37. 37.

    If 54$_{ten}$ = X$_{four}$, find the value of X to 3 decimal point

    1. A.132$_{four}$
    2. B.312$_{four}$
    3. C.123$_{four}$
    4. D.321$_{four}$

    Show answer

    Answer: B. 312$_{four}$

    AI explanation

    Analysis

    The question asks us to convert a number from base ten (decimal) to base four. We are given the value $54_{10}$ and need to find its equivalent in base four.

    Concept: Successive Division

    To convert a whole number from base ten to any other base, we use the successive division method. We divide the number by the target base (in this case, 4) repeatedly and record the remainders. The base-four representation is then written by reading the remainders from the last one obtained to the first (bottom to top).

    Step-by-Step Solution

    1. Divide 54 by 4: $54 \div 4 = 13$ remainder 2
    2. Divide 13 by 4: $13 \div 4 = 3$ remainder 1
    3. Divide 3 by 4: $3 \div 4 = 0$ remainder 3

    Now, we arrange the remainders starting from the last division result (bottom) to the first (top):

    • Last remainder: 3
    • Middle remainder: 1
    • First remainder: 2

    Putting these together, we get $312_4$.

    (Note: The phrase "to 3 decimal point" in the question appears to be a typo, as we are dealing with whole numbers and base conversion. The calculation clearly results in a three-digit integer in base four.)

    Verification

    We can check our answer by converting $312_4$ back to base ten: $(3 \times 4^2) + (1 \times 4^1) + (2 \times 4^0)$ $= (3 \times 16) + (1 \times 4) + (2 \times 1)$ $= 48 + 4 + 2$ $= 54_{10}$ The calculation is correct.

    Exam Tip

    When converting from base ten to another base, always remember that your remainders must be smaller than the base you are converting to. If you ever get a remainder equal to or greater than the base (e.g., a remainder of 4 or 5 when dividing by 4), you know you have made a calculation error!

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38. 38.

    If the probability of death is q and the probability of survival is p, find the probability of one death and one survival in an accident involving two persons

    1. A.p/q
    2. B.pq
    3. C.p - q
    4. D.p + q

    Show answer

    Answer: C. p - q

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39. 39.

    A varies directly as b$^2$ when A = 4, b = 1. Find A when b = 2

    1. A.12
    2. B.14
    3. C.16
    4. D.11

    Show answer

    Answer: C. 16

    AI explanation

    1. Analyze the question

    We are given a relationship of direct variation where $A$ depends on the square of $b$ ($b^2$). We are provided with an initial set of values to find the constant of variation, and then asked to calculate a new value for $A$ when $b$ changes.

    2. Explain the concept

    In direct variation, the relationship is expressed as:

    $A = kb^2$

    Where:

    • $A$ is the variable.
    • $b^2$ is the square of the second variable.
    • $k$ is the constant of variation (a fixed number we must find first).

    3. Show the solution

    Step 1: Find the constant ($k$) Using the values $A = 4$ and $b = 1$:

    $4 = k(1)^2$

    $4 = k(1)$

    $k = 4$

    Step 2: Use the constant to find the new $A$ Now that we know $k = 4$, our formula is $A = 4b^2$. We need to find $A$ when $b = 2$:

    $A = 4(2)^2$

    $A = 4(4)$

    $A = 16$

    4. Identify the answer

    Comparing our result to the given options: A. 12 B. 14 C. 16 D. 11

    The correct option is C.

    5. Tip

    When dealing with variation questions, always remember the "Find $k$ first" rule. Once you have the constant of variation, the rest of the problem is just simple substitution. Also, don't forget to square the value of $b$ before multiplying by $k$—many students mistakenly multiply $k$ by $b$ first and then square the result, which leads to the wrong answer!

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40. 40.

    Solve the simultaneous equation $\frac{\text{x}}{2}$ - $\frac{\text{y}}{5}$ = 1 and y - $\frac{\text{x}}{3}$ = 8

    1. A.x = 10, y = -6
    2. B.x = 10, y = 6
    3. C.x = 6, y = 10
    4. D.x = 10, y = 6

    Show answer

    Answer: C. x = 6, y = 10

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

42. 42.

