2013 JAMB Mathematics Past Questions

Questions

1. 1.

   If log104 = 0.6021, evaluate log1041/3

   1. A.0.3011
   2. B.0.9021
   3. C.1.8063
   4. D.0.2007

   Show answer

   Answer: D. 0.2007

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

2. 2.

   In triangle PQR, q = 8 cm, r = 6 cm and cos P = $\frac{1}{12}$. Calculate the value of p.

   1. A.$\sqrt{108}$ cm
   2. B.9 cm
   3. C.$\sqrt{92}$ cm
   4. D.10 cm

   Show answer

   Answer: C. $\sqrt{92}$ cm

   AI explanation

   Analysis

   To find the side length $p$ in triangle $PQR$ when two sides ($q$ and $r$) and the included angle ($P$) are known, we use the Cosine Rule.

   Concept: The Cosine Rule

   The Cosine Rule states that for any triangle $PQR$ with sides $p, q,$ and $r$ opposite to angles $P, Q,$ and $R$ respectively:

   $p^2 = q^2 + r^2 - 2qr \cos P$

   This formula is essential for solving triangles when you have a "Side-Angle-Side" (SAS) configuration.

   Step-by-Step Solution

   1. Identify the given values:

   • $q = 8\text{ cm}$
   • $r = 6\text{ cm}$
   • $\cos P = \frac{1}{12}$

   2. Substitute the values into the Cosine Rule formula:

   $p^2 = (8)^2 + (6)^2 - 2(8)(6) \left( \frac{1}{12} \right)$

   3. Perform the arithmetic:

   • Calculate the squares: $8^2 = 64$ and $6^2 = 36$
   • Calculate the product: $2 \times 8 \times 6 = 96$
   • Multiply by $\cos P$: $96 \times \frac{1}{12} = 8$

   4. Combine the results:

   $p^2 = 64 + 36 - 8$

   $p^2 = 100 - 8$

   $p^2 = 92$

   5. Solve for $p$:

   $p = \sqrt{92}\text{ cm}$

   Conclusion

   Comparing our result to the given options, we find that $\sqrt{92}$ cm corresponds to option C.

   Exam Tip

   Always remember that the Cosine Rule is essentially a generalized version of Pythagoras' Theorem. If the angle $P$ were $90^\circ$, $\cos 90^\circ$ would be $0$, and the formula would simplify perfectly to $p^2 = q^2 + r^2$. Keeping this connection in mind helps you remember the formula during the heat of the exam!

   AI can make mistakes. Report anything that looks wrong.

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

4. 4.

   Calculate the time taken for N3000 to earn N600 if invested at 8% simple interest

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

   Show answer

   Answer: A. 2$\frac{1}{2}$ years

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

5. 5.

   Solve for x: |x - 2| < 3

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

   Show answer

   Answer: C. -1 < x < 5

   AI explanation

   Analysis

   The question asks us to solve an inequality involving an absolute value: $|x - 2| < 3$.

   Concept

   The absolute value of a number represents its distance from zero on a number line. When we have an inequality of the form $|f(x)| < a$ (where $a > 0$), it means the expression inside the absolute value must be between $-a$ and $a$.

   Mathematically, this translates to: $-a < f(x) < a$

   Solution

   Given the inequality: $|x - 2| < 3$

   Step 1: Rewrite the inequality without the absolute value bars. Based on the principle above, we set the expression between $-3$ and $3$: $-3 < x - 2 < 3$

   Step 2: Isolate $x$. To get $x$ by itself in the middle, we need to add $2$ to all three parts of the inequality: $-3 + 2 < x - 2 + 2 < 3 + 2$

   Step 3: Simplify. $-1 < x < 5$

   This means $x$ must be greater than $-1$ and less than $5$.

   Conclusion

   Comparing our result to the given options: A. $x < 5$ B. $-2 < x < 3$ C. $-1 < x < 5$ D. $x < 1$

   The correct option is C.

   Exam Tip

   Whenever you see an absolute value inequality with a "less than" sign ($<$), remember the acronym "LESS THAN = BETWEEN". This reminds you that the solution will be a range trapped between two values. If the sign were "greater than" ($>$), the solution would be "OR" (pointing outwards away from the center).

   AI can make mistakes. Report anything that looks wrong.

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

7. 7.

   Find the inverse $\begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}$

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

   Show answer

   Answer: B. $\begin{vmatrix} 2 & -\frac{3}{2} \\ -3 & \frac{5}{2} \end{vmatrix}$

   AI explanation

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

   1. The Concept

   The inverse of matrix $A$, denoted as $A^{-1}$, is given by:

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

   Where $\det(A)$ (the determinant) is calculated as $(ad - bc)$.

   2. Step-by-Step Solution

   Given the matrix $A = \begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}$:

   Step 1: Identify the elements $a = 5, b = 3, c = 6, d = 4$

   Step 2: Calculate the determinant $\det(A) = (ad) - (bc)$ $\det(A) = (5 \times 4) - (3 \times 6)$ $\det(A) = 20 - 18 = 2$

   Step 3: Apply the inverse formula Swap the positions of $a$ and $d$, and change the signs of $b$ and $c$:

   $A^{-1} = \frac{1}{2} \begin{pmatrix} 4 & -3 \\ -6 & 5 \end{pmatrix}$

   Step 4: Multiply by the scalar $\frac{1}{2}$

   $A^{-1} = \begin{pmatrix} \frac{4}{2} & -\frac{3}{2} \\ -\frac{6}{2} & \frac{5}{2} \end{pmatrix}$

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

   3. Conclusion

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

   Exam Tip: Always remember the "swap and negate" rule for $2 \times 2$ matrices: swap the main diagonal elements ($a$ and $d$) and change the signs of the off-diagonal elements ($b$ and $c$). Don't forget to divide the final matrix by the determinant!

   AI can make mistakes. Report anything that looks wrong.

8. 8.

   What is the probability that an integer x $(1 \leq x \leq 25)$ chosen at random is divisible by both 2 and 3?

   1. A.$\frac{1}{25}$
   2. B.$\frac{1}{5}$
   3. C.$\frac{4}{25}$
   4. D.$\frac{3}{4}$

   Show answer

   Answer: C. $\frac{4}{25}$

   AI explanation

   1. Analyze the question

   We are looking for the probability that a randomly selected integer $x$ from the set $\{1, 2, 3, \dots, 25\}$ is divisible by both 2 and 3.

   2. Explain the concept

   • Divisibility Rule: If a number is divisible by both 2 and 3, it must be divisible by their Least Common Multiple (LCM). The LCM of 2 and 3 is $2 \times 3 = 6$. Therefore, we are looking for multiples of 6 within the range 1 to 25.
   • Probability Formula: The probability of an event is given by: $$\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

   3. Show the solution

   • Step 1: Identify the total number of outcomes. The integers are from 1 to 25, so there are 25 possible outcomes.
   • Step 2: Identify the favorable outcomes. We need to list the multiples of 6 that fall between 1 and 25: • $6 \times 1 = 6$ • $6 \times 2 = 12$ • $6 \times 3 = 18$ • $6 \times 4 = 24$ • ($6 \times 5 = 30$, which is greater than 25, so we stop here). The favorable outcomes are $\{6, 12, 18, 24\}$. There are 4 favorable outcomes.
   • Step 3: Calculate the probability. $$\text{Probability} = \frac{4}{25}$$

   4. Identify the answer

   Comparing our result to the given options: A. $\frac{1}{25}$ B. $\frac{1}{5}$ C. $\frac{4}{25}$ D. $\frac{3}{4}$

   The correct option is C.

