2014 JAMB Mathematics Past Questions

Questions

1. 1.

   Find y, if $\begin{pmatrix}5 & -6 \\2 & -7\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}7 \\-11\end{pmatrix}$

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

   Show answer

   Answer: C. 3

   AI explanation

   1. Analyze the question

   We are given a matrix equation involving a $2 \times 2$ matrix multiplied by a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$, resulting in another column vector. We need to solve for the value of $y$.

   2. Explain the concept

   To solve this, we perform matrix multiplication. When you multiply a $2 \times 2$ matrix by a $2 \times 1$ column vector, you multiply the rows of the first matrix by the column of the second matrix to form a system of linear equations: $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$

   By equating the resulting matrix to the given vector $\begin{pmatrix} 7 \\ -11 \end{pmatrix}$, we get two simultaneous equations.

   3. Show the solution

   Step 1: Set up the equations Multiply the matrix: Row 1: $5x - 6y = 7$ --- (Equation 1) Row 2: $2x - 7y = -11$ --- (Equation 2)

   Step 2: Solve the simultaneous equations We can use the elimination method. To eliminate $x$, multiply Equation 1 by 2 and Equation 2 by 5: $(5x - 6y = 7) \times 2 \implies 10x - 12y = 14$ --- (Equation 3) $(2x - 7y = -11) \times 5 \implies 10x - 35y = -55$ --- (Equation 4)

   Step 3: Subtract Equation 4 from Equation 3 $(10x - 12y) - (10x - 35y) = 14 - (-55)$ $10x - 10x - 12y + 35y = 14 + 55$ $23y = 69$

   Step 4: Solve for $y$ $y = \frac{69}{23}$ $y = 3$

   4. Identify the answer

   Comparing our result to the options provided: A. 8 B. 5 C. 3 D. 2

   The correct option is C.

   5. Tip

   When solving matrix equations, always check your answer by substituting $x$ and $y$ back into the original equations. If we solve for $x$ using $y=3$: $5x - 6(3) = 7 \implies 5x - 18 = 7 \implies 5x = 25 \implies x = 5$. Checking in Equation 2: $2(5) - 7(3) = 10 - 21 = -11$. It matches perfectly!

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2. 2.

   What is the common ratio of the G.P. $(\sqrt{10} + \sqrt{5}) + (\sqrt{10} + 2\sqrt{5}) + ...$?

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

   Show answer

   Answer: A. $\sqrt{2}$

   AI explanation

   1. Analyze the Question

   We are given the first two terms of a Geometric Progression (G.P.):

   • First term ($a$): $\sqrt{10} + \sqrt{5}$
   • Second term ($ar$): $\sqrt{10} + 2\sqrt{5}$

   We need to find the common ratio ($r$).

   2. Explain the Concept

   In a Geometric Progression, the common ratio ($r$) is found by dividing any term by the term immediately preceding it:

   $r = \frac{\text{Second term}}{\text{First term}} = \frac{ar}{a}$

   3. Show the Solution

   Given: $a = \sqrt{10} + \sqrt{5}$ $ar = \sqrt{10} + 2\sqrt{5}$

   Substitute these into the formula:

   $r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}$

   To simplify this, we can factor out $\sqrt{5}$ from both the numerator and the denominator, since $\sqrt{10} = \sqrt{5} \times \sqrt{2}$:

   Numerator: $\sqrt{10} + 2\sqrt{5} = \sqrt{5}(\sqrt{2} + 2)$ Denominator: $\sqrt{10} + \sqrt{5} = \sqrt{5}(\sqrt{2} + 1)$

   Now, divide them:

   $r = \frac{\sqrt{5}(\sqrt{2} + 2)}{\sqrt{5}(\sqrt{2} + 1)} = \frac{\sqrt{2} + 2}{\sqrt{2} + 1}$

   To rationalize the denominator, multiply the numerator and denominator by the conjugate of the denominator $(\sqrt{2} - 1)$:

   $r = \frac{(\sqrt{2} + 2)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)}$

   Expand the numerator: $(\sqrt{2} \times \sqrt{2}) - (\sqrt{2} \times 1) + (2 \times \sqrt{2}) - (2 \times 1)$ $= 2 - \sqrt{2} + 2\sqrt{2} - 2$ $= \sqrt{2}$

   Expand the denominator: $(\sqrt{2})^2 - (1)^2$ $= 2 - 1 = 1$

   So, $r = \frac{\sqrt{2}}{1} = \sqrt{2}$

   4. Identify the Answer

   The calculated common ratio is $\sqrt{2}$, which matches option A.

   5. Exam Tip

   When you see square roots in a G.P. question, look for common factors immediately. Factoring out the smallest root (in this case, $\sqrt{5}$) often simplifies the expression significantly before you even need to perform complex rationalization.

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3. 3.

   A binary operation * is defined by x * y = xy. If x * 2 = 12 - x, find the possible values of x

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

   Show answer

   Answer: B. 3,-4

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4. 4.

   The mean of 2 - t, 4 + t, 3 - 2t, 2 + t and t - 1 is

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

   Show answer

   Answer: C. 2

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5. 5.

   y varies directly as w2. When y = 8, w = 2. Find y when w = 3

   1. A.18
   2. B.12
   3. C.9
   4. D.6

   Show answer

   Answer: A. 18

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6. 6.

   Find the standard deviation of 5, 4, 3, 2, 1

   1. A.$\sqrt{2}$
   2. B.$\sqrt{3}$
   3. C.$\sqrt{6}$
   4. D.$\sqrt{10}$

   Show answer

   Answer: A. $\sqrt{2}$

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7. 7.

   If the angle of a sector of a circle with radius 10.5 cm is 120°, find the perimeter of the sector. [Take $\pi = \frac{22}{7}$]

   1. A.48 cm
   2. B.40 cm
   3. C.43 cm
   4. D.45 cm

   Show answer

   Answer: C. 43 cm

   AI explanation

   Analysis

   To find the perimeter of a sector, we must remember that a sector is bounded by two radii and an arc. Therefore, the perimeter is the sum of the length of the arc and the two radii.

   Concept

   1. Arc Length Formula: The length of an arc ($L$) is given by: $L = \frac{\theta}{360} \times 2\pi r$ where $\theta$ is the angle of the sector and $r$ is the radius.
   2. Perimeter Formula: The perimeter ($P$) of a sector is: $P = L + 2r$

   Step-by-Step Solution

   Step 1: Calculate the Arc Length ($L$) Given: $\theta = 120^\circ$, $r = 10.5\text{ cm}$, $\pi = \frac{22}{7}$ $L = \frac{120}{360} \times 2 \times \frac{22}{7} \times 10.5$

   Simplify the fraction: $\frac{120}{360} = \frac{1}{3}$

   Now calculate: $L = \frac{1}{3} \times 2 \times \frac{22}{7} \times 10.5$ $L = \frac{1}{3} \times \frac{44}{7} \times 10.5$ $L = \frac{44 \times 10.5}{21}$ $L = \frac{462}{21} = 22\text{ cm}$

   Step 2: Calculate the Perimeter ($P$) $P = L + 2r$ $P = 22 + 2(10.5)$ $P = 22 + 21$ $P = 43\text{ cm}$

   Conclusion

   The perimeter of the sector is 43 cm. This matches option C.

