2019 JAMB Mathematics Past Questions

Questions

1. 1.

   The histogram above represents the number of candidates who did Further Mathematics examination in a school. How many candidates scored more than 40?

   

   1. A.100
   2. B.79
   3. C.150
   4. D.90

   Show answer

   Answer: A. 100

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

2. 2.

   If the universal set μ = {x : 1 ≤ x ≤ 20} and A = {y : multiple of 3} B = |z : odd numbers} Find A ∩ B

   1. A.{1, 3, 6}
   2. B.{3, 5, 9, 12}
   3. C.{3, 9, 15}
   4. D.{2, 3, 9}

   Show answer

   Answer: C. {3, 9, 15}

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

3. 3.

   Find the value of $\frac{(0.5436)^3}{0.017 \times 0.219}$ to 3 significant figures.

   1. A.46.2
   2. B.43.1
   3. C.534
   4. D.431

   Show answer

   Answer: B. 43.1

   AI explanation

   Analysis

   To solve this problem, we need to perform the arithmetic operations in the correct order: first, calculate the cube of the numerator, then calculate the product in the denominator, and finally perform the division. After obtaining the result, we will round it to 3 significant figures.

   Concept

   1. Exponents: $(0.5436)^3 = 0.5436 \times 0.5436 \times 0.5436$
   2. Multiplication: Calculate the product of the two numbers in the denominator.
   3. Division: Divide the result of the cube by the result of the product.
   4. Significant Figures: When rounding to 3 significant figures, we count three digits starting from the first non-zero digit.

   Step-by-Step Solution

   Step 1: Calculate the numerator $(0.5436)^3 = 0.5436 \times 0.5436 \times 0.5436$ $0.5436 \times 0.5436 \approx 0.29550096$ $0.29550096 \times 0.5436 \approx 0.1606343$

   Step 2: Calculate the denominator $0.017 \times 0.219 = 0.003723$

   Step 3: Perform the division $\frac{0.1606343}{0.003723} \approx 43.146467...$

   Step 4: Round to 3 significant figures The first three digits are 4, 3, and 1. The next digit is 4, which is less than 5, so we keep the 1 as it is. Result $\approx 43.1$

   Conclusion

   Comparing our result with the options provided, we see that 43.1 corresponds to option B.

   Exam Tip

   When dealing with decimals in JAMB, if you are not allowed a calculator, try converting the decimals to standard form (e.g., $5.436 \times 10^{-1}$) before multiplying. This helps prevent errors with decimal point placement. Also, always perform a quick "sanity check" by rounding the numbers to one significant figure first to estimate the magnitude of your answer (e.g., $0.5^3 / (0.02 \times 0.2) = 0.125 / 0.004 \approx 31$), which helps you spot if your calculated answer is way off.

   AI can make mistakes. Report anything that looks wrong.

4. 4.

   Score (x) 0 1 2 3 4 5 6 Freq (f) 5 7 3 7 11 6 7 Find the variance

   1. A.3.42
   2. B.4.69
   3. C.4.85
   4. D.3.72

   Show answer

   Answer: D. 3.72

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

5. 5.

   From the cyclic quadrilateral MNOP above, find the value of x.

   1. A.16°
   2. B.25°
   3. C.42°
   4. D.39°

   Show answer

   Answer: C. 42°

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

6. 6.

   Given $\sin 58° = \cos p°$, find p.

   1. A.48°
   2. B.58°
   3. C.32°
   4. D.52°

   Show answer

   Answer: C. 32°

   AI explanation

   Analysis

   The question asks us to find the value of $p$ given the equation $\sin 58° = \cos p°$. This is a classic trigonometry problem involving the relationship between complementary angles.

   Concept: Complementary Angles

   In trigonometry, the sine and cosine functions are "cofunctions." This means that the sine of an angle is equal to the cosine of its complement. The complement of an angle $\theta$ is $(90° - \theta)$.

   The fundamental identity is:

   $\sin \theta = \cos(90° - \theta)$

   or conversely:

   $\cos \theta = \sin(90° - \theta)$

   Step-by-Step Solution

   1. Identify the given equation: $\sin 58° = \cos p°$

   2. Apply the identity: We know that $\sin \theta = \cos(90° - \theta)$. Substituting $58°$ for $\theta$: $\sin 58° = \cos(90° - 58°)$

   3. Simplify the expression: $90° - 58° = 32°$ Therefore, $\sin 58° = \cos 32°$

   4. Compare with the given equation: Since $\sin 58° = \cos p°$ and we found $\sin 58° = \cos 32°$, it follows that: $p = 32$

   Conclusion

   Comparing our result with the options provided, we see that $p = 32°$ corresponds to option C.

   Exam Tip

   Whenever you see $\sin x = \cos y$ in a JAMB question, simply remember that $x + y$ must equal $90°$. In this case: $58° + p° = 90° \implies p = 90° - 58° = 32°$. This shortcut will save you valuable time during the exam!

   AI can make mistakes. Report anything that looks wrong.

7. 7.

   If $f(x) = 3x^3 + 4x^2 + x - 8$, what is the value of f(-2)?

   1. A.-24
   2. B.30
   3. C.-18
   4. D.-50

   Show answer

   Answer: C. -18

   AI explanation

   Analysis

   To find the value of $f(-2)$, we need to perform a process called substitution. This means we replace every instance of the variable $x$ in the given polynomial with the value $-2$ and then simplify the expression following the BODMAS/PEMDAS rule.

   Concept: Polynomial Evaluation

   A polynomial $f(x)$ is evaluated at a specific point $a$ by calculating $f(a)$. When dealing with negative numbers raised to powers, remember:

   • A negative number raised to an odd power remains negative (e.g., $(-2)^3 = -8$).
   • A negative number raised to an even power becomes positive (e.g., $(-2)^2 = 4$).

   Step-by-Step Solution

   Given: $f(x) = 3x^3 + 4x^2 + x - 8$

   Step 1: Substitute $x = -2$ into the equation $f(-2) = 3(-2)^3 + 4(-2)^2 + (-2) - 8$

   Step 2: Evaluate the powers

   • $(-2)^3 = -2 \times -2 \times -2 = -8$
   • $(-2)^2 = -2 \times -2 = 4$

   Now substitute these back: $f(-2) = 3(-8) + 4(4) - 2 - 8$

   Step 3: Perform the multiplications

   • $3 \times -8 = -24$
   • $4 \times 4 = 16$

   Now the expression is: $f(-2) = -24 + 16 - 2 - 8$

   Step 4: Simplify the addition and subtraction

   • $-24 + 16 = -8$
   • $-8 - 2 = -10$
   • $-10 - 8 = -18$

   The value of $f(-2)$ is -18.

   Conclusion

   Comparing our result to the given options: A. -24 B. 30 C. -18 D. -50

   The correct option is C.

   Exam Tip

   When substituting negative numbers into polynomials, always use brackets. This prevents common sign errors, especially when dealing with terms like $-x$ or negative coefficients. Also, double-check your arithmetic for the final addition/subtraction, as that is where most students lose marks in JAMB!

   AI can make mistakes. Report anything that looks wrong.

8. 8.

   Given matrix M = $\begin{vmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{vmatrix}$, find $M^{T} + 2M$

   1. A.$\begin{vmatrix} -4 & 2 & 1\\ 6 & 0 & 5 \\ 0 & 6 & 2 \end{vmatrix}$
   2. B.$\begin{vmatrix} -6 & 0 & 13\\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{vmatrix}$
   3. C.$\begin{vmatrix} 5 & 2 & 6 \\ 0 & 1 & 1\\ 3 & 4 & -7 \end{vmatrix}$
   4. D.$\begin{vmatrix} -4 & 0 & 8 \\ 0 & -2 & -16 \\ 10 & 12 & 6 \end{vmatrix}$

   Show answer

   Answer: B. $\begin{vmatrix} -6 & 0 & 13\\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{vmatrix}$

   AI explanation

   To solve this problem, we need to perform two operations: find the transpose of matrix $M$ ($M^T$) and calculate $2M$, then add them together.

   1. Analyze the components

   Given $M = \begin{pmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{pmatrix}$

   Step A: Find the Transpose ($M^T$) To find the transpose, we swap the rows and columns. The first row becomes the first column, the second row becomes the second column, and the third row becomes the third column. $M^T = \begin{pmatrix} -2 & 0 & 5 \\ 0 & -1 & 6 \\ 4 & 6 & 3 \end{pmatrix}$

   Step B: Find $2M$ To find $2M$, we multiply every element inside matrix $M$ by 2: $2M = 2 \times \begin{pmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{pmatrix} = \begin{pmatrix} -4 & 0 & 8 \\ 0 & -2 & 12 \\ 10 & 12 & 6 \end{pmatrix}$

   2. Perform the Addition ($M^T + 2M$)

   Now, we add the corresponding elements of $M^T$ and $2M$:

   $\begin{pmatrix} -2 & 0 & 5 \\ 0 & -1 & 6 \\ 4 & 6 & 3 \end{pmatrix} + \begin{pmatrix} -4 & 0 & 8 \\ 0 & -2 & 12 \\ 10 & 12 & 6 \end{pmatrix}$

   • Row 1: $(-2 + -4) = -6$, $(0 + 0) = 0$, $(5 + 8) = 13$
   • Row 2: $(0 + 0) = 0$, $(-1 + -2) = -3$, $(6 + 12) = 18$
   • Row 3: $(4 + 10) = 14$, $(6 + 12) = 18$, $(3 + 6) = 9$

   Resulting Matrix: $\begin{pmatrix} -6 & 0 & 13 \\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{pmatrix}$

   3. Conclusion

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

   Exam Tip: When dealing with matrix operations, always double-check your signs (positive/negative). A common mistake in JAMB is adding a negative number incorrectly (e.g., $-1 + -2$ becoming $+1$ instead of $-3$). Take your time with the arithmetic!

   AI can make mistakes. Report anything that looks wrong.

9. 9.

   The angles of a polygon are given by 2x, 5x, x and 4x respectively. The value of x is

   1. A.31°
   2. B.30°
   3. C.26°
   4. D.48°

   Show answer

   Answer: B. 30°

   AI explanation

   1. Analyze the question

   We are given the interior angles of a polygon: $2x$, $5x$, $x$, and $4x$. We need to find the value of $x$. Since there are four angles listed, this is a quadrilateral (a 4-sided polygon).

   2. Explain the concept

   The sum of the interior angles of any polygon is calculated using the formula:

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

   where $n$ is the number of sides.

   For a quadrilateral ($n = 4$):

   $\text{Sum} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ$

   Therefore, the sum of the given angles must equal $360^\circ$.

   3. Show the solution

   • Step 1: Set up the equation by adding all the given angles and equating them to $360^\circ$. $$2x + 5x + x + 4x = 360^\circ$$

   • Step 2: Combine the like terms. $$(2 + 5 + 1 + 4)x = 360^\circ$$ $$12x = 360^\circ$$

   • Step 3: Solve for $x$ by dividing both sides by 12. $$x = \frac{360^\circ}{12}$$ $$x = 30^\circ$$

   4. Identify the answer

   Comparing our result to the options provided: A. 31° B. 30° C. 26° D. 48°

   The calculated value matches option B.

   5. Tip

   Whenever you are given the interior angles of a polygon, always count the number of terms to determine $n$. Remember the "Sum of interior angles" table:

   • Triangle ($n=3$): $180^\circ$
   • Quadrilateral ($n=4$): $360^\circ$
   • Pentagon ($n=5$): $540^\circ$
   • Hexagon ($n=6$): $720^\circ$

   AI can make mistakes. Report anything that looks wrong.

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

11. 11.

    A bricklayer charges ₦1,500 per day for himself and ₦500 per day for his assistant. If a two bedroom flat was built for ₦95,000 and the bricklayer worked 10 days more than his assistant, how much did the assistant receive?

    1. A.N20,000
    2. B.N28,000
    3. C.N31,200
    4. D.N41,000

    Show answer

    Answer: A. N20,000

    AI explanation

    1. Analyze the Question

    We need to find the total amount paid to the assistant. We are given:

    • Bricklayer's daily rate: ₦1,500
    • Assistant's daily rate: ₦500
    • Total cost of the project: ₦95,000
    • Relationship between days worked: The bricklayer worked 10 days more than the assistant.

    2. Explain the Concept

    This is an algebraic word problem. We can represent the number of days the assistant worked as a variable ($x$). Since the bricklayer worked 10 days more, his days will be $(x + 10)$. We then set up an equation where the sum of the payments to both equals the total cost.

    3. Show the Solution

    • Let $x$ = number of days the assistant worked.
    • Then, $x + 10$ = number of days the bricklayer worked.

    Calculate the total cost: (Bricklayer's rate × Bricklayer's days) + (Assistant's rate × Assistant's days) = Total Cost $1,500(x + 10) + 500(x) = 95,000$

    Expand and solve for $x$: $1,500x + 15,000 + 500x = 95,000$ $2,000x + 15,000 = 95,000$

    Subtract 15,000 from both sides: $2,000x = 95,000 - 15,000$ $2,000x = 80,000$

    Divide by 2,000: $x = \frac{80,000}{2,000}$ $x = 40$ days

    The assistant worked for 40 days.

    Calculate the assistant's total pay: Assistant's pay = (Assistant's daily rate) × (Assistant's days) Assistant's pay = $500 \times 40$ Assistant's pay = ₦20,000

    4. Identify the Answer

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

    5. Exam Tip

    When solving "work and pay" problems, always define your variable based on the person who worked the least amount of time. This usually makes the addition in your equation simpler and reduces the chance of making a sign error.

    AI can make mistakes. Report anything that looks wrong.

12. 12.

    If given two points A(3, 12) and B(5, 22) on a x-y plane. Find the equation of the straight line with intercept at 2.