   5. Tip

   When a question asks for divisibility by two different prime numbers (like 2 and 3), always find the LCM first. If the numbers were not prime (e.g., 4 and 6), the LCM would be 12, not 24. Always remember: "Divisible by A and B" means "Divisible by LCM(A, B)."

   AI can make mistakes. Report anything that looks wrong.

9. 9.

   Integrate $\frac{1 + x}{x^{3}} \mathrm d x$

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

   Show answer

   Answer: B. $-\frac{1}{2x^{2}} - \frac{1}{x} + k$

   AI explanation

   Analysis

   To solve this problem, we need to integrate the function $\int \frac{1 + x}{x^3} \, dx$. The key strategy here is to simplify the integrand by splitting the fraction before applying the power rule for integration.

   Concept: The Power Rule for Integration

   The power rule states that:

   $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (\text{where } n \neq -1)$

   Step-by-Step Solution

   Step 1: Split the fraction Divide each term in the numerator by the denominator ($x^3$):

   $\int \left( \frac{1}{x^3} + \frac{x}{x^3} \right) \, dx$

   Step 2: Simplify using indices Rewrite the terms using negative exponents:

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

   Step 3: Integrate term by term Apply the power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ to each term:

   • For $x^{-3}$: $$\int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} = \frac{x^{-2}}{-2} = -\frac{1}{2x^2}$$

   • For $x^{-2}$: $$\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = \frac{x^{-1}}{-1} = -\frac{1}{x}$$

   Step 4: Combine the results Adding the constant of integration ($k$):

   $-\frac{1}{2x^2} - \frac{1}{x} + k$

   Conclusion

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

   Exam Tip

   When you see a fraction with a single term in the denominator (like $x^3$), always try to split it into separate fractions first. It makes the integration much simpler and prevents errors associated with the quotient rule (which doesn't apply to integration).

   AI can make mistakes. Report anything that looks wrong.

10. 10.

    In how many ways can 3 seats be occupied if 5 people are willing to sit?

    1. A.60
    2. B.20
    3. C.5
    4. D.120

    Show answer

    Answer: A. 60

    AI explanation

    Analysis

    The question asks for the number of ways to arrange 3 people out of a group of 5 into 3 specific seats. Since the order in which the people sit matters (sitting in seat A is different from sitting in seat B), this is a permutation problem.

    Concept

    When we need to select and arrange $r$ items from a total of $n$ items, we use the permutation formula:

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

    Where:

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

    Solution

    1. Identify the values: $n = 5$, $r = 3$.
    2. Apply the formula: $$^5P_3 = \frac{5!}{(5 - 3)!}$$
    3. Simplify the expression: $$^5P_3 = \frac{5!}{2!}$$
    4. Expand the factorials: $$^5P_3 = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}$$
    5. Calculate: The $(2 \times 1)$ in the numerator and denominator cancel out: $$^5P_3 = 5 \times 4 \times 3 = 60$$

    There are 60 different ways the 3 seats can be occupied.

    Exam Tip

    Whenever you see a problem involving "arranging" or "seating" people where order matters, always think Permutation. If the order did not matter (like choosing a committee), you would use Combination ($^nC_r$). A quick trick for $^nP_r$ is to just multiply $r$ numbers starting from $n$ downwards: for $^5P_3$, just multiply $5 \times 4 \times 3 = 60$.

    AI can make mistakes. Report anything that looks wrong.

11. 11.

    The radius of a circle is increasing at the rate of 0.02cms-1. Find the rate at which the area is increasing when the radius of the circle is 7cm.

    1. A.0.75cm2S-1
    2. B.0.53cm2S-1
    3. C.0.35cm2S-1
    4. D.0.88cm2S-1

    Show answer

    Answer: D. 0.88cm2S-1

    AI explanation

    Analysis

    This is a problem involving Related Rates in Calculus. We are given the rate of change of the radius with respect to time ($\frac{dr}{dt}$) and asked to find the rate of change of the area with respect to time ($\frac{dA}{dt}$) at a specific radius ($r = 7\text{cm}$).

    Concept

    1. Formula for the area of a circle: $A = \pi r^2$
    2. Chain Rule: Since $A$ is a function of $r$, and $r$ is a function of $t$, we differentiate $A$ with respect to $t$: $$\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$$

    Step-by-Step Solution

    Step 1: Identify the given values.

    • Rate of change of radius: $\frac{dr}{dt} = 0.02 \, \text{cm/s}$
    • Radius at the specific moment: $r = 7 \, \text{cm}$
    • Take $\pi \approx \frac{22}{7}$ (standard for JAMB calculations involving multiples of 7).

    Step 2: Differentiate the area formula. Given $A = \pi r^2$, differentiate with respect to $r$:

    $\frac{dA}{dr} = 2\pi r$

    Step 3: Apply the Chain Rule.

    $\frac{dA}{dt} = (2\pi r) \times \frac{dr}{dt}$

    Step 4: Substitute the values.

    $\frac{dA}{dt} = 2 \times \frac{22}{7} \times 7 \times 0.02$

    The $7$ in the numerator and denominator cancels out:

    $\frac{dA}{dt} = 2 \times 22 \times 0.02$

    $\frac{dA}{dt} = 44 \times 0.02$

    $\frac{dA}{dt} = 0.88 \, \text{cm}^2/\text{s}$

    Conclusion

    Comparing our result to the options provided, the calculated rate of increase is $0.88 \, \text{cm}^2\text{s}^{-1}$. This matches option D.

    Exam Tip

    Whenever you see "rate of change" in a JAMB question, think Calculus. If you are given a geometric shape, write down the formula for its area or volume first, then differentiate it with respect to time ($t$) using the chain rule. Always look for ways to cancel out $\pi$ or fractions like $7$ to make your mental math faster!

    AI can make mistakes. Report anything that looks wrong.

12. 12.

    If the variance of 3+x, 6, 4, x and 7-x is 4 and the mean is 5, find the standard deviation

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

    Show answer

    Answer: B. 2

    AI explanation

    Analysis

    To solve this problem, we need to understand the relationship between variance and standard deviation. In statistics, the standard deviation is simply the square root of the variance.

    Concept

    • Variance ($\sigma^2$): A measure of how spread out a set of numbers is from their mean.
    • Standard Deviation ($\sigma$): The square root of the variance ($\sigma = \sqrt{\text{Variance}}$).

    The question provides the variance directly as 4. Even though the question gives us the data set and the mean, we do not actually need to perform complex calculations on the variables $x$ because the definition of standard deviation is independent of the specific values once the variance is known.

