2024 JAMB Mathematics Past Questions and Answers

Questions

1. 1.

   The ratio of men to women in a 20-member committee is 3:1. How many women must be added to the committee to make the ratio of men to women 3:2?

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

   Show answer

   Answer: B. 5

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

3. 3.

   Convert the number 10111.11$_{two}$ to a mixed number.

   1. A.23$\frac{1}{4}$
   2. B.23$\frac{3}{4}$
   3. C.22$\frac{1}{4}$
   4. D.24$\frac{3}{4}$

   Show answer

   Answer: B. 23$\frac{3}{4}$

   AI explanation

   To convert a number from base two (binary) to base ten (decimal), we use the positional notation method. Each digit is multiplied by 2 raised to the power of its position relative to the binary point.

   1. Analyze the question

   We need to convert $10111.11_2$ into a decimal mixed number. The number consists of an integer part ($10111$) and a fractional part ($0.11$).

   2. Explain the concept

   • Integer part: Starting from the right of the binary point and moving left, the powers of 2 are $2^0, 2^1, 2^2, 2^3, 2^4, \dots$
   • Fractional part: Starting from the left of the binary point and moving right, the powers of 2 are $2^{-1}, 2^{-2}, 2^{-3}, \dots$

   3. Show the solution

   Step 1: Convert the integer part ($10111_2$)

   • $1 \times 2^4 = 1 \times 16 = 16$
   • $0 \times 2^3 = 0 \times 8 = 0$
   • $1 \times 2^2 = 1 \times 4 = 4$
   • $1 \times 2^1 = 1 \times 2 = 2$
   • $1 \times 2^0 = 1 \times 1 = 1$
   • Sum: $16 + 0 + 4 + 2 + 1 = 23$

   Step 2: Convert the fractional part ($0.11_2$)

   • $1 \times 2^{-1} = 1 \times \frac{1}{2} = 0.5$
   • $1 \times 2^{-2} = 1 \times \frac{1}{4} = 0.25$
   • Sum: $0.5 + 0.25 = 0.75$

   Step 3: Combine the parts

   • $23 + 0.75 = 23.75$

   Step 4: Convert to a mixed number

   • $0.75$ is the same as $\frac{75}{100}$, which simplifies to $\frac{3}{4}$.
   • Therefore, $23.75 = 23\frac{3}{4}$.

   4. Identify the answer

   Comparing our result to the given options, $23\frac{3}{4}$ corresponds to option B.

   5. Tip

   When dealing with binary fractions, remember that the first position after the point is always "halves" ($\frac{1}{2}$), the second is "quarters" ($\frac{1}{4}$), and the third is "eighths" ($\frac{1}{8}$). This makes quick mental conversion much easier!

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4. 4.

   Given that P is the set of all prime numbers between 0 and 10, and Q is the set of all odd numbers between 0 and 10. Find the union of elements in P that are not in Q and the elements in Q that are not in P.

   1. A.{2, 9}
   2. B.{ 9}
   3. C.{1, 9}
   4. D.{1, 2, 9}

   Show answer

   Answer: D. {1, 2, 9}

   AI explanation

   Step 1: Analyze the Question

   We are given two sets, $P$ and $Q$, defined within the range of 0 to 10. We need to find the symmetric difference between these two sets, which is represented by the union of $(P \setminus Q)$ and $(Q \setminus P)$.

   Step 2: Define the Sets

   • Set P (Prime numbers between 0 and 10): Prime numbers are numbers greater than 1 with only two factors (1 and itself). $P = \{2, 3, 5, 7\}$
   • Set Q (Odd numbers between 0 and 10): Odd numbers are integers not divisible by 2. $Q = \{1, 3, 5, 7, 9\}$

   Step 3: Solve Step-by-Step

   1. Find elements in P that are not in Q ($P \setminus Q$): These are prime numbers that are not odd. Looking at $P = \{2, 3, 5, 7\}$ and $Q = \{1, 3, 5, 7, 9\}$, the only element in $P$ that is not in $Q$ is 2. $P \setminus Q = \{2\}$

   2. Find elements in Q that are not in P ($Q \setminus P$): These are odd numbers that are not prime. Looking at $Q = \{1, 3, 5, 7, 9\}$ and $P = \{2, 3, 5, 7\}$, the elements in $Q$ that are not in $P$ are 1 and 9. $Q \setminus P = \{1, 9\}$

   3. Find the union of these two results: Union means combining all unique elements from both sets. $\{2\} \cup \{1, 9\} = \{1, 2, 9\}$

   Step 4: Identify the Answer

   Comparing our result $\{1, 2, 9\}$ to the given options, it matches option D.

   Exam Tip

   When dealing with set operations like "elements in P not in Q" (relative complement), always list your sets clearly first. Remember that 1 is not a prime number, which is a common trap in JAMB questions. Checking your set definitions carefully before performing the operation will save you from silly mistakes!

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5. 5.

   In how many ways can 6 people sit around a table

   1. A.120
   2. B.240
   3. C.60
   4. D.30

   Show answer

   Answer: A. 120

   AI explanation

   1. Analyze the question

   The question asks for the number of ways to arrange 6 people around a table. In mathematics, this is a problem of circular permutation. Unlike linear arrangements (where there is a clear start and end), circular arrangements are relative to each other.

   2. Explain the concept

   When arranging $n$ distinct objects in a straight line, the number of ways is $n!$ (n factorial). However, when arranging $n$ objects in a circle, rotating the entire arrangement does not create a new unique arrangement.

   To account for this, we fix one person in a seat to serve as a reference point. Once that 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, $n = 6$.
   • Step 2: Apply the circular permutation formula. $$\text{Ways} = (6 - 1)!$$
   • Step 3: Calculate the factorial. $$\text{Ways} = 5!$$
   • Step 4: Expand the factorial. $$5! = 5 \times 4 \times 3 \times 2 \times 1$$ $$5 \times 4 = 20$$ $$20 \times 3 = 60$$ $$60 \times 2 = 120$$ $$120 \times 1 = 120$$

   4. Identify the answer

   The calculation results in 120. Comparing this to the provided options, it matches option A.

   5. Tip

   Always remember: "Linear is $n!$, Circular is $(n-1)!$." If the question mentions a "round table" or "circular arrangement," immediately subtract 1 from the total number of items before calculating the factorial. This simple trick will save you time in the exam hall!

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6. 6.

   ($\sqrt{a} +\sqrt{8a})^2$ = 54 + b$\sqrt{2}$, a and b are positive integers. Find the value of a and the value of b.

   1. A.a = 24, b = 6
   2. B.a = 6, b = 2
   3. C.a = 6, b = 24
   4. D.a = 2, b = 6

   Show answer

   Answer: C. a = 6, b = 24

   AI explanation

   1. Analyze the question

   We are given the equation $(\sqrt{a} + \sqrt{8a})^2 = 54 + b\sqrt{2}$. We need to expand the left side, simplify it, and compare it to the right side to find the values of the positive integers $a$ and $b$.

   2. Explain the concept

   To solve this, we use the algebraic identity $(x + y)^2 = x^2 + 2xy + y^2$. We also need to remember the properties of surds:

   • $\sqrt{x} \cdot \sqrt{x} = x$
   • $\sqrt{8a} = \sqrt{4 \cdot 2 \cdot a} = 2\sqrt{2a}$

   3. Show the solution

   Step 1: Simplify the expression inside the bracket. $\sqrt{8a} = \sqrt{4 \times 2 \times a} = 2\sqrt{2a}$. So, the expression becomes: $(\sqrt{a} + 2\sqrt{2a})^2$

   Step 2: Expand using $(x + y)^2 = x^2 + 2xy + y^2$. Let $x = \sqrt{a}$ and $y = 2\sqrt{2a}$.