    1. A.y = 5x + 2
    2. B.y = 5x + 3
    3. C.y = 12x + 2
    4. D.y = 22x + 3

    Show answer

    Answer: A. y = 5x + 2

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

13. 13.

    In how many ways can the word MATHEMATICIAN be arranged?

    1. A.6794800 ways
    2. B.2664910 ways
    3. C.6227020800 ways
    4. D.129729600 ways

    Show answer

    Answer: D. 129729600 ways

    AI explanation

    To solve this problem, we use the formula for permutations of a multiset (arrangements of objects where some are identical).

    1. Analyze the question

    We need to find the number of distinct ways to arrange the letters in the word MATHEMATICIAN.

    2. Explain the concept

    When arranging letters, if some letters repeat, we use the formula:

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

    Where:

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

    3. Show the solution

    Step 1: Count the total number of letters ($n$). M-A-T-H-E-M-A-T-I-C-I-A-N Counting them: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Total letters ($n$) = 13

    Step 2: Identify the repeating letters and their frequencies.

    • M: appears 2 times
    • A: appears 3 times
    • T: appears 2 times
    • H: appears 1 time
    • E: appears 1 time
    • I: appears 2 times
    • C: appears 1 time
    • N: appears 1 time

    Step 3: Apply the formula.

    $\text{Arrangements} = \frac{13!}{2! \times 3! \times 2! \times 2!}$

    • $13! = 6,227,020,800$
    • $2! = 2$
    • $3! = 6$

    $\text{Arrangements} = \frac{6,227,020,800}{2 \times 6 \times 2 \times 2}$

    $\text{Arrangements} = \frac{6,227,020,800}{48}$

    $\text{Arrangements} = 129,729,600$

    4. Identify the answer

    Comparing our result to the given options, the correct value is 129,729,600. This matches option D.

    5. Exam Tip

    Always write out the letters and their counts clearly before plugging them into the formula. A common mistake is forgetting to count the "1s" (letters that appear only once), though they don't change the denominator (since $1! = 1$), it helps ensure you haven't missed any letters in your total count of 13!

    AI can make mistakes. Report anything that looks wrong.

14. 14.

    Points X and Y are 20km North and 9km East of point O, respectively. What is the bearing of Y from X? Correct to the nearest degree.

    1. A.24°
    2. B.56°
    3. C.127°
    4. D.156°

    Show answer

    Answer: D. 156°

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

15. 15.

    Find the distance between the points C(2, 2) and D(5, 6).

    1. A.13 units
    2. B.7 units
    3. C.12 units
    4. D.5 units

    Show answer

    Answer: D. 5 units

    AI explanation

    1. Analyze the Question

    We are asked to find the distance between two points on a Cartesian plane: $C(2, 2)$ and $D(5, 6)$.

    2. Explain the Concept

    To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$, we use the Distance Formula, which is derived from the Pythagorean theorem:

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

    3. Show the Solution

    • Step 1: Identify the coordinates. • Point $C(x_1, y_1) = (2, 2)$ • Point $D(x_2, y_2) = (5, 6)$

    • Step 2: Substitute the values into the formula. $$d = \sqrt{(5 - 2)^2 + (6 - 2)^2}$$

    • Step 3: Simplify the expressions inside the brackets. $$d = \sqrt{(3)^2 + (4)^2}$$

    • Step 4: Calculate the squares. $$d = \sqrt{9 + 16}$$

    • Step 5: Add and find the square root. $$d = \sqrt{25}$$ $$d = 5$$

    4. Identify the Answer

    The distance between the two points is 5 units. This corresponds to option D.

    5. Exam Tip

    Whenever you see coordinates involving small integers like 3 and 4, always look out for a 3-4-5 Pythagorean triple. Recognizing these common triangles can save you time during your JAMB exam, as you won't always need to write out the full calculation!

    AI can make mistakes. Report anything that looks wrong.

16. 16.

    The simple interest on ₦8550 for 3 years at x% per annum is ₦4890. Calculate the value of x to the nearest whole number.

    1. A.19%
    2. B.20%
    3. C.25%
    4. D.16.3%

    Show answer

    Answer: A. 19%

    AI explanation

    Analysis

    To solve this problem, we need to use the standard formula for Simple Interest. We are given the principal, the time, and the interest earned, and we need to find the rate ($x$).

    Concept

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

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

    Where:

    • $I$ = Simple Interest (₦4890)
    • $P$ = Principal (₦8550)
    • $R$ = Rate ($x$)
    • $T$ = Time (3 years)

    Step-by-Step Solution

    1. Substitute the given values into the formula: $$4890 = \frac{8550 \times x \times 3}{100}$$

    2. Simplify the right side: $$4890 = \frac{25650 \times x}{100}$$ $$4890 = 256.5 \times x$$

    3. Solve for $x$: $$x = \frac{4890}{256.5}$$ $$x \approx 19.064$$

    4. Round to the nearest whole number: Since 19.064 is closer to 19 than 20, the value of $x$ to the nearest whole number is 19%.

    Conclusion

    Comparing our result to the options provided, the correct value is 19%.

    Exam Tip

    When dealing with "Simple Interest" questions in JAMB, always write out your variables ($P, R, T, I$) first. It prevents you from accidentally swapping the Principal and the Interest, which is a common mistake students make under exam pressure.

    AI can make mistakes. Report anything that looks wrong.

17. 17.

    Solve for x in $\frac{4x - 6}{3} \leq \frac{3 + 2x}{2}$

    1. A.$x \leq 1\frac{1}{2}$
    2. B.$x \leq \frac{21}{2}$
    3. C.$x \geq \frac{21}{2}$
    4. D.$x \geq 1\frac{1}{2}$

    Show answer

    Answer: B. $x \leq \frac{21}{2}$

    AI explanation

    Analysis

    To solve this linear inequality, we need to eliminate the fractions by finding a common denominator and then isolate the variable $x$.

    Concept

    When dealing with inequalities involving fractions, the most efficient method is to multiply the entire inequality by the Least Common Multiple (LCM) of the denominators. In this case, the denominators are 3 and 2, so the LCM is 6.

    Important Rule: When multiplying or dividing an inequality by a negative number, you must reverse the inequality sign. Since we are multiplying by a positive 6, the sign remains the same.

    Step-by-Step Solution

    Step 1: Multiply both sides by 6 to clear the fractions.

    $6 \times \left( \frac{4x - 6}{3} \right) \leq 6 \times \left( \frac{3 + 2x}{2} \right)$

    Step 2: Simplify the expression.

    $2(4x - 6) \leq 3(3 + 2x)$

    Step 3: Expand the brackets.

    $8x - 12 \leq 9 + 6x$

    Step 4: Group the $x$ terms on one side and the constants on the other. Subtract $6x$ from both sides:

    $8x - 6x - 12 \leq 9$

    $2x - 12 \leq 9$

    Add 12 to both sides:

    $2x \leq 9 + 12$

    $2x \leq 21$

    Step 5: Solve for $x$. Divide both sides by 2:

    $x \leq \frac{21}{2}$

    Or, in mixed fraction form:

    $x \leq 10\frac{1}{2}$

    Conclusion

    Comparing our result ($x \leq \frac{21}{2}$) with the provided options, it matches option B.

    Exam Tip

    Always check your answer by picking a number that satisfies your result. For example, if $x = 0$ (which is $\leq 10.5$), the original inequality becomes $\frac{-6}{3} \leq \frac{3}{2}$, which is $-2 \leq 1.5$. Since this is true, your solution is likely correct!

    AI can make mistakes. Report anything that looks wrong.

18. 18.

    Find the equation of the locus of a point A(x, y) which is equidistant from B(0, 2) and C(2, 1)

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

    Show answer

    Answer: C. 4x - 2y = 1

    AI explanation

    1. Analyze the question

    We are looking for the equation of the locus of a point $A(x, y)$ that is equidistant from two fixed points, $B(0, 2)$ and $C(2, 1)$. This means the distance from $A$ to $B$ is equal to the distance from $A$ to $C$.

    2. Explain the concept

    The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

    Since the point $A(x, y)$ is equidistant from $B$ and $C$, then $AB = AC$. To make the calculation easier, we square both sides: $AB^2 = AC^2$

    3. Show the solution

    Step 1: Set up the distance equations. For $AB^2$: $AB^2 = (x - 0)^2 + (y - 2)^2$ $AB^2 = x^2 + (y^2 - 4y + 4)$

    For $AC^2$: $AC^2 = (x - 2)^2 + (y - 1)^2$ $AC^2 = (x^2 - 4x + 4) + (y^2 - 2y + 1)$

    Step 2: Equate $AB^2$ and $AC^2$. $x^2 + y^2 - 4y + 4 = x^2 - 4x + 4 + y^2 - 2y + 1$

    Step 3: Simplify the equation. Subtract $x^2$ and $y^2$ from both sides: $-4y + 4 = -4x + 4 - 2y + 1$

    Combine the constants and variables: $-4y + 4 = -4x - 2y + 5$

    Move all terms involving $x$ and $y$ to one side and constants to the other: $4x - 4y + 2y = 5 - 4$ $4x - 2y = 1$

    4. Identify the answer

    Comparing our result $4x - 2y = 1$ to the given options, it matches option C.

    5. Tip

    When dealing with "equidistant" problems, always square both sides immediately to get rid of the square root sign. This saves time and prevents errors involving radicals. Also, remember that the locus of a point equidistant from two points is simply the perpendicular bisector of the line segment joining those two points!

    AI can make mistakes. Report anything that looks wrong.

19. 19.

    Determine the values for which $x^2 - 7x + 10 \leq 0$

    1. A.2 $\leq$ x $\geq$ 5
    2. B.-2 $\leq$ x $\leq$ 3
    3. C.-2 $\leq$ x $\geq$ 3
    4. D.2 $\leq$ x $\leq$ 5

    Show answer

    Answer: D. 2 $\leq$ x $\leq$ 5

    AI explanation

    Analysis

    To solve the inequality $x^2 - 7x + 10 \leq 0$, we need to find the range of values for $x$ that makes the expression less than or equal to zero. This is a quadratic inequality.

    Concept

    1. Factorize the quadratic expression: We look for two numbers that multiply to give $+10$ and add up to $-7$. These numbers are $-2$ and $-5$.
    2. Find the critical points: Set the factors to zero to find where the expression equals zero.
    3. Test the intervals: Determine which region between the critical points satisfies the "less than or equal to" condition.

    Step-by-Step Solution

    Step 1: Factorize the quadratic $x^2 - 7x + 10 = 0$ $(x - 2)(x - 5) = 0$

    Step 2: Identify the critical points $x - 2 = 0 \implies x = 2$ $x - 5 = 0 \implies x = 5$

    Step 3: Determine the interval We have three regions on the number line: $x < 2$, $2 \leq x \leq 5$, and $x > 5$.

    • If we pick a number between 2 and 5 (e.g., $x = 3$): $(3 - 2)(3 - 5) = (1)(-2) = -2$. Since $-2 \leq 0$, this region satisfies the inequality.
    • If we pick a number less than 2 (e.g., $x = 0$): $(0 - 2)(0 - 5) = (-2)(-5) = +10$. Since $10$ is not $\leq 0$, this region does not satisfy the inequality.
    • If we pick a number greater than 5 (e.g., $x = 6$): $(6 - 2)(6 - 5) = (4)(1) = +4$. Since $4$ is not $\leq 0$, this region does not satisfy the inequality.

    Therefore, the inequality holds true for $2 \leq x \leq 5$.

    Conclusion

    The values for which the expression is less than or equal to zero are between 2 and 5 inclusive.

    Exam Tip: For quadratic inequalities of the form $(x - a)(x - b) \leq 0$ (where $a < b$), the solution is always the interval between the roots: $a \leq x \leq b$. If the inequality were $\geq 0$, the solution would be $x \leq a$ or $x \geq b$.

    AI can make mistakes. Report anything that looks wrong.

20. 20.

    Evaluate $\frac{2\log_{3} 9 \times \log_{3} 81^{-2}}{\log_{5} 625}$

    1. A.4
    2. B.-32
    3. C.-8
    4. D.16

    Show answer

    Answer: C. -8

    AI explanation

    To solve this expression, we need to apply the laws of logarithms, specifically the power rule: $\log_b(x^n) = n \log_b(x)$ and the definition $\log_b(b^n) = n$.

    1. Analyze and Simplify each term

    The expression is: $\frac{2\log_{3} 9 \times \log_{3} 81^{-2}}{\log_{5} 625}$

    • Term 1: $2\log_{3} 9$ Since $9 = 3^2$, we have: $2 \log_{3} (3^2) = 2 \times 2 \log_{3} 3 = 4 \times 1 = \mathbf{4}$

    • Term 2: $\log_{3} 81^{-2}$ Since $81 = 3^4$, then $81^{-2} = (3^4)^{-2} = 3^{-8}$. So, $\log_{3} (3^{-8}) = \mathbf{-8}$

    • Term 3 (Denominator): $\log_{5} 625$ Since $625 = 5^4$, we have: $\log_{5} (5^4) = \mathbf{4}$

    2. Substitute the values back into the expression

    Now, substitute these simplified values into the original fraction:

    $\frac{4 \times (-8)}{4}$

    3. Final Calculation

    $\frac{-32}{4} = -8$

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

    Exam Tip: Whenever you see logarithms with numbers like 9, 81, or 625, always try to express them as powers of the base (e.g., $3^2, 3^4, 5^4$). This makes the $\log_b(b^n) = n$ rule very easy to apply and saves you a lot of time during the exam!

    AI can make mistakes. Report anything that looks wrong.

21. 21.

    If $\begin{vmatrix} 2 & -4 \\ x & 9 \end{vmatrix} = 58$, find the value of x.

    1. A.10
    2. B.30
    3. C.14
    4. D.28

    Show answer

    Answer: A. 10

    AI explanation

    Analysis

    The question asks us to find the value of $x$ in a $2 \times 2$ determinant equation. To solve this, we must apply the standard formula for finding the determinant of a $2 \times 2$ matrix.