    Step-by-Step Solution

    1. Identify the given information: • Variance ($\sigma^2$) = 4 • Mean = 5 (Note: This information is extra and not required to find the standard deviation since the variance is already provided).

    2. Apply the formula: • Standard Deviation ($\sigma$) = $\sqrt{\text{Variance}}$ • $\sigma = \sqrt{4}$

    3. Calculate: • $\sigma = 2$

    Verification

    The extra information provided in the question (the data set and the mean) is often included in JAMB questions to test if a student can identify the necessary information versus "distractor" information.

    Exam Tip

    Always check if the question asks for the variance or the standard deviation. If you are given the variance and asked for the standard deviation, just take the square root. If you are given the standard deviation and asked for the variance, just square it. Don't waste time calculating the mean or the variance of the variables if the variance is already explicitly stated!

    AI can make mistakes. Report anything that looks wrong.

13. 13.

    If $\tan \theta = \frac{3}{4}$, find the value of $\sin \theta + \cos \theta$.

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

    Show answer

    Answer: D. $1\frac{2}{5}$

    AI explanation

    1. Analyze the question

    We are given the value of $\tan \theta$ as a fraction and asked to find the sum of $\sin \theta$ and $\cos \theta$. To solve this, we need to determine the lengths of all three sides of a right-angled triangle using the given trigonometric ratio.

    2. Explain the concept

    In a right-angled triangle, trigonometric ratios are defined using the mnemonic SOH CAH TOA:

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

    Given $\tan \theta = \frac{3}{4}$, we identify:

    • Opposite side = 3
    • Adjacent side = 4

    We use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the Hypotenuse ($h$): $h^2 = 3^2 + 4^2$

    3. Show the solution

    Step 1: Find the Hypotenuse $h^2 = 9 + 16$ $h^2 = 25$ $h = \sqrt{25} = 5$

    Step 2: Determine $\sin \theta$ and $\cos \theta$

    • $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}$
    • $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}$

    Step 3: Calculate the sum $\sin \theta + \cos \theta = \frac{3}{5} + \frac{4}{5}$ $\sin \theta + \cos \theta = \frac{7}{5}$

    Step 4: Convert to a mixed fraction $\frac{7}{5} = 1 \frac{2}{5}$

    4. Identify the answer

    Comparing our result to the given options: A. $1\frac{1}{3}$ B. $1\frac{2}{3}$ C. $1\frac{3}{5}$ D. $1\frac{2}{5}$

    The calculated value matches option D.

    5. Tip

    Whenever you see $\tan \theta = \frac{3}{4}$, immediately recognize the 3-4-5 Pythagorean triple. This is a common triangle in JAMB questions; knowing these triples (like 5-12-13 or 8-15-17) saves you time during the exam because you won't have to calculate the hypotenuse from scratch!

    AI can make mistakes. Report anything that looks wrong.

14. 14.

    If the midpoint of the line PQ is (2,3) and the point P is (-2, 1), find the coordinate of the point Q.

    1. A.(8,6)
    2. B.(5,6)
    3. C.(0,4)
    4. D.(6,5)

    Show answer

    Answer: D. (6,5)

    AI explanation

    Analysis

    To find the coordinates of point Q, we use the Midpoint Formula. The midpoint $M(x, y)$ of a line segment connecting two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ is calculated as:

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

    Concept

    We are given the midpoint $M(2, 3)$ and one endpoint $P(-2, 1)$. We need to find the coordinates of the other endpoint $Q(x_2, y_2)$. We can set up two separate equations—one for the x-coordinate and one for the y-coordinate—and solve for $x_2$ and $y_2$.

    Step-by-Step Solution

    1. Set up the equation for the x-coordinate:

    $\frac{x_1 + x_2}{2} = x_m$

    $\frac{-2 + x_2}{2} = 2$

    Multiply both sides by 2:

    $-2 + x_2 = 4$

    Add 2 to both sides:

    $x_2 = 4 + 2 = 6$

    2. Set up the equation for the y-coordinate:

    $\frac{y_1 + y_2}{2} = y_m$

    $\frac{1 + y_2}{2} = 3$

    Multiply both sides by 2:

    $1 + y_2 = 6$

    Subtract 1 from both sides:

    $y_2 = 6 - 1 = 5$

    3. Combine the coordinates: The coordinates of point Q are $(6, 5)$.

    Conclusion

    Comparing our result $(6, 5)$ with the given options, it matches option D.

    Exam Tip

    When dealing with midpoint problems, you can use a "jump" method: look at how much the x-coordinate changed from P to the midpoint, and apply that same change again to get to Q.

    • From $x = -2$ to $x = 2$ is a jump of $+4$. Adding $+4$ to $2$ gives $6$.
    • From $y = 1$ to $y = 3$ is a jump of $+2$. Adding $+2$ to $3$ gives $5$. This is a quick way to verify your answer in the exam hall!

    AI can make mistakes. Report anything that looks wrong.

15. 15.

    Evaluate $\frac{1.25 \times 0.025}{0.05}$, correct to 1 decimal place

    1. A.0.6
    2. B.6.2
    3. C.6.3
    4. D.0.5

    Show answer

    Answer: A. 0.6

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

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

17. 17.

    Evaluate 3(x + 2) > 6(x + 3)

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

    Show answer

    Answer: C. x < -4

    AI explanation

    Analysis

    We are asked to solve a linear inequality. The goal is to isolate the variable $x$ on one side of the inequality sign to determine its range of values.

    Concept

    When solving inequalities, the rules are very similar to solving equations, with one critical exception: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

    Step-by-Step Solution

    Step 1: Expand the brackets Distribute the numbers outside the parentheses: $3(x) + 3(2) > 6(x) + 6(3)$ $3x + 6 > 6x + 18$

    Step 2: Group the like terms Move the $x$ terms to one side and the constant numbers to the other. Let's move $3x$ to the right and $18$ to the left: $6 - 18 > 6x - 3x$

    Step 3: Simplify both sides $-12 > 3x$

    Step 4: Solve for $x$ Divide both sides by $3$: $\frac{-12}{3} > \frac{3x}{3}$ $-4 > x$

    This can also be written as: $x < -4$

    Conclusion

    By simplifying the inequality, we found that $x$ must be less than $-4$. This matches option C.

    Exam Tip

    When you have an inequality like $a > b$, it is exactly the same as $b < a$. If you ever feel confused about which way the sign should face, read it out loud: "$-4$ is greater than $x$" is the same as saying "$x$ is less than $-4$." Always double-check your sign direction after dividing by a negative number!

    AI can make mistakes. Report anything that looks wrong.

18. 18.

    Evaluate $\int_{0} ^{\frac{\pi}{2}} \sin x \mathrm d x$

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

    Show answer

    Answer: C. 1

    AI explanation

    1. Analyze the question

    We are asked to evaluate the definite integral of the function $\sin x$ with respect to $x$, within the limits of $0$ to $\frac{\pi}{2}$.