   • $x^2 = (\sqrt{a})^2 = a$
   • $y^2 = (2\sqrt{2a})^2 = 4 \times 2a = 8a$
   • $2xy = 2(\sqrt{a})(2\sqrt{2a}) = 4\sqrt{2a^2} = 4a\sqrt{2}$

   Step 3: Combine the terms. $a + 8a + 4a\sqrt{2} = 9a + 4a\sqrt{2}$

   Step 4: Compare with the given equation. $9a + 4a\sqrt{2} = 54 + b\sqrt{2}$

   By comparing the rational parts and the irrational parts:

   • Rational part: $9a = 54 \implies a = \frac{54}{9} \implies \mathbf{a = 6}$
   • Irrational part: $4a\sqrt{2} = b\sqrt{2} \implies b = 4a$

   Substitute $a = 6$ into the equation for $b$:

   • $b = 4(6) \implies \mathbf{b = 24}$

   4. Identify the answer

   Comparing our results ($a=6, b=24$) with the given options, we find that it matches option C.

   5. Tip

   When dealing with surds in equations, always simplify the square roots (e.g., $\sqrt{8a} \to 2\sqrt{2a}$) before expanding. This makes it much easier to group the "rational" and "irrational" parts on both sides of the equation.

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7. 7.

   The average age of the four female teachers in a school is 40 and the average age of eight male teachers in the school is 25. Calculate the average age of the teachers in the school.

   1. A.20
   2. B.31
   3. C.32
   4. D.30

   Show answer

   Answer: D. 30

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8. 8.

   How many proper and improper subsets are there in the set K = { a, b, c, d, e}?

   1. A.10
   2. B.25
   3. C.32
   4. D.50

   Show answer

   Answer: C. 32

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9. 9.

   Which of the following is a measure of central tendency?

   1. A.Standard deviation
   2. B.Range
   3. C.Variance
   4. D.Mean

   Show answer

   Answer: D. Mean

   AI explanation

   1. Analyze the question

   The question asks you to identify which of the given statistical terms represents a "measure of central tendency." In statistics, measures of central tendency are values that describe the center or typical value of a dataset.

   2. Explain the concept

   • Measures of Central Tendency: These are statistical measures used to find the "middle" or "average" of a set of data. The three most common measures are the Mean (average), the Median (middle value), and the Mode (most frequent value).
   • Measures of Dispersion (Spread): These describe how spread out the data is. Examples include Range, Variance, and Standard Deviation. These do not describe the center; they describe the variation or distance between data points.

   3. Show the solution

   • Option A (Standard deviation): This measures how much the data deviates from the mean. It is a measure of dispersion, not central tendency.
   • Option B (Range): This is the difference between the highest and lowest values. It is a measure of dispersion.
   • Option C (Variance): This is the average of the squared differences from the mean. It is a measure of dispersion.
   • Option D (Mean): This is calculated by summing all values and dividing by the number of values. It is the most common measure of central tendency.

   4. Identify the answer

   Comparing the options to the definitions above, the Mean is the only measure of central tendency listed.

   5. Exam Tip

   Remember the acronym "M-M-M" for Central Tendency: Mean, Median, and Mode. If you see any of these three in an exam question about central tendency, that is your target! Everything else (Range, Variance, Standard Deviation) relates to how "spread out" the data is.

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10. 10.

    Evaluate the determinant of the matrix M = $\begin{bmatrix}3 & 4 & 6\\2 & 1 & -1\\-1 & 3 & 5\end{bmatrix}$

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

    Show answer

    Answer: C. 30

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11. 11.

    Differentiate y = (5x + 1)$^4$

    1. A.20 (5x + 1)$^3$
    2. B.4(5x + 1)$^3$
    3. C.5(5x + 1)$^3$
    4. D.(5x)$^3$

    Show answer

    Answer: A. 20 (5x + 1)$^3$

    AI explanation

    To solve this problem, we need to apply the Chain Rule in calculus. The Chain Rule is used when we have a function nested inside another function, such as $y = [f(x)]^n$.

    1. Analyze the Question

    We are asked to find the derivative of $y = (5x + 1)^4$ with respect to $x$. This is written as $\frac{dy}{dx}$.

    2. Explain the Concept

    The Chain Rule states that if $y = u^n$, then:

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

    In simpler terms:

    1. Bring the power down to the front (multiply by the exponent).
    2. Keep the expression inside the bracket exactly as it is and subtract 1 from the exponent.
    3. Multiply by the derivative of the expression inside the bracket.

    3. Show the Solution

    Given: $y = (5x + 1)^4$

    Step 1: Differentiate the outer function. Treat $(5x + 1)$ as a single unit ($u$). The derivative of $u^4$ is $4u^3$. So, we have: $4(5x + 1)^3$

    Step 2: Differentiate the inner function. Now, differentiate the expression inside the bracket $(5x + 1)$ with respect to $x$. The derivative of $5x$ is $5$, and the derivative of $1$ is $0$. So, the derivative of $(5x + 1)$ is $5$.

    Step 3: Combine the results. Multiply the result from Step 1 by the result from Step 2:

    $\frac{dy}{dx} = 4(5x + 1)^3 \cdot 5$

    $\frac{dy}{dx} = 20(5x + 1)^3$

    4. Identify the Answer

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

    5. Exam Tip

    Whenever you see a bracket raised to a power, always remember the "Inside-Outside" rule: Differentiate the outside, keep the inside the same, then multiply by the derivative of the inside. Don't forget that last step—many students lose marks by forgetting to multiply by the derivative of the inner function!

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12. 12.

    If p * q = 2p + pq + q, find p  when ( p * 2) - (p * 1) = 40

    1. A.39
    2. B.40
    3. C.41
    4. D.42

    Show answer

    Answer: A. 39

    AI explanation

    Analysis

    The problem defines a binary operation: $p * q = 2p + pq + q$. We are given the equation $(p * 2) - (p * 1) = 40$ and need to solve for $p$.

    Concept

    A binary operation is simply a rule that tells you how to combine two numbers. To solve this, we must evaluate the two expressions $(p * 2)$ and $(p * 1)$ separately using the provided formula, substitute them back into the equation, and solve for $p$.

    Step-by-Step Solution

    Step 1: Evaluate $(p * 2)$ Using the formula $p * q = 2p + pq + q$, substitute $q = 2$: $(p * 2) = 2p + p(2) + 2$ $(p * 2) = 2p + 2p + 2$ $(p * 2) = 4p + 2$

    Step 2: Evaluate $(p * 1)$ Using the same formula, substitute $q = 1$: $(p * 1) = 2p + p(1) + 1$ $(p * 1) = 2p + p + 1$ $(p * 1) = 3p + 1$

    Step 3: Substitute into the given equation The equation is $(p * 2) - (p * 1) = 40$. $(4p + 2) - (3p + 1) = 40$

    Step 4: Solve for $p$ Remove the brackets (be careful with the negative sign): $4p + 2 - 3p - 1 = 40$ Group the like terms: $(4p - 3p) + (2 - 1) = 40$ $p + 1 = 40$ $p = 40 - 1$ $p = 39$

    Conclusion

    By substituting the values into the binary operation formula and simplifying the resulting linear equation, we find that $p = 39$.

    Exam Tip: When dealing with binary operations involving brackets, always expand the expressions inside the brackets first before performing the subtraction or addition. Also, watch out for the negative sign when subtracting an expression—it must be distributed to every term inside the second bracket!

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13. 13.

    Solve 16$^x$ = 0.25

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

    Show answer

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

    AI explanation

    Analysis

    The equation given is an exponential equation: $16^x = 0.25$. To solve for $x$, we need to express both sides of the equation using the same base.