    Concept: Determinant of a $2 \times 2$ Matrix

    For any $2 \times 2$ matrix given as: $A = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$

    The determinant is calculated using the formula: $\text{det}(A) = (a \times d) - (b \times c)$

    Step-by-Step Solution

    Given the matrix: $\begin{vmatrix} 2 & -4 \\ x & 9 \end{vmatrix} = 58$

    Step 1: Apply the determinant formula $(2 \times 9) - (-4 \times x) = 58$

    Step 2: Simplify the multiplication $18 - (-4x) = 58$ $18 + 4x = 58$

    Step 3: Isolate the term with $x$ Subtract 18 from both sides of the equation: $4x = 58 - 18$ $4x = 40$

    Step 4: Solve for $x$ Divide both sides by 4: $x = \frac{40}{4}$ $x = 10$

    Conclusion

    Comparing our result to the provided options, the value of $x$ is 10.

    Exam Tip

    When dealing with determinants involving negative numbers, always be extra careful with the signs. Remember that subtracting a negative number (like $-(-4x)$) is the same as adding a positive number ($+4x$). This is a common place where students lose marks due to simple sign errors!

    AI can make mistakes. Report anything that looks wrong.

22. 22.

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

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

    Show answer

    Answer: C. 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.

    • When two straight lines intersect, they form angles at the point of intersection.
    • If a point is equidistant from two intersecting lines, it must lie on a line that divides the angle between those two lines into two equal parts. This line is known as the angle bisector.
    • Because two intersecting lines create two pairs of vertically opposite angles, there are actually two such bisectors (the internal and external angle bisectors), both of which are straight lines passing through the point of intersection.

    Step-by-Step Solution

    1. Visualize the intersection: Imagine two lines crossing 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 geometric property: The set of points that are equidistant from two intersecting lines are the lines that bisect the angles formed by the intersection.
    4. Evaluate the options: • A "bisector of the two lines" is vague. • A "line parallel to the two lines" would not be equidistant to both unless the lines were parallel themselves (which they are not). • An "angle bisector" specifically describes the line that maintains equal distance from the two intersecting sides of an angle. • A "perpendicular bisector" refers to a line that cuts a line segment into two equal parts at a 90-degree angle, which is not the case here.

    Conclusion

    The locus of a point equidistant from two intersecting lines is the pair of lines that bisect the angles formed by the intersection. Among the given options, "angle bisector" is the mathematically correct term.

    Exam Tip

    Whenever you see the phrase "equidistant from two lines," immediately think of angle bisectors. If the question mentions "equidistant from two points," think of the perpendicular bisector. Don't confuse the two!

    AI can make mistakes. Report anything that looks wrong.

23. 23.

    Evaluate $\frac{2\sin 30 + 5\tan 60}{\sin 60}$, leaving your answer in surd form.

    1. A.$\frac{2\sqrt{3}}{3} + 10$
    2. B.$\frac{3\sqrt{2} - 1}{5}$
    3. C.$\frac{3\sqrt{2} + 1}{5}$
    4. D.$\frac{2\sqrt{3}}{3} - 10$

    Show answer

    Answer: A. $\frac{2\sqrt{3}}{3} + 10$

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

24. 24.

    If $P = (\frac{Q(R - T)}{15})^{\frac{1}{3}}$, make T the subject of the formula.

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

    Show answer

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

    AI explanation

    To make $T$ the subject of the formula, we need to isolate it by performing inverse operations step-by-step.

    1. Analyze the equation

    We are given:

    $P = \left( \frac{Q(R - T)}{15} \right)^{\frac{1}{3}}$

    Our goal is to move everything away from $T$ until it stands alone on one side of the equation.

    2. Step-by-step solution

    Step 1: Remove the cube root (the power of $1/3$) To eliminate the exponent of $\frac{1}{3}$, we raise both sides of the equation to the power of 3:

    $P^3 = \left[ \left( \frac{Q(R - T)}{15} \right)^{\frac{1}{3}} \right]^3$

    $P^3 = \frac{Q(R - T)}{15}$

    Step 2: Clear the fraction Multiply both sides by 15 to remove the denominator:

    $15P^3 = Q(R - T)$

    Step 3: Isolate the term containing $T$ Divide both sides by $Q$:

    $\frac{15P^3}{Q} = R - T$

    Step 4: Solve for $T$ To get $T$ by itself, we can move $T$ to the left side (making it positive) and move $\frac{15P^3}{Q}$ to the right side (making it negative):

    $T = R - \frac{15P^3}{Q}$

    3. Conclusion

    Comparing our result to the given options, we can see that our derived formula matches option B.

    4. Exam Tip

    When dealing with "change of subject" questions, always remember the Reverse Order of Operations (SADMEP). Usually, you undo addition/subtraction first, then multiplication/division, then exponents/roots. However, because the term $(R-T)$ was trapped inside a bracket with a power, we had to deal with the power first!

    AI can make mistakes. Report anything that looks wrong.

25. 25.

    Find the value of k in the equation: $\sqrt{28} + \sqrt{112} - \sqrt{k} = \sqrt{175}$

    1. A.$\sqrt{28}$
    2. B.7
    3. C.28
    4. D.$\sqrt{7}$

    Show answer

    Answer: B. 7

    AI explanation

    To solve this problem, we need to simplify each square root by expressing the numbers under the radicals as products of perfect squares.

    1. Analyze the question

    We are given the equation: $\sqrt{28} + \sqrt{112} - \sqrt{k} = \sqrt{175}$. Our goal is to isolate $\sqrt{k}$ and solve for $k$.

    2. Explain the concept

    To simplify a square root, we look for the largest perfect square factor (e.g., 4, 9, 16, 25, 36, 49, 64, 81, 100) that divides the number. For example, $\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}$.

    3. Show the solution

    Step 1: Simplify each radical term.

    • $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$
    • $\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}$
    • $\sqrt{175} = \sqrt{25 \times 7} = 5\sqrt{7}$

    Step 2: Substitute these back into the original equation. $2\sqrt{7} + 4\sqrt{7} - \sqrt{k} = 5\sqrt{7}$

    Step 3: Combine the terms on the left side. $(2 + 4)\sqrt{7} - \sqrt{k} = 5\sqrt{7}$ $6\sqrt{7} - \sqrt{k} = 5\sqrt{7}$

    Step 4: Solve for $\sqrt{k}$. Subtract $6\sqrt{7}$ from both sides: $-\sqrt{k} = 5\sqrt{7} - 6\sqrt{7}$ $-\sqrt{k} = -1\sqrt{7}$ $\sqrt{k} = \sqrt{7}$

    Step 5: Find $k$. Square both sides of the equation: $(\sqrt{k})^2 = (\sqrt{7})^2$ $k = 7$

    4. Identify the answer

    Comparing our result to the given options, $k = 7$ matches option B.

    5. Exam Tip

    When dealing with surds in JAMB, always try to express all terms in the equation in terms of the smallest possible surd (in this case, $\sqrt{7}$). This makes adding and subtracting them as easy as adding simple algebraic terms like $2x + 4x$.

    AI can make mistakes. Report anything that looks wrong.

26. 26.

    If a fair coin is tossed 3 times, what is the probability of getting at least two heads?

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

    Show answer

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

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

27. 27.

    Given the matrix $A = \begin{vmatrix} 3 & -2 \\ 1 & 6 \end{vmatrix}$. Find the inverse of matrix A.

    1. A.$\begin{vmatrix} 6 & 2 \\ 1 & 6 \end{vmatrix}$
    2. B.$\begin{vmatrix} \frac{2}{11} & \frac{1}{12}\\ \frac{3}{20} & \frac{1}{10} \end{vmatrix}$
    3. C.$\begin{vmatrix} -3 & 2 \\ -1 & -6 \end{vmatrix}$
    4. D.$\begin{vmatrix} \frac{3}{10} & \frac{1}{10} \\ \frac{-1}{20} & \frac{3}{20}\end{vmatrix}$

    Show answer

    Answer: D. $\begin{vmatrix} \frac{3}{10} & \frac{1}{10} \\ \frac{-1}{20} & \frac{3}{20}\end{vmatrix}$

    AI explanation

    To find the inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, we use the formula:

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

    where $\det(A) = (ad - bc)$ is the determinant of the matrix.

    Step 1: Calculate the determinant of A

    Given $A = \begin{pmatrix} 3 & -2 \\ 1 & 6 \end{pmatrix}$:

    • $a = 3, b = -2, c = 1, d = 6$
    • $\det(A) = (3 \times 6) - (-2 \times 1)$
    • $\det(A) = 18 - (-2)$
    • $\det(A) = 18 + 2 = 20$

    Step 2: Apply the inverse formula

    Now, swap the elements on the leading diagonal ($3$ and $6$) and change the signs of the elements on the other diagonal ($-2$ and $1$):

    • The adjugate matrix is $\begin{pmatrix} 6 & 2 \\ -1 & 3 \end{pmatrix}$

    Now, multiply by $\frac{1}{\det(A)}$:

    • $A^{-1} = \frac{1}{20} \begin{pmatrix} 6 & 2 \\ -1 & 3 \end{pmatrix}$
    • $A^{-1} = \begin{pmatrix} \frac{6}{20} & \frac{2}{20} \\ \frac{-1}{20} & \frac{3}{20} \end{pmatrix}$

    Step 3: Simplify the fractions

    • $\frac{6}{20} = \frac{3}{10}$
    • $\frac{2}{20} = \frac{1}{10}$
    • The matrix becomes $\begin{pmatrix} \frac{3}{10} & \frac{1}{10} \\ \frac{-1}{20} & \frac{3}{20} \end{pmatrix}$

    Comparing this result to your options, it matches option D.

    Exam Tip: Always remember the "Swap and Negate" rule for $2 \times 2$ matrices: Swap the positions of the top-left and bottom-right numbers, and change the signs of the top-right and bottom-left numbers. Don't forget to divide the final matrix by the determinant!

    AI can make mistakes. Report anything that looks wrong.

28. 28.

    If S = (4t + 3)(t - 2), find ds/dt when t = 5 secs.

    1. A.50 units per sec
    2. B.35 units per sec
    3. C.22 units per sec
    4. D.13 units per sec

    Show answer

    Answer: B. 35 units per sec

    AI explanation

    1. Analyze the question

    We are given a function $S$ in terms of $t$, defined as $S = (4t + 3)(t - 2)$. We need to find the derivative of $S$ with respect to $t$ ($\frac{dS}{dt}$) and then evaluate that derivative at $t = 5$.

    2. Explain the concept

    To solve this, we use the Product Rule of differentiation or expand the brackets first. The expansion method is often simpler for polynomials.

    • Expansion: Multiply the two binomials to get a standard polynomial.
    • Power Rule: If $y = ax^n$, then $\frac{dy}{dx} = n \cdot ax^{n-1}$.
    • Differentiation: Once expanded, we differentiate term by term.

    3. Show the solution

    Step 1: Expand the expression for $S$ $S = (4t + 3)(t - 2)$ $S = 4t(t) + 4t(-2) + 3(t) + 3(-2)$ $S = 4t^2 - 8t + 3t - 6$ $S = 4t^2 - 5t - 6$

    Step 2: Differentiate $S$ with respect to $t$ Using the power rule: $\frac{dS}{dt} = \frac{d}{dt}(4t^2) - \frac{d}{dt}(5t) - \frac{d}{dt}(6)$ $\frac{dS}{dt} = (2 \times 4)t^{2-1} - (1 \times 5)t^{1-1} - 0$ $\frac{dS}{dt} = 8t - 5$

    Step 3: Evaluate at $t = 5$ Substitute $t = 5$ into the derivative: $\frac{dS}{dt} = 8(5) - 5$ $\frac{dS}{dt} = 40 - 5$ $\frac{dS}{dt} = 35$

    4. Identify the answer

    The result is 35 units per sec. Comparing this to the given options: A. 50 B. 35 C. 22 D. 13

    The correct option is B.

    5. Tip

    When you see a product of two brackets, always check if expanding them is faster than using the Product Rule formula $[u \frac{dv}{dt} + v \frac{du}{dt}]$. For simple polynomials like this, expansion saves time and reduces the risk of errors during your JAMB exam!

    AI can make mistakes. Report anything that looks wrong.

29. 29.

    Integrate $\int (4x^{-3} - 7x^2 + 5x - 6) \mathrm d x$.

    1. A.$-2x^{-2} - \frac{7}{3}x^3 + \frac{5}{2} x^2 - 6x$
    2. B.$2x^2 + \frac{7}{3} x^3 - 5x + 6$
    3. C.$12x^2 + 14x - 5$
    4. D.$-12x^{-4} - 14x + 5$

    Show answer

    Answer: A. $-2x^{-2} - \frac{7}{3}x^3 + \frac{5}{2} x^2 - 6x$

    AI explanation

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

    1. Analyze the Concept

    The power rule states that for any term $ax^n$ (where $n \neq -1$):

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

    Essentially, you add 1 to the exponent and then divide the term by that new exponent. We apply this rule to each term in the expression individually.

    2. Step-by-Step Solution

    We are integrating: $\int (4x^{-3} - 7x^2 + 5x^1 - 6x^0) \, dx$

    • Term 1: $4x^{-3}$ Add 1 to the exponent: $-3 + 1 = -2$. Divide by the new exponent: $\frac{4x^{-2}}{-2} = -2x^{-2}$

    • Term 2: $-7x^2$ Add 1 to the exponent: $2 + 1 = 3$. Divide by the new exponent: $-\frac{7x^3}{3} = -\frac{7}{3}x^3$

    • Term 3: $5x^1$ Add 1 to the exponent: $1 + 1 = 2$. Divide by the new exponent: $\frac{5x^2}{2} = \frac{5}{2}x^2$

    • Term 4: $-6$ (which is $-6x^0$) Add 1 to the exponent: $0 + 1 = 1$. Divide by the new exponent: $-\frac{6x^1}{1} = -6x$

    3. Combine the results

    Putting these together, we get:

    $-2x^{-2} - \frac{7}{3}x^3 + \frac{5}{2}x^2 - 6x + C$

    Comparing this result to your given options, it matches Option A.