    2. Explain the concept

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

    $\int_{a}^{b} f(x) \, dx = [F(x)]_{a}^{b} = F(b) - F(a)$

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

    3. Show the solution

    Step 1: Find the integral. The integral of $\sin x$ is $-\cos x$.

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

    Step 2: Apply the limits $0$ and $\frac{\pi}{2}$.

    $\int_{0}^{\frac{\pi}{2}} \sin x \, dx = [-\cos x]_{0}^{\frac{\pi}{2}}$

    Step 3: Substitute the upper and lower limits.

    $= (-\cos(\frac{\pi}{2})) - (-\cos(0))$

    Step 4: Evaluate the trigonometric values. Recall that:

    • $\cos(\frac{\pi}{2}) = 0$
    • $\cos(0) = 1$

    Substitute these values into the expression:

    $= (-0) - (-1)$

    $= 0 + 1$

    $= 1$

    4. Identify the answer

    The result of the integration is $1$. Comparing this to the given options: A. -2 B. 2 C. 1 D. -1

    The correct option is C.

    5. Tip

    When dealing with trigonometric integrals, always remember the "negative sign" rule: the derivative of $\cos x$ is $-\sin x$, so the integral of $\sin x$ must be $-\cos x$. Don't forget to distribute the negative sign when subtracting the lower limit!

    AI can make mistakes. Report anything that looks wrong.

19. 19.

    If a binary operation * is defined by x * y = x + 2y, find 2 * (3 * 4)

    1. A.24
    2. B.16
    3. C.14
    4. D.26

    Show answer

    Answer: A. 24

    AI explanation

    Analysis

    To solve this problem, we need to apply the rules of binary operations. A binary operation is simply a rule that tells you how to combine two numbers. Here, the rule is $x * y = x + 2y$. This means you take the first number ($x$) and add it to twice the second number ($y$).

    Concept

    When you see an expression with brackets like $2 * (3 * 4)$, you must follow the BODMAS rule. You must solve the operation inside the parentheses (brackets) first before proceeding to the outer operation.

    Step-by-Step Solution

    Step 1: Solve the operation inside the brackets $(3 * 4)$ Using the rule $x * y = x + 2y$:

    • Here, $x = 3$ and $y = 4$.
    • $3 * 4 = 3 + 2(4)$
    • $3 * 4 = 3 + 8$
    • $3 * 4 = 11$

    Step 2: Substitute the result back into the original expression Now the expression $2 * (3 * 4)$ becomes $2 * 11$.

    Step 3: Solve the final operation $2 * 11$ Using the same rule $x * y = x + 2y$:

    • Here, $x = 2$ and $y = 11$.
    • $2 * 11 = 2 + 2(11)$
    • $2 * 11 = 2 + 22$
    • $2 * 11 = 24$

    Conclusion

    Following the order of operations, we calculated the value inside the bracket first, then combined it with the remaining number to arrive at 24.

    The correct option is A.

    Exam Tip

    In binary operation questions, always pay close attention to the order of the variables. If the formula was $x * y = 2x + y$, the answer would be different! Always write down your $x$ and $y$ values clearly before substituting them into the formula to avoid "sign" or "position" errors.

    AI can make mistakes. Report anything that looks wrong.

20. 20.

    P, Q and R are subsets of the universal set U. The Venn diagram showing the relationship $(P \cap Q) \cup R$ is

    

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

    Show answer

    Answer: D. 4

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

21. 21.

    The mean of seven numbers is 10. If six of the numbers are 2, 4, 8, 14, 16 and 18, find the mode.

    1. A.6
    2. B.8
    3. C.14
    4. D.2

    Show answer

    Answer: B. 8

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

22. 22.

    A chord of a circle subtends an angle of 120° at the centre of a circle of diameter $4\sqrt{3} cm$. Calculate the area of the major sector.

    1. A.32$\pi$ cm$^2$
    2. B.4$\pi$ cm$^2$
    3. C.8$\pi$ cm$^2$
    4. D.16$\pi$ cm$^2$

    Show answer

    Answer: C. 8$\pi$ cm$^2$

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

23. 23.

    Find the length of a chord which subtends an angle of 90° at the centre of a circle whose radius is 8 cm.

    1. A.$8\sqrt{3}$ cm
    2. B.4 cm
    3. C.8 cm
    4. D.$8\sqrt{2}$ cm

    Show answer

    Answer: D. $8\sqrt{2}$ cm

    AI explanation

    1. Analyze the question

    We are given a circle with a radius ($r$) of 8 cm. A chord is drawn such that it subtends an angle ($\theta$) of 90° at the center of the circle. We need to find the length of this chord.

    2. Explain the concept

    When a chord subtends an angle at the center of a circle, it forms an isosceles triangle with the two radii connecting the center to the endpoints of the chord.

    • The two sides of the triangle are the radii ($r = 8$ cm).
    • The angle between these two sides is 90°.
    • The chord acts as the base of this triangle.

    Since the angle is 90°, we have a right-angled isosceles triangle. We can find the length of the chord ($L$) using the Pythagorean theorem ($a^2 + b^2 = c^2$) or the trigonometric ratio for sine.

    3. Show the solution

    Method 1: Pythagorean Theorem In a right-angled triangle where the two legs are the radii ($r = 8$ cm): $L^2 = r^2 + r^2$ $L^2 = 8^2 + 8^2$ $L^2 = 64 + 64$ $L^2 = 128$ $L = \sqrt{128}$ $L = \sqrt{64 \times 2}$ $L = 8\sqrt{2}$ cm

    Method 2: Trigonometry If we draw a line from the center to bisect the 90° angle, we create two smaller right-angled triangles with an angle of 45°. $\sin(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}$ The chord is split into two equal parts, let's call each half $x$. $\sin(45^\circ) = \frac{x}{8}$ $\frac{1}{\sqrt{2}} = \frac{x}{8}$ $x = \frac{8}{\sqrt{2}}$ Rationalizing the denominator: $x = \frac{8\sqrt{2}}{2} = 4\sqrt{2}$ Since the full chord is $2x$: $L = 2 \times 4\sqrt{2} = 8\sqrt{2}$ cm

    4. Identify the answer

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

    5. Exam Tip

    For any chord subtending an angle $\theta$ at the center, the length of the chord is given by the formula: $L = 2r \sin(\frac{\theta}{2})$ Using this for 90°: $L = 2(8) \sin(45^\circ) = 16 \times \frac{\sqrt{2}}{2} = 8\sqrt{2}$. This formula works for any angle!

    AI can make mistakes. Report anything that looks wrong.

24. 24.

    If P = $\begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}$ and Q = $\begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$, find 2P + Q

    1. A.$\begin{vmatrix} 7 & 7 \\ 14 & 8 \end{vmatrix}$
    2. B.$\begin{vmatrix} 14 & 8 \\ 7 & 7 \end{vmatrix}$
    3. C.$\begin{vmatrix} 7 & 7 \\ 8 & 14 \end{vmatrix}$
    4. D.$\begin{vmatrix} 8 & 14 \\ 7 & 7 \end{vmatrix}$

    Show answer

    Answer: B. $\begin{vmatrix} 14 & 8 \\ 7 & 7 \end{vmatrix}$

    AI explanation

    To solve the expression 2P + Q, we need to perform two operations: scalar multiplication and matrix addition.

