2011 JAMB Mathematics Past Questions

Questions

1. 1.

   If the numbers M, N, Q are in the ratio 5:4:3, find the value of $\frac{2N - Q}{M}$

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

   Show answer

   Answer: C. 1

   AI explanation

   Analysis

   To solve this problem, we need to use the properties of ratios. When quantities are given in a ratio, we can represent them as multiples of a common constant.

   Concept

   If $M:N:Q = 5:4:3$, we can assign a common multiplier, let's call it $x$, to each term:

   • $M = 5x$
   • $N = 4x$
   • $Q = 3x$

   By substituting these values into the given expression, the variable $x$ will cancel out, allowing us to find the numerical value of the expression.

   Solution

   Step 1: Substitute the values into the expression. The expression is:

   $\frac{2N - Q}{M}$

   Substitute $M = 5x$, $N = 4x$, and $Q = 3x$:

   $\frac{2(4x) - 3x}{5x}$

   Step 2: Simplify the numerator.

   $2(4x) = 8x$

   So, the expression becomes:

   $\frac{8x - 3x}{5x}$

   Step 3: Perform the subtraction.

   $8x - 3x = 5x$

   Now we have:

   $\frac{5x}{5x}$

   Step 4: Divide. Since $x$ is a common factor (and $x \neq 0$), they cancel out:

   $\frac{5}{5} = 1$

   Conclusion

   The value of the expression is 1. Comparing this to the given options: A. 2 B. 3 C. 1 D. 4

   The correct option is C.

   Exam Tip

   Whenever you see a ratio problem where you are asked to find the value of a fraction involving those variables, you don't need to find the actual values of the numbers. Simply treat the ratio parts as the actual values (e.g., let $M=5, N=4, Q=3$). This saves time during your JAMB exam!

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2. 2.

   $\begin{array}{c|c} Class Intervals & 0 - 2 & 3 - 5 & 6 - 8 & 9 - 11 & \\ \hline Frequency & 3 & 2 & 5 & 3 &\end{array}$ Find the mode of the above distribution.

   1. A.9
   2. B.8
   3. C.10
   4. D.7

   Show answer

   Answer: D. 7

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3. 3.

   A binary operation $\oplus$ om real numbers is defined by x $\oplus$ y = xy + x + y for two real numbers x and y. Find the value of 3 $\oplus$ - $\frac{2}{3}$.

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

   Show answer

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

   AI explanation

   1. Analyze the Question

   We are given a binary operation defined as $x \oplus y = xy + x + y$. We need to evaluate this operation for the specific values $x = 3$ and $y = -\frac{2}{3}$.

   2. Explain the Concept

   A binary operation is simply a rule that tells you how to combine two numbers to get a third number. In this case, the rule is:

   • Multiply the two numbers ($xy$)
   • Add the first number ($x$)
   • Add the second number ($y$)

   To solve this, we substitute the given values into the formula and follow the BODMAS rule (multiplication before addition).

   3. Show the Solution

   Given: $x \oplus y = xy + x + y$ Substitute $x = 3$ and $y = -\frac{2}{3}$:

   $3 \oplus (-\frac{2}{3}) = (3 \times -\frac{2}{3}) + 3 + (-\frac{2}{3})$

   Step 1: Perform the multiplication $3 \times (-\frac{2}{3}) = -2$

   Step 2: Substitute back into the expression $-2 + 3 - \frac{2}{3}$

   Step 3: Simplify the addition $(-2 + 3) - \frac{2}{3}$ $= 1 - \frac{2}{3}$

   Step 4: Subtract the fraction To subtract, convert $1$ to a fraction with a denominator of $3$: $\frac{3}{3} - \frac{2}{3} = \frac{1}{3}$

   4. Identify the Answer

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

   The calculated value matches option B.

   5. Tip

   When dealing with binary operations involving fractions, always simplify the multiplication part first. It often results in a whole number that makes the subsequent addition or subtraction much easier to handle. Don't rush to find a common denominator until you have simplified the entire expression as much as possible!

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4. 4.

   The sum of four consecutive integers is 34. Find the least of these numbers

   1. A.7
   2. B.6
   3. C.8
   4. D.5

   Show answer

   Answer: A. 7

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

6. 6.

   Find $\int$ cos4 x dx

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

   Show answer

   Answer: D. $\frac{1}{4}$ sin 4x + k

   AI explanation

   To solve the integral $\int \cos(4x) \, dx$, we use the standard rule for integrating trigonometric functions of the form $\cos(ax)$.

   1. Analyze the question

   We are asked to find the indefinite integral of the function $f(x) = \cos(4x)$ with respect to $x$.

   2. Explain the concept

   The integration rule for a cosine function with a linear argument $ax$ is:

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

   where $k$ is the constant of integration. This is derived from the reverse of the chain rule in differentiation. Since the derivative of $\sin(4x)$ is $4\cos(4x)$, we must divide by $4$ to balance the equation when integrating $\cos(4x)$.

   3. Show the solution

   • Step 1: Identify the constant $a$ in the expression $\cos(4x)$. Here, $a = 4$.
   • Step 2: Apply the integration formula: $$\int \cos(4x) \, dx = \frac{1}{4} \sin(4x) + k$$
   • Step 3: Compare this result with the given options.

   4. Identify the answer

   The result $\frac{1}{4} \sin(4x) + k$ matches option D.

   5. Exam Tip

   Whenever you are integrating a trigonometric function like $\sin(ax)$ or $\cos(ax)$, always remember to divide by the coefficient of $x$. A common mistake is to multiply by the coefficient instead of dividing. Always check your answer by differentiating it; if you differentiate $\frac{1}{4} \sin(4x)$, you get $\frac{1}{4} \cdot 4 \cos(4x) = \cos(4x)$, which confirms the answer is correct.

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7. 7.

   Make R the subject of the formula if T = $\frac {KR^2 + M}{3}$

   1. A.$\sqrt\frac{3T - K}{M}$
   2. B.$\sqrt\frac{3T - M}{K}$
   3. C.$\sqrt\frac{3T + K}{M}$
   4. D.$\sqrt\frac{3T - K}{M}$

   Show answer

   Answer: B. $\sqrt\frac{3T - M}{K}$

   AI explanation

   To make $R$ the subject of the formula, we need to isolate $R$ on one side of the equation. Let’s break this down step-by-step.

   1. Analyze the equation

   We are given: $T = \frac{KR^2 + M}{3}$

   Our goal is to move all other variables ($T, K,$ and $M$) to the other side, leaving $R$ by itself.

   2. Step-by-step Solution

   Step 1: Eliminate the fraction Multiply both sides of the equation by 3 to remove the denominator: $3 \times T = \left( \frac{KR^2 + M}{3} \right) \times 3$ $3T = KR^2 + M$

   Step 2: Isolate the term containing $R$ Subtract $M$ from both sides of the equation: $3T - M = KR^2$

   Step 3: Isolate $R^2$ Divide both sides by $K$: $\frac{3T - M}{K} = R^2$

   Step 4: Solve for $R$ Take the square root of both sides to get $R$: $R = \sqrt{\frac{3T - M}{K}}$

   3. Conclusion

   Comparing our result with the given options: A. $\sqrt\frac{3T - K}{M}$ B. $\sqrt\frac{3T - M}{K}$ C. $\sqrt\frac{3T + K}{M}$ D. $\sqrt\frac{3T - K}{M}$

   The derived expression matches option B.