    Simplify $\sqrt{75}$ - $\sqrt{12}$ + $\sqrt{27}$

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

    Show answer

    Answer: D. $6\sqrt{3}$

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43. 43.

    A binary operation * is defined on the set X = {1, 2, 3, 4, 5, 6} as a*b = ab + a + b. Compute 1 * 3

    1. A.14
    2. B.4
    3. C.7
    4. D.5

    Show answer

    Answer: C. 7

    AI explanation

    1. Analyze the question

    You are given a binary operation defined by the rule $a * b = ab + a + b$. You need to find the result of the operation when $a = 1$ and $b = 3$.

    2. Explain the concept

    A binary operation is simply a rule for combining two elements of a set to produce a third element. In this case, the rule tells you to:

    1. Multiply the two numbers ($a \times b$).
    2. Add the first number ($a$).
    3. Add the second number ($b$).
    4. Sum these three results together.

    3. Show the solution

    Given the definition: $a * b = ab + a + b$

    Substitute $a = 1$ and $b = 3$ into the formula: $1 * 3 = (1 \times 3) + 1 + 3$

    Now, perform the arithmetic: $1 * 3 = 3 + 1 + 3$ $1 * 3 = 7$

    4. Identify the answer

    Comparing our result to the given options: A. 14 B. 4 C. 7 D. 5

    The correct option is C.

    5. Tip

    When dealing with binary operations in JAMB, always write out the formula clearly and substitute your values inside parentheses. This prevents "sign errors" or confusion, especially when the operation involves negative numbers or more complex algebraic expressions.

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44. 44.

    Which of the following angles cannot be constructed using a protractor, a compass, and a sharpened pencil?

    1. A.135°
    2. B.90°
    3. C.145°
    4. D.60°

    Show answer

    Answer: C. 145°

    AI explanation

    Analysis of the Question

    The question asks us to identify which of the given angles cannot be constructed using only a ruler (straightedge) and a pair of compasses. In geometry, "construction" refers to drawing precise geometric figures using only these two tools.

    Concept Explanation

    In Euclidean geometry, an angle can be constructed using a compass and straightedge if and only if it can be expressed as a combination of bisections (dividing by 2) and the construction of equilateral triangles (60°).

    Specifically:

    1. Basic angles: We can easily construct 60° (equilateral triangle), 90° (perpendicular bisector), and 45° (bisecting 90°).
    2. Combinations: By adding or bisecting these, we can construct angles like 30°, 15°, 75°, 105°, 120°, 135°, etc.
    3. The Rule: Any angle that is a multiple of 3° can be constructed. However, angles that require dividing an angle into parts other than powers of 2 (like 7, 9, or 11 parts) generally cannot be constructed. Specifically, angles that are not multiples of 3° (like 145°) are impossible to construct using only a compass and straightedge.

    Step-by-Step Solution

    Let's evaluate the options:

    • A. 135°: This is $90° + 45°$. Since both 90° and 45° are constructible, 135° is constructible.
    • B. 90°: This is the basic perpendicular angle, which is the foundation of most constructions. It is constructible.
    • C. 145°: To construct 145°, we would need to be able to construct 5°. Constructing 5° is equivalent to trisecting a 15° angle, which is mathematically proven to be impossible using only a compass and straightedge. Therefore, 145° is not constructible.
    • D. 60°: This is the standard angle formed by an equilateral triangle. It is constructible.

    Conclusion

    The angle 145° cannot be constructed because it is not a multiple of 3° and cannot be derived from the bisection of standard constructible angles.

    Exam Tip: Remember that most "constructible" angles in JAMB questions are multiples of 15° (e.g., 15, 30, 45, 60, 75, 90, 105, 120, 135, 150). If you see an angle that isn't a multiple of 15° (or at least 3°), it is highly likely to be the "cannot be constructed" option.

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45. 45.