    Concept: Laws of Indices

    1. Expressing decimals as fractions: $0.25$ is the same as $\frac{25}{100}$, which simplifies to $\frac{1}{4}$.
    2. Negative Indices: Recall that $\frac{1}{a^n} = a^{-n}$.
    3. Common Base: Both $16$ and $4$ are powers of $2$. Specifically, $16 = 2^4$ and $4 = 2^2$.

    Step-by-Step Solution

    Step 1: Rewrite the equation with fractions

    $16^x = \frac{1}{4}$

    Step 2: Express both sides using base 2 Since $16 = 2^4$:

    $(2^4)^x = \frac{1}{2^2}$

    Step 3: Apply the laws of indices On the left side, use the power of a power rule $(a^m)^n = a^{m \times n}$:

    $2^{4x} = \frac{1}{2^2}$

    On the right side, use the negative index rule $\frac{1}{a^n} = a^{-n}$:

    $2^{4x} = 2^{-2}$

    Step 4: Equate the exponents Since the bases are now the same ($2$), we can equate the exponents:

    $4x = -2$

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

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

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

    Conclusion

    Comparing our result with the given options, the correct value is $-\frac{1}{2}$.

    Exam Tip

    Whenever you see a decimal like $0.25$, $0.5$, or $0.125$ in an exponential equation, immediately convert it to a fraction ($\frac{1}{4}$, $\frac{1}{2}$, or $\frac{1}{8}$). It makes it much easier to spot the common base!

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14. 14.

    Simplify $\frac{ 5 + \sqrt{7}}{3 + \sqrt{7}}$

    1. A.17 - $\sqrt{7}$
    2. B.4 - $\sqrt{7}$
    3. C.7 - $\sqrt{7}$
    4. D.15 - $\sqrt{7}$

    Show answer

    Answer: B. 4 - $\sqrt{7}$

    AI explanation

    To simplify an expression involving a square root in the denominator, we use a process called rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

    1. Analyze the question

    We are given: $\frac{5 + \sqrt{7}}{3 + \sqrt{7}}$ The conjugate of the denominator $(3 + \sqrt{7})$ is $(3 - \sqrt{7})$.

    2. Explain the concept

    When you multiply a binomial of the form $(a + \sqrt{b})$ by its conjugate $(a - \sqrt{b})$, you get a difference of two squares: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b$. This removes the square root from the denominator, making the expression easier to work with.

    3. Show the solution

    Multiply the numerator and denominator by $(3 - \sqrt{7})$:

    $\frac{5 + \sqrt{7}}{3 + \sqrt{7}} \times \frac{3 - \sqrt{7}}{3 - \sqrt{7}}$

    Step 1: Expand the numerator $(5 + \sqrt{7})(3 - \sqrt{7}) = 5(3) - 5(\sqrt{7}) + 3(\sqrt{7}) - (\sqrt{7})(\sqrt{7})$ $= 15 - 5\sqrt{7} + 3\sqrt{7} - 7$ $= (15 - 7) + (-5\sqrt{7} + 3\sqrt{7})$ $= 8 - 2\sqrt{7}$

    Step 2: Expand the denominator $(3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2$ $= 9 - 7$ $= 2$

    Step 3: Combine and simplify

    $\frac{8 - 2\sqrt{7}}{2}$

    Divide both terms in the numerator by 2:

    $\frac{8}{2} - \frac{2\sqrt{7}}{2} = 4 - \sqrt{7}$

    4. Identify the answer

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

    5. Exam Tip

    Whenever you see a surd in the denominator, don't panic! Always look for the conjugate. Remember that the conjugate is simply the same two terms with the middle sign flipped. If the denominator is $(a + \sqrt{b})$, multiply by $(a - \sqrt{b})$. If it is $(a - \sqrt{b})$, multiply by $(a + \sqrt{b})$.

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15. 15.

    Identify the equation of the shaded region in the given graph

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

    Show answer

    Answer: B. x < 1

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16. 16.

    Find the roots of x$^3$ - 19x - 30=0

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

    Show answer

    Answer: C. 5, -3 and -2

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17. 17.

    y is inversely proportional to x and y = 4, when x = $\frac{1}{2}$, find x when y = 10

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

    Show answer

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

    AI explanation

    1. Analyze the question

    We are given that $y$ is inversely proportional to $x$. We are provided with an initial pair of values ($y=4, x=\frac{1}{2}$) to find the constant of proportionality, and then asked to find the new value of $x$ when $y=10$.

    2. Explain the concept

    When two variables are inversely proportional, it means that as one increases, the other decreases in such a way that their product remains constant. Mathematically, this is expressed as:

    $y \propto \frac{1}{x} \implies y = \frac{k}{x}$

    Where $k$ is the constant of proportionality. To solve this, we first find $k$ using the given values, then use that $k$ to find the unknown variable.

    3. Show the solution

    Step 1: Find the constant $k$ Using the formula $y = \frac{k}{x}$:

    $4 = \frac{k}{\frac{1}{2}}$

    To solve for $k$, multiply both sides by $\frac{1}{2}$:

    $k = 4 \times \frac{1}{2}$

    $k = 2$

    Step 2: Find $x$ when $y = 10$ Now that we know $k = 2$, our equation is $y = \frac{2}{x}$. Substitute $y = 10$ into the equation:

    $10 = \frac{2}{x}$

    Multiply both sides by $x$:

    $10x = 2$

    Divide both sides by 10:

    $x = \frac{2}{10}$

    Simplify the fraction:

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

    4. Identify the answer

    Comparing our result to the options provided: A. 2 B. 10 C. $\frac{1}{5}$ D. $\frac{1}{10}$

    The correct option is C.

    5. Tip

    In inverse proportion problems, always remember the "Product Rule": $x_1y_1 = x_2y_2$. Using this shortcut: $(4) \times (\frac{1}{2}) = (10) \times (x)$ $2 = 10x$ $x = \frac{2}{10} = \frac{1}{5}$ This method is much faster for JAMB exams where time management is key!

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18. 18.

    If tan$\theta$ = $\frac{8}{15}$, simplify $\frac{ Sin\theta - Cos\theta}{Sin^2\theta - Sin\theta}$

    1. A.$\frac{-119}{72}$
    2. B.$\frac{119}{72}$
    3. C.$\frac{-7}{17}$
    4. D.$\frac{8}{17}$

    Show answer

    Answer: B. $\frac{119}{72}$

    AI explanation

    1. Analyze the question

    We are given $\tan\theta = \frac{8}{15}$ and asked to simplify the expression $\frac{\sin\theta - \cos\theta}{\sin^2\theta - \sin\theta}$.

    2. Explain the concept

    • Trigonometric Ratios: In a right-angled triangle, $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$. Given $\tan\theta = \frac{8}{15}$, we can identify the Opposite side = 8 and the Adjacent side = 15.
    • Pythagoras Theorem: To find the Hypotenuse ($h$), use $h^2 = \text{Opp}^2 + \text{Adj}^2$.
    • SOH CAH TOA: $\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$ and $\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$.

    3. Show the solution

    Step 1: Find the Hypotenuse $h^2 = 8^2 + 15^2$ $h^2 = 64 + 225 = 289$ $h = \sqrt{289} = 17$

    Step 2: Determine $\sin\theta$ and $\cos\theta$ $\sin\theta = \frac{8}{17}$ $\cos\theta = \frac{15}{17}$

    Step 3: Substitute into the expression Expression = $\frac{\sin\theta - \cos\theta}{\sin^2\theta - \sin\theta}$

    Numerator: $\frac{8}{17} - \frac{15}{17} = -\frac{7}{17}$

    Denominator: $(\frac{8}{17})^2 - \frac{8}{17}$ $= \frac{64}{289} - \frac{8}{17}$ To subtract, find a common denominator (289): $= \frac{64}{289} - \frac{8 \times 17}{289} = \frac{64 - 136}{289} = -\frac{72}{289}$

    Step 4: Divide the Numerator by the Denominator $\frac{-\frac{7}{17}}{-\frac{72}{289}} = \frac{7}{17} \times \frac{289}{72}$ Since $289 \div 17 = 17$: $= \frac{7 \times 17}{72} = \frac{119}{72}$

    4. Identify the answer

    The calculated value is $\frac{119}{72}$, which matches option B.