    Exam Tip

    When integrating, always check your work by differentiating your answer. If you differentiate $-2x^{-2} - \frac{7}{3}x^3 + \frac{5}{2}x^2 - 6x$, you will arrive back at the original expression $(4x^{-3} - 7x^2 + 5x - 6)$. This is a great way to guarantee you haven't made a sign error during the exam!

    AI can make mistakes. Report anything that looks wrong.

30. 30.

    If the volume of a frustrum is given as $V = \frac{\pi h}{3} (R^2 + Rr + r^2)$, find $\frac{\mathrm d V}{\mathrm d R}$.

    1. A.$\frac{\pi h}{3} (2R + r)$
    2. B.$2R + r + \frac{\pi h}{3}$
    3. C.$\frac{\pi h}{3} (2R^2 + r + 2r)$
    4. D.$\frac{2R^2}{3} \pi h$

    Show answer

    Answer: A. $\frac{\pi h}{3} (2R + r)$

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

31. 31.

    Integrate $\int_{-1} ^{2} (2x^2 + x) \mathrm {d} x$

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

    Show answer

    Answer: C. $7\frac{1}{2}$

    AI explanation

    1. Analyze the question

    We are required to evaluate a definite integral of the polynomial function $(2x^2 + x)$ over the interval $[-1, 2]$.

    2. Explain the concept

    To solve this, we use the Power Rule for Integration, which states:

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

    For a definite integral $\int_{a}^{b} f(x) \, dx$, we find the antiderivative $F(x)$ and then calculate $F(b) - F(a)$.

    3. Show the solution

    Step 1: Find the indefinite integral

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

    Step 2: Apply the limits from -1 to 2

    $\left[ \frac{2x^3}{3} + \frac{x^2}{2} \right]_{-1}^{2}$

    Step 3: Substitute the upper limit (2)

    $F(2) = \frac{2(2)^3}{3} + \frac{(2)^2}{2} = \frac{2(8)}{3} + \frac{4}{2} = \frac{16}{3} + 2$

    Convert 2 to a fraction with denominator 3: $2 = \frac{6}{3}$

    $F(2) = \frac{16}{3} + \frac{6}{3} = \frac{22}{3}$

    Step 4: Substitute the lower limit (-1)

    $F(-1) = \frac{2(-1)^3}{3} + \frac{(-1)^2}{2} = \frac{2(-1)}{3} + \frac{1}{2} = -\frac{2}{3} + \frac{1}{2}$

    Find a common denominator (6):

    $F(-1) = -\frac{4}{6} + \frac{3}{6} = -\frac{1}{6}$

    Step 5: Calculate $F(2) - F(-1)$

    $\frac{22}{3} - \left( -\frac{1}{6} \right) = \frac{22}{3} + \frac{1}{6}$

    Convert $\frac{22}{3}$ to a fraction with denominator 6: $\frac{44}{6}$

    $\frac{44}{6} + \frac{1}{6} = \frac{45}{6}$

    Simplify the fraction by dividing by 3:

    $\frac{45 \div 3}{6 \div 3} = \frac{15}{2} = 7\frac{1}{2}$

    4. Identify the answer

    The result is $7\frac{1}{2}$, which matches option C.

    5. Tip

    When dealing with definite integrals involving negative limits, be extremely careful with the signs during the subtraction step ($F(b) - F(a)$). A common mistake is to lose a negative sign when subtracting a negative value, which leads to the wrong answer. Always use brackets to keep your signs organized!

    AI can make mistakes. Report anything that looks wrong.

32. 32.

    If 2$^{x + y}$ = 16 and 4$^{x - y} = \frac{1}{32}$, find the values of x and y.

    1. A.x = $\frac{3}{4}$, y = $\frac{11}{4}$
    2. B.x = $\frac{3}{4}$, y = $\frac{13}{4}$
    3. C.x = $\frac{2}{3}$, y = $\frac{4}{5}$
    4. D.x = $\frac{2}{3}$, y = $\frac{13}{4}$

    Show answer

    Answer: B. x = $\frac{3}{4}$, y = $\frac{13}{4}$

    AI explanation

    To solve this problem, we need to express both sides of the equations with the same base so that we can equate the exponents.

    1. Analyze the equations

    We are given: (i) $2^{x + y} = 16$ (ii) $4^{x - y} = \frac{1}{32}$

    2. Simplify Equation (i)

    Since $16 = 2^4$, we can rewrite the equation as: $2^{x + y} = 2^4$ Equating the exponents: $x + y = 4$ --- (Equation 1)

    3. Simplify Equation (ii)

    We know that $4 = 2^2$ and $32 = 2^5$. Substituting these into the equation: $(2^2)^{x - y} = \frac{1}{2^5}$ $2^{2(x - y)} = 2^{-5}$ Equating the exponents: $2(x - y) = -5$ $x - y = -\frac{5}{2}$ $x - y = -2.5$ --- (Equation 2)

    4. Solve the simultaneous equations

    We now have:

    1. $x + y = 4$
    2. $x - y = -2.5$

    Add Equation 1 and Equation 2: $(x + y) + (x - y) = 4 + (-2.5)$ $2x = 1.5$ $x = \frac{1.5}{2} = 0.75$ or $\frac{3}{4}$

    Now, substitute $x = \frac{3}{4}$ into Equation 1: $\frac{3}{4} + y = 4$ $y = 4 - \frac{3}{4}$ $y = \frac{16 - 3}{4}$ $y = \frac{13}{4}$

    5. Conclusion

    The values are $x = \frac{3}{4}$ and $y = \frac{13}{4}$. This matches option B.

    Exam Tip: When dealing with indices, always try to reduce the numbers to their smallest prime base (like 2, 3, or 5). Remember that $\frac{1}{a^n} = a^{-n}$; this is a common trap in JAMB questions that tests your knowledge of negative indices.

    AI can make mistakes. Report anything that looks wrong.

33. 33.

    $\frac{\frac{2}{3} \div \frac{4}{5}}{\frac{1}{4} + \frac{3}{5} - \frac{1}{3}}$

    1. A.$\frac{31}{50}$
    2. B.$\frac{20}{31}$
    3. C.$\frac{31}{20}$
    4. D.$\frac{50}{31}$

    Show answer

    Answer: D. $\frac{50}{31}$

    AI explanation

    To solve this expression, we will simplify the numerator and the denominator separately before performing the final division.

    1. Simplify the Numerator

    The numerator is: $\frac{2}{3} \div \frac{4}{5}$

    When dividing fractions, we multiply by the reciprocal of the divisor:

    $\frac{2}{3} \times \frac{5}{4} = \frac{10}{12}$

    Simplifying $\frac{10}{12}$ by dividing both numerator and denominator by 2 gives: Numerator = $\frac{5}{6}$

    2. Simplify the Denominator

    The denominator is: $\frac{1}{4} + \frac{3}{5} - \frac{1}{3}$

    To add and subtract these, we find the Least Common Multiple (LCM) of 4, 5, and 3, which is 60.

    • $\frac{1}{4} = \frac{15}{60}$
    • $\frac{3}{5} = \frac{36}{60}$
    • $\frac{1}{3} = \frac{20}{60}$

    Now, combine them:

    $\frac{15 + 36 - 20}{60} = \frac{51 - 20}{60} = \frac{31}{60}$

    Denominator = $\frac{31}{60}$

    3. Final Division

    Now, divide the simplified numerator by the simplified denominator:

    $\frac{5}{6} \div \frac{31}{60}$

    Again, multiply by the reciprocal:

    $\frac{5}{6} \times \frac{60}{31}$

    We can simplify before multiplying: 6 goes into 60 ten times:

    $5 \times \frac{10}{31} = \frac{50}{31}$

    Conclusion

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

    Exam Tip: When dealing with complex fractions, always handle the numerator and denominator as two separate problems first. This prevents "fraction fatigue" and reduces the likelihood of making a silly calculation error in the middle of the process.

    AI can make mistakes. Report anything that looks wrong.

34. 34.

    Express $(0.0439 \div 3.62)$ as a fraction.

    1. A.$\frac{21}{100}$
    2. B.$\frac{21}{1000}$
    3. C.$\frac{12}{1000}$
    4. D.$\frac{12}{100}$

    Show answer

    Answer: C. $\frac{12}{1000}$

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

35. 35.

    Find the equation of a line perpendicular to the line 4y = 7x + 3 which passes through (-3, 1)

    1. A.7y + 4x + 5 = 0
    2. B.7y - 4x - 5 = 0
    3. C.3y - 5x + 2 = 0
    4. D.3y + 5x - 2 = 0

    Show answer

    Answer: A. 7y + 4x + 5 = 0

    AI explanation

    1. Analyze the question

    We need to find the equation of a line that is perpendicular to a given line ($4y = 7x + 3$) and passes through a specific point $(-3, 1)$.

    2. Explain the concept

    • Slope-Intercept Form: A line equation in the form $y = mx + c$ has a slope of $m$.
    • Perpendicular Slopes: If two lines are perpendicular, the product of their slopes is $-1$ ($m_1 \times m_2 = -1$). This means the slope of the new line will be the negative reciprocal of the original slope ($m_2 = -1 / m_1$).
    • Point-Slope Form: Once we have the new slope ($m$) and a point $(x_1, y_1)$, we use the formula: $y - y_1 = m(x - x_1)$.

    3. Show the solution

    Step 1: Find the slope of the given line. The equation is $4y = 7x + 3$. Divide by 4 to get it into $y = mx + c$ form: $y = \frac{7}{4}x + \frac{3}{4}$ The slope ($m_1$) is $\frac{7}{4}$.

    Step 2: Find the slope of the perpendicular line ($m_2$). Since $m_1 \times m_2 = -1$: $\frac{7}{4} \times m_2 = -1$ $m_2 = -\frac{4}{7}$

    Step 3: Use the point-slope formula with point $(-3, 1)$. $y - 1 = -\frac{4}{7}(x - (-3))$ $y - 1 = -\frac{4}{7}(x + 3)$

    Step 4: Simplify the equation. Multiply both sides by 7 to clear the fraction: $7(y - 1) = -4(x + 3)$ $7y - 7 = -4x - 12$

    Rearrange to the form $Ay + Bx + C = 0$: $7y + 4x - 7 + 12 = 0$ $7y + 4x + 5 = 0$

    4. Identify the answer

    Comparing our result ($7y + 4x + 5 = 0$) to the given options, it matches option A.

    5. Tip

    When dealing with perpendicular lines, remember the "flip and switch" rule: flip the fraction of the slope and switch the sign (e.g., $\frac{7}{4}$ becomes $-\frac{4}{7}$). This saves you time during your JAMB exam!

    AI can make mistakes. Report anything that looks wrong.

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

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

38. 38.

    In a committee of 5, which must be selected from 4 males and 3 females. In how many ways can the members be chosen if it were to include 2 females?

    1. A.144 ways
    2. B.15 ways
    3. C.185 ways
    4. D.12 ways

    Show answer

    Answer: D. 12 ways

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

39. 39.

    A binary operation $\otimes$ is defined by $m \otimes n = mn + m - n$ on the set of real numbers, for all m, n $\in$ R. Find the value of 3 $\otimes$ (2 $\otimes$ 4).

    1. A.6
    2. B.25
    3. C.15
    4. D.18

    Show answer

    Answer: C. 15

    AI explanation

    To solve this problem, we need to apply the given binary operation rule step by step, following the order of operations (BODMAS), which requires us to solve the expression inside the parentheses first.

    1. Analyze the operation

    The rule is defined as: $m \otimes n = mn + m - n$

    We need to find the value of: $3 \otimes (2 \otimes 4)$

    2. Solve the inner bracket: $(2 \otimes 4)$

    Using the rule $m \otimes n = mn + m - n$, where $m = 2$ and $n = 4$: $(2 \otimes 4) = (2 \times 4) + 2 - 4$ $(2 \otimes 4) = 8 + 2 - 4$ $(2 \otimes 4) = 10 - 4$ $(2 \otimes 4) = 6$

    3. Solve the final expression: $3 \otimes 6$

    Now substitute the result back into the original expression: $3 \otimes (2 \otimes 4) = 3 \otimes 6$

    Using the rule again, where $m = 3$ and $n = 6$: $3 \otimes 6 = (3 \times 6) + 3 - 6$ $3 \otimes 6 = 18 + 3 - 6$ $3 \otimes 6 = 21 - 6$ $3 \otimes 6 = 15$

    4. Conclusion

    Comparing our result to the options provided: A. 6 B. 25 C. 15 D. 18

    The calculated value is 15, which matches option C.

    Exam Tip: When dealing with binary operations involving brackets, always treat the bracketed part as a single number first. Never try to distribute the operation across the brackets, as binary operations are not always distributive!

    AI can make mistakes. Report anything that looks wrong.

40. 40.

    Rationalize $\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}}$

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

    Show answer

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

    AI explanation

    Analysis

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

    Concept

    Rationalization removes the square root from the denominator. We use the difference of two squares identity: $(a - b)(a + b) = a^2 - b^2$.