   Exam Tip

   When rearranging formulas, always follow the reverse of the BODMAS rule. Deal with additions or subtractions first (moving them to the other side), then multiplications or divisions, and finally powers or roots. This systematic approach prevents common errors when moving terms across the equals sign.

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8. 8.

   Find the value of x at the minimum point of the curve y = x3 + x2 - x + 1

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

   Show answer

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

   AI explanation

   To find the minimum point of the curve $y = x^3 + x^2 - x + 1$, we follow the principles of calculus used in finding stationary points.

   1. Analyze the question

   We are looking for the value of $x$ where the function reaches its minimum. This is a stationary point problem where we must find the derivative, set it to zero, and use the second derivative test to confirm if it is a minimum.

   2. Explain the concept

   • First Derivative ($\frac{dy}{dx}$): This represents the slope of the curve. At any stationary point (maximum or minimum), the slope is zero.
   • Second Derivative ($\frac{d^2y}{dx^2}$): This helps us classify the stationary point. • If $\frac{d^2y}{dx^2} > 0$, the point is a minimum. • If $\frac{d^2y}{dx^2} < 0$, the point is a maximum.

   3. Show the solution

   Step 1: Find the first derivative. Given $y = x^3 + x^2 - x + 1$ $\frac{dy}{dx} = 3x^2 + 2x - 1$

   Step 2: Set the derivative to zero to find stationary points. $3x^2 + 2x - 1 = 0$

   Factorize the quadratic equation: $3x^2 + 3x - x - 1 = 0$ $3x(x + 1) - 1(x + 1) = 0$ $(3x - 1)(x + 1) = 0$

   So, $x = \frac{1}{3}$ or $x = -1$.

   Step 3: Find the second derivative to classify the points. $\frac{d^2y}{dx^2} = \frac{d}{dx}(3x^2 + 2x - 1) = 6x + 2$

   Step 4: Test the points.

   • For $x = -1$: $\frac{d^2y}{dx^2} = 6(-1) + 2 = -4$. Since $-4 < 0$, this is a maximum.
   • For $x = \frac{1}{3}$: $\frac{d^2y}{dx^2} = 6(\frac{1}{3}) + 2 = 2 + 2 = 4$. Since $4 > 0$, this is a minimum.

   4. Identify the answer

   The value of $x$ at the minimum point is $\frac{1}{3}$. This matches option A.

   5. Tip

   When dealing with cubic functions ($ax^3 + bx^2 + cx + d$), the stationary points are always found by solving the quadratic equation resulting from the first derivative. Always remember: "Positive second derivative = Minimum (think 'positive/happy face'), Negative second derivative = Maximum (think 'negative/sad face')."

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9. 9.

   Rationalize $\frac{2 - \sqrt5}{3 - \sqrt5}$

   1. A.$\frac{1 - \sqrt5}{2}$
   2. B.$\frac{1 - \sqrt5}{4}$
   3. C.$\frac{ \sqrt5 - 1}{2}$
   4. D.$\frac{1 + \sqrt5}{4}$

   Show answer

   Answer: B. $\frac{1 - \sqrt5}{4}$

   AI explanation

   Analysis

   To rationalize a fraction with a radical (surd) in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $(3 - \sqrt{5})$ is $(3 + \sqrt{5})$.

   Concept

   The goal is to eliminate the square root from the denominator. We use the difference of two squares identity: $(a - b)(a + b) = a^2 - b^2$

   Step-by-Step Solution

   Step 1: Set up the multiplication

   $\frac{2 - \sqrt{5}}{3 - \sqrt{5}} \times \frac{3 + \sqrt{5}}{3 + \sqrt{5}}$

   Step 2: Expand the numerator $(2 - \sqrt{5})(3 + \sqrt{5})$ $= 2(3) + 2(\sqrt{5}) - 3(\sqrt{5}) - (\sqrt{5})(\sqrt{5})$ $= 6 + 2\sqrt{5} - 3\sqrt{5} - 5$ $= (6 - 5) + (2\sqrt{5} - 3\sqrt{5})$ $= 1 - \sqrt{5}$

   Step 3: Expand the denominator $(3 - \sqrt{5})(3 + \sqrt{5})$ $= 3^2 - (\sqrt{5})^2$ $= 9 - 5$ $= 4$

   Step 4: Combine the results

   $\frac{1 - \sqrt{5}}{4}$

   Conclusion

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

   Exam Tip: When rationalizing, always remember to change the sign in the middle of the denominator to get the conjugate. If the denominator is $(a - \sqrt{b})$, the conjugate is $(a + \sqrt{b})$. If it is $(a + \sqrt{b})$, the conjugate is $(a - \sqrt{b})$. This simple sign flip is the key to clearing the surd!

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10. 10.

    The seconds term of a geometric series is 4 while the fourth term is 16. Find the sum of the first five terms

    1. A.60
    2. B.62
    3. C.54
    4. D.64

    Show answer

    Answer: B. 62

    AI explanation

    Analysis

    To solve this problem, we need to use the properties of a Geometric Progression (GP). A GP is defined by its first term ($a$) and its common ratio ($r$). The $n^{th}$ term of a GP is given by the formula: $T_n = ar^{n-1}$

    Concept

    1. Identify the given terms: • Second term ($T_2$) = $ar^{2-1} = ar = 4$ --- (Equation 1) • Fourth term ($T_4$) = $ar^{4-1} = ar^3 = 16$ --- (Equation 2)

    2. Find $r$ and $a$: Divide Equation 2 by Equation 1: $\frac{ar^3}{ar} = \frac{16}{4}$ $r^2 = 4$ $r = \sqrt{4} = 2$ (We take the positive value for this series)

    Substitute $r = 2$ into Equation 1: $a(2) = 4$ $a = 2$

    3. Calculate the sum of the first five terms ($S_n$): The formula for the sum of the first $n$ terms of a GP is: $S_n = \frac{a(r^n - 1)}{r - 1}$

    Solution

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

    $S_5 = \frac{2(2^5 - 1)}{2 - 1}$ $S_5 = \frac{2(32 - 1)}{1}$ $S_5 = 2 \times 31$ $S_5 = 62$

    Conclusion

    Comparing our result with the options provided, we see that 62 corresponds to option B.

    Exam Tip: When dealing with geometric series in JAMB, if the numbers are small (like the first five terms), you can also list them out manually to verify your answer:

    • $T_1 = a = 2$
    • $T_2 = ar = 4$
    • $T_3 = ar^2 = 8$
    • $T_4 = ar^3 = 16$
    • $T_5 = ar^4 = 32$ Sum = $2 + 4 + 8 + 16 + 32 = 62$. This is a great way to double-check your work under pressure!

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11. 11.

    If log318 + log33 - log3x = 3, Find x.

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

    Show answer

    Answer: B. 2

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12. 12.

    Evaluate $\int^{1}_{0}$(3 - 2x)dx

    1. A.33m
    2. B.5
    3. C.2
    4. D.6

    Show answer

    Answer: C. 2

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13. 13.