    5. Tip

    When dealing with trigonometric ratios, always draw a quick right-angled triangle. It helps you visualize the sides and prevents errors when calculating the hypotenuse or determining the signs of the ratios.

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19. 19.

    Find the perimeter of a triangle whose vertices pass through ( 3, 2), (4, 5) and (6, 2) in surd form.

    1. A.3 + $\sqrt{12} + \sqrt{13}$
    2. B.3 - $\sqrt{10} + \sqrt{13}$
    3. C.3 + $\sqrt{10} + \sqrt{13}$
    4. D.3 - $\sqrt{12} + \sqrt{13}$

    Show answer

    Answer: C. 3 + $\sqrt{10} + \sqrt{13}$

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20. 20.

    Differentiate Cos25º - Sin 25º

    1. A.- ( Sin25º + Cos25º)
    2. B.- ( Sec25 + Cot25)
    3. C.- ( Cot25 + Sec25)
    4. D.- ( Sin25 + Sec25)

    Show answer

    Answer: A. - ( Sin25º + Cos25º)

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21. 21.

    In how many ways can a committee of 5 be selected from a group of 7 males and 3 females, if the committee must have one female?

    1. A.105ways
    2. B.135 ways
    3. C.210 ways
    4. D.231 ways

    Show answer

    Answer: A. 105ways

    AI explanation

    1. Analyze the Question

    We need to form a committee of 5 people from a total group of 10 people (7 males and 3 females). The condition is that the committee must have exactly one female.

    2. Explain the Concept

    This is a combination problem because the order in which the committee members are selected does not matter. The formula for combinations is:

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

    To satisfy the condition of having exactly one female in a committee of 5:

    • We must choose 1 female from the 3 available females.
    • We must choose the remaining 4 members from the 7 available males (since $5 - 1 = 4$).

    3. Show the Solution

    Step 1: Choose 1 female from 3

    $^3C_1 = \frac{3!}{1!(3-1)!} = \frac{3}{1} = 3 \text{ ways}$

    Step 2: Choose 4 males from 7

    $^7C_4 = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2 \times 1}$

    $^7C_4 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35 \text{ ways}$

    Step 3: Calculate the total number of ways Since both actions must happen to form the committee, we multiply the results:

    $\text{Total ways} = (^3C_1) \times (^7C_4)$

    $\text{Total ways} = 3 \times 35 = 105 \text{ ways}$

    4. Identify the Answer

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

    5. Exam Tip

    When a question says "must have one," it often implies "exactly one." If the question had said "at least one," you would have had to calculate the scenarios for 1, 2, and 3 females separately and add them together. Always read carefully to see if it implies "exactly" or "at least"!

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22. 22.

    Make q the subject of the relation t = $\sqrt{(\frac{pq}{r} - r^2q)}$

    1. A.q = $\frac{rt^2}{p - r^3}$
    2. B.q = $\frac{p - r^3}{rt^2}$
    3. C.q = $\frac{t^2}{p - r^3}$
    4. D.q = $\frac{rt}{p - r^3}$

    Show answer

    Answer: A. q = $\frac{rt^2}{p - r^3}$

    AI explanation

    Analysis

    The goal is to isolate $q$ in the given equation: $t = \sqrt{\frac{pq}{r} - r^2q}$

    Concept

    To make a variable the subject of a formula, we need to perform algebraic operations (squaring, factoring, and dividing) to isolate that variable on one side of the equation.

    Step-by-Step Solution

    Step 1: Eliminate the square root Square both sides of the equation to remove the radical: $t^2 = \frac{pq}{r} - r^2q$

    Step 2: Clear the fraction To make the equation easier to handle, multiply every term by $r$ to eliminate the denominator: $rt^2 = pq - r^3q$

    Step 3: Factor out $q$ Since $q$ is common to both terms on the right side, factor it out: $rt^2 = q(p - r^3)$

    Step 4: Isolate $q$ Divide both sides by $(p - r^3)$ to get $q$ by itself: $q = \frac{rt^2}{p - r^3}$

    Conclusion

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

    Exam Tip

    When dealing with formulas involving square roots, always square both sides first to simplify the expression. Also, whenever you see a variable appearing in multiple terms on one side, factoring is almost always your next logical step to isolate that variable!

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23. 23.

    A bag contains 7 red and 4 black identical balls. Two balls were picked at random from the bag and replaced each time. Find the probability the two balls were of same colour.

    1. A.$\frac{54}{121}$
    2. B.$\frac{65}{121}$
    3. C.$\frac{29}{55}$
    4. D.$\frac{27}{55}$

    Show answer

    Answer: B. $\frac{65}{121}$

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24. 24.

    Find the amount if simple interest is paid yearly at 5% per annum for 3 years, on a principal of # 200,000.00

    1. A.# 23, 000.00
    2. B.# 30,000.00
    3. C.# 300,000.00
    4. D.# 230,000.00

    Show answer

    Answer: D. # 230,000.00

    AI explanation

    Analysis

    To solve this problem, we need to distinguish between Simple Interest and the Amount. The question asks for the "amount," which is the total sum of the principal and the interest earned over the given period.

    Concept

    1. Simple Interest (SI) is calculated using the formula: $$SI = \frac{P \times R \times T}{100}$$ Where: • $P$ = Principal (the initial amount) = #200,000 • $R$ = Rate per annum = 5% • $T$ = Time in years = 3 years

    2. Amount (A) is the total value after the interest is added: $$Amount = Principal + Simple Interest$$

    Step-by-Step Solution

    Step 1: Calculate the Simple Interest

    $SI = \frac{200,000 \times 5 \times 3}{100}$

    $SI = \frac{200,000 \times 15}{100}$

    $SI = 2,000 \times 15$

    $SI = 30,000$

    Step 2: Calculate the Total Amount

    $Amount = Principal + SI$

    $Amount = 200,000 + 30,000$

    $Amount = 230,000$

    Conclusion

    The total amount after 3 years is #230,000.00. This matches option D.

    Exam Tip

    Always read the question carefully to see if it asks for the Interest or the Amount. Many students calculate the interest (#30,000) and stop there, choosing option B by mistake. Remember: Amount = Principal + Interest.

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25. 25.

    If A = $\begin{pmatrix}3 & 2 & 1\\4 & 2 & -1\end{pmatrix}$ and B = $\begin{pmatrix}1 & 4\\0 & 1\\3 & 2\end{pmatrix}$. Find A$^T$ + B, ( where T means transpose)

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

    Show answer

    Answer: D. $\begin{pmatrix}4 & 8\\2 & 3\\4 & 1\end{pmatrix}$

    AI explanation

    To solve this problem, we need to perform two operations: finding the transpose of matrix A and then adding it to matrix B.

    1. Analyze the Question

    • Matrix A is a $2 \times 3$ matrix: $\begin{pmatrix}3 & 2 & 1\\4 & 2 & -1\end{pmatrix}$
    • Matrix B is a $3 \times 2$ matrix: $\begin{pmatrix}1 & 4\\0 & 1\\3 & 2\end{pmatrix}$
    • We need to calculate $A^T + B$.

    2. Explain the Concept

    • Transpose ($A^T$): To find the transpose of a matrix, you swap its rows and columns. The first row of A becomes the first column of $A^T$, and the second row of A becomes the second column of $A^T$.
    • Matrix Addition: You can only add matrices of the same dimensions. Since $A$ is $2 \times 3$, $A^T$ will be $3 \times 2$. Since $B$ is also $3 \times 2$, we can add them by summing their corresponding elements.