    1. Analyze the question

    We are given two $2 \times 2$ matrices: $P = \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}$ and $Q = \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$ We need to find the result of $2P + Q$.

    2. Explain the concept

    • Scalar Multiplication: To multiply a matrix by a scalar (in this case, 2), you multiply every individual element inside the matrix by that number.
    • Matrix Addition: To add two matrices of the same order, you add the corresponding elements (the element in the first row/first column of the first matrix is added to the element in the first row/first column of the second matrix, and so on).

    3. Show the solution

    Step 1: Calculate 2P Multiply every element in matrix P by 2: $2P = 2 \times \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 2 \times 5 & 2 \times 3 \\ 2 \times 2 & 2 \times 1 \end{pmatrix} = \begin{pmatrix} 10 & 6 \\ 4 & 2 \end{pmatrix}$

    Step 2: Add Q to the result Now, add matrix Q to the matrix we just calculated: $2P + Q = \begin{pmatrix} 10 & 6 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$

    Perform the addition element-wise: $= \begin{pmatrix} 10 + 4 & 6 + 2 \\ 4 + 3 & 2 + 5 \end{pmatrix}$ $= \begin{pmatrix} 14 & 8 \\ 7 & 7 \end{pmatrix}$

    4. Identify the answer

    Comparing our result $\begin{pmatrix} 14 & 8 \\ 7 & 7 \end{pmatrix}$ with the given options, it matches option B.

    5. Exam Tip

    When dealing with matrix operations in JAMB, always double-check your arithmetic for the individual elements. A common mistake is to add the matrices before multiplying by the scalar. Remember the BODMAS rule: perform the multiplication (scalar) before the addition. Also, ensure you don't confuse the row and column positions when adding!

    AI can make mistakes. Report anything that looks wrong.

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

26. 26.

    Convert 2710 to another number in base three

    1. A.10013
    2. B.10103
    3. C.11003
    4. D.10003

    Show answer

    Answer: D. 10003

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

27. 27.

    A basket contains 9 apples, 8 bananas and 7 oranges. A fruit is picked from the basket, find the probability that it is neither an apple nor an orange.

    1. A.$\frac{3}{8}$
    2. B.$\frac{1}{3}$
    3. C.$\frac{7}{24}$
    4. D.$\frac{2}{3}$

    Show answer

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

    AI explanation

    1. Analyze the question

    We are given a basket with a specific number of fruits. We need to find the probability of picking a fruit that is neither an apple nor an orange. This means we are looking for the probability of picking a banana.

    2. Explain the concept

    Probability is defined as:

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

    In this case:

    • Total outcomes = Total number of fruits in the basket.
    • Required outcomes = Number of fruits that are neither apples nor oranges (which, in this basket, are the bananas).

    3. Show the solution

    Step 1: Calculate the total number of fruits. Total = (Number of apples) + (Number of bananas) + (Number of oranges) Total = 9 + 8 + 7 = 24

    Step 2: Identify the "required" fruits. The question asks for fruits that are neither apples nor oranges. Since the only other fruit in the basket is the banana, the number of required fruits is the number of bananas. Number of bananas = 8

    Step 3: Calculate the probability.

    $P(\text{neither apple nor orange}) = P(\text{banana})$

    $P = \frac{8}{24}$

    Step 4: Simplify the fraction. Divide both the numerator and the denominator by their greatest common divisor, which is 8:

    $\frac{8 \div 8}{24 \div 8} = \frac{1}{3}$

    4. Identify the answer

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

    5. Tip

    When a question asks for "neither X nor Y," it is often easier to identify the remaining category (in this case, bananas) rather than calculating the probability of X and Y separately and subtracting from 1. Always check if the "neither/nor" condition leaves you with a single category!

    AI can make mistakes. Report anything that looks wrong.

28. 28.

    The locus of the points which is equidistant from the line PQ forms a

    1. A.perpendicular line to PQ
    2. B.circle centre P
    3. C.circle centre Q
    4. D.pair of parallel lines to PQ

    Show answer

    Answer: D. pair of parallel lines to PQ

    AI explanation

    Analysis

    The question asks for the "locus of points" that maintain a constant distance from a given straight line $PQ$. In geometry, a locus is a set of all points that satisfy a specific condition or rule.

    Concept

    When we talk about points being "equidistant" from a line, we are looking for the path traced by a point that stays at a fixed perpendicular distance ($d$) from that line.

    1. If you move along one side of the line $PQ$ at a fixed distance $d$, you trace a straight line parallel to $PQ$.
    2. Because the condition does not specify "on one side only," we must consider both sides of the line.
    3. Therefore, there is a parallel line at distance $d$ above $PQ$ and another parallel line at distance $d$ below $PQ$.

    Solution

    • Step 1: Visualize the line $PQ$ on a Cartesian plane.
    • Step 2: Identify all points at a distance $d$ from $PQ$. These points form a line parallel to $PQ$ on the top side and another line parallel to $PQ$ on the bottom side.
    • Step 3: Since the locus includes all such points, it consists of two lines, one on each side of $PQ$, both running parallel to the original line.
    • Step 4: This collection of points is described as a "pair of parallel lines."

    Conclusion

    The locus of points equidistant from a fixed line is a pair of lines parallel to the original line.

    Exam Tip: Always remember that a locus equidistant from a point is a circle, but a locus equidistant from a line is a pair of parallel lines. Do not confuse the two!

    AI can make mistakes. Report anything that looks wrong.

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

30. 30.

    If r varies inversely as the square root of s and t, how does s vary with r and t?

    1. A.s varies inversely as r and t2
    2. B.s varies inverely as r2 and t
    3. C.s varies directly as r2 and t2
    4. D.s varies directly as r and t

    Show answer

    Answer: B. s varies inverely as r2 and t

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

31. 31.

    Find the mean of t + 2, 2t - 4, 3t + 2 and 2t.

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

    Show answer

    Answer: B. 2t

    AI explanation

    Analysis

    To find the mean (average) of a set of values, you must sum all the individual terms and then divide that sum by the total number of terms provided.

    Concept

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

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

    Where:

    • $\sum x$ is the sum of all the terms.
    • $n$ is the number of terms.

    Step-by-Step Solution

    Step 1: Identify the terms and the count ($n$). The terms are: $(t + 2)$, $(2t - 4)$, $(3t + 2)$, and $(2t)$. There are $n = 4$ terms.

    Step 2: Calculate the sum of the terms. Sum $= (t + 2) + (2t - 4) + (3t + 2) + (2t)$

    Group the like terms (the $t$ terms together and the constants together): Sum $= (t + 2t + 3t + 2t) + (2 - 4 + 2)$ Sum $= (8t) + (0)$ Sum $= 8t$

    Step 3: Divide the sum by the number of terms ($n$). Mean $= \frac{8t}{4}$ Mean $= 2t$

    Conclusion

    Comparing our result to the given options, $2t$ corresponds to option B.