    Step-by-Step Solution

    Step 1: Set up the multiplication

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

    Step 2: Expand the numerator Using the FOIL method (First, Outer, Inner, Last): $(\sqrt{2} + \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{2})^2 + \sqrt{2}\sqrt{3} + \sqrt{3}\sqrt{2} + (\sqrt{3})^2$ $= 2 + \sqrt{6} + \sqrt{6} + 3$ $= 5 + 2\sqrt{6}$

    Step 3: Expand the denominator Using the difference of two squares: $(\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{2})^2 - (\sqrt{3})^2$ $= 2 - 3$ $= -1$

    Step 4: Combine and simplify

    $\frac{5 + 2\sqrt{6}}{-1}$

    Divide each term by $-1$: $= -5 - 2\sqrt{6}$

    Conclusion

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

    Exam Tip

    When rationalizing, always check the sign of your denominator after squaring. Many students forget that $(\sqrt{2})^2 - (\sqrt{3})^2$ results in $2 - 3 = -1$, not $1$. If you get a negative denominator, remember to distribute that negative sign to every term in your numerator!

    AI can make mistakes. Report anything that looks wrong.

41. 41.

    If M varies directly as N and inversely as the root of P. Given that M = 3, N = 5 and P = 25. Find the value of P when M = 2 and N = 6.

    1. A.36
    2. B.63
    3. C.47
    4. D.81

    Show answer

    Answer: D. 81

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

42. 42.

    The angle of elevation of the top of a tree from a point on the ground 60m away from the foot of the tree is 78°. Find the height of the tree correct to the nearest whole number.

    1. A.148m
    2. B.382m
    3. C.282m
    4. D.248m

    Show answer

    Answer: C. 282m

    AI explanation

    1. Analyze the question

    We are given:

    • The distance from the foot of the tree to the point on the ground (adjacent side) = $60\text{m}$
    • The angle of elevation ($\theta$) = $78^\circ$
    • We need to find the height of the tree (opposite side).

    2. Explain the concept

    This is a trigonometry problem involving a right-angled triangle. We use the SOH CAH TOA mnemonic:

    • SOH: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$
    • CAH: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
    • TOA: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

    Since we have the Adjacent side and need the Opposite side, we use the Tangent (tan) ratio.

    3. Show the solution

    Step 1: Set up the formula

    $\tan(\theta) = \frac{\text{Height of tree}}{\text{Distance from foot}}$

    $\tan(78^\circ) = \frac{h}{60}$

    Step 2: Solve for $h$

    $h = 60 \times \tan(78^\circ)$

    Step 3: Calculate the value Using a calculator, $\tan(78^\circ) \approx 4.7046$

    $h = 60 \times 4.7046$

    $h = 282.276$

    Step 4: Round to the nearest whole number Rounding $282.276$ to the nearest whole number gives $282\text{m}$.

    4. Identify the answer

    Comparing our result to the given options: A. 148m B. 382m C. 282m D. 248m

    The correct option is C.

    5. Exam Tip

    When dealing with "angle of elevation" problems, always draw a quick sketch of a right-angled triangle. Remember that the "foot of the tree" is the base, and the "height" is the vertical side. If your calculator gives you a very strange number, ensure it is set to DEG (Degrees) mode and not RAD (Radians) mode!

    AI can make mistakes. Report anything that looks wrong.

43. 43.

    Simplify 81$^{\frac{-3}{4}}$ x 25$^{\frac{1}{2}}$ x 243$^{\frac{2}{5}}$

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

    Show answer

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

    AI explanation

    To simplify the expression $81^{-\frac{3}{4}} \times 25^{\frac{1}{2}} \times 243^{\frac{2}{5}}$, we need to express each base as a power of its prime factors.

    1. Analyze the components

    • $81$: $81 = 3^4$
    • $25$: $25 = 5^2$
    • $243$: $243 = 3^5$

    2. Substitute and apply the Power of a Power rule

    The rule states that $(a^m)^n = a^{m \times n}$. Let's apply this to each term:

    • First term: $(3^4)^{-\frac{3}{4}} = 3^{4 \times (-\frac{3}{4})} = 3^{-3}$ • Recall that $x^{-n} = \frac{1}{x^n}$, so $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$

    • Second term: $(5^2)^{\frac{1}{2}} = 5^{2 \times \frac{1}{2}} = 5^1 = 5$

    • Third term: $(3^5)^{\frac{2}{5}} = 3^{5 \times \frac{2}{5}} = 3^2 = 9$

    3. Combine the results

    Now, multiply the simplified values together:

    $\frac{1}{27} \times 5 \times 9$

    We can simplify this by multiplying $5 \times 9$ first:

    $\frac{1}{27} \times 45$

    Now, divide both 45 and 27 by their greatest common divisor, which is 9:

    $\frac{45 \div 9}{27 \div 9} = \frac{5}{3}$

    4. Conclusion

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

    Exam Tip: When dealing with fractional exponents, always look for the prime base that allows the denominator of the exponent to cancel out. For example, seeing a denominator of "4" in the exponent should immediately make you look for a way to express the base as something to the power of 4 (like $81 = 3^4$). This makes the calculation much faster!

    AI can make mistakes. Report anything that looks wrong.

44. 44.

    Find the value of x and y in the simultaneous equation: 3x + y = 21; xy = 30

    1. A.x = 3 or 7, y = 12 or 8
    2. B.x = 6 or 1, y = 11 or 5
    3. C.x = 2 or 5, y = 15 or 6
    4. D.x = 1 or 5, y = 10 or 7

    Show answer

    Answer: C. x = 2 or 5, y = 15 or 6

    AI explanation

    Analysis

    We are given a system of two equations:

    1. $3x + y = 21$
    2. $xy = 30$

    We need to find the values of $x$ and $y$ that satisfy both equations simultaneously.

    Concept

    This is a classic simultaneous equation problem where one equation is linear and the other is non-linear (quadratic). The best approach is the substitution method: express one variable in terms of the other using the linear equation and substitute it into the second equation.

    Step-by-Step Solution

    Step 1: Express $y$ in terms of $x$ using equation (1). From $3x + y = 21$: $y = 21 - 3x$ --- (Equation 3)

    Step 2: Substitute Equation (3) into Equation (2). $x(21 - 3x) = 30$ $21x - 3x^2 = 30$

    Step 3: Rearrange into a standard quadratic equation ($ax^2 + bx + c = 0$). Move all terms to one side: $3x^2 - 21x + 30 = 0$

    Divide the entire equation by 3 to simplify: $x^2 - 7x + 10 = 0$

    Step 4: Factorize the quadratic equation. We look for two numbers that multiply to $10$ and add to $-7$. These numbers are $-2$ and $-5$. $(x - 2)(x - 5) = 0$

    So, $x = 2$ or $x = 5$.

    Step 5: Find the corresponding values for $y$. Using $y = 21 - 3x$:

    • If $x = 2$: $y = 21 - 3(2) = 21 - 6 = 15$
    • If $x = 5$: $y = 21 - 3(5) = 21 - 15 = 6$

    Thus, the pairs are $(x=2, y=15)$ and $(x=5, y=6)$.

    Conclusion

    Comparing our results to the options provided: Option C states $x = 2$ or $5$ and $y = 15$ or $6$. This matches our calculated values perfectly.

    Exam Tip

    When you have a multiple-choice question like this, you can save time by testing the options. Simply plug the values of $x$ and $y$ into the equations. For example, if you test $x=2$ and $y=15$: $3(2) + 15 = 6 + 15 = 21$ (Correct) $2 \times 15 = 30$ (Correct) This confirms Option C is the right choice without needing to solve the full quadratic equation!

    AI can make mistakes. Report anything that looks wrong.

45. 45.

    In the figure above, |CD| is the base of the triangle CDE. Find the area of the figure to the nearest whole number.

    

    1. A.56 cm$^2$
    2. B.24 cm$^2$
    3. C.42 cm$^2$
    4. D.34 cm$^2$

    Show answer

    Answer: D. 34 cm$^2$

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

46. 46.

    Calculate the volume of the regular three dimensional figure drawn above, where < ABC = 90° (a right- angled triangle).

    

    1. A.394 cm$^3$
    2. B.425 cm$^3$
    3. C.268 cm$^3$
    4. D.540 cm$^3$

    Show answer

    Answer: B. 425 cm$^3$

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

47. 47.

    The nth term of a sequence is given by 2$^{2n - 1}$. Find the sum of the first four terms.

    1. A.74
    2. B.32
    3. C.42
    4. D.170

    Show answer

    Answer: D. 170

    AI explanation

    1. Analyze the question

    We are given the general formula for the $n^{th}$ term of a sequence: $T_n = 2^{2n - 1}$. We need to find the sum of the first four terms ($S_4$), which means calculating $T_1, T_2, T_3,$ and $T_4$ and adding them together.

    2. Explain the concept

    A sequence is an ordered list of numbers. To find the sum of the first $n$ terms, we evaluate the formula for each value of $n$ from 1 to $n$ and find their total.

    • For $n=1$: $T_1 = 2^{2(1) - 1}$
    • For $n=2$: $T_2 = 2^{2(2) - 1}$
    • And so on.

    3. Show the solution

    Let's calculate each term individually:

    • For $n = 1$: $T_1 = 2^{2(1) - 1} = 2^{2 - 1} = 2^1 = 2$

    • For $n = 2$: $T_2 = 2^{2(2) - 1} = 2^{4 - 1} = 2^3 = 8$

    • For $n = 3$: $T_3 = 2^{2(3) - 1} = 2^{6 - 1} = 2^5 = 32$

    • For $n = 4$: $T_4 = 2^{2(4) - 1} = 2^{8 - 1} = 2^7 = 128$

    Now, find the sum ($S_4$): $S_4 = T_1 + T_2 + T_3 + T_4$ $S_4 = 2 + 8 + 32 + 128$ $S_4 = 10 + 32 + 128$ $S_4 = 42 + 128$ $S_4 = 170$

    4. Identify the answer

    Comparing our result to the given options, the correct sum is 170, which corresponds to option D.

    5. Exam Tip

    When dealing with exponential sequences, always double-check your powers. A common mistake is to multiply the base by the exponent instead of raising the base to the power. Also, notice that this is a Geometric Progression (GP) where the first term $a = 2$ and the common ratio $r = 4$. You could also use the sum formula for a GP: $S_n = \frac{a(r^n - 1)}{r - 1}$. $S_4 = \frac{2(4^4 - 1)}{4 - 1} = \frac{2(256 - 1)}{3} = \frac{2(255)}{3} = 2 \times 85 = 170$.

    AI can make mistakes. Report anything that looks wrong.

48. 48.

    If 2x$^2$ + x - 3 divides x - 2, find the remainder.

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

    Show answer

    Answer: A. 7

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

49. 49.

    If $6x^3 + 2x^2 - 5x + 1$ divides $x^2 - x - 1$, find the remainder.

    1. A.9x + 9
    2. B.2x + 6
    3. C.6x + 8
    4. D.5x - 3

    Show answer

    Answer: A. 9x + 9

    AI explanation

    To solve this problem, we need to perform polynomial long division. We are dividing the polynomial $P(x) = 6x^3 + 2x^2 - 5x + 1$ by the divisor $D(x) = x^2 - x - 1$.

    1. Analyze the question

    We are dividing a cubic polynomial by a quadratic polynomial. The result will be a linear quotient and a linear remainder (since the degree of the remainder must be less than the degree of the divisor).

    2. Explain the concept

    Polynomial long division follows the same logic as numerical long division:

    • Divide the leading term of the dividend by the leading term of the divisor.
    • Multiply the entire divisor by the result.
    • Subtract the result from the current dividend.
    • Repeat until the degree of the remainder is less than the degree of the divisor.

    3. Show the solution

    Step 1: Divide the first terms Divide $6x^3$ by $x^2$: $6x^3 / x^2 = 6x$

    Step 2: Multiply and Subtract Multiply $6x$ by $(x^2 - x - 1)$: $6x(x^2 - x - 1) = 6x^3 - 6x^2 - 6x$

    Subtract this from the original polynomial: $(6x^3 + 2x^2 - 5x + 1) - (6x^3 - 6x^2 - 6x)$ $= (6x^3 - 6x^3) + (2x^2 - (-6x^2)) + (-5x - (-6x)) + 1$ $= 8x^2 + x + 1$

    Step 3: Divide the new leading term Divide $8x^2$ by $x^2$: $8x^2 / x^2 = 8$

    Step 4: Multiply and Subtract Multiply $8$ by $(x^2 - x - 1)$: $8(x^2 - x - 1) = 8x^2 - 8x - 8$

    Subtract this from the current remainder ($8x^2 + x + 1$): $(8x^2 + x + 1) - (8x^2 - 8x - 8)$ $= (8x^2 - 8x^2) + (x - (-8x)) + (1 - (-8))$ $= 9x + 9$

    4. Identify the answer

    The remainder is $9x + 9$. This matches option A.

    5. Exam Tip

    When performing polynomial subtraction, be very careful with the signs, especially when subtracting a negative term (e.g., $-(-6x^2)$ becomes $+6x^2$). A common mistake in JAMB exams is losing a negative sign during this step. Always write out the subtraction clearly to avoid "silly" errors.

    AI can make mistakes. Report anything that looks wrong.

50. 50.

    Differentiate $\frac{2x}{\sin x}$ with respect to x.

    1. A.$2 \cot x \sec x (1 + \tan x)$
    2. B.$2 \csc x - x \cot x$
    3. C.$2x \csc x + \tan x$
    4. D.$2\csc x(1 - x\cot x)$

    Show answer

    Answer: D. $2\csc x(1 - x\cot x)$

    AI explanation

    1. Analyze the Question

    We are asked to find the derivative of the function $y = \frac{2x}{\sin x}$ with respect to $x$. Since this is a quotient of two functions, we must use the Quotient Rule.