    3. Show the Solution

    Step 1: Find $A^T$ Given $A = \begin{pmatrix}3 & 2 & 1\\4 & 2 & -1\end{pmatrix}$ Swap rows for columns: $A^T = \begin{pmatrix}3 & 4\\2 & 2\\1 & -1\end{pmatrix}$

    Step 2: Add $A^T$ and B $A^T + B = \begin{pmatrix}3 & 4\\2 & 2\\1 & -1\end{pmatrix} + \begin{pmatrix}1 & 4\\0 & 1\\3 & 2\end{pmatrix}$

    Perform the addition element by element:

    • Row 1: $(3+1) = 4$ and $(4+4) = 8$
    • Row 2: $(2+0) = 2$ and $(2+1) = 3$
    • Row 3: $(1+3) = 4$ and $(-1+2) = 1$

    Resulting matrix = $\begin{pmatrix}4 & 8\\2 & 3\\4 & 1\end{pmatrix}$

    4. Identify the Answer

    Comparing our result with the given options, the correct matrix matches option D.

    5. Exam Tip

    When dealing with matrix transposes, always double-check your dimensions. If you start with a $2 \times 3$ matrix, its transpose must be $3 \times 2$. If the dimensions don't match the matrix you are adding it to, you have likely made a mistake in the transpose step!

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26. 26.

    Let A = $\begin{pmatrix}2 & -4 & 3\\5 & 1 & 0\end{pmatrix}$ and B =  $\begin{pmatrix}1 & 4 & -2\\-3 & 3 & -1\end{pmatrix}$. Find A + 2B

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

    Show answer

    Answer: A. $\begin{pmatrix}4 & 4 & -1\\-1 & 7 & -2\end{pmatrix}$

    AI explanation

    Analysis

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

    1. Scalar Multiplication: Multiply every element in matrix B by 2.
    2. Matrix Addition: Add the resulting matrix (2B) to matrix A.

    Concept

    • Scalar Multiplication: If $k$ is a scalar and $B$ is a matrix, $kB$ is found by multiplying each individual entry in $B$ by $k$.
    • Matrix Addition: Two matrices can only be added if they have the same dimensions. You add them by summing the corresponding elements (the element in row $i$, column $j$ of matrix A is added to the element in row $i$, column $j$ of matrix 2B).

    Step-by-Step Solution

    Step 1: Calculate 2B Given $B = \begin{pmatrix}1 & 4 & -2 \\ -3 & 3 & -1\end{pmatrix}$, multiply each element by 2: $2B = 2 \times \begin{pmatrix}1 & 4 & -2 \\ -3 & 3 & -1\end{pmatrix} = \begin{pmatrix}2(1) & 2(4) & 2(-2) \\ 2(-3) & 2(3) & 2(-1)\end{pmatrix}$ $2B = \begin{pmatrix}2 & 8 & -4 \\ -6 & 6 & -2\end{pmatrix}$

    Step 2: Calculate A + 2B Now, add matrix A to the result of 2B: $A + 2B = \begin{pmatrix}2 & -4 & 3 \\ 5 & 1 & 0\end{pmatrix} + \begin{pmatrix}2 & 8 & -4 \\ -6 & 6 & -2\end{pmatrix}$

    Perform the addition element by element:

    • Row 1, Col 1: $2 + 2 = 4$
    • Row 1, Col 2: $-4 + 8 = 4$
    • Row 1, Col 3: $3 + (-4) = -1$
    • Row 2, Col 1: $5 + (-6) = -1$
    • Row 2, Col 2: $1 + 6 = 7$
    • Row 2, Col 3: $0 + (-2) = -2$

    Resulting matrix: $\begin{pmatrix}4 & 4 & -1 \\ -1 & 7 & -2\end{pmatrix}$

    Conclusion

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

    Tip

    When dealing with matrix addition or subtraction, always double-check your signs, especially when adding negative numbers. A common mistake is to add the absolute values instead of performing the algebraic sum (e.g., $3 + (-4)$ is $-1$, not $7$).

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

28. 28.

    The scores of students in a test are recorded as follows: 4, 3, 3, 2, 1, 2, 5, 7, 8, 3, and 5. Find the mode of the mark.

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

    Show answer

    Answer: B. 3

    AI explanation

    Analysis

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

    Concept

    To find the mode, follow these simple steps:

    1. Arrange the data: It is often easier to count the frequency of each number if you list them in ascending order (though not strictly necessary).
    2. Count the frequency: Determine how many times each unique number appears in the list.
    3. Identify the highest frequency: The number that appears the most times is your mode.

    Solution

    Given data: 4, 3, 3, 2, 1, 2, 5, 7, 8, 3, 5

    Let's list the frequency of each score:

    • 1: appears 1 time
    • 2: appears 2 times (2, 2)
    • 3: appears 3 times (3, 3, 3)
    • 4: appears 1 time
    • 5: appears 2 times (5, 5)
    • 7: appears 1 time
    • 8: appears 1 time

    Comparing the frequencies:

    • 1 occurs once.
    • 2 occurs twice.
    • 3 occurs three times.
    • 4 occurs once.
    • 5 occurs twice.
    • 7 occurs once.
    • 8 occurs once.

    Since the number 3 appears more frequently than any other number, it is the mode.

    Conclusion

    Exam Tip

    When you see a list of numbers in an exam, quickly tally them on your rough sheet. Don't just scan with your eyes, as it is very easy to miss one digit and get the wrong answer. A quick frequency table (Score | Tally) will save you from silly mistakes!

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29. 29.

    A boy bought Oranges at the rate of #24.00 for 5 and sold it at the rate of # 30.00 for 4 Oranges. Find the profit made of the ones sold

    1. A.# 24.00
    2. B.#30.00
    3. C.# 10.80
    4. D.#20.00

    Show answer

    Answer: C. # 10.80

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30. 30.

    The weights of 15 students in a class are given as 25, 30, 32, 30, 42, 45, 48, 50, 52, 51, 42, 38, 40, and 42. What is the mode of the given data?

    1. A.51
    2. B.45
    3. C.42
    4. D.25

    Show answer

    Answer: C. 42

    AI explanation

    Analysis

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

    Concept

    To find the mode, follow these steps:

    1. List all the unique numbers present in the data.
    2. Count how many times each number appears (the frequency).
    3. Identify the number with the highest frequency.

    Step-by-Step Solution

    First, let's list the weights provided: 25, 30, 32, 30, 42, 45, 48, 50, 52, 51, 42, 38, 40, 42.

    Note: You mentioned there are 15 students, but only 14 weights were listed. Let's count the frequency of the numbers provided:

    • 25: appears 1 time
    • 30: appears 2 times
    • 32: appears 1 time
    • 42: appears 3 times
    • 45: appears 1 time
    • 48: appears 1 time
    • 50: appears 1 time
    • 52: appears 1 time
    • 51: appears 1 time
    • 38: appears 1 time
    • 40: appears 1 time

    Comparing the frequencies:

    • 30 appears 2 times.
    • 42 appears 3 times.

    Since 42 has the highest frequency, it is the mode of the data set.

    Conclusion

    The number 42 appears more frequently than any other value in the list.

    Exam Tip

    When dealing with a long list of numbers in an exam, always arrange them in ascending order (from smallest to largest) first. This makes it much easier to spot the frequency of each number and prevents you from accidentally skipping a value.

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31. 31.

    If B varies inversely as c$^{\frac{1}{3}}$ and C = 27 when B = 2, find the value of the constant of proportionality K.

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

    Show answer

    Answer: B. 6

    AI explanation

    1. Analyze the question

    We are given that $B$ varies inversely as $C^{\frac{1}{3}}$. We are provided with a pair of values ($C = 27$ and $B = 2$) and asked to find the constant of proportionality, $k$.