   Exam Tip

   Always remember that a sector is like a slice of pizza. When asked for the "perimeter," students often calculate only the arc length (the crust) and forget to add the two straight sides (the radii). Don't fall for that trap! Always add $2r$ to your arc length.

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8. 8.

   Evaluate $\int \sin 2x dx$

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

   Show answer

   Answer: C. $-\frac{1}{2}$cos 2x + k

   AI explanation

   1. Analyze the question

   You are asked to find the indefinite integral of the trigonometric function $\sin 2x$ with respect to $x$.

   2. Explain the concept

   To solve this, we use the rule for integrating trigonometric functions of the form $\sin(ax)$, where $a$ is a constant. The general rule is:

   $\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + k$

   where $k$ is the constant of integration.

   This rule is derived from the reverse of the chain rule in differentiation. Since the derivative of $\cos(ax)$ is $-a \sin(ax)$, the integral of $\sin(ax)$ must involve $-\frac{1}{a} \cos(ax)$.

   3. Show the solution

   • Step 1: Identify the constant $a$ in the expression $\sin 2x$. Here, $a = 2$.
   • Step 2: Apply the integration formula: $$\int \sin 2x dx$$
   • Step 3: Integrate $\sin(2x)$. The integral of $\sin$ is $-\cos$. Because the argument is $2x$, we must divide by the derivative of $2x$ (which is $2$): $$= -\frac{1}{2} \cos(2x) + k$$

   4. Identify the answer

   Comparing our result to the given options: A. $\cos 2x + k$ B. $\frac{1}{2}\cos 2x + k$ C. $-\frac{1}{2}\cos 2x + k$ D. $-\cos 2x + k$

   The correct option is C.

   5. Tip

   Memory Trick: Remember the "C-Negative" rule for trigonometry integration. Whenever your integral results in a function starting with "co-" (like cosine), the result is negative.

   • Integral of $\sin x = -\cos x$
   • Integral of $\cos x = \sin x$

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9. 9.

   A woman bought a grinder for N60,000. She sold it at a loss of 15%. How much did she sell it?

   1. A.N53,000
   2. B.N52,000
   3. C.N51,000
   4. D.N50,000

   Show answer

   Answer: C. N51,000

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10. 10.

    If gt2 - k - w = 0, make g the subject of the formula

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

    Show answer

    Answer: A. $\frac{k + w}{t^2}$

    AI explanation

    Analysis

    The goal is to rearrange the equation $gt^2 - k - w = 0$ so that $g$ stands alone on one side of the equals sign. This is a classic algebraic manipulation problem often found in JAMB Mathematics.

    Concept

    To make a variable the subject of a formula, you must isolate it by performing inverse operations on both sides of the equation.

    1. Move all terms not containing the target variable ($g$) to the other side.
    2. Divide by the coefficient of the target variable to leave it by itself.

    Step-by-Step Solution

    Step 1: Write down the original equation.

    $gt^2 - k - w = 0$

    Step 2: Move the terms $-k$ and $-w$ to the right side of the equation. When you move a term across the equals sign, its sign changes (from negative to positive).

    $gt^2 = k + w$

    Step 3: Isolate $g$. Since $g$ is being multiplied by $t^2$, you must divide both sides of the equation by $t^2$ to isolate $g$.

    $\frac{gt^2}{t^2} = \frac{k + w}{t^2}$

    Step 4: Simplify.

    $g = \frac{k + w}{t^2}$

    Conclusion

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

    Exam Tip

    When moving terms across the equals sign, always remember the "Sign Flip" rule:

    • Addition becomes Subtraction ($+$ becomes $-$)
    • Subtraction becomes Addition ($-$ becomes $+$)
    • Multiplication becomes Division ($\times$ becomes $\div$)
    • Division becomes Multiplication ($\div$ becomes $\times$)

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11. 11.

    Find the value of k if y - 1 is a factor of y3 + 4y2 + ky - 6

    1. A.-6
    2. B.-4
    3. C.O
    4. D.1

    Show answer

    Answer: D. 1

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12. 12.

    Find the value of 1101112 + 101002

    1. A.11010112
    2. B.10010012
    3. C.10010112
    4. D.10011112

    Show answer

    Answer: C. 10010112

    AI explanation

    Analysis

    This is a problem involving binary addition (base 2). In binary, the only digits allowed are 0 and 1. The rules for addition are:

    • 0 + 0 = 0
    • 0 + 1 = 1
    • 1 + 1 = 10 (write 0, carry 1)
    • 1 + 1 + 1 (including a carry) = 11 (write 1, carry 1)

    Step-by-Step Solution

    To solve this, we align the numbers vertically and add them column by column from right to left:

        1 1 0 1 1 1
    +     1 0 1 0 0
    ---------------

    Step 1 (Rightmost column): 1 + 0 = 1 Step 2 (Second column): 1 + 0 = 1 Step 3 (Third column): 1 + 1 = 10. Write 0, carry 1. Step 4 (Fourth column): 0 + 0 + (carry 1) = 1 Step 5 (Fifth column): 1 + 1 = 10. Write 0, carry 1. Step 6 (Sixth column): 1 + (carry 1) = 10. Write 10.

    Putting it all together:

      (1) (1)   (1)
        1 1 0 1 1 1
    +     1 0 1 0 0
    ---------------
      1 0 0 1 0 1 1

    Verification

    • $110111_2 = (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 32 + 16 + 0 + 4 + 2 + 1 = 55_{10}$
    • $10100_2 = (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (0 \times 2^0) = 16 + 0 + 4 + 0 + 0 = 20_{10}$
    • $55 + 20 = 75_{10}$

    Now, convert $75_{10}$ back to binary:

    • $75 \div 2 = 37$ remainder 1
    • $37 \div 2 = 18$ remainder 1
    • $18 \div 2 = 9$ remainder 0
    • $9 \div 2 = 4$ remainder 1
    • $4 \div 2 = 2$ remainder 0
    • $2 \div 2 = 1$ remainder 0
    • $1 \div 2 = 0$ remainder 1

    Reading the remainders from bottom to top, we get 1001011.

    Conclusion

    The result of the addition is $1001011_2$. This matches option C.

    Exam Tip: When adding binary numbers, always keep your columns straight. If you get confused, convert the binary numbers to base 10 (decimal), add them, and then convert your final answer back to base 2. It takes a little longer, but it is a foolproof way to check your work!

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13. 13.

    The gradient of a line joining (x,4) and (1,2) is $\frac{1}{2}$. Find the value of x

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

    Show answer

    Answer: A. 5

    AI explanation

    Analysis

    To find the value of $x$, we need to use the formula for the gradient (slope) of a straight line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$.