    Find the remainder when X3 - 2X2 + 3X - 3 is divided by X2 + 1

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

    Show answer

    Answer: A. 2X - 1

    AI explanation

    To solve this problem, we will use the Polynomial Long Division method. This is the most reliable way to find the remainder when dividing one polynomial by another.

    1. Analyze the Question

    We are dividing the polynomial $P(x) = x^3 - 2x^2 + 3x - 3$ by the divisor $D(x) = x^2 + 1$. We need to find the remainder.

    2. Explain the Concept

    When dividing polynomials, we seek a quotient $Q(x)$ and a remainder $R(x)$ such that: $P(x) = D(x) \cdot Q(x) + R(x)$ The degree of the remainder $R(x)$ must be less than the degree of the divisor $D(x)$. Since our divisor is of degree 2 ($x^2$), our remainder will be of the form $ax + b$.

    3. Show the Solution

    Step 1: Divide the first term of the dividend by the first term of the divisor. Divide $x^3$ by $x^2$: $x^3 \div x^2 = x$ (This is the first term of our quotient).

    Step 2: Multiply the divisor by this term and subtract from the dividend. $(x^2 + 1) \cdot x = x^3 + x$ Subtracting this from the original polynomial: $(x^3 - 2x^2 + 3x - 3) - (x^3 + 0x^2 + x)$ $= -2x^2 + 2x - 3$

    Step 3: Divide the first term of the new expression by the first term of the divisor. Divide $-2x^2$ by $x^2$: $-2x^2 \div x^2 = -2$ (This is the second term of our quotient).

    Step 4: Multiply the divisor by this term and subtract. $(x^2 + 1) \cdot (-2) = -2x^2 - 2$ Subtracting this from our current expression: $(-2x^2 + 2x - 3) - (-2x^2 + 0x - 2)$ $= 2x - 3 + 2$ $= 2x - 1$

    Since the degree of $2x - 1$ (which is 1) is less than the degree of $x^2 + 1$ (which is 2), we stop here. The remainder is $2x - 1$.

    4. Identify the Answer

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

    The correct option is A.

    5. Exam Tip

    When performing polynomial division, always ensure your polynomials are written in descending order of powers (e.g., $x^3, x^2, x^1, x^0$). If a term is missing (like if there was no $x$ term), use a placeholder like $0x$ to keep your columns aligned and avoid silly subtraction errors!

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14. 14.

    Raial has 7 different posters to be hanged in her bedroom, living room and kitchen. Assuming she has plans to place at least a poster in each of the 3 rooms, how many choices does she have?

    1. A.49
    2. B.170
    3. C.21
    4. D.210

    Show answer

    Answer: D. 210

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15. 15.

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

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

    Show answer

    Answer: B. x $\geq$ -1

    AI explanation

    Analysis

    To solve the inequality $-6(x + 3) \leq 4(x - 2)$, we need to expand the brackets, collect like terms, and isolate the variable $x$. A crucial rule to remember is that when you divide or multiply an inequality by a negative number, the inequality sign must be reversed.

    Concept

    1. Expansion: Use the distributive property to remove the brackets.
    2. Grouping: Move all terms containing $x$ to one side and constant terms to the other.
    3. Division: Divide by the coefficient of $x$. If the coefficient is negative, flip the inequality sign ($\leq$ becomes $\geq$).

    Step-by-Step Solution

    Step 1: Expand the brackets $-6 \times x - 6 \times 3 \leq 4 \times x - 4 \times 2$ $-6x - 18 \leq 4x - 8$

    Step 2: Collect like terms Move $4x$ to the left side and $-18$ to the right side: $-6x - 4x \leq -8 + 18$

    Step 3: Simplify both sides $-10x \leq 10$

    Step 4: Solve for $x$ Divide both sides by $-10$. Because we are dividing by a negative number, the inequality sign $\leq$ changes to $\geq$: $x \geq \frac{10}{-10}$ $x \geq -1$

    Conclusion

    Comparing our result ($x \geq -1$) with the given options, it matches option B.

    Exam Tip: Always double-check your sign when dividing by a negative number! Many students lose marks here by forgetting to flip the inequality sign. If you are ever unsure, pick a number that satisfies your final answer (e.g., $x = 0$) and plug it back into the original equation to see if it holds true.

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16. 16.

    Find the derivative of $\frac {\sin\theta}{\cos\theta}$

    1. A.sec2 $\theta$
    2. B.tan $\theta$ cosec $\theta$
    3. C.cosec $\theta$sec $\theta$
    4. D.cosec2$\theta$

    Show answer

    Answer: A. sec2 $\theta$

    AI explanation

    Analysis

    The question asks for the derivative of the trigonometric expression $\frac{\sin\theta}{\cos\theta}$ with respect to $\theta$.

    Concept

    1. Trigonometric Identity: Recall that $\frac{\sin\theta}{\cos\theta} = \tan\theta$.
    2. Differentiation Rule: The derivative of $\tan\theta$ with respect to $\theta$ is a standard derivative in calculus: $$\frac{d}{d\theta}(\tan\theta) = \sec^2\theta$$
    3. Alternative Method (Quotient Rule): If you prefer to use the quotient rule $\frac{d}{d\theta}(\frac{u}{v}) = \frac{v\frac{du}{d\theta} - u\frac{dv}{d\theta}}{v^2}$: • Let $u = \sin\theta \implies \frac{du}{d\theta} = \cos\theta$ • Let $v = \cos\theta \implies \frac{dv}{d\theta} = -\sin\theta$ • Applying the formula: $$\frac{(\cos\theta)(\cos\theta) - (\sin\theta)(-\sin\theta)}{\cos^2\theta}$$ $$\frac{\cos^2\theta + \sin^2\theta}{\cos^2\theta}$$ • Since $\cos^2\theta + \sin^2\theta = 1$ (Pythagorean identity): $$\frac{1}{\cos^2\theta} = \sec^2\theta$$

    Solution

    Both the direct derivative of the identity and the quotient rule lead to the same result:

    $\frac{d}{d\theta}(\tan\theta) = \sec^2\theta$

    Comparing this to the provided options: A. $\sec^2\theta$ B. $\tan\theta \csc\theta$ C. $\csc\theta \sec\theta$ D. $\csc^2\theta$

    The result matches option A.

    Tip

    Always simplify trigonometric expressions before differentiating if possible. Converting $\frac{\sin\theta}{\cos\theta}$ to $\tan\theta$ first saves you the time and potential errors associated with the quotient rule!

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17. 17.

    A solid metal cube of side 3 cm is placed in a rectangular tank of dimension 3, 4 and 5 cm. What volume of water can the tank now hold

    1. A.48 cm3
    2. B.33 cm3
    3. C.60 cm3
    4. D.27 cm3

    Show answer

    Answer: B. 33 cm3

    AI explanation

    1. Analyze the question

    We are given a rectangular tank with dimensions 3 cm, 4 cm, and 5 cm. A solid metal cube with a side length of 3 cm is placed inside this tank. We need to find the remaining volume available in the tank to hold water after the cube has occupied some of the space.