    2. Explain the concept

    In mathematics, when a variable $B$ varies inversely as another variable $X$, it is written as:

    $B \propto \frac{1}{X} \quad \text{or} \quad B = \frac{k}{X}$

    where $k$ is the constant of proportionality.

    In this specific problem, $X = C^{\frac{1}{3}}$, so our equation becomes:

    $B = \frac{k}{C^{\frac{1}{3}}}$

    3. Show the solution

    Step 1: Set up the equation.

    $B = \frac{k}{C^{\frac{1}{3}}}$

    Step 2: Substitute the given values into the equation. We are told $B = 2$ and $C = 27$.

    $2 = \frac{k}{27^{\frac{1}{3}}}$

    Step 3: Simplify the cube root. Recall that $27^{\frac{1}{3}}$ is the same as $\sqrt[3]{27}$. Since $3 \times 3 \times 3 = 27$, then $\sqrt[3]{27} = 3$. So, the equation becomes:

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

    Step 4: Solve for $k$. Multiply both sides by 3:

    $k = 2 \times 3$

    $k = 6$

    4. Identify the answer

    Comparing our result ($k = 6$) with the given options, the correct choice is B.

    5. Tip

    Whenever you see a fractional exponent like $x^{\frac{1}{n}}$, remember it is just a fancy way of writing the $n^{th}$ root ($\sqrt[n]{x}$). If you are dealing with large numbers, try to express them as powers of smaller numbers (e.g., $27 = 3^3$) to make the calculation much easier!

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32. 32.

    Find the 7$^{th}$ term of the sequence -10, 50, -250 ...........

    1. A.165250
    2. B.156250
    3. C.-156250
    4. D.-165,250

    Show answer

    Answer: C. -156250

    AI explanation

    1. Analyze the question

    We are given a sequence: -10, 50, -250, ... and we need to find the 7th term. First, we must identify the type of sequence. By dividing consecutive terms:

    • $50 \div (-10) = -5$
    • $(-250) \div 50 = -5$

    Since there is a common ratio ($r = -5$), this is a Geometric Progression (G.P.).

    2. Explain the concept

    The formula for the $n^{th}$ term of a Geometric Progression is:

    $T_n = a \cdot r^{(n-1)}$

    Where:

    • $a$ = the first term
    • $r$ = the common ratio
    • $n$ = the position of the term we want to find

    3. Show the solution

    From the sequence:

    • $a = -10$
    • $r = -5$
    • $n = 7$

    Now, substitute these values into the formula:

    $T_7 = -10 \cdot (-5)^{(7-1)}$

    $T_7 = -10 \cdot (-5)^6$

    Calculate $(-5)^6$: Since the power (6) is an even number, the result will be positive. $(-5)^6 = 5^6 = 15,625$

    Now, multiply by the first term ($a$):

    $T_7 = -10 \cdot 15,625$

    $T_7 = -156,250$

    4. Identify the answer

    Comparing our result to the given options: A. 165250 B. 156250 C. -156250 D. -165,250

    The correct option is C.

    5. Exam Tip

    When dealing with sequences where the ratio is negative, remember the Alternating Sign Rule:

    • If the ratio is negative, the signs of the terms will alternate (negative, positive, negative, positive...).
    • Since the 1st term is negative, all odd-numbered terms (1st, 3rd, 5th, 7th) will be negative, and all even-numbered terms will be positive. This quick check helps you eliminate options A and B immediately!

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33. 33.

    PQR is a triangle such that |PQ| = |QR| = 8cm and QPR = 60º. Find the area of $\angle$ PQR

    1. A.16$\sqrt{3}cm^2$
    2. B.16$\sqrt{2}cm^2$
    3. C.8$\sqrt{3}cm^2$
    4. D.16$\sqrt{2}cm^2$

    Show answer

    Answer: A. 16$\sqrt{3}cm^2$

    AI explanation

    1. Analyze the question

    We are given a triangle $PQR$ where two sides are equal ($|PQ| = |QR| = 8\text{ cm}$). This means $\triangle PQR$ is an isosceles triangle. We are also given that $\angle QPR = 60^\circ$. We need to find the area of the triangle.

    2. Explain the concept

    • Isosceles Triangle Property: In an isosceles triangle, the angles opposite the equal sides are equal. Since $|PQ| = |QR|$, then $\angle QRP = \angle QPR = 60^\circ$.
    • Sum of Angles in a Triangle: The sum of angles in any triangle is $180^\circ$. Therefore, $\angle PQR = 180^\circ - (60^\circ + 60^\circ) = 60^\circ$.
    • Equilateral Triangle: Since all three angles are $60^\circ$, the triangle is actually an equilateral triangle with all sides equal to $8\text{ cm}$.
    • Area Formula: The area of a triangle given two sides ($a$ and $b$) and the included angle ($\theta$) is: $$\text{Area} = \frac{1}{2}ab \sin(\theta)$$

    3. Show the solution

    • Step 1: Identify the sides and the included angle. We have side $PQ = 8\text{ cm}$, side $QR = 8\text{ cm}$, and the included angle $\angle PQR = 60^\circ$.
    • Step 2: Apply the area formula. $$\text{Area} = \frac{1}{2} \times |PQ| \times |QR| \times \sin(\angle PQR)$$ $$\text{Area} = \frac{1}{2} \times 8 \times 8 \times \sin(60^\circ)$$
    • Step 3: Substitute the value of $\sin(60^\circ)$. Recall that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$. $$\text{Area} = \frac{1}{2} \times 64 \times \frac{\sqrt{3}}{2}$$ $$\text{Area} = 32 \times \frac{\sqrt{3}}{2}$$ $$\text{Area} = 16\sqrt{3}\text{ cm}^2$$

    4. Identify the answer

    Comparing our result to the given options, $16\sqrt{3}\text{ cm}^2$ corresponds to option A.

    5. Tip

    Whenever you see a triangle with two equal sides and one angle of $60^\circ$, immediately recognize it as an equilateral triangle. You can then use the shortcut formula for the area of an equilateral triangle: $\text{Area} = \frac{\sqrt{3}}{4} \times (\text{side})^2$. Using the shortcut: $\frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3}$. This saves you time in the exam hall!

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34. 34.

    The mean of the numbers 13, 16, x, 18, 21, 2x, 35, is 22. Find the value of x

    1. A.16
    2. B.17
    3. C.23
    4. D.25

    Show answer

    Answer: B. 17

    AI explanation

    Analysis

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

    Concept

    The formula for the mean is:

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

    Given:

    • The numbers are: $13, 16, x, 18, 21, 2x, 35$
    • The total count of numbers ($n$) = $7$
    • The mean = $22$

    Step-by-Step Solution

    Step 1: Set up the equation.

    $\frac{13 + 16 + x + 18 + 21 + 2x + 35}{7} = 22$

    Step 2: Simplify the numerator by adding the constants and the $x$ terms separately.

    • Constants: $13 + 16 + 18 + 21 + 35 = 103$
    • $x$ terms: $x + 2x = 3x$

    So, the equation becomes:

    $\frac{103 + 3x}{7} = 22$

    Step 3: Solve for $x$. Multiply both sides by $7$ to clear the fraction:

    $103 + 3x = 22 \times 7$

    $103 + 3x = 154$

    Subtract $103$ from both sides:

    $3x = 154 - 103$

    $3x = 51$

    Divide by $3$:

    $x = \frac{51}{3}$

    $x = 17$

    Conclusion

    By substituting $x = 17$ back into the set, we can verify the mean: $(13 + 16 + 17 + 18 + 21 + 34 + 35) / 7 = 154 / 7 = 22$.