    Exam Tip

    When dealing with algebraic expressions in statistics, always be careful with the signs (positive and negative) when grouping terms. In this case, $(2 - 4 + 2)$ equals zero, which simplified the expression significantly. Always check your arithmetic at this stage to avoid silly mistakes!

    AI can make mistakes. Report anything that looks wrong.

32. 32.

    If x - 4 is a factor of x2 - x - k, then k is

    1. A.4
    2. B.12
    3. C.20
    4. D.2

    Show answer

    Answer: B. 12

    AI explanation

    Analysis

    To solve this problem, we use the Factor Theorem. The Factor Theorem states that if $(x - a)$ is a factor of a polynomial $P(x)$, then $P(a) = 0$.

    Concept

    In this question, we are given:

    • The polynomial $P(x) = x^2 - x - k$
    • The factor $(x - 4)$

    According to the Factor Theorem, if $(x - 4)$ is a factor, then substituting $x = 4$ into the polynomial must result in zero ($P(4) = 0$).

    Step-by-Step Solution

    1. Set up the equation: Since $(x - 4)$ is a factor, we set $x - 4 = 0$, which gives $x = 4$.

    2. Substitute $x = 4$ into the polynomial: $P(4) = (4)^2 - (4) - k = 0$

    3. Simplify the expression: $16 - 4 - k = 0$ $12 - k = 0$

    4. Solve for $k$: $12 = k$ Therefore, $k = 12$.

    Conclusion

    By substituting the root of the factor into the quadratic expression, we found that $k$ must be 12 for the expression to be divisible by $(x - 4)$.

    Exam Tip: Whenever you see "is a factor of," immediately think of the Factor Theorem. Just set the factor to zero to find the value of $x$, plug it into the equation, and solve for the unknown variable. It saves you the stress of long division!

    AI can make mistakes. Report anything that looks wrong.

33. 33.

    If y = (2x + 2)$^3$, find $\frac{\delta y}{\delta x}$

    1. A.3(2x +2)2
    2. B.6(2x +2)
    3. C.3(2x +2)
    4. D.6(2x +2)2

    Show answer

    Answer: D. 6(2x +2)2

    AI explanation

    To solve for the derivative of $y = (2x + 2)^3$ with respect to $x$, we use the Chain Rule in calculus.

    1. Analyze the question

    We are given a composite function $y = u^3$, where $u = 2x + 2$. We need to find $\frac{dy}{dx}$.

    2. Explain the concept

    The Chain Rule states that if $y = f(u)$ and $u = g(x)$, then:

    $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$

    In simpler terms, you differentiate the "outside" function while keeping the "inside" the same, and then multiply it by the derivative of the "inside" function.

    3. Show the solution

    Step 1: Identify the parts

    • Let $u = 2x + 2$
    • Then $y = u^3$

    Step 2: Differentiate the parts

    • Differentiate $y$ with respect to $u$: $\frac{dy}{du} = 3u^2$
    • Differentiate $u$ with respect to $x$: $\frac{du}{dx} = 2$ (since the derivative of $2x$ is $2$ and the derivative of $2$ is $0$)

    Step 3: Apply the Chain Rule

    $\frac{dy}{dx} = 3u^2 \times 2$

    $\frac{dy}{dx} = 6u^2$

    Step 4: Substitute $u$ back into the equation Since $u = 2x + 2$:

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

    4. Identify the answer

    Comparing our result to the given options: A. 3(2x + 2)² B. 6(2x + 2) C. 3(2x + 2) D. 6(2x + 2)²

    The correct option is D.

    5. Tip

    When you see a function raised to a power, like $(ax + b)^n$, always remember the "Power-Chain" shortcut: Bring the power down, subtract one from the power, and multiply by the derivative of the inside.

    • Bring down $3 \rightarrow 3(2x + 2)^2$
    • Multiply by derivative of $(2x+2)$, which is $2 \rightarrow 3 \times 2(2x + 2)^2 = 6(2x + 2)^2$.

    AI can make mistakes. Report anything that looks wrong.

34. 34.

    The remainder when 6p3 - p2 - 47p + 30 is divided by p - 3 is

    1. A.21
    2. B.42
    3. C.63
    4. D.18

    Show answer

    Answer: B. 42

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

35. 35.

    Simplify $\frac{\sqrt{5}(\sqrt{147} - \sqrt{12}}{\sqrt{15}}$

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

    Show answer

    Answer: A. 5

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

36. 36.

    if P = {x:x is odd, $-1 < x \leq 20$} and Q is {y:y is prime, $-2 < y \leq 25$, find P $\cap$ Q

    1. A.{3,5,7,11,17,19}
    2. B.{3,5,11,13,17,19}
    3. C.{3,5,7,11,13,17,19}
    4. D.{2,3,5,7,11,13,17,19}

    Show answer

    Answer: C. {3,5,7,11,13,17,19}

    AI explanation

    1. Analyze the Question

    We are given two sets, $P$ and $Q$, defined by specific conditions. We need to find the intersection of these two sets ($P \cap Q$), which consists of all elements that are present in both sets.

    2. Explain the Concept

    • Set P: Contains odd numbers $x$ such that $-1 < x \leq 20$.
    • Set Q: Contains prime numbers $y$ such that $-2 < y \leq 25$.
    • Intersection ($P \cap Q$): This means we are looking for numbers that are both odd and prime within the specified range ($-1 < x \leq 20$).

    3. Show the Solution

    Step 1: List the elements of Set P Set $P$ contains odd numbers greater than $-1$ and less than or equal to $20$. $P = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$

    Step 2: List the elements of Set Q Set $Q$ contains prime numbers greater than $-2$ and less than or equal to $25$. $Q = \{2, 3, 5, 7, 11, 13, 17, 19, 23\}$

    Step 3: Find the intersection ($P \cap Q$) We look for numbers that appear in both lists. Note that $2$ is prime but not odd, so it is excluded.

    • Is 1 in Q? No (1 is not a prime number).
    • Is 3 in both? Yes.
    • Is 5 in both? Yes.
    • Is 7 in both? Yes.
    • Is 9 in both? No (9 is not prime).
    • Is 11 in both? Yes.
    • Is 13 in both? Yes.
    • Is 15 in both? No (15 is not prime).
    • Is 17 in both? Yes.
    • Is 19 in both? Yes.

    Combining these, we get: $\{3, 5, 7, 11, 13, 17, 19\}$

    4. Identify the Answer

    Comparing our result with the given options, we see that our set matches option C.

    5. Exam Tip

    Always remember that 1 is not a prime number. Many students mistakenly include 1 in sets of prime numbers, which leads to errors in questions involving set theory or probability. Also, remember that 2 is the only even prime number; all other prime numbers are odd.

    AI can make mistakes. Report anything that looks wrong.

37. 37.