    Concept

    The gradient ($m$) of a line is defined as the change in $y$ divided by the change in $x$:

    $m = \frac{y_2 - y_1}{x_2 - x_1}$

    In this problem, we are given:

    • Point 1: $(x_1, y_1) = (x, 4)$
    • Point 2: $(x_2, y_2) = (1, 2)$
    • Gradient ($m$) = $\frac{1}{2}$

    Solution

    1. Substitute the given values into the gradient formula: $$\frac{1}{2} = \frac{2 - 4}{1 - x}$$

    2. Simplify the numerator: $$\frac{1}{2} = \frac{-2}{1 - x}$$

    3. Cross-multiply to solve for $x$: $$1(1 - x) = 2(-2)$$ $$1 - x = -4$$

    4. Isolate $x$: $$-x = -4 - 1$$ $$-x = -5$$ $$x = 5$$

    Conclusion

    By substituting the coordinates into the gradient formula and solving the resulting linear equation, we find that $x = 5$.

    Exam Tip: When dealing with gradient problems, always be careful with the signs of the coordinates. A common mistake is to subtract the coordinates in the wrong order (e.g., $y_1 - y_2$ on top but $x_2 - x_1$ on the bottom). Always maintain the same order for both: $\frac{y_2 - y_1}{x_2 - x_1}$ or $\frac{y_1 - y_2}{x_1 - x_2}$.

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

15. 15.

    Evaluate the inequality $\frac{x}{2} + \frac{3}{4} \leq \frac{5x}{6} - \frac{7}{12}$

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

    Show answer

    Answer: A. $x \geq 4$

    AI explanation

    Analysis

    To solve this linear inequality, we need to isolate $x$. The best approach is to clear the fractions first by finding the Least Common Multiple (LCM) of the denominators (2, 4, 6, and 12).

    Concept

    When dealing with inequalities involving fractions, multiplying the entire inequality by the LCM of the denominators simplifies the expression into a standard linear inequality. Remember the golden rule: If you multiply or divide an inequality by a negative number, you must reverse the inequality sign.

    Step-by-Step Solution

    Step 1: Find the LCM of the denominators. The denominators are 2, 4, 6, and 12. The LCM is 12.

    Step 2: Multiply every term by 12 to clear the fractions.

    $12 \left( \frac{x}{2} \right) + 12 \left( \frac{3}{4} \right) \leq 12 \left( \frac{5x}{6} \right) - 12 \left( \frac{7}{12} \right)$

    Step 3: Simplify each term.

    $6(x) + 3(3) \leq 2(5x) - 1(7)$

    $6x + 9 \leq 10x - 7$

    Step 4: Group the like terms. Subtract $6x$ from both sides:

    $9 \leq 4x - 7$

    Add 7 to both sides:

    $9 + 7 \leq 4x$

    $16 \leq 4x$

    Step 5: Solve for $x$. Divide both sides by 4:

    $\frac{16}{4} \leq x$

    $4 \leq x$

    This is the same as:

    $x \geq 4$

    Conclusion

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

    Exam Tip

    When you have an inequality with many fractions, don't try to find a common denominator for the fractions themselves (which can get messy). Instead, multiply the entire equation by the LCM of all denominators immediately. This turns the problem into a simple whole-number equation, which is much faster and less prone to calculation errors!

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16. 16.

    A man donates 10% of his monthly net earnings to his church. If it amounts to N4,500, what is his net monthly income?

    1. A.N40,500
    2. B.N45,000
    3. C.N52,500
    4. D.N62,000

    Show answer

    Answer: B. N45,000

    AI explanation

    1. Analyze the question

    We are given that 10% of a man's total net monthly income is equal to N4,500. We need to find the total monthly income (100%).

    2. Explain the concept

    This is a basic percentage problem. If we let the total monthly income be represented by $x$, the problem states that: $10\% \text{ of } x = 4,500$

    In mathematics, "of" means multiplication, and percentage means "per hundred". So, we can write the equation as: $\frac{10}{100} \times x = 4,500$

    3. Show the solution

    • Step 1: Set up the equation. $0.10 \times x = 4,500$

    • Step 2: Isolate $x$ by dividing both sides by 0.10. $x = \frac{4,500}{0.10}$

    • Step 3: Perform the division. Dividing by 0.10 is the same as multiplying by 10. $x = 4,500 \times 10$ $x = 45,000$

    The man's net monthly income is N45,000.

    4. Identify the answer

    Comparing our result to the given options: A. N40,500 B. N45,000 C. N52,500 D. N62,000

    The correct option is B.

    5. Tip

    When dealing with percentages like 10%, you don't always need a complex formula. Just remember that 10% is simply the total divided by 10. Therefore, to find the total, just multiply the given amount by 10. If the question had said 20%, you would divide the amount by 20 and multiply by 100 (or simply multiply by 5). This mental math shortcut saves time during JAMB exams!

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17. 17.

    The 4th term of an A.P. is 13 while the 10th term is 31. Find the 24th term.

    1. A.89
    2. B.75
    3. C.73
    4. D.69

    Show answer

    Answer: C. 73

    AI explanation

    1. Analyze the question

    We are given two specific terms of an Arithmetic Progression (A.P.): the 4th term ($T_4 = 13$) and the 10th term ($T_{10} = 31$). We need to find the 24th term ($T_{24}$).

    2. Explain the concept

    In an Arithmetic Progression, the $n^{th}$ term is calculated using the formula:

    $T_n = a + (n - 1)d$

    Where:

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

    3. Show the solution

    Step 1: Set up equations based on the given information. For the 4th term ($n=4$): $a + (4 - 1)d = 13$ $a + 3d = 13$ --- (Equation 1)

    For the 10th term ($n=10$): $a + (10 - 1)d = 31$ $a + 9d = 31$ --- (Equation 2)

    Step 2: Solve for $d$ by subtracting Equation 1 from Equation 2. $(a + 9d) - (a + 3d) = 31 - 13$ $6d = 18$ $d = \frac{18}{6}$ $d = 3$

    Step 3: Solve for $a$ using Equation 1. $a + 3(3) = 13$ $a + 9 = 13$ $a = 13 - 9$ $a = 4$

    Step 4: Find the 24th term ($T_{24}$). $T_{24} = a + (24 - 1)d$ $T_{24} = 4 + (23 \times 3)$ $T_{24} = 4 + 69$ $T_{24} = 73$

    4. Identify the answer

    Comparing our result to the options provided: A. 89 B. 75 C. 73 D. 69

    The correct option is C.

    5. Tip

    In A.P. problems, if you are given two terms, you can find the common difference ($d$) quickly using the slope formula: $d = \frac{T_n - T_m}{n - m}$. Here: $d = \frac{31 - 13}{10 - 4} = \frac{18}{6} = 3$. This saves time during the actual JAMB exam!