    2. Explain the concept

    To solve this, we use the principle of displacement:

    • Total Volume of the Tank: The capacity of a rectangular container is calculated as $Length \times Width \times Height$.
    • Volume of the Cube: The volume of a cube is calculated as $side^3$ (or $side \times side \times side$).
    • Remaining Volume: Once the solid cube is placed inside, it occupies space that water can no longer fill. Therefore, the volume of water the tank can hold is the Total Volume of the Tank minus the Volume of the Cube.

    3. Show the solution

    Step 1: Calculate the total volume of the rectangular tank.

    $\text{Volume of tank} = L \times W \times H$

    $\text{Volume of tank} = 3\text{ cm} \times 4\text{ cm} \times 5\text{ cm} = 60\text{ cm}^3$

    Step 2: Calculate the volume of the metal cube.

    $\text{Volume of cube} = \text{side}^3$

    $\text{Volume of cube} = 3\text{ cm} \times 3\text{ cm} \times 3\text{ cm} = 27\text{ cm}^3$

    Step 3: Calculate the remaining volume for water.

    $\text{Volume of water} = \text{Total Volume} - \text{Volume of Cube}$

    $\text{Volume of water} = 60\text{ cm}^3 - 27\text{ cm}^3 = 33\text{ cm}^3$

    4. Identify the answer

    Comparing our result to the given options: A. 48 cm³ B. 33 cm³ C. 60 cm³ D. 27 cm³

    The correct option is B.

    5. Tip

    When solving geometry problems involving "displacement" (like dropping an object into a container), always remember: Water Volume = Container Volume - Submerged Object Volume. Also, ensure all your units are consistent (in this case, everything is in cm, so the result is correctly in cm³). You've got this!

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18. 18.

    Evaluate $\begin{vmatrix} 4 & 2 & -1 \\ 2 & 3 & -1 \\ -1 & 1 & 3 \end{vmatrix}$

    1. A.25
    2. B.45
    3. C.15
    4. D.55

    Show answer

    Answer: A. 25

    AI explanation

    To evaluate the determinant of a $3 \times 3$ matrix, we use the expansion method along the first row.

    1. Analyze the question

    We are given the matrix:

    $A = \begin{vmatrix} 4 & 2 & -1 \\ 2 & 3 & -1 \\ -1 & 1 & 3 \end{vmatrix}$

    We need to find the determinant, denoted as $|A|$.

    2. Explain the concept

    The determinant of a $3 \times 3$ matrix $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$ is calculated as:

    $a(ei - fh) - b(di - fg) + c(dh - eg)$

    Remember the sign pattern for the first row is $(+, -, +)$.

    3. Show the solution

    Let's expand along the first row: $|A| = 4 \begin{vmatrix} 3 & -1 \\ 1 & 3 \end{vmatrix} - 2 \begin{vmatrix} 2 & -1 \\ -1 & 3 \end{vmatrix} + (-1) \begin{vmatrix} 2 & 3 \\ -1 & 1 \end{vmatrix}$

    Step 1: Calculate the $2 \times 2$ determinants

    • First part: $4[(3 \times 3) - (-1 \times 1)] = 4[9 - (-1)] = 4[10] = 40$
    • Second part: $-2[(2 \times 3) - (-1 \times -1)] = -2[6 - 1] = -2[5] = -10$
    • Third part: $-1[(2 \times 1) - (3 \times -1)] = -1[2 - (-3)] = -1[5] = -5$

    Step 2: Combine the results $|A| = 40 - 10 - 5$ $|A| = 25$

    4. Identify the answer

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

    5. Exam Tip

    When calculating determinants, be extremely careful with the signs of negative numbers inside the brackets (e.g., $-(-1)$ becomes $+1$). A common mistake is losing a negative sign during the multiplication of the $2 \times 2$ minors. Always write out the small brackets clearly to avoid these "silly" errors!

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19. 19.

    Simplify ($\sqrt2 + \frac{1}{\sqrt3})(\sqrt2 - \frac{1}{\sqrt3}$)

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

    Show answer

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

    AI explanation

    Analysis

    The expression given is $(\sqrt{2} + \frac{1}{\sqrt{3}})(\sqrt{2} - \frac{1}{\sqrt{3}})$. This is a classic algebraic problem that follows the Difference of Two Squares pattern.

    Concept

    The difference of two squares identity states that:

    $(a + b)(a - b) = a^2 - b^2$

    In this problem:

    • $a = \sqrt{2}$
    • $b = \frac{1}{\sqrt{3}}$

    Step-by-Step Solution

    Step 1: Apply the identity Using the formula $a^2 - b^2$:

    $(\sqrt{2})^2 - (\frac{1}{\sqrt{3}})^2$

    Step 2: Evaluate the squares

    • $(\sqrt{2})^2 = 2$
    • $(\frac{1}{\sqrt{3}})^2 = \frac{1^2}{(\sqrt{3})^2} = \frac{1}{3}$

    Step 3: Subtract the values Now, we have:

    $2 - \frac{1}{3}$

    Step 4: Find a common denominator To subtract, express 2 as a fraction with a denominator of 3:

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

    So,

    $\frac{6}{3} - \frac{1}{3} = \frac{6 - 1}{3} = \frac{5}{3}$

    Conclusion

    The result of the simplification is $\frac{5}{3}$. Comparing this to the options provided, it matches option B.

    Exam Tip

    Whenever you see an expression in the form $(x + y)(x - y)$, don't waste time using the "FOIL" method (multiplying term by term). Simply square the first term, square the second term, and subtract them. It saves time and reduces the risk of calculation errors!

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20. 20.

    The inverse of matrix N = $\begin{vmatrix} 2 & 3 \\ 1 & 4\end{vmatrix}$ is

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

    Show answer

    Answer: B. $\frac{1}{5}$ $\begin{vmatrix} 4 & -3 \\ -1 & 2\end{vmatrix}$

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21. 21.

    In how many was can the letters of the word ELATION be arranged?

    1. A.6!
    2. B.7!
    3. C.5!
    4. D.8!

    Show answer

    Answer: B. 7!

    AI explanation

    1. Analyze the question

    The question asks for the number of ways to arrange the letters of the word ELATION. This is a problem of permutations, specifically dealing with the arrangement of distinct objects.

    2. Explain the concept

    When you are asked to arrange a set of objects (or letters), you use the concept of factorials.

    • If all the items in a set are distinct (none of them repeat), the number of ways to arrange $n$ items is given by $n!$ (n factorial).
    • $n! = n \times (n-1) \times (n-2) \times \dots \times 1$.

    First, let's count the letters in the word ELATION: E, L, A, T, I, O, N. There are 7 letters in total.

    Next, check for repetitions:

    • E: 1
    • L: 1
    • A: 1
    • T: 1
    • I: 1
    • O: 1
    • N: 1 Since all 7 letters are unique (distinct), we do not need to divide by any repeating factorials.

    3. Show the solution

    • Total number of letters ($n$) = 7
    • Since all letters are distinct, the number of arrangements is $7!$
    • $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
    • $7! = 5040$

    Looking at the provided options, the question asks for the answer in factorial notation rather than the calculated numerical value.

    4. Identify the answer

    Comparing our result ($7!$) with the given options: A. 6! B. 7! C. 5! D. 8!