    The set {1,2,3,4,5} is equivalent to

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

    Show answer

    Answer: D. {4,3,1,5,2}

    AI explanation

    Analysis

    The question asks us to identify which set is "equivalent" to the set $\{1, 2, 3, 4, 5\}$. In set theory, two sets are considered equivalent if they have the exact same number of elements (the same cardinality).

    Concept Explanation

    • Equal Sets: Two sets are equal if they contain exactly the same elements, regardless of order. For example, $\{1, 2\}$ is equal to $\{2, 1\}$.
    • Equivalent Sets: Two sets $A$ and $B$ are equivalent (written as $A \sim B$) if they have the same number of elements, meaning there is a one-to-one correspondence between them.
    • Cardinality ($n$): The number of elements in a set. The set $\{1, 2, 3, 4, 5\}$ has 5 elements, so its cardinality $n = 5$.

    Step-by-Step Solution

    Let's count the number of elements in the given set and each of the options:

    1. Given Set: $\{1, 2, 3, 4, 5\}$ has 5 elements.
    2. Option A: $\{2, 3, 1, 4\}$ has 4 elements. (Not equivalent)
    3. Option B: $\{1, 2, 3, 4, 5, 5\}$. In set theory, we do not count duplicate elements. This set is effectively $\{1, 2, 3, 4, 5\}$, which has 5 elements. However, in many JAMB contexts, "equivalent" is used to test the understanding of one-to-one mapping. Let's look at D.
    4. Option C: $\{1, 2, 3, 4\}$ has 4 elements. (Not equivalent)
    5. Option D: $\{4, 3, 1, 5, 2\}$ has 5 elements.

    Comparing the options, Option D contains the exact same elements as the original set, just rearranged. While Option B contains the same values, the inclusion of a repeated '5' technically makes it the same set as the original, but Option D is a perfect mapping of the original set.

    Exam Tip

    When you see the word "equivalent" in a JAMB question, always count the number of elements first. If two sets have the same number of elements, they are equivalent. If they have the exact same elements, they are equal (and therefore also equivalent).

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46. 46.

    Integrate y = 4x$^3$ + 2x + cos x.

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

    Show answer

    Answer: B. x$^4$ + x$^2$ + sin x + C

    AI explanation

    Analysis

    To solve this problem, we need to find the indefinite integral of the function $y = 4x^3 + 2x + \cos x$. We will apply the basic rules of integration term by term.

    Concept: The Power Rule and Trigonometric Integration

    1. The Power Rule for Integration: To integrate a term of the form $ax^n$, we use the formula: $$\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + C$$ (where $n \neq -1$)
    2. Trigonometric Integration: The integral of $\cos x$ is a standard derivative rule in reverse: $$\int \cos x \, dx = \sin x + C$$

    Step-by-Step Solution

    We are integrating the expression: $\int (4x^3 + 2x + \cos x) \, dx$

    Step 1: Integrate $4x^3$ Using the power rule:

    $\int 4x^3 \, dx = \frac{4x^{3+1}}{3+1} = \frac{4x^4}{4} = x^4$

    Step 2: Integrate $2x$ Using the power rule (remember that $x$ is $x^1$):

    $\int 2x^1 \, dx = \frac{2x^{1+1}}{1+1} = \frac{2x^2}{2} = x^2$

    Step 3: Integrate $\cos x$ Using the standard trigonometric integral:

    $\int \cos x \, dx = \sin x$

    Step 4: Combine the results and add the constant of integration ($C$)

    $x^4 + x^2 + \sin x + C$

    Conclusion

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

    Exam Tip: Always remember to add the constant of integration ($+ C$) whenever you are solving an indefinite integral. In JAMB, examiners often include options without the "$+ C$" to trick students who forget this fundamental rule!

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47. 47.

    Express $\sqrt[4]{0.16}$ in standard form

    1. A.2 $\times 10^{-\frac{1}{2}}$
    2. B.2 $\times 10^2$
    3. C.2 $\times 10^{-2}$
    4. D.2 $\times 10^{\frac{1}{2}}$

    Show answer

    Answer: A. 2 $\times 10^{-\frac{1}{2}}$

    AI explanation

    Analysis

    To solve this problem, we need to simplify the fourth root of a decimal number and then convert the resulting value into standard form (scientific notation).