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18. 18.

    Evaluate $\int (2x + 3)^{\frac{1}{2}} \delta x$

    1. A.$\frac{1}{12} (2x + 3)^6 + k$
    2. B.$\frac{1}{3} (2x + 3)^{\frac{1}{2}} + k$
    3. C.$\frac{1}{3} (2x + 3)^{\frac{3}{2}} + k$
    4. D.$\frac{1}{12} (2x + 3)^{\frac{3}{4}} + k$

    Show answer

    Answer: C. $\frac{1}{3} (2x + 3)^{\frac{3}{2}} + k$

    AI explanation

    To solve the integral $\int (2x + 3)^{\frac{1}{2}} \, dx$, we use the method of Integration by Substitution (or the reverse chain rule).

    1. Analyze the Concept

    When integrating a function of the form $(ax + b)^n$, the rule is:

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

    Where:

    • $a$ is the coefficient of $x$ (in this case, $2$).
    • $n$ is the power (in this case, $\frac{1}{2}$).
    • $k$ is the constant of integration.

    2. Step-by-Step Solution

    Step 1: Identify the components.

    • $n = \frac{1}{2}$
    • $a = 2$

    Step 2: Apply the integration formula. Increase the power by 1:

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

    Divide by the new power and multiply by the derivative of the inner function (which is $2$):

    $\int (2x + 3)^{\frac{1}{2}} \, dx = \frac{(2x + 3)^{\frac{3}{2}}}{2 \times (\frac{3}{2})} + k$

    Step 3: Simplify the expression. The $2$ in the denominator and the $2$ in the denominator of the fraction $\frac{3}{2}$ cancel out:

    $2 \times \frac{3}{2} = 3$

    So, the expression becomes:

    $\frac{(2x + 3)^{\frac{3}{2}}}{3} + k$

    Which can be written as:

    $\frac{1}{3} (2x + 3)^{\frac{3}{2}} + k$

    3. Conclusion

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

    Exam Tip: Whenever you see a linear expression inside a power (like $2x+3$), always remember to divide by the coefficient of $x$. Many students forget to divide by the $2$, which leads to common errors. Always check if your answer differentiates back to the original function to verify your work!

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19. 19.

    If $\begin{vmatrix}-x & 12 \\-1 & 4 \end{vmatrix} = - 12$, find x.

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

    Show answer

    Answer: D. 6

    AI explanation

    1. Analyze the question

    You are given a $2 \times 2$ determinant equation and asked to solve for the unknown variable $x$. The determinant of a matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is calculated as $(ad - bc)$.

    2. Explain the concept

    For any $2 \times 2$ matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$, the determinant is found by multiplying the elements of the leading diagonal ($a$ and $d$) and subtracting the product of the elements of the other diagonal ($b$ and $c$). Formula: Determinant = $(a \times d) - (b \times c)$

    3. Show the solution

    Given the matrix: $\begin{vmatrix} -x & 12 \\ -1 & 4 \end{vmatrix} = -12$

    Step 1: Apply the determinant formula $(-x \times 4) - (12 \times -1) = -12$

    Step 2: Simplify the expressions $-4x - (-12) = -12$ $-4x + 12 = -12$

    Step 3: Isolate the term with $x$ Subtract 12 from both sides: $-4x = -12 - 12$ $-4x = -24$

    Step 4: Solve for $x$ Divide both sides by -4: $x = \frac{-24}{-4}$ $x = 6$

    4. Identify the answer

    Comparing our result to the given options: A. -6 B. -2 C. 3 D. 6

    The correct option is D.

    5. Tip

    When dealing with determinants involving negative signs, always use brackets for your multiplication. A common mistake in JAMB exams is forgetting that subtracting a negative number (like $-(-12)$) turns it into a positive addition ($+12$). Always double-check your signs!

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20. 20.

    Find the value of $\begin{vmatrix}0 & 3 & 2 \\1 & 7 & 8 \\0 & 5 & 4\end{vmatrix}$.

    1. A.12
    2. B.10
    3. C.-1
    4. D.-2

    Show answer

    Answer: D. -2

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21. 21.

    P varies directly as Q and inversely as R. When Q = 36 and R = 16, P = 27. Find the relation between P, Q and R.

    1. A.$P = \frac{Q}{12R}$
    2. B.$P = \frac{12Q}{R}$
    3. C.$P = 12QR$
    4. D.$P = \frac{12}{QR}$

    Show answer

    Answer: B. $P = \frac{12Q}{R}$

    AI explanation

    1. Analyze the question

    We are given a variation problem involving three variables: $P$, $Q$, and $R$. We need to find the mathematical relationship (the formula) connecting them based on the provided values.

    2. Explain the concept

    • Direct Variation: If $P$ varies directly as $Q$, it means $P \propto Q$ (or $P = kQ$).
    • Inverse Variation: If $P$ varies inversely as $R$, it means $P \propto \frac{1}{R}$ (or $P = \frac{k}{R}$).
    • Combined Variation: When combined, $P$ varies directly as $Q$ and inversely as $R$, expressed as: $$P = \frac{kQ}{R}$$where$k$ is the constant of proportionality.

    3. Show the solution

    Step 1: Set up the equation. Using the combined variation formula:

    $P = \frac{kQ}{R}$

    Step 2: Substitute the given values to find $k$. We are given $P = 27$, $Q = 36$, and $R = 16$.

    $27 = \frac{k \times 36}{16}$

    Step 3: Solve for $k$. Multiply both sides by 16:

    $27 \times 16 = 36k$

    $432 = 36k$

    Divide both sides by 36:

    $k = \frac{432}{36}$

    $k = 12$

    Step 4: Write the final relation. Substitute $k = 12$ back into the original equation:

    $P = \frac{12Q}{R}$

    4. Identify the answer

    Comparing our result to the given options, the correct relation is $P = \frac{12Q}{R}$. This matches option B.

    5. Exam Tip

    When dealing with variation questions, always remember: "Direct" means the variable goes on the numerator (top), and "Inverse" means the variable goes on the denominator (bottom). This simple rule will help you set up your equation in seconds!

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

23. 23.

    Solve for x in 8x-2 = 2/25

    1. A.4
    2. B.6
    3. C.8
    4. D.10

    Show answer

    Answer: D. 10

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24. 24.

    $\begin{array}{c|c} Values & 0 & 1 & 2 & 3 & 4 \\ \hline Frequency & 1 & 2 & 2 & 1 & 9 \end{array}$ Find the mode of the distribution above

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

    Show answer

    Answer: D. 4

    AI explanation

    Analysis of the Question

    The question asks for the mode of a given frequency distribution. In statistics, the mode is defined as the value that appears most frequently in a data set.

    Concept Explanation

    To find the mode from a frequency table, you do not need to perform complex calculations. You simply look at the "Frequency" row to identify the highest number. The "Value" corresponding to that highest frequency is your mode.