    The correct option is B.

    5. Tip

    Always check for repeating letters first! If the word had been "BALL" (4 letters, with two Ls), the formula would be $\frac{4!}{2!}$. Because "ELATION" has no repeating letters, it is a straightforward $n!$ calculation.

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22. 22.

    In how many ways can five people sit round a circular table?

    1. A.24
    2. B.60
    3. C.12
    4. D.120

    Show answer

    Answer: A. 24

    AI explanation

    1. Analyze the Question

    The question asks for the number of ways to arrange $n$ distinct objects (people) around a circular table. In circular permutations, the arrangement is considered the same if everyone shifts one position to the left or right, as the relative positions remain unchanged.

    2. Explain the Concept

    When arranging objects in a straight line (linear permutation), the number of ways to arrange $n$ objects is $n!$ (n factorial).

    However, in a circular arrangement, we fix one person in a seat to break the symmetry. Once one person is fixed, the remaining $(n - 1)$ people can be arranged in the remaining seats in $(n - 1)!$ ways.

    The formula for circular permutation is:

    $\text{Number of ways} = (n - 1)!$

    3. Show the Solution

    • Step 1: Identify the number of people ($n$). Here, $n = 5$.
    • Step 2: Apply the circular permutation formula: $$\text{Ways} = (n - 1)!$$
    • Step 3: Substitute the value of $n$: $$\text{Ways} = (5 - 1)!$$ $$\text{Ways} = 4!$$
    • Step 4: Calculate the factorial: $$4! = 4 \times 3 \times 2 \times 1$$ $$4 \times 3 = 12$$ $$12 \times 2 = 24$$ $$24 \times 1 = 24$$

    4. Identify the Answer

    The calculation gives us 24. Comparing this to the provided options, it matches option A.

    5. Exam Tip

    Always remember the difference between linear and circular arrangements:

    • Linear: $n!$ (e.g., 5 people in a row = $5! = 120$)
    • Circular: $(n - 1)!$ (e.g., 5 people at a table = $4! = 24$) A quick way to remember this is that in a circle, you "lose" one position because there is no "start" or "end" seat.

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23. 23.

    A chord of circle of radius 7cm is 5cm from the centre of the circle.What is the length of the chord?

    1. A.4√6 cm
    2. B.3√6 cm
    3. C.6√6 cm
    4. D.2√6 cm

    Show answer

    Answer: A. 4√6 cm

    AI explanation

    Analysis

    To find the length of the chord, we need to visualize the circle. We are given:

    • The radius of the circle ($r$) = 7 cm.
    • The perpendicular distance from the center to the chord ($d$) = 5 cm.

    Concept

    When a line is drawn from the center of a circle perpendicular to a chord, it bisects the chord (divides it into two equal parts). This creates a right-angled triangle where:

    • The hypotenuse is the radius of the circle.
    • One leg is the distance from the center to the chord.
    • The other leg is half the length of the chord ($x$).

    We use the Pythagorean theorem: $a^2 + b^2 = c^2$.

    Solution

    1. Set up the equation: Let $x$ be half the length of the chord. $x^2 + d^2 = r^2$ $x^2 + 5^2 = 7^2$

    2. Solve for $x$: $x^2 + 25 = 49$ $x^2 = 49 - 25$ $x^2 = 24$ $x = \sqrt{24}$

    3. Simplify the radical: $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ cm.

    4. Find the full length of the chord: Since $x$ is only half the chord, the full length is $2x$: Length $= 2 \times (2\sqrt{6}) = 4\sqrt{6}$ cm.

    Conclusion

    The length of the chord is $4\sqrt{6}$ cm. Comparing this to the given options, it matches option A.

    Exam Tip

    Whenever you see a problem involving a chord and the distance from the center, immediately sketch a right-angled triangle. Remember that the radius is always the hypotenuse. If you forget the Pythagorean theorem, just remember: (Half-chord)² + (Distance)² = (Radius)².

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24. 24.

    If 2q35 = 778, find q

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

    Show answer

    Answer: A. 2

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

26. 26.

    In a right angled triangle, if tan $\theta$ = $\frac{3}{4}$. What is cos$\theta$ - sin$\theta$?

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

    Show answer

    Answer: C. $\frac{1}{5}$

    AI explanation

    1. Analyze the question

    We are given the value of $\tan \theta$ in a right-angled triangle and asked to find the value of the expression $(\cos \theta - \sin \theta)$.

    2. Explain the concept

    In a right-angled triangle, we use the SOH CAH TOA mnemonic:

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

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

    • Opposite = 3
    • Adjacent = 4

    To find the Hypotenuse, we use the Pythagoras theorem: $\text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2$

    3. Show the solution

    Step 1: Find the Hypotenuse. $H^2 = 3^2 + 4^2$ $H^2 = 9 + 16$ $H^2 = 25$ $H = \sqrt{25} = 5$

    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 $\cos \theta - \sin \theta$. $\cos \theta - \sin \theta = \frac{4}{5} - \frac{3}{5}$ $\cos \theta - \sin \theta = \frac{4 - 3}{5} = \frac{1}{5}$

    4. Identify the answer

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

    5. Tip

    Whenever you see a trigonometric ratio involving 3 and 4, remember the 3-4-5 Pythagorean triple. It is a common pattern in JAMB questions that saves you time from calculating the hypotenuse from scratch!

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27. 27.

    Solve the inequality x2 + 2x > 15.

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

    Show answer

    Answer: D. x > 3 or x < -5

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28. 28.

    The derivatives of (2x + 1)(3x + 1) is

    1. A.12x + 1
    2. B.6x + 5
    3. C.6x + 1
    4. D.12x + 5

    Show answer

    Answer: D. 12x + 5

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29. 29.

    Solve for x and y respectively in the simultaneous equations -2x - 5y = 3. x + 3y = 0

    1. A.-3, -9
    2. B.9, -3
    3. C.-9,3
    4. D.3, -9

    Show answer

    Answer: C. -9,3

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30. 30.

    A man invested N5,000 for 9 months at 4%. What is the simple interest?

    1. A.N150
    2. B.N220
    3. C.N130
    4. D.N250

    Show answer

    Answer: A. N150

    AI explanation

    Analysis

    To find the simple interest, we need to use the standard simple interest formula. We are given the principal, the rate, and the time period.

    Concept

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

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

    Where:

    • $P$ (Principal) = The amount invested = N5,000
    • $R$ (Rate) = The interest rate per annum = 4%
    • $T$ (Time) = The duration of the investment in years. Since the time is given in months, we must convert it to years by dividing by 12 (because there are 12 months in a year).

    Step-by-Step Solution

    1. Convert the time to years: $T = \frac{9 \text{ months}}{12 \text{ months/year}} = \frac{3}{4} \text{ year}$ (or 0.75 years).

    2. Apply the formula: $I = \frac{5000 \times 4 \times (\frac{3}{4})}{100}$

    3. Simplify the calculation: First, multiply the principal by the rate: $5000 \times 4 = 20,000$

    Next, multiply by the time ($T = 0.75$): $20,000 \times 0.75 = 15,000$

    Finally, divide by 100: $I = \frac{15,000}{100} = 150$

    The simple interest is N150.