    Exam Tip

    When dealing with mean questions, always count the number of items carefully. A common mistake is to miscount the items in the list (e.g., forgetting to count $x$ or $2x$ as individual entries). Also, remember that $2x$ is a single term in the sum, not two separate numbers.

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35. 35.

    Subtract 14256$_{seven}$ from 20045$_{seven}$

    1. A.1256$_{seven}$
    2. B.2456$_{seven}$
    3. C.1356$_{seven}$
    4. D.2465$_{seven}$

    Show answer

    Answer: B. 2456$_{seven}$

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

37. 37.

    If x is inversely proportional to y and x = 9 when y = 4, find the law containing x and y

    1. A.x = $\frac{16}{y}$
    2. B.x = $\frac{36}{y}$
    3. C.x = $\frac{y}{36}$
    4. D.x = $\frac{13}{y}$

    Show answer

    Answer: B. x = $\frac{36}{y}$

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38. 38.

    Find the angle subtended by an arc 33cm long of a circle of diameter 28cm.

    1. A.105$^0$
    2. B.115$^0$
    3. C.120$^0$
    4. D.135$^0$

    Show answer

    Answer: D. 135$^0$

    AI explanation

    Analysis

    To find the angle subtended by an arc at the center of a circle, we use the formula for arc length: Arc Length ($L$) = $\frac{\theta}{360^\circ} \times 2\pi r$ Where:

    • $L$ = length of the arc
    • $\theta$ = angle subtended at the center
    • $r$ = radius of the circle
    • $\pi$ = approximately $\frac{22}{7}$

    Concept

    The arc length is a fraction of the total circumference of the circle. Since the total circumference is $2\pi r$ (or $\pi d$), the arc length is simply the ratio of the angle to the full circle ($360^\circ$) multiplied by the circumference.

    Step-by-Step Solution

    1. Identify the given values:

    • Arc length ($L$) = $33\text{ cm}$
    • Diameter ($d$) = $28\text{ cm}$
    • Radius ($r$) = $\frac{d}{2} = \frac{28}{2} = 14\text{ cm}$

    2. Set up the equation: Using the formula $L = \frac{\theta}{360^\circ} \times 2\pi r$: $33 = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times 14$

    3. Simplify the right side:

    • First, simplify $2 \times \frac{22}{7} \times 14$: $2 \times 22 \times 2 = 88$
    • Now the equation is: $33 = \frac{\theta}{360} \times 88$

    4. Solve for $\theta$:

    • Multiply both sides by 360: $33 \times 360 = \theta \times 88$
    • $11,880 = 88\theta$
    • $\theta = \frac{11,880}{88}$
    • $\theta = 135^\circ$

    Conclusion

    The calculation confirms that the angle subtended by the arc is $135^\circ$.

    Exam Tip

    When dealing with circles in JAMB, always check if you are given the diameter or the radius. A common mistake is using the diameter directly in the $2\pi r$ formula. Remember: $2r = d$, so you can also write the formula as $L = \frac{\theta}{360} \times \pi d$. Using this shortcut, $33 = \frac{\theta}{360} \times \frac{22}{7} \times 28$, which simplifies to $33 = \frac{\theta}{360} \times 88$, saving you time!

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

40. 40.

    P(x, 4) and Q( 10, 8) are two points joined by a straight line in a plane. If the midpoint of the line is (9, 6), find the value of x.

    1. A.12
    2. B.9
    3. C.8
    4. D.7

    Show answer

    Answer: C. 8

    AI explanation

    1. Analyze the question

    We are given two points, $P(x, 4)$ and $Q(10, 8)$, and their midpoint $M(9, 6)$. We need to find the unknown coordinate $x$.

    2. Explain the concept

    In coordinate geometry, the midpoint $M(x_m, y_m)$ of a line segment connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the midpoint formula:

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

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

    Since we are looking for $x$, we only need to focus on the formula for the x-coordinate.

    3. Show the solution

    Given:

    • $P = (x_1, y_1) = (x, 4)$
    • $Q = (x_2, y_2) = (10, 8)$
    • Midpoint $M = (x_m, y_m) = (9, 6)$

    Using the formula for the x-coordinate of the midpoint:

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

    Substitute the known values into the equation:

    $9 = \frac{x + 10}{2}$

    Now, solve for $x$:

    1. Multiply both sides by 2 to clear the fraction: $$9 \times 2 = x + 10$$ $$18 = x + 10$$

    2. Subtract 10 from both sides: $$x = 18 - 10$$ $$x = 8$$

    (Self-check: If we check the y-coordinates, $\frac{4 + 8}{2} = \frac{12}{2} = 6$. This matches our midpoint y-coordinate, confirming our calculation is correct.)

    4. Identify the answer

    The value of $x$ is 8, which corresponds to option C.

    5. Tip

    When dealing with midpoint questions, always remember that the midpoint is simply the average of the x-coordinates and the average of the y-coordinates. If you ever forget the formula, just think: "Add them up and divide by two!"

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41. 41.

    From the top of a building 10m high, the angle of elevation of a fruit on top of a tree 25m is 30º. Calculate the horizontal distance between the building and the tree.

    1. A.15$\sqrt{3}$m
    2. B.5$\sqrt{3}$m
    3. C.25$\sqrt{3}$m
    4. D.30$\sqrt{3}$m

    Show answer

    Answer: A. 15$\sqrt{3}$m

    AI explanation

    Analysis

    To solve this problem, we need to visualize the geometry of the situation. We have a building and a tree standing on horizontal ground. We are looking for the horizontal distance between them.

    Concept: Trigonometry (Right-Angled Triangles)

    When dealing with angles of elevation, we use the tangent ratio:

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

    In this scenario:

    • The height of the building is $10\text{m}$.
    • The height of the tree is $25\text{m}$.
    • The angle of elevation from the top of the building to the top of the tree is $30^\circ$.

    Step-by-Step Solution

    1. Find the vertical difference: Since the observer is at the top of the $10\text{m}$ building looking at the top of the $25\text{m}$ tree, the "opposite" side of our triangle is the difference in their heights: $$\text{Opposite} = 25\text{m} - 10\text{m} = 15\text{m}$$

    2. Identify the variables: • $\text{Opposite} = 15\text{m}$ • $\text{Angle} (\theta) = 30^\circ$ • $\text{Adjacent} (\text{horizontal distance}) = x$

    3. Apply the formula: $$\tan(30^\circ) = \frac{15}{x}$$

    4. Solve for $x$: We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. $$\frac{1}{\sqrt{3}} = \frac{15}{x}$$ $$x = 15 \times \sqrt{3}$$ $$x = 15\sqrt{3}\text{m}$$

    Conclusion

    The horizontal distance between the building and the tree is $15\sqrt{3}\text{m}$. This matches option A.

    Exam Tip: Always remember to subtract the height of the observer (the building) from the height of the object (the tree) when calculating angles of elevation from a point above the ground. If you don't subtract, you will get the wrong "opposite" side!

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42. 42.

    Find the variance of a group of data whose standard deviation is 12.34 to the nearest whole number.

    1. A.150
    2. B.151
    3. C.152
    4. D.153

    Show answer

    Answer: C. 152

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43. 43.

    Find x if $\frac{16^{2 + x}}{4} = 64^x$

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

    Show answer

    Answer: A. 3

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44. 44.

    U varies directly as the square root of V when U = 24, V = 9, find the value of V when U = 16.

    1. A.0.32
    2. B.4
    3. C.6
    4. D.0.25

    Show answer

    Answer: B. 4

    AI explanation

    1. Analyze the Question

    The problem involves direct variation. We are told that $U$ varies directly as the square root of $V$. We are given an initial set of values ($U=24, V=9$) to find the constant of variation, and then asked to find $V$ when $U=16$.