    Step-by-Step Solution

    1. Examine the table: • Value 0 has a frequency of 1 • Value 1 has a frequency of 2 • Value 2 has a frequency of 2 • Value 3 has a frequency of 1 • Value 4 has a frequency of 9

    2. Identify the highest frequency: Comparing the frequencies (1, 2, 2, 1, 9), the largest number is 9.

    3. Match the frequency to the value: The value associated with the frequency of 9 is 4.

    Since 4 appears more times (9 times) than any other value in the distribution, it is the mode.

    Conclusion

    The mode of the distribution is 4.

    Exam Tip

    Always be careful not to confuse the mode with the mean or median.

    • Mode: Most frequent (look for the biggest number in the frequency row).
    • Median: The middle value (arrange data in order first).
    • Mean: The average (sum of all values divided by total frequency).

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25. 25.

    A cylindrical tank has a capacity of 6160m3. What is the depth of the tank if the radius of its base is 28cm?

    1. A.8.0m
    2. B.7.5m
    3. C.5.0m
    4. D.2.5m

    Show answer

    Answer: D. 2.5m

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26. 26.

    What is the solution of $\frac{x - 5}{x + 3} < -1$?

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

    Show answer

    Answer: A. -3 < x < 1

    AI explanation

    Analysis

    To solve the inequality $\frac{x - 5}{x + 3} < -1$, we must move all terms to one side to compare the expression to zero. We cannot simply multiply both sides by $(x + 3)$ because we do not know if $(x + 3)$ is positive or negative, and multiplying by a negative number would flip the inequality sign.

    Concept

    When solving rational inequalities:

    1. Move all terms to one side to get $f(x) < 0$.
    2. Find the critical values (where the numerator is zero and where the denominator is zero).
    3. Use a sign chart or test intervals to determine where the expression is negative.

    Step-by-Step Solution

    Step 1: Move -1 to the left side.

    $\frac{x - 5}{x + 3} + 1 < 0$

    Step 2: Find a common denominator.

    $\frac{x - 5 + 1(x + 3)}{x + 3} < 0$

    $\frac{x - 5 + x + 3}{x + 3} < 0$

    $\frac{2x - 2}{x + 3} < 0$

    Step 3: Identify critical values.

    • Numerator: $2x - 2 = 0 \implies x = 1$
    • Denominator: $x + 3 = 0 \implies x = -3$

    Step 4: Test the intervals. Our critical values divide the number line into three regions: $x < -3$, $-3 < x < 1$, and $x > 1$.

    • Test $x = -4$ (Region $x < -3$): $\frac{2(-4) - 2}{-4 + 3} = \frac{-10}{-1} = 10$. (Positive, so this is not the solution).
    • Test $x = 0$ (Region $-3 < x < 1$): $\frac{2(0) - 2}{0 + 3} = \frac{-2}{3}$. (Negative, so this is the solution).
    • Test $x = 2$ (Region $x > 1$): $\frac{2(2) - 2}{2 + 3} = \frac{2}{5}$. (Positive, so this is not the solution).

    The inequality holds true when $-3 < x < 1$.

    Conclusion

    Comparing our result to the given options, the correct range is $-3 < x < 1$. This matches option A.

    Exam Tip

    When dealing with rational inequalities, never cross-multiply unless you are absolutely certain of the sign of the denominator. Always bring everything to one side and set it to zero; this prevents the common mistake of forgetting to flip the inequality sign!

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27. 27.

    A number is chosen at random from 10 to 30 both inclusive. What is the probability that the number is divisible by 3?

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

    Show answer

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

    AI explanation

    1. Analyze the question

    We need to find the probability that a number chosen at random from the set of integers $\{10, 11, 12, \dots, 30\}$ is divisible by 3.

    2. Explain the concept

    Probability is defined as:

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

    • Total outcomes: Since the range is "inclusive," we include both 10 and 30. To find the total count of numbers in a range $[a, b]$, we use the formula: $(b - a) + 1$.
    • Favorable outcomes: These are the numbers within that range that are multiples of 3.

    3. Show the solution

    Step 1: Determine the total number of outcomes. The range is from 10 to 30 inclusive. Total numbers = $(30 - 10) + 1 = 20 + 1 = 21$.

    Step 2: List the numbers divisible by 3 in this range. We look for multiples of 3 starting from 10: 12, 15, 18, 21, 24, 27, 30.

    Step 3: Count the favorable outcomes. Counting the list above: 1 (12), 2 (15), 3 (18), 4 (21), 5 (24), 6 (27), 7 (30). There are 7 favorable outcomes.

    Step 4: Calculate the probability.

    $\text{Probability} = \frac{7}{21}$

    Simplifying the fraction by dividing both numerator and denominator by 7:

    $\text{Probability} = \frac{1}{3}$

    4. Identify the answer

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

    )*

    5. Exam Tip

    When a question says "inclusive," always remember to add 1 to the difference of the numbers. If you had calculated $30 - 10 = 20$, you would have missed one number, leading to an incorrect probability of $7/20$. Always write out the sequence if the range is small to avoid silly mistakes!

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28. 28.

    How many sides has a regular polygon whose interior angle is 135°?

    1. A.12
    2. B.10
    3. C.9
    4. D.8

    Show answer

    Answer: D. 8

    AI explanation

    Analysis

    To find the number of sides of a regular polygon when the interior angle is given, we use the relationship between interior and exterior angles.

    Concept

    1. Interior + Exterior Angle = 180°: In any polygon, the interior angle and its corresponding exterior angle lie on a straight line, so they sum up to 180°.
    2. Sum of Exterior Angles: The sum of all exterior angles of any convex polygon is always 360°.
    3. Formula for Exterior Angle: For a regular polygon with $n$ sides, each exterior angle is given by: $$\text{Exterior Angle} = \frac{360^\circ}{n}$$

    Step-by-Step Solution

    Step 1: Find the exterior angle. Since the interior angle is 135°, we calculate the exterior angle:

    $\text{Exterior Angle} = 180^\circ - 135^\circ$

    $\text{Exterior Angle} = 45^\circ$

    Step 2: Use the exterior angle formula to find $n$ (the number of sides). Using the formula $\text{Exterior Angle} = \frac{360^\circ}{n}$:

    $45^\circ = \frac{360^\circ}{n}$

    Step 3: Solve for $n$. Multiply both sides by $n$:

    $45n = 360$

    Divide by 45:

    $n = \frac{360}{45}$

    $n = 8$

    The polygon has 8 sides (it is an octagon).

    Conclusion

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

    Exam Tip

    Whenever you are given the interior angle, always convert it to the exterior angle first. It is much easier to divide 360 by the exterior angle than to use the more complex interior angle formula: $\text{Interior Angle} = \frac{(n-2) \times 180}{n}$. Saving time on calculations helps you finish your JAMB paper faster!

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29. 29.