    2. Explain the Concept

    The Quotient Rule states that if you have a function $y = \frac{u}{v}$, its derivative is given by:

    $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$

    In this problem:

    • $u = 2x \implies \frac{du}{dx} = 2$
    • $v = \sin x \implies \frac{dv}{dx} = \cos x$

    3. Show the Solution

    Substitute these values into the Quotient Rule formula:

    $\frac{dy}{dx} = \frac{(\sin x)(2) - (2x)(\cos x)}{(\sin x)^2}$

    Simplify the expression:

    $\frac{dy}{dx} = \frac{2\sin x - 2x\cos x}{\sin^2 x}$

    Now, split the fraction into two parts:

    $\frac{dy}{dx} = \frac{2\sin x}{\sin^2 x} - \frac{2x\cos x}{\sin^2 x}$

    Simplify each term:

    • For the first term: $\frac{2\sin x}{\sin^2 x} = \frac{2}{\sin x} = 2\csc x$
    • For the second term: $\frac{2x\cos x}{\sin^2 x} = 2x \cdot \left(\frac{\cos x}{\sin x}\right) \cdot \left(\frac{1}{\sin x}\right) = 2x \cot x \csc x$

    So, the derivative is:

    $\frac{dy}{dx} = 2\csc x - 2x\csc x \cot x$

    Factor out $2\csc x$:

    $\frac{dy}{dx} = 2\csc x(1 - x\cot x)$

    4. Identify the Answer

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

    5. Exam Tip

    When dealing with trigonometric functions in calculus, always remember your basic identities: $\frac{1}{\sin x} = \csc x$ and $\frac{\cos x}{\sin x} = \cot x$. Recognizing these early makes simplifying your final answer much easier and helps you match it to the JAMB options quickly!

    AI can make mistakes. Report anything that looks wrong.

51. 51.

    Evaluate $(\frac{6}{0.32} \div \frac{2}{0.084})^{-1}$ correct to 1 decimal place.

    1. A.1.3
    2. B.2.5
    3. C.4.6
    4. D.3.2

    Show answer

    Answer: A. 1.3

    AI explanation

    Step 1: Analyze the question

    We need to evaluate the expression $(\frac{6}{0.32} \div \frac{2}{0.084})^{-1}$ and round the result to one decimal place.

    Step 2: Explain the concept

    To solve this, we follow the order of operations (BODMAS):

    1. Solve the divisions inside the parentheses first.
    2. Remember that dividing by a fraction is the same as multiplying by its reciprocal (flip the second fraction).
    3. Apply the exponent $-1$, which means finding the reciprocal of the final result ($x^{-1} = \frac{1}{x}$).

    Step 3: Show the solution

    Part A: Simplify the fractions inside the parentheses First, let's simplify the two divisions:

    • $\frac{6}{0.32} = \frac{600}{32} = 18.75$
    • $\frac{2}{0.084} = \frac{2000}{84} \approx 23.8095$

    Part B: Perform the division Now, we divide the first result by the second: $18.75 \div 23.8095$

    Using the reciprocal method for accuracy: $\frac{6}{0.32} \times \frac{0.084}{2}$ $= \frac{6}{2} \times \frac{0.084}{0.32}$ $= 3 \times \frac{84}{320}$ $= 3 \times 0.2625 = 0.7875$

    Part C: Apply the exponent $-1$ The expression is $(0.7875)^{-1}$, which is $\frac{1}{0.7875}$: $\frac{1}{0.7875} \approx 1.2698$

    Part D: Round to 1 decimal place Rounding $1.2698$ to one decimal place gives 1.3.

    Step 4: Identify the answer

    Comparing our result to the given options, we see that 1.3 matches option A.

    Step 5: Exam Tip

    When dealing with decimals in JAMB, convert them to whole numbers by multiplying the numerator and denominator by powers of 10 (e.g., $10, 100, 1000$). This eliminates decimal errors and makes the division much easier to handle without a calculator.

    AI can make mistakes. Report anything that looks wrong.

52. 52.

    In the circle above, with centre O and radius 7 cm. Find the length of the arc AB, when < AOB = 57°.

    

    1. A.5.32 cm
    2. B.4.39 cm
    3. C.7.33 cm
    4. D.6.97 cm

    Show answer

    Answer: D. 6.97 cm

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

53. 53.

    Find the polynomial if given q(x) = x$^2$ - x - 5, d(x) = 3x - 1 and r(x) = 7.

    1. A.3x$^3$ - 4x$^2$ - 14x + 12
    2. B.3x$^2$ + 3x - 7
    3. C.3x$^3$ + 4x$^2$ + 14x - 12
    4. D.3x$^2$ - 3x + 4

    Show answer

    Answer: A. 3x$^3$ - 4x$^2$ - 14x + 12

    AI explanation

    1. Analyze the question

    You are given the quotient $q(x)$, the divisor $d(x)$, and the remainder $r(x)$ of a polynomial division. You need to find the original polynomial, which we can call $P(x)$.

    2. Explain the concept

    In polynomial division, the relationship between the dividend $P(x)$, the divisor $d(x)$, the quotient $q(x)$, and the remainder $r(x)$ is defined by the Division Algorithm:

    $P(x) = d(x) \cdot q(x) + r(x)$

    To find the polynomial, we simply need to multiply the divisor by the quotient and then add the remainder.

    3. Show the solution

    Given:

    • $q(x) = x^2 - x - 5$
    • $d(x) = 3x - 1$
    • $r(x) = 7$

    Step 1: Multiply $d(x)$ and $q(x)$ $P(x) = (3x - 1)(x^2 - x - 5) + 7$

    Expand the brackets: $= 3x(x^2 - x - 5) - 1(x^2 - x - 5) + 7$ $= (3x^3 - 3x^2 - 15x) - (x^2 - x - 5) + 7$

    Step 2: Simplify the expression $= 3x^3 - 3x^2 - 15x - x^2 + x + 5 + 7$

    Group the like terms: $= 3x^3 + (-3x^2 - x^2) + (-15x + x) + (5 + 7)$ $= 3x^3 - 4x^2 - 14x + 12$

    4. Identify the answer

    Comparing our result with the given options, the correct polynomial is $3x^3 - 4x^2 - 14x + 12$. This matches Option A.

    5. Tip

    When multiplying polynomials, always be careful with the negative signs. A common mistake in JAMB exams is forgetting to distribute the negative sign to every term inside the second bracket (e.g., $-1 \times -5$ becomes $+5$). Always double-check your signs before adding the remainder!

    AI can make mistakes. Report anything that looks wrong.

54. 54.

    A man bought a car newly for ₦1,250,000. He had a crash with the car and later sold it at the rate of ₦1,085,000. What is the percentage gain or loss of the man?

    1. A.43.7% loss
    2. B.13.2% gain
    3. C.13.2% loss
    4. D.43.7% gain

    Show answer

    Answer: C. 13.2% loss

    AI explanation

    Analysis

    To solve this problem, we need to determine whether the man made a profit (gain) or incurred a loss.

    • Cost Price (CP): The amount the man originally paid for the car = ₦1,250,000.
    • Selling Price (SP): The amount the man sold the car for = ₦1,085,000.

    Since the Selling Price is less than the Cost Price, the man incurred a loss.

    Concept

    The formula for calculating the percentage loss is:

    $\text{Percentage Loss} = \left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100\%$

    Where:

    $\text{Loss} = \text{Cost Price} - \text{Selling Price}$

    Step-by-Step Solution

    Step 1: Calculate the actual loss in Naira.

    $\text{Loss} = 1,250,000 - 1,085,000$

    $\text{Loss} = 165,000$

    Step 2: Calculate the percentage loss.

    $\text{Percentage Loss} = \left( \frac{165,000}{1,250,000} \right) \times 100\%$

    Step 3: Simplify the calculation. Cancel out the zeros:

    $\text{Percentage Loss} = \left( \frac{165}{1,250} \right) \times 100\%$

    $\text{Percentage Loss} = \frac{165}{12.5}$

    $\text{Percentage Loss} = 13.2\%$

    Since the selling price was lower than the cost price, this is a 13.2% loss.

    Conclusion

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

    Exam Tip

    Always check if the Selling Price is higher or lower than the Cost Price before calculating. If $SP < CP$, it is a loss; if $SP > CP$, it is a gain. This simple check helps you eliminate two of the four options immediately!

    AI can make mistakes. Report anything that looks wrong.

55. 55.

    Simplify $\frac{0.0839 \times 6.381}{5.44}$ to 2 significant figures.

    1. A.0.2809
    2. B.2.51
    3. C.3.5
    4. D.0.098

    Show answer

    Answer: D. 0.098

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

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

57. 57.

    If $\begin{vmatrix} 2 & -5 & 3 \\ x & 1 & 4 \\ 0 & 3 & 2 \end{vmatrix} = 132$, find the value of x.

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

    Show answer

    Answer: B. 8

    AI explanation

    Analysis

    To find the value of $x$, we need to calculate the determinant of the given $3 \times 3$ matrix and equate it to $132$.

    Concept: Determinant of a $3 \times 3$ Matrix

    For a matrix $A = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$, the determinant is calculated by expanding along the first row:

    $\text{Det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

    Step-by-Step Solution

    Given the matrix:

    $\begin{vmatrix} 2 & -5 & 3 \\ x & 1 & 4 \\ 0 & 3 & 2 \end{vmatrix} = 132$

    1. Expand along the first row: $= 2 \begin{vmatrix} 1 & 4 \\ 3 & 2 \end{vmatrix} - (-5) \begin{vmatrix} x & 4 \\ 0 & 2 \end{vmatrix} + 3 \begin{vmatrix} x & 1 \\ 0 & 3 \end{vmatrix} = 132$

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

    • First part: $2[(1 \times 2) - (4 \times 3)] = 2[2 - 12] = 2[-10] = -20$
    • Second part: $+5[(x \times 2) - (4 \times 0)] = 5[2x - 0] = 10x$
    • Third part: $+3[(x \times 3) - (1 \times 0)] = 3[3x - 0] = 9x$

    3. Combine the results: $-20 + 10x + 9x = 132$

    4. Solve for $x$: $19x - 20 = 132$ $19x = 132 + 20$ $19x = 152$ $x = \frac{152}{19}$ $x = 8$

    Conclusion

    The value of $x$ is 8. Comparing this to the provided options, it matches option B.

    Exam Tip

    When calculating determinants, always be very careful with the signs of the middle term. Remember the sign pattern for a $3 \times 3$ matrix is:

    $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$

    Since the middle element in the first row was $-5$, the formula $-b$ makes it $-(-5) = +5$. Missing this sign change is the most common mistake students make in JAMB Mathematics!

    AI can make mistakes. Report anything that looks wrong.

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

59. 59.

    The weight of a day-old chick was measured to be 0.21g. If the actual weight of the chick is 0.18g, what was the percentage error in the measurement?

    1. A.15.5%
    2. B.18.2%
    3. C.14.8%
    4. D.16.7%

    Show answer

    Answer: D. 16.7%

    AI explanation

    Analysis

    To solve this problem, we need to calculate the percentage error. Percentage error is a measure of how far off a measurement is from the true or actual value, expressed as a percentage.

    Concept

    The formula for percentage error is:

    $\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{Actual Value}}{\text{Actual Value}} \right) \times 100\%$

    • Measured Value: The value obtained during the measurement (0.21g).
    • Actual Value: The true or correct weight (0.18g).
    • Error: The difference between the measured and actual values.

    Step-by-Step Solution

    1. Find the absolute error: $$\text{Error} = \text{Measured Value} - \text{Actual Value}$$ $$\text{Error} = 0.21\text{g} - 0.18\text{g} = 0.03\text{g}$$

    2. Calculate the percentage error: $$\text{Percentage Error} = \left( \frac{0.03}{0.18} \right) \times 100\%$$

    3. Simplify the fraction: $$\frac{0.03}{0.18} = \frac{3}{18} = \frac{1}{6}$$

    4. Convert to percentage: $$\frac{1}{6} \times 100\% = 16.666...\%$$

    Rounding this to one decimal place gives 16.7%.

    Conclusion

    Comparing our result to the given options, 16.7% matches option D.

    Exam Tip

    When calculating percentage error, always remember that the denominator is the Actual Value (the "true" value), not the measured value. A common mistake students make is dividing by the measured value (0.21), which would lead to the wrong answer. Always keep the "True" value at the bottom!

    AI can make mistakes. Report anything that looks wrong.

60. 60.

    If $4\sin^2 x - 3 = 0$, find the value of x, when 0° $\leq$ x $\leq$ 90°

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

    Show answer

    Answer: C. 60°

    AI explanation

    1. Analyze the question

    We are given a trigonometric equation $4\sin^2 x - 3 = 0$ and asked to find the value of $x$ within the range $0^\circ \leq x \leq 90^\circ$.

    2. Explain the concept

    To solve this, we need to isolate the trigonometric function ($\sin x$) and then use the inverse sine function (or our knowledge of special angles) to find the value of $x$. Remember that $\sin^2 x$ is the same as $(\sin x)^2$.

    3. Show the solution

    Step 1: Isolate $\sin^2 x$ Start with the equation: $4\sin^2 x - 3 = 0$

    Add 3 to both sides: $4\sin^2 x = 3$

    Divide both sides by 4: $\sin^2 x = \frac{3}{4}$

    Step 2: Solve for $\sin x$ Take the square root of both sides: $\sqrt{\sin^2 x} = \sqrt{\frac{3}{4}}$ $\sin x = \frac{\sqrt{3}}{\sqrt{4}}$ $\sin x = \frac{\sqrt{3}}{2}$

    Step 3: Find the value of $x$ We need to find the angle $x$ whose sine is $\frac{\sqrt{3}}{2}$. From the standard trigonometric ratios for special angles: $\sin 60^\circ = \frac{\sqrt{3}}{2}$

    Since $60^\circ$ falls within the given range ($0^\circ \leq x \leq 90^\circ$), this is our solution.

    4. Identify the answer

    Comparing our result to the options provided: A. 90° B. 45° C. 60° D. 30°

    The correct option is C.