    Concept

    1. Fourth Root: $\sqrt[4]{x}$ is the same as $x^{1/4}$.
    2. Decimal to Fraction: Convert $0.16$ to a fraction to make the root easier to calculate: $0.16 = \frac{16}{100} = \frac{4}{25}$ or $\frac{16}{100} = \frac{16}{10^2}$.
    3. Standard Form: A number in standard form is written as $a \times 10^n$, where $1 \le a < 10$.

    Step-by-Step Solution

    Step 1: Express the decimal as a fraction.

    $0.16 = \frac{16}{100}$

    So, we are looking for $\sqrt[4]{\frac{16}{100}}$.

    Step 2: Apply the root to the numerator and denominator.

    $\sqrt[4]{\frac{16}{100}} = \frac{\sqrt[4]{16}}{\sqrt[4]{100}}$

    Step 3: Simplify the numerator. Since $2^4 = 16$, then $\sqrt[4]{16} = 2$.

    Step 4: Simplify the denominator. $\sqrt[4]{100} = (10^2)^{1/4} = 10^{2 \times \frac{1}{4}} = 10^{1/2}$.

    Step 5: Combine the results. The expression becomes:

    $\frac{2}{10^{1/2}}$

    Step 6: Convert to standard form. Using the law of indices, $\frac{1}{a^n} = a^{-n}$:

    $\frac{2}{10^{1/2}} = 2 \times 10^{-1/2}$

    Conclusion

    Comparing our result $2 \times 10^{-1/2}$ to the given options, it matches Option A.

    Exam Tip: When dealing with roots of decimals, always convert them to fractions with denominators that are powers of 10 (like 100, 10000). It makes applying the laws of indices much cleaner and prevents errors with decimal point placement!

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48. 48.

    Given that Cos A = $\frac{12}{13}$ for 0 ≤ A ≤ 90º, find Tan A.

    1. A.$\frac{5}{12}$
    2. B.$\frac{12}{13}$
    3. C.$\frac{5}{13}$
    4. D.$\frac{13}{12}$

    Show answer

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

    AI explanation

    1. Analyze the question

    We are given the value of $\cos A$ in a right-angled triangle (since $0^\circ \le A \le 90^\circ$, it lies in the first quadrant). We need to find the value of $\tan A$.

    2. Explain the concept

    In trigonometry, for a right-angled triangle:

    • $\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
    • $\tan A = \frac{\text{Opposite}}{\text{Adjacent}}$

    We can use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the missing side (the Opposite side) when we know the Adjacent and the Hypotenuse.

    3. Show the solution

    • Step 1: Identify the given values. $\cos A = \frac{12}{13} = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ So, Adjacent = 12 and Hypotenuse = 13.

    • Step 2: Find the Opposite side using Pythagoras theorem. $\text{Opposite}^2 + \text{Adjacent}^2 = \text{Hypotenuse}^2$ $\text{Opposite}^2 + 12^2 = 13^2$ $\text{Opposite}^2 + 144 = 169$ $\text{Opposite}^2 = 169 - 144$ $\text{Opposite}^2 = 25$ $\text{Opposite} = \sqrt{25} = 5$

    • Step 3: Calculate $\tan A$. $\tan A = \frac{\text{Opposite}}{\text{Adjacent}}$ $\tan A = \frac{5}{12}$

    4. Identify the answer

    Comparing our result to the given options, the correct value is $\frac{5}{12}$. This matches option A.

    5. Tip

    Always remember the "SOH CAH TOA" mnemonic:

    • SOH: Sine = Opposite / Hypotenuse
    • CAH: Cosine = Adjacent / Hypotenuse
    • TOA: Tangent = Opposite / Adjacent

    Also, keep in mind the common Pythagorean triples like (5, 12, 13) and (3, 4, 5). Recognizing these can save you valuable time during the JAMB exam!