    Find the minimum value of y = x2 - 2x - 3

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

    Show answer

    Answer: D. -4

    AI explanation

    1. Analyze the question

    We are asked to find the minimum value of the quadratic function $y = x^2 - 2x - 3$. Since the coefficient of $x^2$ is positive ($+1$), the parabola opens upwards, meaning it has a minimum point at its vertex.

    2. Explain the concept

    For a quadratic equation in the form $y = ax^2 + bx + c$, the $x$-coordinate of the vertex (where the minimum or maximum occurs) is given by the formula:

    $x = -\frac{b}{2a}$

    Once you find the value of $x$, you substitute it back into the original equation to find the corresponding $y$-value, which is the minimum value of the function.

    Alternatively, you can use the method of completing the square or differentiation.

    3. Show the solution

    Method 1: Using the vertex formula Given: $y = x^2 - 2x - 3$ Here, $a = 1$, $b = -2$, and $c = -3$.

    • Step 1: Find the $x$-coordinate of the vertex. $$x = -\frac{b}{2a} = -\frac{-2}{2(1)} = \frac{2}{2} = 1$$

    • Step 2: Substitute $x = 1$ into the equation to find $y$. $$y = (1)^2 - 2(1) - 3$$ $$y = 1 - 2 - 3$$ $$y = -1 - 3$$ $$y = -4$$

    Method 2: Using Differentiation

    • Step 1: Find the derivative $\frac{dy}{dx}$ and set it to zero. $$\frac{dy}{dx} = 2x - 2$$ $$2x - 2 = 0 \implies 2x = 2 \implies x = 1$$
    • Step 2: Substitute $x = 1$ into the original equation. $$y = (1)^2 - 2(1) - 3 = -4$$

    4. Identify the answer

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

    The calculated minimum value is -4, which corresponds to option D.

    5. Tip

    When you see a quadratic equation and need the minimum or maximum value, remember that the "value" requested is the $y$-coordinate, not the $x$-coordinate. Many students stop after finding $x = 1$ and mistakenly pick option B. Always substitute your $x$ back into the original equation to get the final answer!

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

31. 31.

    Evaluate Log28 + Log216 - Log24

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

    Show answer

    Answer: C. 5

    AI explanation

    Analysis

    The question requires the application of the laws of logarithms to simplify a single expression. Specifically, we will use the Product Law and the Quotient Law of logarithms.

    Concept

    The laws of logarithms state that for any base $a$:

    1. Product Law: $\log_a(x) + \log_a(y) = \log_a(x \cdot y)$
    2. Quotient Law: $\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)$

    Step-by-Step Solution

    Step 1: Apply the Product Law to the first two terms The expression is: $\log_2(8) + \log_2(16) - \log_2(4)$

    Combine the first two terms: $\log_2(8 \times 16) - \log_2(4)$ $\log_2(128) - \log_2(4)$

    Step 2: Apply the Quotient Law Now, combine the remaining terms: $\log_2\left(\frac{128}{4}\right)$ $\log_2(32)$

    Step 3: Evaluate the logarithm We need to find the power to which $2$ must be raised to get $32$. $2^1 = 2$ $2^2 = 4$ $2^3 = 8$ $2^4 = 16$ $2^5 = 32$

    Therefore, $\log_2(32) = 5$.

    Conclusion

    The calculation leads us to the value of 5, which corresponds to option C.

    Exam Tip

    When dealing with logarithms in JAMB, always look to simplify the numbers into powers of the base first (e.g., $8=2^3$, $16=2^4$, $4=2^2$). You can then use the power rule $\log_a(x^n) = n \log_a(x)$ to solve it even faster!

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32. 32.

    In how many ways can a team of 3 girls be selected from 7 girls?

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

    Show answer

    Answer: C. $\frac{7!}{3!4!}$

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

34. 34.

    If P = {1,2,3,4,5} and P $\cup$ Q = {1,2,3,4,5,6,7}, list the elements in Q

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

    Show answer

    Answer: C. {6,7}

    AI explanation

    Analysis

    The question asks us to find the set Q, given the set P and the union of P and Q ($P \cup Q$).

    Concept

    The union of two sets ($P \cup Q$) contains all the elements that are in set P, in set Q, or in both. To find the elements of Q, we look for any elements present in the union ($P \cup Q$) that are not already accounted for in set P.

    Step-by-Step Solution

    1. Identify the given sets: • $P = \{1, 2, 3, 4, 5\}$ • $P \cup Q = \{1, 2, 3, 4, 5, 6, 7\}$

    2. Compare the sets: • We see that the elements $\{1, 2, 3, 4, 5\}$ are already in set P. • The elements $\{6, 7\}$ are present in the union ($P \cup Q$) but are missing from set P.

    3. Determine Q: • Since $P \cup Q$ must contain all elements from both sets, the elements $\{6, 7\}$ must belong to set Q to complete the union. • Therefore, $Q = \{6, 7\}$.

    Conclusion

    Comparing our result to the options provided:

    • A. {6}
    • B. {7}
    • C. {6, 7}
    • D. {5, 6}

    The correct set is {6, 7}, which corresponds to option C.

    Exam Tip

    When dealing with set operations, think of the union ($\cup$) as a "marriage" of sets. If you have the result of the marriage and one partner, you can easily find the other partner by identifying who is "missing" from the first partner's list but present in the final result!

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35. 35.

    If $\sin\theta = \frac{12}{13}$, find the value of $1 + \cos\theta$

    1. A.$\frac{25}{13}$
    2. B.$\frac{18}{13}$
    3. C.$\frac{8}{13}$
    4. D.$\frac{5}{13}$

    Show answer

    Answer: B. $\frac{18}{13}$

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36. 36.

    Factorize 2y2 - 15xy + 18x2

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

    Show answer

    Answer: B. (2y - 3x) (y - 6x)

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37. 37.

    Calculate the mid point of the line segment y - 4x + 3 = 0, which lies between the x-axis and y-axis.

    1. A.$\begin{pmatrix} 3 & -3 \\ 8 & 2 \end{pmatrix}$
    2. B.$\begin{pmatrix} 3 & 3 \\ 8 & 2 \end{pmatrix}$
    3. C.$\begin{pmatrix} -2 & 2 \\ 2 & 2 \end{pmatrix}$
    4. D.$\begin{pmatrix} -2 & 3 \\ 3 & 2 \end{pmatrix}$

    Show answer

    Answer: A. $\begin{pmatrix} 3 & -3 \\ 8 & 2 \end{pmatrix}$

    AI explanation

    1. Analyze the Question

    We are given the equation of a line $y - 4x + 3 = 0$. We need to find the midpoint of the segment of this line that is intercepted between the x-axis and the y-axis.

    2. Explain the Concept

    • Intercepts: To find where a line crosses the axes: • The x-intercept occurs where $y = 0$. • The y-intercept occurs where $x = 0$.
    • Midpoint Formula: If you have two points $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M$ is calculated as: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

    3. Show the Solution

    Step 1: Find the x-intercept. Set $y = 0$ in the equation $y - 4x + 3 = 0$: $0 - 4x + 3 = 0$ $-4x = -3$ $x = \frac{3}{4}$ So, the first point is $P_1 = (\frac{3}{4}, 0)$.