    Conclusion

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

    Exam Tip

    Always check the unit of time in interest questions! If the time is given in months, divide by 12; if it is given in days, divide by 365 (or 360, depending on the specific exam convention, though 365 is standard). Never plug the number of months directly into the formula as "T" without converting it to years first.

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31. 31.

    If x varies directly as square root of y and x = 81 when y = 9, Find x when y = 1$\frac{7}{9}$

    1. A.20$\frac{1}{4}$
    2. B.27
    3. C.2$\frac{1}{4}$
    4. D.36

    Show answer

    Answer: D. 36

    AI explanation

    1. Analyze the question

    We are given a variation problem where $x$ varies directly as the square root of $y$. We are provided with an initial set of values ($x=81, y=9$) to find the constant of variation, and then asked to find the new value of $x$ when $y = 1\frac{7}{9}$.

    2. Explain the concept

    • Direct Variation: When $x$ varies directly as $\sqrt{y}$, it is written mathematically as $x \propto \sqrt{y}$.
    • Constant of Variation: To turn this into an equation, we introduce a constant $k$: $$x = k\sqrt{y}$$
    • Steps: • Find $k$ using the first set of values. • Use the value of $k$ to solve for $x$ using the second value of $y$.

    3. Show the solution

    Step 1: Find the constant $k$ Given $x = 81$ and $y = 9$:

    $81 = k\sqrt{9}$

    $81 = k \times 3$

    $k = \frac{81}{3}$

    $k = 27$

    Step 2: Find $x$ when $y = 1\frac{7}{9}$ First, convert the mixed fraction to an improper fraction:

    $y = 1\frac{7}{9} = \frac{(1 \times 9) + 7}{9} = \frac{16}{9}$

    Now, substitute $k = 27$ and $y = \frac{16}{9}$ into the equation $x = k\sqrt{y}$:

    $x = 27 \times \sqrt{\frac{16}{9}}$

    $x = 27 \times \frac{\sqrt{16}}{\sqrt{9}}$

    $x = 27 \times \frac{4}{3}$

    $x = 9 \times 4$

    $x = 36$

    4. Identify the answer

    Comparing our result to the options provided: A. 20$\frac{1}{4}$ B. 27 C. 2$\frac{1}{4}$ D. 36

    The correct option is D.

    5. Tip

    When dealing with square roots in variation problems, always convert mixed fractions to improper fractions first. It makes finding the square root much easier because the numerator and denominator will often be perfect squares (like 16 and 9 in this case).

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32. 32.

    Find the probability that a number picked at random from the set(43, 44, 45, ..., 60) is a prime number.

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

    Show answer

    Answer: C. $\frac{2}{9}$

    AI explanation

    1. Analyze the Question

    We need to find the probability of picking a prime number from the set of integers ranging from 43 to 60 inclusive. Probability is defined as:

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

    2. Explain the Concept

    • Total outcomes: To find the number of integers in a range $[a, b]$, use the formula: $(b - a) + 1$.
    • Prime numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. We need to carefully check each number in the set to see if it is divisible by any smaller prime numbers (like 2, 3, 5, or 7).

    3. Show the Solution

    Step 1: Find the total number of elements in the set. The set is $\{43, 44, 45, \dots, 60\}$. Total count = $(60 - 43) + 1 = 17 + 1 = 18$.

    Step 2: Identify the prime numbers in the set. Let's list the numbers and check them:

    • 43: Prime (Yes)
    • 44: Even (No)
    • 45: Divisible by 5 (No)
    • 46: Even (No)
    • 47: Prime (Yes)
    • 48: Even (No)
    • 49: $7 \times 7$ (No)
    • 50: Even (No)
    • 51: $3 \times 17$ (No)
    • 52: Even (No)
    • 53: Prime (Yes)
    • 54: Even (No)
    • 55: Divisible by 5 (No)
    • 56: Even (No)
    • 57: $3 \times 19$ (No)
    • 58: Even (No)
    • 59: Prime (Yes)
    • 60: Even (No)

    The prime numbers are $\{43, 47, 53, 59\}$. Number of favorable outcomes = 4.

    Step 3: Calculate the probability.

    $\text{Probability} = \frac{4}{18}$

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

    $\text{Probability} = \frac{2}{9}$

    4. Identify the Answer

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

    5. Exam Tip

    When counting numbers in a range, students often forget to add the "+ 1". Always remember: if you are counting from 1 to 10, there are $(10 - 1) + 1 = 10$ numbers, not 9! Also, for numbers like 51 and 57, always check for divisibility by 3 by adding the digits (e.g., $5+1=6$, which is divisible by 3, so 51 is not prime).

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33. 33.

    Find the equation of a line perpendicular to line 2y = 5x + 4 which passes through (4, 2).

    1. A.5y - 2x -18 = 0
    2. B.5y + 2x - 18 = 0
    3. C.5y - 2x + 18 = 0
    4. D.5y + 2x - 2 = 0

    Show answer

    Answer: B. 5y + 2x - 18 = 0

    AI explanation

    1. Analyze the question

    We need to find the equation of a line that is perpendicular to a given line ($2y = 5x + 4$) and passes through a specific point $(4, 2)$.

    2. Explain the concept

    • Slope-Intercept Form: A line equation in the form $y = mx + c$ has a slope of $m$.
    • Perpendicular Slopes: If two lines are perpendicular, the product of their slopes is $-1$. That is, $m_1 \times m_2 = -1$, or $m_2 = -\frac{1}{m_1}$.
    • Point-Slope Formula: To find the equation of a line passing through $(x_1, y_1)$ with slope $m$, we use: $y - y_1 = m(x - x_1)$.

    3. Show the solution

    Step 1: Find the slope of the given line. The given equation is $2y = 5x + 4$. Divide by 2 to get it into $y = mx + c$ form: $y = \frac{5}{2}x + 2$ The slope ($m_1$) is $\frac{5}{2}$.

    Step 2: Find the slope of the perpendicular line ($m_2$). Since $m_1 \times m_2 = -1$: $\frac{5}{2} \times m_2 = -1$ $m_2 = -\frac{2}{5}$

    Step 3: Use the point-slope formula with point (4, 2). $y - y_1 = m_2(x - x_1)$ $y - 2 = -\frac{2}{5}(x - 4)$

    Step 4: Simplify the equation. Multiply both sides by 5 to clear the fraction: $5(y - 2) = -2(x - 4)$ $5y - 10 = -2x + 8$

    Rearrange to match the format of the options (bringing all terms to one side): $5y + 2x - 10 - 8 = 0$ $5y + 2x - 18 = 0$

    4. Identify the answer

    Comparing our result to the given options, the correct equation is $5y + 2x - 18 = 0$. This matches option B.

    5. Tip

    When dealing with perpendicular lines, a quick trick to find the new slope is to "flip and negate" the original slope. If the original slope is $\frac{a}{b}$, the perpendicular slope is $-\frac{b}{a}$. Also, remember that for any line $Ax + By + C = 0$, a perpendicular line will always take the form $Bx - Ay + K = 0$.

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34. 34.