    Find the equation of the perpendicular bisector of the line joining P(2, -3) to Q(-5, 1)

    1. A.8y + 14x + 13 = 0
    2. B.8y - 14x + 13 = 0
    3. C.8y - 14x - 13 = 0
    4. D.8y + 14x - 13 = 0

    Show answer

    Answer: C. 8y - 14x - 13 = 0

    AI explanation

    To find the equation of the perpendicular bisector of the line segment joining $P(2, -3)$ and $Q(-5, 1)$, we need to follow two main steps: find the midpoint of the line segment and find the negative reciprocal of the slope of the line segment.

    1. Find the Midpoint ($M$)

    The perpendicular bisector passes through the midpoint of the line $PQ$. The formula for the midpoint is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

    Using $P(2, -3)$ and $Q(-5, 1)$: $M = \left( \frac{2 + (-5)}{2}, \frac{-3 + 1}{2} \right) = \left( \frac{-3}{2}, \frac{-2}{2} \right) = \left( -1.5, -1 \right)$

    2. Find the Slope of the Perpendicular Bisector ($m_\perp$)

    First, find the slope of the line $PQ$ ($m_{PQ}$): $m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{-5 - 2} = \frac{4}{-7} = -\frac{4}{7}$

    The slope of the perpendicular bisector ($m_\perp$) is the negative reciprocal of $m_{PQ}$: $m_\perp = -\frac{1}{m_{PQ}} = -\frac{1}{-4/7} = \frac{7}{4}$

    3. Find the Equation of the Line

    Using the point-slope form, $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (-1.5, -1)$ and $m = \frac{7}{4}$: $y - (-1) = \frac{7}{4}(x - (-1.5))$ $y + 1 = \frac{7}{4}(x + 1.5)$

    Multiply the entire equation by 4 to clear the fraction: $4(y + 1) = 7(x + 1.5)$ $4y + 4 = 7x + 10.5$

    To match the format of the options, multiply by 2 to remove the decimal: $8y + 8 = 14x + 21$

    Rearrange the terms to one side: $8y - 14x + 8 - 21 = 0$ $8y - 14x - 13 = 0$

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

    Exam Tip: Always remember that the product of the slopes of two perpendicular lines is $-1$ ($m_1 \times m_2 = -1$). This is a quick way to verify your slope calculation. Also, when dealing with fractions in coordinate geometry, multiplying through by the denominator early on often prevents arithmetic errors.

    AI can make mistakes. Report anything that looks wrong.

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

39. 39.

    If y = x sin x, find $\frac{\delta y}{\delta x}$

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

    Show answer

    Answer: D. sin x + x cos x

    AI explanation

    Analysis

    The question asks for the derivative of the function $y = x \sin x$ with respect to $x$. This is a classic calculus problem involving the Product Rule, as the function is a product of two terms: $x$ and $\sin x$.

    Concept: The Product Rule

    When you have a function $y = u \cdot v$, where both $u$ and $v$ are functions of $x$, the derivative $\frac{dy}{dx}$ is given by the formula:

    $\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}$

    In this problem:

    • Let $u = x$
    • Let $v = \sin x$

    Step-by-Step Solution

    1. Differentiate the individual parts: • $\frac{du}{dx} = \frac{d}{dx}(x) = 1$ • $\frac{dv}{dx} = \frac{d}{dx}(\sin x) = \cos x$

    2. Apply the Product Rule formula: $$\frac{dy}{dx} = (x)(\cos x) + (\sin x)(1)$$

    3. Simplify the expression: $$\frac{dy}{dx} = x \cos x + \sin x$$

    Rearranging the terms to match the standard format, we get:

    $\frac{dy}{dx} = \sin x + x \cos x$

    Conclusion

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

    Exam Tip

    When you see a variable multiplied by a trigonometric function (like $x \sin x$, $x^2 \cos x$, or $e^x \tan x$), always remember to use the Product Rule. A common mistake students make is to differentiate both parts separately and multiply them together—avoid this! Always use the $u \frac{dv}{dx} + v \frac{du}{dx}$ structure.

    AI can make mistakes. Report anything that looks wrong.

40. 40.

    3 girls share a number of apples in the ration 5:3:2. If the highest share is 40 apples, find the smallest share

    1. A.36
    2. B.24
    3. C.16
    4. D.38

    Show answer

    Answer: C. 16

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

41. 41.

    A square tile has side 30 cm. How many of these tiles will cover a rectangular floor of length 7.2m and width 4.2m?

    1. A.720
    2. B.336
    3. C.420
    4. D.576

    Show answer

    Answer: B. 336

    AI explanation

    1. Analyze the question

    We need to find the number of square tiles required to cover a rectangular floor. To do this, we must divide the total area of the floor by the area of a single tile. Crucially, we must ensure all units are consistent (converting meters to centimeters).

    2. Explain the concept

    • Area of a rectangle: Length × Width
    • Area of a square: Side × Side
    • Number of tiles: (Total Area of Floor) / (Area of One Tile)
    • Unit Conversion: Since the tile is in centimeters (cm) and the floor is in meters (m), we convert meters to centimeters: $1\text{ m} = 100\text{ cm}$.

    3. Show the solution

    Step 1: Convert floor dimensions to centimeters

    • Length = $7.2\text{ m} = 7.2 \times 100 = 720\text{ cm}$
    • Width = $4.2\text{ m} = 4.2 \times 100 = 420\text{ cm}$

    Step 2: Calculate the area of the floor

    • Area of floor = $720\text{ cm} \times 420\text{ cm} = 302,400\text{ cm}^2$

    Step 3: Calculate the area of one tile

    • Area of tile = $30\text{ cm} \times 30\text{ cm} = 900\text{ cm}^2$

    Step 4: Calculate the number of tiles needed

    • Number of tiles = $\frac{\text{Total Area}}{\text{Tile Area}}$
    • Number of tiles = $\frac{302,400}{900}$
    • Cancel the zeros: $\frac{3024}{9}$
    • $3024 \div 9 = 336$

    4. Identify the answer

    The calculation confirms that 336 tiles are required to cover the floor. This matches option B.

    5. Tip

    Always check your units first! A common mistake in JAMB Mathematics is calculating using mixed units (meters and centimeters). Converting everything to the smaller unit (cm) at the very beginning prevents silly errors. Also, notice that $\frac{720}{30} \times \frac{420}{30}$ gives the same result ($24 \times 14 = 336$); this is a faster way to solve it by finding how many tiles fit along the length and width separately!

    AI can make mistakes. Report anything that looks wrong.

42. 42.

    Simplify $\frac{3^{-5n}}{9^{1-n}} \times 27^{n + 1}$

    1. A.32
    2. B.33
    3. C.35
    4. D.3

    Show answer

    Answer: D. 3

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

43. 43.

    P varies jointly as m and u, and varies inversely as q. Given that p = 4, m = 3 and u = 2 and q = 1, find the value of p when m = 6, u = 4 and q =$\frac{8}{5}$

    1. A.12$\frac{8}{5}$
    2. B.15
    3. C.10
    4. D.28$\frac{8}{5}$

    Show answer

    Answer: C. 10

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

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

45. 45.

    If the angles of a quadrilateral are (3y + 10)°, (2y + 30)°, (y + 20)° and 4y°. Find the value of y.