    Step 2: Find the y-intercept. Set $x = 0$ in the equation $y - 4x + 3 = 0$: $y - 4(0) + 3 = 0$ $y + 3 = 0$ $y = -3$ So, the second point is $P_2 = (0, -3)$.

    Step 3: Calculate the midpoint. Using $P_1 = (\frac{3}{4}, 0)$ and $P_2 = (0, -3)$: $x_{mid} = \frac{\frac{3}{4} + 0}{2} = \frac{3}{4} \div 2 = \frac{3}{8}$ $y_{mid} = \frac{0 + (-3)}{2} = -\frac{3}{2}$

    The midpoint is $(\frac{3}{8}, -\frac{3}{2})$.

    4. Identify the Answer

    Looking at the options provided, Option A is written in a matrix-like notation $\begin{pmatrix} 3 & -3 \\ 8 & 2 \end{pmatrix}$. While the notation is unconventional for a coordinate point, the values $3/8$ and $-3/2$ correspond to the components presented in Option A.

    5. Tip

    When dealing with intercepts, always remember: "To find the x-intercept, kill the y (set $y=0$); to find the y-intercept, kill the x (set $x=0$)." This simple trick saves time in JAMB exams!

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38. 38.

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

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

    Show answer

    Answer: A. 3$\sqrt{6} - 7$

    AI explanation

    To simplify the expression $\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}}$, we use the method of rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

    1. Identify the Conjugate

    The denominator is $\sqrt{2} + \sqrt{3}$. Its conjugate is $\sqrt{2} - \sqrt{3}$.

    2. Set up the Multiplication

    Multiply the numerator and denominator by $(\sqrt{2} - \sqrt{3})$:

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

    3. Expand the Numerator

    Using the FOIL method (First, Outer, Inner, Last): $(2\sqrt{2} - \sqrt{3})(\sqrt{2} - \sqrt{3})$ $= (2\sqrt{2} \times \sqrt{2}) - (2\sqrt{2} \times \sqrt{3}) - (\sqrt{3} \times \sqrt{2}) + (\sqrt{3} \times \sqrt{3})$ $= (2 \times 2) - 2\sqrt{6} - \sqrt{6} + 3$ $= 4 - 3\sqrt{6} + 3$ $= 7 - 3\sqrt{6}$

    4. Expand the Denominator

    Using the difference of two squares formula $(a+b)(a-b) = a^2 - b^2$: $(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})$ $= (\sqrt{2})^2 - (\sqrt{3})^2$ $= 2 - 3$ $= -1$

    5. Combine and Simplify

    Now, place the expanded numerator over the expanded denominator:

    $\frac{7 - 3\sqrt{6}}{-1}$

    Divide each term by $-1$: $= \frac{7}{-1} - \frac{3\sqrt{6}}{-1}$ $= -7 + 3\sqrt{6}$ $= 3\sqrt{6} - 7$

    Comparing this to the given options, it matches option A.

    Exam Tip: When rationalizing, always remember that the denominator will result in a simple integer (or a negative integer). If you end up with a square root in your denominator, you haven't finished the process! Also, be very careful with your signs when subtracting the terms in the numerator.

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39. 39.

    If y = cos 3x, find $\frac{\delta y}{\delta x}$

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

    Show answer

    Answer: D. -3 sin 3x

    AI explanation

    Analysis

    The question asks for the derivative of the trigonometric function $y = \cos 3x$ with respect to $x$. This is a classic calculus problem involving the Chain Rule.

    Concept: The Chain Rule

    When you have a composite function like $y = \cos(u)$, where $u$ is a function of $x$ (in this case, $u = 3x$), you use the chain rule:

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

    1. Derivative of $\cos(u)$: The derivative of $\cos(u)$ with respect to $u$ is $-\sin(u)$.
    2. Derivative of $3x$: The derivative of $3x$ with respect to $x$ is $3$.

    Step-by-Step Solution

    1. Identify the components: Let $u = 3x$. Then $y = \cos(u)$.

    2. Differentiate each part: $\frac{dy}{du} = -\sin(u)$ $\frac{du}{dx} = 3$

    3. Apply the Chain Rule: $\frac{dy}{dx} = (-\sin(u)) \cdot (3)$

    4. Substitute $u = 3x$ back into the equation: $\frac{dy}{dx} = -3 \sin(3x)$

    Conclusion

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

    Exam Tip: Always remember the "negative" sign when differentiating cosine functions. A common mistake students make is forgetting that the derivative of $\cos(x)$ is $-\sin(x)$, not $+\sin(x)$. Also, don't forget to multiply by the derivative of the "inside" function (the 3 in $3x$).

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40. 40.

    Find the median of 5,9,1,10,3,8,9,2,4,5,5,5,7,3 and 6

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

    Show answer

    Answer: B. 5

    AI explanation

    Analysis

    To find the median of a set of numbers, you must first arrange them in ascending order (from the smallest to the largest). The median is the middle value in the ordered list.

    Concept

    If the total number of values ($n$) is odd, the median is the value at the position $\frac{n+1}{2}$. If $n$ is even, the median is the average of the two middle values.

    Step-by-Step Solution

    Step 1: List the given numbers 5, 9, 1, 10, 3, 8, 9, 2, 4, 5, 5, 5, 7, 3, 6

    Step 2: Count the numbers There are 15 numbers in the set ($n = 15$). Since 15 is an odd number, there will be exactly one middle value.

    Step 3: Arrange the numbers in ascending order 1, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 9, 9, 10

    Step 4: Locate the middle position Using the formula $\frac{n+1}{2}$: $\frac{15 + 1}{2} = \frac{16}{2} = 8$ The median is the 8th number in the ordered list.

    Step 5: Identify the 8th number 1st: 1 2nd: 2 3rd: 3 4th: 3 5th: 4 6th: 5 7th: 5 8th: 5 9th: 5 ...and so on.

    The 8th number is 5.

    Conclusion

    The median of the set is 5.

    Exam Tip

    Always double-check your count before you start ordering the numbers. A common mistake in JAMB exams is skipping a number while rewriting the list. After you arrange them, count the original list and your new list to ensure you have the same number of items (in this case, 15).

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

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

43. 43.

    If log7.5 = 0.8751, evaluate 2 log75 + log750

    1. A.6.6252
    2. B.6.6253
    3. C.66.252
    4. D.66.253

    Show answer

    Answer: B. 6.6253

    AI explanation

    1. Analyze the question

    We are given that $\log 7.5 = 0.8751$. We need to evaluate the expression $2 \log 75 + \log 750$ using the properties of logarithms.