    Simplify $\frac{3\frac{2}{3} \times \frac{5}{6} \times \frac{2}{3}}{\frac{11}{15} \times \frac{3}{4} \times \frac{2}{27}}$

    1. A.$5\frac{2}{3}$
    2. B.30
    3. C.$4\frac{1}{3}$
    4. D.50

    Show answer

    Answer: D. 50

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35. 35.

    The pie chart shows the distribution of courses offered by students. What percentage of the students offer English?

    1. A.30%
    2. B.25%
    3. C.35%
    4. D.20%

    Show answer

    Answer: D. 20%

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36. 36.

    The perpendicular bisector of a line XY is the locus of a point

    1. A.whose distance from X is always twice its distance from Y
    2. B.whose distance from Y is always twice its distance from X.
    3. C.which moves on the line XY
    4. D.which is equidistant from the points X and Y

    Show answer

    Answer: D. which is equidistant from the points X and Y

    AI explanation

    Analysis of the Question

    The question asks for the definition of the perpendicular bisector of a line segment $XY$ in terms of the locus of a point. A "locus" is simply the path or set of all points that satisfy a specific geometric condition.

    Explaining the Concept

    In geometry, the perpendicular bisector of a line segment $XY$ is a line that:

    1. Passes through the midpoint of $XY$.
    2. Is perpendicular (at a $90^\circ$ angle) to $XY$.

    The defining property of every point on this perpendicular bisector is that it is equidistant from the two endpoints $X$ and $Y$. If you pick any point $P$ on this bisector, the distance from $P$ to $X$ will always be equal to the distance from $P$ to $Y$ ($PX = PY$).

    Step-by-Step Solution

    • Option A & B: These describe a different locus (an Apollonian circle), where the ratio of distances is constant but not equal. This is not the definition of a perpendicular bisector.
    • Option C: A point moving on the line $XY$ would simply be tracing the line segment itself, not its perpendicular bisector.
    • Option D: This matches the geometric definition of a perpendicular bisector. By definition, the set of all points that are the same distance from two fixed points $X$ and $Y$ forms a straight line that perpendicularly bisects the segment connecting those two points.

    Conclusion

    The perpendicular bisector is the locus of points that are equidistant from the two fixed points $X$ and $Y$.

    Exam Tip

    Whenever you see the word "locus" in a JAMB question, think of it as a "rule" or "condition."

    • If the rule involves being equidistant from two points, the answer is always the perpendicular bisector.
    • If the rule involves being equidistant from two intersecting lines, the answer is the angle bisector.

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37. 37.

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

    1. A.-6
    2. B.6
    3. C.-12
    4. D.12

    Show answer

    Answer: A. -6

    AI explanation

    1. Analyze the question

    We are given an equation involving two $2 \times 2$ determinants. We need to evaluate both determinants, set them equal to each other, and solve for the unknown variable $x$.

    2. Explain the concept

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

    $\text{Determinant} = (ad) - (bc)$

    This means you multiply the elements on the leading diagonal and subtract the product of the elements on the other diagonal.

    3. Show the solution

    Step 1: Evaluate the left-hand side (LHS)

    $\begin{vmatrix} 2 & 3 \\ 5 & 3x \end{vmatrix} = (2 \times 3x) - (3 \times 5)$

    $= 6x - 15$

    Step 2: Evaluate the right-hand side (RHS)

    $\begin{vmatrix} 4 & 1 \\ 3 & 2x \end{vmatrix} = (4 \times 2x) - (1 \times 3)$

    $= 8x - 3$

    Step 3: Set the LHS equal to the RHS and solve for $x$

    $6x - 15 = 8x - 3$

    To isolate $x$, move the $x$ terms to one side and the constants to the other:

    $-15 + 3 = 8x - 6x$

    $-12 = 2x$

    Divide both sides by 2:

    $x = \frac{-12}{2}$

    $x = -6$

    4. Identify the answer

    Comparing our result to the given options: A. -6 B. 6 C. -12 D. 12

    The correct option is A.

    5. Tip

    When dealing with determinants, always remember the "cross-multiplication" rule: (Top-Left $\times$ Bottom-Right) minus (Top-Right $\times$ Bottom-Left). A common mistake students make is forgetting the minus sign between the two products—don't let that happen to you!

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38. 38.

    What is the size of each interior angle of a 12-sided regular polygon?

    1. A.120o
    2. B.150o
    3. C.30o
    4. D.180o

    Show answer

    Answer: B. 150o

    AI explanation

    1. Analyze the question

    You are asked to find the size of each interior angle of a regular polygon with 12 sides (a dodecagon).

    2. Explain the concept

    To solve this, we use the properties of regular polygons:

    • Sum of interior angles: The sum of all interior angles of a polygon with $n$ sides is given by the formula: $(n - 2) \times 180^\circ$.
    • Size of one interior angle: Since it is a regular polygon (all angles are equal), we divide the sum by the number of sides ($n$): $$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$
    • Alternative method: You can also find the exterior angle first using the formula $\frac{360^\circ}{n}$, then subtract that from $180^\circ$ (since interior and exterior angles lie on a straight line and sum to $180^\circ$).

    3. Show the solution

    Using the first method where $n = 12$:

    • Step 1: Calculate the sum of interior angles. $$\text{Sum} = (12 - 2) \times 180^\circ$$ $$\text{Sum} = 10 \times 180^\circ = 1800^\circ$$

    • Step 2: Divide by the number of sides to get the size of one angle. $$\text{Interior Angle} = \frac{1800^\circ}{12}$$ $$\text{Interior Angle} = 150^\circ$$

    • Alternative method check: $$\text{Exterior Angle} = \frac{360^\circ}{12} = 30^\circ$$ $$\text{Interior Angle} = 180^\circ - 30^\circ = 150^\circ$$

    Both methods confirm the result.

    4. Identify the answer

    The calculated size of each interior angle is $150^\circ$, which corresponds to option B.

    5. Tip

    When dealing with regular polygons in JAMB, it is often faster to calculate the exterior angle first ($\frac{360}{n}$) and subtract it from $180^\circ$. It involves smaller numbers and reduces the risk of calculation errors compared to multiplying by $(n-2)$ first!

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39. 39.

    Simplify $(\frac{16}{81})^{\frac{1}{4}} \div (\frac{9}{16})^{-\frac{1}{2}}$

    1. A.$\frac{2}{3}$
    2. B.$\frac{1}{2}$
    3. C.$\frac{8}{9}$
    4. D.$\frac{1}{3}$

    Show answer

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

    AI explanation

    To solve this expression, we need to apply the laws of indices, specifically the rules for fractional exponents and negative exponents.

    1. Analyze the expression

    The expression is: $(\frac{16}{81})^{\frac{1}{4}} \div (\frac{9}{16})^{-\frac{1}{2}}$

    2. Explain the concepts

    • Fractional Exponents: $x^{\frac{1}{n}}$ is the same as the $n^{th}$ root of $x$. For example, $x^{\frac{1}{4}}$ means $\sqrt[4]{x}$.
    • Negative Exponents: $x^{-n} = (\frac{1}{x})^n$. This means if you have a fraction raised to a negative power, you flip the fraction (find the reciprocal) to make the exponent positive: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.

    3. Step-by-Step Solution

    Step 1: Simplify the first part $(\frac{16}{81})^{\frac{1}{4}}$ We look for a number that, when raised to the power of 4, gives 16 and 81.