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49. 49.

    Calculate the interior angle of a 5 - sided regular polygon

    1. A.120º
    2. B.90º
    3. C.108º
    4. D.180º

    Show answer

    Answer: C. 108º

    AI explanation

    Analysis

    The question asks for the measure of a single interior angle of a regular polygon with 5 sides (a pentagon).

    Concept

    To find the interior angle of any regular polygon, we use two main steps:

    1. Find the sum of interior angles: The formula for the sum of interior angles of a polygon with $n$ sides is $(n - 2) \times 180^\circ$.
    2. Find a single interior angle: Since it is a regular polygon (all angles are equal), we divide the sum by the number of sides ($n$): $$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$

    Solution

    Given that $n = 5$:

    Step 1: Calculate the sum of the interior angles.

    $\text{Sum} = (5 - 2) \times 180^\circ$

    $\text{Sum} = 3 \times 180^\circ$

    $\text{Sum} = 540^\circ$

    Step 2: Calculate the measure of one interior angle.

    $\text{Interior Angle} = \frac{540^\circ}{5}$

    $\text{Interior Angle} = 108^\circ$

    Alternatively, you can find the exterior angle first:

    $\text{Exterior Angle} = \frac{360^\circ}{n} = \frac{360^\circ}{5} = 72^\circ$

    Since the interior and exterior angles lie on a straight line, they sum to $180^\circ$:

    $\text{Interior Angle} = 180^\circ - 72^\circ = 108^\circ$

    Conclusion

    The calculation confirms that the interior angle of a regular pentagon is $108^\circ$.

    Exam Tip

    For regular polygons, it is often faster to calculate the exterior angle first ($\frac{360}{n}$) and subtract it from $180^\circ$ than to calculate the total sum of interior angles. This saves time and reduces the risk of calculation errors!

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50. 50.

    The word HANDIER can be arranged in how many ways

    1. A.3080
    2. B.2650
    3. C.4050
    4. D.5040

    Show answer

    Answer: D. 5040

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51. 51.

    A banker spent $\frac{1}{5}$ of his salary on shirts,  $\frac{1}{3}$ of the remainder on transport, and kept the rest for contingencies. What fraction was left

    1. A.$\frac{6}{15}$
    2. B.$\frac{8}{15}$
    3. C.$\frac{7}{15}$
    4. D.$\frac{4}{5}$

    Show answer

    Answer: B. $\frac{8}{15}$

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52. 52.

    Solve the inequality 2x + 3 > 5x + 8

    1. A.x > 1$\frac{2}{3}$
    2. B.x > -1$\frac{2}{3}$
    3. C.x < -1$\frac{2}{3}$
    4. D.x < 1$\frac{2}{3}$

    Show answer

    Answer: C. x < -1$\frac{2}{3}$

    AI explanation

    Analysis

    The goal is to solve for $x$ in the linear inequality $2x + 3 > 5x + 8$. Solving an inequality is very similar to solving an equation, but you must be careful when multiplying or dividing by a negative number, as that flips the inequality sign.

    Concept

    To solve this, we need to group the terms containing $x$ on one side and the constant terms on the other side.

    Step-by-Step Solution

    Step 1: Group the terms Subtract $5x$ from both sides of the inequality to bring all $x$ terms to the left: $2x - 5x + 3 > 8$ $-3x + 3 > 8$

    Step 2: Isolate the $x$ term Subtract $3$ from both sides: $-3x > 8 - 3$ $-3x > 5$

    Step 3: Solve for $x$ Divide both sides by $-3$. Important Rule: Whenever you divide or multiply an inequality by a negative number, you must reverse the inequality sign ($>$ becomes $<$).

    $x < \frac{5}{-3}$

    Step 4: Simplify the fraction $x < -\frac{5}{3}$

    Converting the improper fraction to a mixed fraction: $5 \div 3 = 1$ remainder $2$ So, $x < -1\frac{2}{3}$

    Conclusion

    Comparing our result ($x < -1\frac{2}{3}$) with the given options, it matches option C.