    2. Explain the concept

    To solve this, we need to express $75$ and $750$ in terms of $7.5$ using the laws of logarithms:

    • Product Rule: $\log(a \times b) = \log a + \log b$
    • Power Rule: $n \log a = \log(a^n)$
    • Base 10 property: $\log(10) = 1$ and $\log(100) = 2$

    Since $75 = 7.5 \times 10$ and $750 = 7.5 \times 100$, we can rewrite the expression.

    3. Show the solution

    Step 1: Expand the terms using the product rule.

    • $\log 75 = \log(7.5 \times 10) = \log 7.5 + \log 10$
    • $\log 750 = \log(7.5 \times 100) = \log 7.5 + \log 100$

    Step 2: Substitute these into the original expression. The expression is $2 \log 75 + \log 750$. $= 2(\log 7.5 + \log 10) + (\log 7.5 + \log 100)$

    Step 3: Simplify using known values. We know $\log 10 = 1$ and $\log 100 = 2$. $= 2(\log 7.5 + 1) + (\log 7.5 + 2)$ $= 2 \log 7.5 + 2 + \log 7.5 + 2$ $= 3 \log 7.5 + 4$

    Step 4: Calculate the final value. Substitute $\log 7.5 = 0.8751$: $= 3(0.8751) + 4$ $= 2.6253 + 4$ $= 6.6253$

    4. Identify the answer

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

    5. Tip

    When dealing with log problems involving decimals like $7.5, 75, 750$, always try to break them down into the given value multiplied by powers of $10$ ($10, 100, 1000$). This makes the calculation much faster and avoids the need for a calculator!

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44. 44.

    Find the mid point of S(-5, 4) and T(-3, -2)

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

    Show answer

    Answer: C. -4, 1

    AI explanation

    1. Analyze the question

    You are asked to find the midpoint of a line segment connecting two points in a Cartesian plane: $S(-5, 4)$ and $T(-3, -2)$.

    2. Explain the concept

    To find the midpoint $(x_m, y_m)$ of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, we use the Midpoint Formula:

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

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

    Essentially, you are finding the average of the x-coordinates and the average of the y-coordinates.

    3. Show the solution

    Given the points:

    • $S(x_1, y_1) = (-5, 4)$
    • $T(x_2, y_2) = (-3, -2)$

    Step 1: Find the x-coordinate of the midpoint ($x_m$)

    $x_m = \frac{-5 + (-3)}{2}$

    $x_m = \frac{-5 - 3}{2}$

    $x_m = \frac{-8}{2} = -4$

    Step 2: Find the y-coordinate of the midpoint ($y_m$)

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

    $y_m = \frac{4 - 2}{2}$

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

    The midpoint is $(-4, 1)$.

    4. Identify the answer

    Comparing our result $(-4, 1)$ to the given options: A. -4, 2 B. 4, -2 C. -4, 1 D. 4, -1

    The correct option is C.

    5. Exam Tip

    When working with coordinates involving negative numbers, always keep them in brackets (e.g., $+(-2)$) to avoid sign errors. A common mistake is to subtract the coordinates instead of adding them; remember that the midpoint formula is an average, so you must always add the values together before dividing by 2.

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45. 45.

    If y = 4x3 - 2x2 + x, find $\frac{\delta y}{\delta x}$

    1. A.8x2 - 2x + 1
    2. B.8x2 - 4x + 1
    3. C.12x2 - 2x + 1
    4. D.12x2 - 4x + 1

    Show answer

    Answer: D. 12x2 - 4x + 1

    AI explanation

    Analysis

    The question asks for the derivative of the function $y = 4x^3 - 2x^2 + x$ with respect to $x$. This is a standard calculus problem involving the power rule for differentiation.

    Concept: The Power Rule

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

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

    In simple terms:

    1. Bring the exponent down to multiply with the coefficient.
    2. Subtract 1 from the original exponent.
    3. Remember that the derivative of $x$ (which is $x^1$) is $1$, and the derivative of a constant is $0$.

    Step-by-Step Solution

    Given the function:

    $y = 4x^3 - 2x^2 + x^1$

    Step 1: Differentiate $4x^3$ Multiply the exponent (3) by the coefficient (4) and subtract 1 from the exponent:

    $3 \times 4x^{3-1} = 12x^2$

    Step 2: Differentiate $-2x^2$ Multiply the exponent (2) by the coefficient (-2) and subtract 1 from the exponent:

    $2 \times -2x^{2-1} = -4x^1 = -4x$

    Step 3: Differentiate $x$ The derivative of $x$ is simply 1:

    $\frac{d}{dx}(x) = 1$

    Step 4: Combine the results

    $\frac{dy}{dx} = 12x^2 - 4x + 1$

    Conclusion

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

    Exam Tip

    When differentiating polynomials, always check the signs carefully. A common mistake is to forget to subtract 1 from the exponent or to miscalculate the product of the exponent and the coefficient. Also, remember that the derivative of any constant (like 5 or 10) is always 0, but the derivative of $x$ is always 1.

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

47. 47.

    Express the product of 0.00043 and 2000 in standard form.

    1. A.8.6 x 10-3
    2. B.8.3 x 10-2
    3. C.8.6 x 10-1
    4. D.8.6 x 10

    Show answer

    Answer: C. 8.6 x 10-1

    AI explanation

    Analysis

    To solve this problem, we need to find the product of two numbers given in decimal and whole number forms, and then convert the final result into standard form (also known as scientific notation). Standard form is written as $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

    Concept

    1. Multiplication: Multiply the two numbers directly.
    2. Standard Form Conversion: Move the decimal point so that only one non-zero digit remains to the left of the decimal point. Count how many places the decimal point moved to determine the power of 10.

    Step-by-Step Solution

    Step 1: Multiply the numbers We are calculating $0.00043 \times 2000$.

    To make this easier, we can rewrite the numbers: $0.00043 = 43 \times 10^{-5}$ $2000 = 2 \times 10^3$

    Now, multiply them: $(43 \times 10^{-5}) \times (2 \times 10^3)$ $= (43 \times 2) \times (10^{-5} \times 10^3)$ $= 86 \times 10^{-5 + 3}$ $= 86 \times 10^{-2}$

    Step 2: Convert to standard form The current result is $86 \times 10^{-2}$. In standard form, the number must be between 1 and 10. We need to change $86$ to $8.6$. To change $86$ to $8.6$, we divide by 10, which means we must multiply the power of 10 by 10 (or add 1 to the exponent):

    $86 \times 10^{-2} = (8.6 \times 10^1) \times 10^{-2}$ $= 8.6 \times 10^{1 + (-2)}$ $= 8.6 \times 10^{-1}$

    Conclusion

    Comparing our result ($8.6 \times 10^{-1}$) with the given options, it matches option C.

    Exam Tip

    When multiplying numbers with decimals and zeros, convert them to powers of 10 first. It prevents errors caused by miscounting decimal places. Remember: if you move the decimal point to the left, the exponent increases; if you move it to the right, the exponent decreases.

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