    5. Tip

    Always memorize the "Special Angles" table for $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ and $90^\circ$. A quick way to remember the sine values is:

    • $\sin 0^\circ = \sqrt{0}/2 = 0$
    • $\sin 30^\circ = \sqrt{1}/2 = 1/2$
    • $\sin 45^\circ = \sqrt{2}/2$
    • $\sin 60^\circ = \sqrt{3}/2$
    • $\sin 90^\circ = \sqrt{4}/2 = 1$

    AI can make mistakes. Report anything that looks wrong.

61. 61.

    Each of the interior angles of a regular polygon is 140°. Calculate the sum of all the interior angles of the polygon.

    1. A.1080°
    2. B.1260°
    3. C.2160°
    4. D.1800°

    Show answer

    Answer: B. 1260°

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

62. 62.

    Score (x) 0 1 2 3 4 5 6 Freq (f) 5 7 3 7 11 6 7 Find the mean of the data.

    1. A.3.26
    2. B.4.91
    3. C.6.57
    4. D.3.0

    Show answer

    Answer: A. 3.26

    AI explanation

    To find the mean of a frequency distribution, we use the formula:

    $\text{Mean } (\bar{x}) = \frac{\sum fx}{\sum f}$

    Where:

    • $x$ is the score
    • $f$ is the frequency
    • $\sum fx$ is the sum of the products of each score and its frequency
    • $\sum f$ is the total number of frequencies

    Step 1: Create a frequency table and calculate $fx$

    | Score ($x$) | Frequency ($f$) | $f \times x$ | | :--- | :--- | :--- | | 0 | 5 | $0 \times 5 = 0$ | | 1 | 7 | $1 \times 7 = 7$ | | 2 | 3 | $2 \times 3 = 6$ | | 3 | 7 | $3 \times 7 = 21$ | | 4 | 11 | $4 \times 11 = 44$ | | 5 | 6 | $5 \times 6 = 30$ | | 6 | 7 | $6 \times 7 = 42$ |

    Step 2: Calculate the totals

    • Total Frequency ($\sum f$): $5 + 7 + 3 + 7 + 11 + 6 + 7 = 46$
    • Total Sum of products ($\sum fx$): $0 + 7 + 6 + 21 + 44 + 30 + 42 = 150$

    Step 3: Calculate the Mean

    $\bar{x} = \frac{150}{46}$

    $\bar{x} \approx 3.2608...$

    Rounding to two decimal places, we get 3.26.

    Conclusion

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

    Exam Tip: When dealing with frequency tables in JAMB, always double-check your addition of the frequencies ($\sum f$) first. A small error in the total frequency will throw off your entire calculation. If you have time, perform the multiplication $f \times x$ twice to ensure no arithmetic slips occurred.

    AI can make mistakes. Report anything that looks wrong.

63. 63.

    A binary operation Δ is defined by a Δ b = a + 3b + 2. Find (3 Δ 2) Δ 5

    1. A.35
    2. B.59
    3. C.28
    4. D.87

    Show answer

    Answer: C. 28

    AI explanation

    1. Analyze the question

    We are given a binary operation defined as $a \Delta b = a + 3b + 2$. We need to evaluate the expression $(3 \Delta 2) \Delta 5$. This is a two-step problem where we first solve the operation inside the bracket, then use that result to solve the final operation.

    2. Explain the concept

    A binary operation is simply a rule that tells you how to combine two numbers. In this case, the rule is: "Take the first number, add three times the second number, and then add 2." We must follow the order of operations (BODMAS) to ensure accuracy.

    3. Show the solution

    Step 1: Solve the expression inside the bracket $(3 \Delta 2)$ Using the formula $a \Delta b = a + 3b + 2$:

    • Here, $a = 3$ and $b = 2$.
    • $3 \Delta 2 = 3 + 3(2) + 2$
    • $3 \Delta 2 = 3 + 6 + 2$
    • $3 \Delta 2 = 11$

    Step 2: Solve the final expression $(11 \Delta 5)$ Now, we substitute the result from Step 1 into the original equation:

    • Here, $a = 11$ and $b = 5$.
    • $11 \Delta 5 = 11 + 3(5) + 2$
    • $11 \Delta 5 = 11 + 15 + 2$
    • $11 \Delta 5 = 28$

    4. Identify the answer

    Comparing our result to the given options: A. 35 B. 59 C. 28 D. 87

    The correct option is C.

    5. Tip

    When dealing with binary operations in JAMB, always solve the bracketed part first. If you have multiple operations, treat the result of the first operation as a single number and plug it back into the definition. Don't rush the arithmetic—a simple sign error here is the most common reason students lose marks!

    AI can make mistakes. Report anything that looks wrong.

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

65. 65.

    Make q the subject of the formula in the equation $\frac{mn}{a^2} - \frac{pq}{b^2} = 1$

    1. A.$q = \frac{b^2(mn - a^2)}{a^2 p}$
    2. B.$q = \frac{m^2 n - a^2}{p^2}$
    3. C.$q = \frac{mn - 2b^2}{a^2}$
    4. D.$q = \frac{b^2 (n^2 - ma^2)}{n}$

    Show answer

    Answer: A. $q = \frac{b^2(mn - a^2)}{a^2 p}$

    AI explanation

    Analysis

    We are given the equation:

    $\frac{mn}{a^2} - \frac{pq}{b^2} = 1$

    Our goal is to isolate $q$ on one side of the equation.

    Concept

    To make a variable the subject of a formula, we use algebraic manipulation (addition, subtraction, multiplication, and division) to move all other terms to the opposite side of the equation until the desired variable stands alone.

    Step-by-Step Solution

    Step 1: Isolate the term containing $q$. Subtract $\frac{mn}{a^2}$ from both sides:

    $-\frac{pq}{b^2} = 1 - \frac{mn}{a^2}$

    Step 2: Simplify the right side. Find a common denominator ($a^2$) for the right side:

    $-\frac{pq}{b^2} = \frac{a^2 - mn}{a^2}$

    Step 3: Eliminate the negative sign. Multiply both sides by $-1$ to make the $q$ term positive:

    $\frac{pq}{b^2} = -\left(\frac{a^2 - mn}{a^2}\right)$

    Distributing the negative sign inside the parenthesis:

    $\frac{pq}{b^2} = \frac{mn - a^2}{a^2}$

    Step 4: Solve for $q$. Multiply both sides by $b^2$:

    $pq = \frac{b^2(mn - a^2)}{a^2}$

    Finally, divide both sides by $p$:

    $q = \frac{b^2(mn - a^2)}{a^2 p}$

    Conclusion

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

    Exam Tip

    When dealing with fractions in algebraic equations, always try to clear the fractions by multiplying through by the Least Common Multiple (LCM) or by grouping terms with the same denominator. This makes the equation much easier to handle and reduces the chance of sign errors!

    AI can make mistakes. Report anything that looks wrong.

66. 66.

    If P varies inversely as the square root of q, where p = 3 and q = 16, find the value of q when p = 4.

    1. A.12
    2. B.8
    3. C.9
    4. D.16

    Show answer

    Answer: C. 9

    AI explanation

    1. Analyze the Question

    We are given a relationship of inverse variation between two variables, $p$ and $q$. Specifically, $p$ varies inversely as the square root of $q$. We are given an initial set of values to find the constant of variation, and then asked to find a new value for $q$ when $p$ changes.

    2. Explain the Concept

    In mathematics, when $p$ varies inversely as the square root of $q$, the relationship is expressed as:

    $p \propto \frac{1}{\sqrt{q}}$

    To turn this into an equation, we introduce a constant of variation, $k$:

    $p = \frac{k}{\sqrt{q}}$

    Our goal is to first find $k$ using the given values, then use that $k$ to solve for the unknown $q$.

    3. Show the Solution

    Step 1: Find the constant $k$ Given $p = 3$ and $q = 16$:

    $3 = \frac{k}{\sqrt{16}}$

    Since $\sqrt{16} = 4$:

    $3 = \frac{k}{4}$

    Multiply both sides by 4:

    $k = 3 \times 4 = 12$

    Step 2: Find $q$ when $p = 4$ Now that we know $k = 12$, our formula is $p = \frac{12}{\sqrt{q}}$. Substitute $p = 4$:

    $4 = \frac{12}{\sqrt{q}}$

    Multiply both sides by $\sqrt{q}$ and divide by 4:

    $\sqrt{q} = \frac{12}{4}$

    $\sqrt{q} = 3$

    To solve for $q$, square both sides:

    $q = 3^2$

    $q = 9$

    4. Identify the Answer

    Comparing our result to the given options: A. 12 B. 8 C. 9 D. 16

    The correct option is C.

    5. Exam Tip

    When dealing with inverse variation, always remember: "If one goes up, the other goes down." Since $p$ increased from 3 to 4, we should expect $q$ to decrease from its original value of 16. Since 9 is the only option smaller than 16, you could have narrowed your choices down quickly!

    AI can make mistakes. Report anything that looks wrong.

67. 67.

    If y = 8x$^3$ - 3x$^2$ + 7x - 1, find $\frac{\mathrm d^2 y}{\mathrm d x^2}$.

    1. A.48x - 6
    2. B.11x$^2$ + 6x - 7
    3. C.32x + 7
    4. D.24x$^2$ - 6x + 7

    Show answer

    Answer: A. 48x - 6

    AI explanation

    Analysis

    The question asks for the second derivative ($\frac{\mathrm d^2 y}{\mathrm d x^2}$) of the given polynomial function $y = 8x^3 - 3x^2 + 7x - 1$. To find the second derivative, we must differentiate the function with respect to $x$ twice.

    Concept: The Power Rule

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

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

    The derivative of a constant (a number without $x$) is always $0$.

    Step-by-Step Solution

    Step 1: Find the first derivative ($\frac{\mathrm d y}{\mathrm d x}$) We differentiate each term of $y = 8x^3 - 3x^2 + 7x - 1$ individually:

    • For $8x^3$: $3 \times 8x^{3-1} = 24x^2$
    • For $-3x^2$: $2 \times -3x^{2-1} = -6x$
    • For $7x$: $1 \times 7x^{1-1} = 7x^0 = 7$
    • For $-1$: The derivative is $0$

    So, the first derivative is:

    $\frac{\mathrm d y}{\mathrm d x} = 24x^2 - 6x + 7$

    Step 2: Find the second derivative ($\frac{\mathrm d^2 y}{\mathrm d x^2}$) Now, we differentiate the result from Step 1:

    • For $24x^2$: $2 \times 24x^{2-1} = 48x$
    • For $-6x$: $1 \times -6x^{1-1} = -6$
    • For $7$: The derivative is $0$

    Combining these, we get:

    $\frac{\mathrm d^2 y}{\mathrm d x^2} = 48x - 6$

    Conclusion

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

    Exam Tip: When dealing with polynomials, remember that every time you differentiate, the power of $x$ decreases by 1. If you are asked for the $n^{th}$ derivative, you simply repeat the process $n$ times. Always double-check your signs (positive/negative) during the multiplication step!

    AI can make mistakes. Report anything that looks wrong.

68. 68.

    If $y = 6x^3 + 2x^{-2} - x^{-3}$, find $\frac{\mathrm d y}{\mathrm d x}$.

    1. A.$\frac{\mathrm d y}{\mathrm d x} = 15x^2 - 4x^{-2} - 3x^{-2}$
    2. B.$\frac{\mathrm d y}{\mathrm d x} = 6x + 4x^{-1} - 3x^{-4}$
    3. C.$\frac{\mathrm d y}{\mathrm d x} = 18x^2 - 4x^{-3} + 3x^{-4}$
    4. D.$\frac{\mathrm d y}{\mathrm d x} = 12x^2 + 4x^{-1} - 3x^{-2}$

    Show answer

    Answer: C. $\frac{\mathrm d y}{\mathrm d x} = 18x^2 - 4x^{-3} + 3x^{-4}$

    AI explanation

    Analysis

    To find $\frac{\mathrm{d}y}{\mathrm{d}x}$, we need to apply the Power Rule of Differentiation. The power rule states that if $y = ax^n$, then $\frac{\mathrm{d}y}{\mathrm{d}x} = n \cdot ax^{n-1}$.

    Concept

    The Power Rule is the fundamental tool for differentiating polynomials and terms with negative or fractional exponents. You simply:

    1. Bring the current exponent down to multiply with the coefficient.
    2. Subtract 1 from the current exponent.

    Step-by-Step Solution

    Given the function: $y = 6x^3 + 2x^{-2} - x^{-3}$

    We differentiate each term individually:

    1. For $6x^3$: • Bring down the 3: $3 \times 6 = 18$ • Subtract 1 from the exponent: $3 - 1 = 2$ • Result: $18x^2$

    2. For $2x^{-2}$: • Bring down the -2: $-2 \times 2 = -4$ • Subtract 1 from the exponent: $-2 - 1 = -3$ • Result: $-4x^{-3}$

    3. For $-x^{-3}$: • Bring down the -3: $-3 \times -1 = +3$ • Subtract 1 from the exponent: $-3 - 1 = -4$ • Result: $+3x^{-4}$

    Combining these results, we get: $\frac{\mathrm{d}y}{\mathrm{d}x} = 18x^2 - 4x^{-3} + 3x^{-4}$

    Conclusion

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

    Exam Tip

    When dealing with negative exponents, be very careful with your signs during subtraction. Many students mistakenly calculate $-2 - 1$ as $-1$ instead of $-3$. Always remember: "Subtracting 1 from a negative number makes it more negative."

    AI can make mistakes. Report anything that looks wrong.

69. 69.

    A factory worker earns ₦50,000 per month out of which he spends 15% on his children's education, ₦13,600 on Food, 3% on electricity and uses the rest for his personal purpose. How much does he have left?

    1. A.N21,850
    2. B.N18,780
    3. C.N27,400
    4. D.N32,500

    Show answer

    Answer: C. N27,400

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

70. 70.