    Exam Tip

    Always double-check your inequality sign when dealing with negative coefficients. A common mistake in JAMB is forgetting to flip the sign when dividing by a negative number, which would lead you to choose the wrong direction (like option B).

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53. 53.

    The chord of a circle of radius 17 cm is 30 cm long. Calculate the distance of the chord from the centre of the circle.

    1. A.15cm
    2. B.47cm
    3. C.8cm
    4. D.169cm

    Show answer

    Answer: C. 8cm

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54. 54.

    Y is partly constant and partly varies as x. When x = 3, y = 7 and when x = 5, y = 11. Find the constants of variation.

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

    Show answer

    Answer: B. 1,2

    AI explanation

    Analysis

    The problem describes a relationship where $y$ is the sum of a constant and a variable part that changes with $x$. Mathematically, this is expressed as: $y = k + cx$ Where:

    • $k$ is the constant.
    • $c$ is the constant of variation (the rate at which $y$ changes with $x$).

    Concept

    To solve this, we need to create a system of linear equations using the provided pairs of values $(x, y)$ and solve for the two unknowns, $k$ and $c$.

    Step-by-Step Solution

    Step 1: Set up the equations based on the given data.

    • When $x = 3, y = 7$: $7 = k + 3c$ --- (Equation 1)
    • When $x = 5, y = 11$: $11 = k + 5c$ --- (Equation 2)

    Step 2: Solve the system of equations. We can eliminate $k$ by subtracting Equation 1 from Equation 2: $(k + 5c) - (k + 3c) = 11 - 7$ $k - k + 5c - 3c = 4$ $2c = 4$ $c = \frac{4}{2}$ $c = 2$

    Step 3: Substitute the value of $c$ back into Equation 1 to find $k$. $7 = k + 3(2)$ $7 = k + 6$ $k = 7 - 6$ $k = 1$

    The constants are $k = 1$ and $c = 2$.

    Conclusion

    The constants of variation are 1 and 2. Comparing this to your options, this corresponds to option B.

    Exam Tip

    When you see the phrase "partly constant and partly varies as x," always write down $y = k + cx$ immediately. If the question had said "partly constant and partly varies as the square of x," your equation would have been $y = k + cx^2$. Always look for those keywords!

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55. 55.

    If cos $\theta$ = $\frac{\text{x}}{\text{y}}$, find tan $\theta$ in terms of x and y

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

    Show answer

    Answer: A. $\frac{\sqrt{y^2 - x^2}}{x}$

    AI explanation

    Analysis

    To solve this problem, we need to use the basic trigonometric ratios in a right-angled triangle. We are given the value of $\cos \theta$ and asked to find $\tan \theta$.

    Concept: SOH-CAH-TOA

    In a right-angled triangle:

    • $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
    • $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

    We are given $\cos \theta = \frac{x}{y}$. This means:

    • Adjacent side = $x$
    • Hypotenuse = $y$

    To find $\tan \theta$, we need the Opposite side. We can find this using the Pythagorean theorem:

    $\text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2$

    $y^2 = x^2 + \text{Opposite}^2$

    Step-by-Step Solution

    1. Find the Opposite side: Rearrange the Pythagorean theorem to solve for the Opposite side: $$\text{Opposite}^2 = y^2 - x^2$$ $$\text{Opposite} = \sqrt{y^2 - x^2}$$

    2. Calculate $\tan \theta$: Using the formula $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$: $$\tan \theta = \frac{\sqrt{y^2 - x^2}}{x}$$

    Conclusion

    Comparing our result to the given options, the correct expression is $\frac{\sqrt{y^2 - x^2}}{x}$. This matches option A.

    Exam Tip

    Whenever you are given one trigonometric ratio (like $\sin, \cos,$ or $\tan$) and asked for another, always sketch a quick right-angled triangle. Label the sides based on the ratio given, use Pythagoras to find the missing third side, and then write down the required ratio. This visual method prevents silly mistakes with square roots!

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