    • $16 = 2^4$
    • $81 = 3^4$ So, $(\frac{2^4}{3^4})^{\frac{1}{4}} = (\frac{2}{3})^{4 \times \frac{1}{4}} = (\frac{2}{3})^1 = \mathbf{\frac{2}{3}}$

    Step 2: Simplify the second part $(\frac{9}{16})^{-\frac{1}{2}}$ First, remove the negative sign by flipping the fraction: $(\frac{9}{16})^{-\frac{1}{2}} = (\frac{16}{9})^{\frac{1}{2}}$ Now, calculate the square root ($\frac{1}{2}$ power) of the fraction:

    • $\sqrt{16} = 4$
    • $\sqrt{9} = 3$ So, $(\frac{16}{9})^{\frac{1}{2}} = \mathbf{\frac{4}{3}}$

    Step 3: Perform the division Now substitute the simplified values back into the original expression: $\frac{2}{3} \div \frac{4}{3}$ To divide by a fraction, multiply by its reciprocal: $\frac{2}{3} \times \frac{3}{4}$ The 3s cancel out: $\frac{2}{4} = \mathbf{\frac{1}{2}}$

    4. Identify the answer

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

    5. Exam Tip

    When you see a negative exponent on a fraction, always flip the fraction first to make the exponent positive before doing any other calculation. It prevents errors with signs and makes the numbers much easier to work with!

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40. 40.

    The midpoint of P(x, y) and Q(8, 6) is (5, 8). Find x and y.

    1. A.(2, 10)
    2. B.(2, 8)
    3. C.(2, 12)
    4. D.(2, 6)

    Show answer

    Answer: A. (2, 10)

    AI explanation

    Analysis

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

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

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

    Concept

    In this problem, we are given:

    • Point $P = (x, y)$
    • Point $Q = (8, 6)$
    • Midpoint $M = (5, 8)$

    We need to substitute these values into the midpoint formula to solve for the unknown variables $x$ and $y$.

    Step-by-Step Solution

    Step 1: Solve for $x$ Using the formula for the x-coordinate:

    $5 = \frac{x + 8}{2}$

    Multiply both sides by 2:

    $10 = x + 8$

    Subtract 8 from both sides:

    $x = 10 - 8$

    $x = 2$

    Step 2: Solve for $y$ Using the formula for the y-coordinate:

    $8 = \frac{y + 6}{2}$

    Multiply both sides by 2:

    $16 = y + 6$

    Subtract 6 from both sides:

    $y = 16 - 6$

    $y = 10$

    Step 3: Combine the results The coordinates of point $P$ are $(x, y) = (2, 10)$.

    Conclusion

    Comparing our result $(2, 10)$ with the given options, it matches option A.

    Exam Tip

    When dealing with midpoint questions, you can always verify your answer by calculating the average of your result and the given point. If the average equals the midpoint provided in the question, your answer is definitely correct!

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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.

    T varies inversely as the cube of R. When R = 3, T = $\frac{2}{81}$, find T when R = 2

    1. A.$\frac{1}{18}$
    2. B.$\frac{1}{12}$
    3. C.$\frac{1}{24}$
    4. D.$\frac{1}{6}$

    Show answer

    Answer: B. $\frac{1}{12}$

    AI explanation

    1. Analyze the question

    We are given a relationship of inverse variation: $T$ varies inversely as the cube of $R$. We are provided with an initial set of values ($R=3, T=\frac{2}{81}$) and asked to find the value of $T$ when $R=2$.

    2. Explain the concept

    In inverse variation, the relationship is expressed as:

    $T \propto \frac{1}{R^3}$

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

    $T = \frac{k}{R^3}$

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

    3. Show the solution

    Step 1: Find the constant $k$ Using the values $T = \frac{2}{81}$ and $R = 3$:

    $\frac{2}{81} = \frac{k}{3^3}$

    $\frac{2}{81} = \frac{k}{27}$

    Now, solve for $k$ by multiplying both sides by 27:

    $k = \frac{2 \times 27}{81}$

    Since $81 = 27 \times 3$, we can simplify:

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

    Step 2: Find $T$ when $R = 2$ Now that we know $k = \frac{2}{3}$, our formula is:

    $T = \frac{2/3}{R^3}$

    Substitute $R = 2$:

    $T = \frac{2/3}{2^3}$

    $T = \frac{2/3}{8}$

    $T = \frac{2}{3} \times \frac{1}{8}$

    $T = \frac{2}{24}$

    $T = \frac{1}{12}$

    4. Identify the answer

    Comparing our result to the given options: A. $\frac{1}{18}$ B. $\frac{1}{12}$ C. $\frac{1}{24}$ D. $\frac{1}{6}$

    The correct option is B.

    5. Tip

    When dealing with inverse variation, always remember the "Product Rule": if $T \propto \frac{1}{R^3}$, then $T_1 \times R_1^3 = T_2 \times R_2^3$. This allows you to solve the problem without explicitly finding $k$ if you prefer!

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43. 43.

    $\begin{array}{c|c} Class Interval & 3 - 5 & 6 - 8 & 9 - 11 \\ \hline Frequency & 2 & 2 & 2 \end{array}$. Find the standard deviation of the above distribution.

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

    Show answer

    Answer: B. √6

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44. 44.

    Factorize completely 9y2 - 16X2

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

    Show answer

    Answer: D. (3y - 4x)(3y + 4x)

    AI explanation

    Analysis

    The expression given is $9y^2 - 16x^2$. This is a classic algebraic expression that fits the Difference of Two Squares pattern.

    Concept: Difference of Two Squares

    In algebra, the difference of two squares follows the identity:

    $a^2 - b^2 = (a - b)(a + b)$

    To solve this, we need to express both terms in the given expression as perfect squares.

    Step-by-Step Solution

    1. Identify the squares: • The first term is $9y^2$. Since $9 = 3^2$, we can write this as $(3y)^2$. • The second term is $16x^2$. Since $16 = 4^2$, we can write this as $(4x)^2$.

    2. Rewrite the expression: Substituting these back into the original expression, we get: $$(3y)^2 - (4x)^2$$

    3. Apply the identity: Using the formula $a^2 - b^2 = (a - b)(a + b)$, where $a = 3y$ and $b = 4x$: $$(3y - 4x)(3y + 4x)$$

    Conclusion

    Comparing our result $(3y - 4x)(3y + 4x)$ with the given options, it matches option D.

    Exam Tip

    Whenever you see a binomial (two terms) with a minus sign between them, always check if both terms are perfect squares (like 1, 4, 9, 16, 25, 36, etc.). If they are, you can immediately use the $(a-b)(a+b)$ shortcut to save time during your JAMB exam!

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45. 45.

    Find the sum of the first 18 terms of the series 3, 6, 9,..., 36.

    1. A.505
    2. B.513
    3. C.433
    4. D.635

    Show answer

    Answer: B. 513

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46. 46.

    A man walks 100 m due West from a point X to Y, he then walks 100 m due North to a point Z. Find the bearing of X from Z.

    1. A.195o
    2. B.135o
    3. C.225o
    4. D.045o

    Show answer

    Answer: B. 135o

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