    Find the value of x for $\frac{2 + 2x}{3} - 2 \geq \frac{4x - 6}{5}$

    1. A.x $\geq$ -5
    2. B.x $\geq$ -1
    3. C.x $\leq$ -1
    4. D.x $\leq$ 3

    Show answer

    Answer: C. x $\leq$ -1

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

71. 71.

    If P(2, m) is the midpoint of the line joining Q(m, n) and R(n, -4), find the values of m and n.

    1. A.m = 0, n = 4
    2. B.m = 4, n = 0
    3. C.m = 2, n = 2
    4. D.m = -2, n = 4

    Show answer

    Answer: A. m = 0, n = 4

    AI explanation

    Analysis

    To solve this problem, we use the Midpoint Formula. If a point $P(x, y)$ is the midpoint of a line segment joining two points $Q(x_1, y_1)$ and $R(x_2, y_2)$, the coordinates of the midpoint are given by:

    $x = \frac{x_1 + x_2}{2} \quad \text{and} \quad y = \frac{y_1 + y_2}{2}$

    Concept

    In this question, we are given:

    • Midpoint $P = (2, m)$
    • Point $Q = (m, n)$
    • Point $R = (n, -4)$

    By substituting these values into the midpoint formula, we create a system of two equations that we can solve simultaneously.

    Step-by-Step Solution

    Step 1: Set up the equations for the x and y coordinates. For the x-coordinate:

    $2 = \frac{m + n}{2}$

    Multiply both sides by 2:

    $4 = m + n \quad \dots \text{(Equation 1)}$

    For the y-coordinate:

    $m = \frac{n + (-4)}{2}$

    Multiply both sides by 2:

    $2m = n - 4 \quad \dots \text{(Equation 2)}$

    Step 2: Solve the system of equations. From Equation 1, we can express $n$ in terms of $m$:

    $n = 4 - m$

    Substitute this into Equation 2:

    $2m = (4 - m) - 4$

    $2m = -m$

    $2m + m = 0$

    $3m = 0$

    $m = 0$

    Step 3: Find the value of n. Substitute $m = 0$ back into Equation 1:

    $4 = 0 + n$

    $n = 4$

    Thus, $m = 0$ and $n = 4$.

    Conclusion

    Comparing our result ($m = 0, n = 4$) with the given options, it matches option A.

    Exam Tip

    When dealing with midpoint questions, always write out your $x_1, y_1, x_2, y_2$ clearly before plugging them into the formula. This prevents "sign errors" (like forgetting the negative sign in $-4$), which is the most common reason students lose marks in coordinate geometry.

    AI can make mistakes. Report anything that looks wrong.

72. 72.

    Tade bought 200 mangoes at 4 for ₦2.50. 30 out of the mangoes got spoilt and the remaining were sold at 2 for ₦2.40. Find the percentage profit or loss.

    1. A.43.6% loss
    2. B.35% profit
    3. C.63.2% profit
    4. D.28% loss

    Show answer

    Answer: C. 63.2% profit

    AI explanation

    1. Analyze the question

    We need to determine the total cost price (CP) of the mangoes, the total selling price (SP) after accounting for the spoilt ones, and then calculate the percentage profit or loss using the formula:

    $\text{Percentage Profit/Loss} = \frac{\text{Profit or Loss}}{\text{Cost Price}} \times 100\%$

    2. Explain the concept

    • Cost Price (CP): The total amount spent to acquire the goods.
    • Selling Price (SP): The total amount received from selling the goods.
    • Profit: Occurs when SP > CP (Profit = SP - CP).
    • Loss: Occurs when CP > SP (Loss = CP - SP).

    3. Show the solution

    Step 1: Calculate the Total Cost Price (CP) Tade bought 200 mangoes at 4 for ₦2.50.

    • Number of groups of 4 in 200 = $200 \div 4 = 50$ groups.
    • Total CP = $50 \times ₦2.50 = ₦125.00$.

    Step 2: Calculate the Total Selling Price (SP)

    • Number of good mangoes = $200 - 30 = 170$ mangoes.
    • They were sold at 2 for ₦2.40.
    • Number of groups of 2 in 170 = $170 \div 2 = 85$ groups.
    • Total SP = $85 \times ₦2.40 = ₦204.00$.

    Step 3: Determine Profit or Loss Since the SP (₦204.00) is greater than the CP (₦125.00), Tade made a profit.

    • Profit = $₦204.00 - ₦125.00 = ₦79.00$.

    Step 4: Calculate Percentage Profit

    • Percentage Profit = $\frac{\text{Profit}}{\text{CP}} \times 100\%$
    • Percentage Profit = $\frac{79}{125} \times 100\%$
    • Percentage Profit = $0.632 \times 100\% = 63.2\%$.

    4. Identify the answer

    Comparing our result to the options provided, we see that 63.2% profit matches option C.

    5. Tip

    When dealing with "items per price" questions, always find the cost/selling price of a single unit first if the numbers are complex, or group them as we did above. Also, always double-check if the question asks for the percentage of the Cost Price (which is standard) or the Selling Price, as this is a common trap in JAMB questions!

    AI can make mistakes. Report anything that looks wrong.

73. 73.

    If the 3rd and 7th terms of a G.P are 9 and 1/9 respectively. Find the common ratio.

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

    Show answer

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

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

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

75. 75.

    If $\sin x = \frac{4}{5}$, find $\frac{1 + \cot^2 x}{\csc^2 x - 1}$.

    1. A.$\frac{13}{2}$
    2. B.$\frac{25}{9}$
    3. C.$\frac{3}{13}$
    4. D.$\frac{4}{11}$

    Show answer

    Answer: B. $\frac{25}{9}$

    AI explanation

    1. Analyze the question

    We are given $\sin x = \frac{4}{5}$ and asked to evaluate the expression $\frac{1 + \cot^2 x}{\csc^2 x - 1}$.

    2. Explain the concept

    To solve this, we use fundamental trigonometric identities:

    • Pythagorean Identity: $1 + \cot^2 x = \csc^2 x$
    • Rearranged Identity: $\csc^2 x - 1 = \cot^2 x$

    By substituting these identities into the expression, we can simplify the problem significantly before plugging in the value of $\sin x$.

    3. Show the solution

    Step 1: Simplify the expression using identities. The expression is:

    $\frac{1 + \cot^2 x}{\csc^2 x - 1}$

    Using the identity $1 + \cot^2 x = \csc^2 x$, the numerator becomes $\csc^2 x$. Using the identity $\csc^2 x - 1 = \cot^2 x$, the denominator becomes $\cot^2 x$.

    So, the expression simplifies to:

    $\frac{\csc^2 x}{\cot^2 x}$

    Step 2: Express in terms of $\sin x$ and $\cos x$. Recall that $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$.

    $\frac{\csc^2 x}{\cot^2 x} = \frac{\frac{1}{\sin^2 x}}{\frac{\cos^2 x}{\sin^2 x}} = \frac{1}{\sin^2 x} \times \frac{\sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$

    Step 3: Find $\cos x$ and solve. Given $\sin x = \frac{4}{5}$, we use the identity $\sin^2 x + \cos^2 x = 1$:

    $\left(\frac{4}{5}\right)^2 + \cos^2 x = 1$

    $\frac{16}{25} + \cos^2 x = 1$

    $\cos^2 x = 1 - \frac{16}{25} = \frac{9}{25}$

    Now, substitute $\cos^2 x$ into our simplified expression ($\frac{1}{\cos^2 x}$):

    $\frac{1}{\frac{9}{25}} = \frac{25}{9}$

    4. Identify the answer

    Comparing our result $\frac{25}{9}$ with the given options, it matches option B.

    5. Tip

    Whenever you see trigonometric expressions involving squares like $\csc^2 x$ or $\cot^2 x$, always check if you can simplify them using the Pythagorean identities ($1 + \tan^2 x = \sec^2 x$ or $1 + \cot^2 x = \csc^2 x$) before substituting the numerical values. It saves time and reduces the chance of calculation errors!

    AI can make mistakes. Report anything that looks wrong.

76. 76.

    $\frac{d}{dx} [\log (4x^3 - 2x)]$ is equal to

    1. A.$\frac{12x - 2}{4x^2}$
    2. B.$\frac{43x^2 - 2x}{7x}$
    3. C.$\frac{4x^2 - 2}{7x + 6}$
    4. D.$\frac{12x^2 - 2}{4x^3 - 2x}$

    Show answer

    Answer: D. $\frac{12x^2 - 2}{4x^3 - 2x}$

    AI explanation

    Analysis

    The question asks for the derivative of a logarithmic function with respect to $x$. Specifically, we need to apply the rules of differentiation for logarithms.

    Concept: Differentiation of Logarithmic Functions

    To differentiate a function of the form $y = \ln(u)$, where $u$ is a function of $x$, we use the Chain Rule. The formula is:

    $\frac{d}{dx} [\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx}$

    In this problem, the function is $\log(4x^3 - 2x)$. In standard calculus contexts for JAMB, $\log$ without a base specified is typically treated as the natural logarithm ($\ln$).

    Step-by-Step Solution

    Step 1: Identify $u$ Let $u = 4x^3 - 2x$.

    Step 2: Differentiate $u$ with respect to $x$ Using the power rule ($\frac{d}{dx} ax^n = anx^{n-1}$):

    $\frac{du}{dx} = \frac{d}{dx}(4x^3) - \frac{d}{dx}(2x)$

    $\frac{du}{dx} = (4 \times 3)x^{3-1} - 2$

    $\frac{du}{dx} = 12x^2 - 2$

    Step 3: Apply the Chain Rule formula Substitute $u$ and $\frac{du}{dx}$ into the formula $\frac{1}{u} \cdot \frac{du}{dx}$:

    $\frac{d}{dx} [\ln(4x^3 - 2x)] = \frac{1}{4x^3 - 2x} \cdot (12x^2 - 2)$

    Step 4: Simplify the expression

    $\frac{12x^2 - 2}{4x^3 - 2x}$

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

    Exam Tip

    Whenever you see a derivative of a logarithm, remember the "Rule of Reciprocals": The derivative is simply the derivative of the inside part divided by the inside part itself. Don't let the "log" symbol intimidate you; just focus on differentiating the expression inside the bracket and placing it over the original expression.

    AI can make mistakes. Report anything that looks wrong.

77. 77.

    In a class of 50 students, 40 students offered Physics and 30 offered Biology. How many offered both Physics and Biology?

    1. A.42
    2. B.20
    3. C.70
    4. D.54

    Show answer

    Answer: B. 20

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

78. 78.

    If $25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}$, find the value of x.

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

    Show answer

    Answer: A. x = -4

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

79. 79.

    Find the probability that a number selected at random from 21 to 34 is a multiple of 3

    1. A.$\frac{3}{11}$
    2. B.$\frac{2}{9}$
    3. C.$\frac{5}{14}$
    4. D.$\frac{5}{13}$

    Show answer

    Answer: C. $\frac{5}{14}$

    AI explanation

    1. Analyze the question

    We need to find the probability of picking a multiple of 3 from the set of integers ranging from 21 to 34 (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 all integers from 21 to 34.
    • Favorable outcomes: The count of numbers within that range that are divisible by 3.

    3. Show the solution

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

    $\text{Total} = (34 - 21) + 1$

    $\text{Total} = 13 + 1 = 14$

    There are 14 possible numbers in total.

    Step 2: Identify the multiples of 3 in the range. Let's list the numbers from 21 to 34 and identify the multiples of 3:

    • 21 (3 × 7) - Yes
    • 22
    • 23
    • 24 (3 × 8) - Yes
    • 25
    • 26
    • 27 (3 × 9) - Yes
    • 28
    • 29
    • 30 (3 × 10) - Yes
    • 31
    • 32
    • 33 (3 × 11) - Yes
    • 34

    The multiples are {21, 24, 27, 30, 33}. Counting these, we have 5 favorable outcomes.

    Step 3: Calculate the probability.

    $\text{Probability} = \frac{5}{14}$

    4. Identify the answer

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

    5. Tip

    When counting numbers in an inclusive range (e.g., "from X to Y"), always remember to add 1 to the difference $(Y - X)$. A common mistake students make is calculating $34 - 21 = 13$ and forgetting to include the starting number, which leads to an incorrect denominator!

    AI can make mistakes. Report anything that looks wrong.

80. 80.

    Solve the inequality: -7 $\leq$ 9 - 8x < 16 - x

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

    Show answer

    Answer: D. -1 < x $\leq$ 2

    AI explanation

    To solve the inequality $-7 \leq 9 - 8x < 16 - x$, we need to split it into two separate inequalities and solve them individually, then find the intersection of their solution sets.

    Step 1: Solve the first part

    $-7 \leq 9 - 8x$

    Subtract 9 from both sides: $-7 - 9 \leq -8x$ $-16 \leq -8x$

    Now, divide both sides by -8. Remember: When you divide or multiply an inequality by a negative number, the inequality sign must be reversed. $\frac{-16}{-8} \geq x$ $2 \geq x$ (or $x \leq 2$)

    Step 2: Solve the second part

    $9 - 8x < 16 - x$

    Group the $x$ terms on one side and the constants on the other. Add $8x$ to both sides: $9 < 16 - x + 8x$ $9 < 16 + 7x$

    Subtract 16 from both sides: $9 - 16 < 7x$ $-7 < 7x$

    Divide both sides by 7: $-1 < x$ (or $x > -1$)

    Step 3: Combine the results

    We have two conditions:

    1. $x \leq 2$
    2. $x > -1$

    Combining these, we get: $-1 < x \leq 2$

    Our calculated result, $-1 < x \leq 2$, matches this option perfectly.

    Exam Tip: When dealing with compound inequalities, always remember the "Golden Rule": if you multiply or divide by a negative number, flip the inequality sign immediately! It is the most common place where students lose marks in JAMB Mathematics.

    AI can make mistakes. Report anything that looks wrong.