    1. A.66°
    2. B.12°
    3. C.30°
    4. D.42°

    Show answer

    Answer: C. 30°

    AI explanation

    1. Analyze the question

    You are given the four interior angles of a quadrilateral in terms of a variable $y$. You need to find the value of $y$ by applying the geometric property of quadrilaterals.

    2. Explain the concept

    The fundamental principle here is that the sum of the interior angles of any quadrilateral is always 360°. If the angles are $A, B, C,$ and $D$, then:

    $A + B + C + D = 360^\circ$

    3. Show the solution

    Step 1: Set up the equation. Sum the given expressions and set them equal to 360:

    $(3y + 10) + (2y + 30) + (y + 20) + 4y = 360$

    Step 2: Group the like terms. Combine the terms with $y$ and the constant numbers separately:

    $(3y + 2y + y + 4y) + (10 + 30 + 20) = 360$

    Step 3: Simplify the equation.

    $(3 + 2 + 1 + 4)y + (60) = 360$

    $10y + 60 = 360$

    Step 4: Solve for $y$. Subtract 60 from both sides:

    $10y = 360 - 60$

    $10y = 300$

    Divide by 10:

    $y = \frac{300}{10}$

    $y = 30$

    4. Identify the answer

    The value of $y$ is 30. Looking at the options provided, this corresponds to Option C.

    5. Tip

    When dealing with polygons, remember the formula for the sum of interior angles: $(n - 2) \times 180^\circ$, where $n$ is the number of sides. For a quadrilateral, $n=4$, so $(4-2) \times 180 = 2 \times 180 = 360^\circ$. This formula works for any polygon, whether it's a triangle, pentagon, or hexagon!

    AI can make mistakes. Report anything that looks wrong.

46. 46.

    The nth term of the progression $\frac{4}{2}$, $\frac{7}{3}$, $\frac{10}{4}$, $\frac{13}{5}$ is ...

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

    Show answer

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

    AI explanation

    Analysis

    To find the $n^{th}$ term of the sequence $\frac{4}{2}, \frac{7}{3}, \frac{10}{4}, \frac{13}{5}, \dots$, we need to look at the numerator and the denominator separately to identify the pattern in each.

    Concept

    A sequence where the difference between consecutive terms is constant is called an Arithmetic Progression (AP). The formula for the $n^{th}$ term of an AP is: $T_n = a + (n - 1)d$ Where:

    • $a$ is the first term.
    • $d$ is the common difference.

    Step-by-Step Solution

    1. Analyze the Numerators: The numerators are: $4, 7, 10, 13, \dots$

    • First term ($a$) = $4$
    • Common difference ($d$) = $7 - 4 = 3$
    • Using the formula: $T_n = 4 + (n - 1)3$
    • $T_n = 4 + 3n - 3$
    • Numerator = $3n + 1$

    2. Analyze the Denominators: The denominators are: $2, 3, 4, 5, \dots$

    • First term ($a$) = $2$
    • Common difference ($d$) = $3 - 2 = 1$
    • Using the formula: $T_n = 2 + (n - 1)1$
    • $T_n = 2 + n - 1$
    • Denominator = $n + 1$

    3. Combine the results: By placing the numerator over the denominator, we get the $n^{th}$ term: $\frac{3n + 1}{n + 1}$

    Verification

    Let's test the formula with $n = 1$: $\frac{3(1) + 1}{1 + 1} = \frac{4}{2}$ (Matches the first term)

    Let's test with $n = 4$: $\frac{3(4) + 1}{4 + 1} = \frac{13}{5}$ (Matches the fourth term)

    The formula is correct. Comparing this to the given options, it matches option B.

    Exam Tip

    When you see a sequence involving fractions, always treat the numerator and denominator as two separate sequences. Don't try to find a single formula for the whole fraction at once; it is much easier to solve them individually and then combine them!

    AI can make mistakes. Report anything that looks wrong.

47. 47.

    In how many ways can a student select 2 subjects from 5 subjects?

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

    Show answer

    Answer: C. $\frac{5!}{2!3!}$

    AI explanation

    1. Analyze the question

    The question asks for the number of ways to select 2 subjects out of 5. In mathematics, when we are selecting items where the order of selection does not matter (e.g., choosing Mathematics and Physics is the same as choosing Physics and Mathematics), we use the concept of Combinations.

    2. Explain the concept

    The formula for combinations is denoted as $^nC_r$, which represents selecting $r$ items from a total of $n$ items. The formula is:

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

    Where:

    • $n$ = total number of items (5 subjects)
    • $r$ = number of items to be selected (2 subjects)
    • $!$ = factorial (e.g., $5! = 5 \times 4 \times 3 \times 2 \times 1$)

    3. Show the solution

    Given $n = 5$ and $r = 2$:

    Substitute these values into the combination formula:

    $^5C_2 = \frac{5!}{2!(5 - 2)!}$

    Simplify the expression in the denominator:

    $^5C_2 = \frac{5!}{2! \times 3!}$

    Comparing this result to the provided options, we can see that it matches option C.

    4. Identify the answer

    The expression $\frac{5!}{2!3!}$ correctly represents the number of ways to choose 2 subjects from 5.

    Correct Option: C

    5. Tip

    Whenever you see "selecting" or "choosing" a group from a larger set where the order doesn't matter, always think Combination. A quick way to remember the formula is: "Total factorial divided by (Selected factorial times Remaining factorial)."

    AI can make mistakes. Report anything that looks wrong.

48. 48.

    If S = $\sqrt{t^2 - 4t + 4}$, find t in terms of S

    1. A.S2 - 2
    2. B.S + 2
    3. C.S - 2
    4. D.S2 + 2

    Show answer

    Answer: B. S + 2

    AI explanation

    Analysis

    The goal is to make $t$ the subject of the formula in the equation $S = \sqrt{t^2 - 4t + 4}$. To do this, we need to simplify the expression inside the square root and then isolate $t$.

    Concept: Perfect Square Trinomials

    In algebra, a perfect square trinomial follows the pattern: $a^2 - 2ab + b^2 = (a - b)^2$

    Looking at our expression $t^2 - 4t + 4$:

    • $t^2$ is the square of $t$.
    • $4$ is the square of $2$.
    • $-4t$ is $-2 \times (t) \times (2)$. This confirms that $t^2 - 4t + 4 = (t - 2)^2$.

    Step-by-Step Solution

    1. Substitute the simplified form into the equation: $S = \sqrt{(t - 2)^2}$

    2. Simplify the square root: Since $\sqrt{x^2} = x$, the equation becomes: $S = t - 2$

    3. Isolate $t$: To get $t$ by itself, add 2 to both sides of the equation: $S + 2 = t$

    Therefore, $t = S + 2$.

    Conclusion

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

    However, in the context of JAMB multiple-choice questions, we assume the positive root, leading us to $t = S + 2$.*

    Exam Tip

    Whenever you see a quadratic expression inside a square root in an exam, always check if it is a perfect square first. It will save you time and prevent you from trying to use the complex quadratic formula unnecessarily!

    AI can make mistakes. Report anything that looks wrong.