    2. Explain the Concept

    When a variable $U$ varies directly as the square root of $V$, it is expressed mathematically as:

    $U \propto \sqrt{V}$

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

    $U = k\sqrt{V}$

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

    3. Show the Solution

    Step 1: Find the constant $k$ Using $U = 24$ and $V = 9$:

    $24 = k\sqrt{9}$

    Since $\sqrt{9} = 3$:

    $24 = 3k$

    Divide both sides by 3:

    $k = \frac{24}{3} = 8$

    Step 2: Find $V$ when $U = 16$ Now that we know $k = 8$, our formula is $U = 8\sqrt{V}$. Substitute $U = 16$:

    $16 = 8\sqrt{V}$

    Divide both sides by 8:

    $2 = \sqrt{V}$

    To solve for $V$, square both sides of the equation:

    $2^2 = (\sqrt{V})^2$

    $4 = V$

    4. Identify the Answer

    The value of $V$ is 4. This corresponds to option B.

    5. Exam Tip

    In variation problems, always find the constant $k$ first. Think of $k$ as the "bridge" that connects the two variables. Once you have $k$, you can solve for any missing value in that relationship. Also, remember that "varies directly" means the variables move in the same direction, while "varies inversely" means they move in opposite directions.

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45. 45.

    From a class of 5 girls and 7 boys, a committee consisting of 2 girls and 3 boys is to be formed. How many ways can this be done?

    1. A.350 ways
    2. B.420 ways
    3. C.550 ways
    4. D.720 ways

    Show answer

    Answer: A. 350 ways

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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.

    Given P = $\begin{bmatrix}1 & 2\\2 & 3\end{bmatrix}$, find P$^2$ - 4P - I where I is the identity matrix

    1. A.$\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$
    2. B.$\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}$
    3. C.$\begin{bmatrix}-1 & 0\\0 & -1\end{bmatrix}$
    4. D.$\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$

    Show answer

    Answer: B. $\begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}$

    AI explanation

    1. Analyze the Question

    We are given a $2 \times 2$ matrix $P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$. We need to evaluate the expression $P^2 - 4P - I$, where $I$ is the identity matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

    2. Explain the Concept

    • Matrix Multiplication ($P^2$): To find $P^2$, we multiply matrix $P$ by itself ($P \times P$). Remember the row-by-column rule: multiply the elements of the row of the first matrix by the corresponding elements of the column of the second matrix and sum them.
    • Scalar Multiplication ($4P$): Multiply every element inside matrix $P$ by the scalar 4.
    • Matrix Subtraction: Subtract corresponding elements of the matrices.

    3. Show the Solution

    Step 1: Calculate $P^2$ $P^2 = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ $= \begin{bmatrix} (1 \times 1 + 2 \times 2) & (1 \times 2 + 2 \times 3) \\ (2 \times 1 + 3 \times 2) & (2 \times 2 + 3 \times 3) \end{bmatrix}$ $= \begin{bmatrix} (1 + 4) & (2 + 6) \\ (2 + 6) & (4 + 9) \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix}$

    Step 2: Calculate $4P$ $4P = 4 \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix}$

    Step 3: Evaluate $P^2 - 4P - I$ $= \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix} - \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

    Perform the subtraction element by element: Top-left: $5 - 4 - 1 = 0$ Top-right: $8 - 8 - 0 = 0$ Bottom-left: $8 - 8 - 0 = 0$ Bottom-right: $13 - 12 - 1 = 0$

    Result: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

    4. Identify the Answer

    The result is the zero matrix. Comparing this to the given options, the correct choice is B.

    5. Tip

    When dealing with matrix expressions like $P^2 - 4P - I$, always double-check your arithmetic in the multiplication step. A single sign error in $P^2$ will ruin the entire calculation. Also, remember that $I$ is always $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ for a $2 \times 2$ matrix; don't mistake it for a zero matrix!

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48. 48.

    Calculate the standard deviation of the following scores 5, 4, 6, 7, and 8

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

    Show answer

    Answer: A. $\sqrt{2}$

    AI explanation

    To calculate the standard deviation of a set of numbers, we follow a systematic process.

    1. Analyze the question

    We are given the dataset: 5, 4, 6, 7, 8. We need to find the standard deviation, which measures how spread out the numbers are from the mean.

    2. Explain the concept

    The formula for standard deviation ($\sigma$) is:

    $\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}$

    Where:

    • $x$ = each individual score
    • $\bar{x}$ = the mean (average) of the scores
    • $n$ = the number of scores
    • $\sum$ = the sum of the values

    3. Show the solution

    Step 1: Find the mean ($\bar{x}$)

    $\bar{x} = \frac{5 + 4 + 6 + 7 + 8}{5} = \frac{30}{5} = 6$

    Step 2: Subtract the mean from each score and square the result $(x - \bar{x})^2$

    • $(5 - 6)^2 = (-1)^2 = 1$
    • $(4 - 6)^2 = (-2)^2 = 4$
    • $(6 - 6)^2 = (0)^2 = 0$
    • $(7 - 6)^2 = (1)^2 = 1$
    • $(8 - 6)^2 = (2)^2 = 4$

    Step 3: Find the sum of these squares

    $1 + 4 + 0 + 1 + 4 = 10$

    Step 4: Divide by the number of scores ($n=5$)

    $\text{Variance} = \frac{10}{5} = 2$

    Step 5: Take the square root to find the standard deviation

    $\sigma = \sqrt{2}$

    4. Identify the answer

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

    5. Exam Tip

    When calculating standard deviation, always remember that Variance = (Standard Deviation)². If you find the variance first, simply take the square root at the very end. Also, if your calculated mean is a whole number, the subtraction step becomes much faster and less prone to errors!

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49. 49.

    A binary operation * is defined on a set of real numbers by x*y = x$^y$ for all values of x and y, if x * 2 = x, find the possible values of x.

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

    Show answer

    Answer: C. 0, 1

    AI explanation

    Analysis

    The problem defines a binary operation $x * y = x^y$. We are given the condition $x * 2 = x$ and asked to find the possible values of $x$ that satisfy this equation.

    Concept

    A binary operation is simply a rule for combining two numbers. In this case, the rule is to raise the first number to the power of the second number. To solve for $x$, we substitute the given values into the defined operation and solve the resulting algebraic equation.

    Step-by-Step Solution

    1. Apply the definition: Given $x * y = x^y$, substitute $y = 2$ into the expression: $x * 2 = x^2$

    2. Set up the equation: We are told that $x * 2 = x$. Therefore: $x^2 = x$

    3. Solve for $x$: Bring all terms to one side to form a quadratic equation: $x^2 - x = 0$

    4. Factorize the equation: Factor out $x$: $x(x - 1) = 0$

    5. Find the roots: For the product to be zero, either: $x = 0$ or $x - 1 = 0 \implies x = 1$

    The possible values for $x$ are $0$ and $1$.

    Conclusion

    Comparing our result with the provided options, we see that $0$ and $1$ correspond to option C.

    Exam Tip

    When you have an equation like $x^2 = x$, never divide both sides by $x$ to get $x = 1$. If you do that, you lose the solution $x = 0$. Always move all terms to one side and factorize to ensure you capture all possible roots!

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50. 50.

    The sum to infinity of a GP is 100, find its first term if the common ratio is -$\frac{1}{2}$

    1. A.150
    2. B.155
    3. C.160
    4. D.165

    Show answer

    Answer: A. 150

    AI explanation

    1. Analyze the question

    We are given the sum to infinity ($S_\infty$) of a Geometric Progression (GP) and the common ratio ($r$). We need to find the first term ($a$).

    2. Explain the concept

    For a Geometric Progression where the common ratio $|r| < 1$, the sum to infinity is calculated using the formula:

    $S_\infty = \frac{a}{1 - r}$

    Where:

    • $S_\infty$ = Sum to infinity
    • $a$ = First term
    • $r$ = Common ratio

    3. Show the solution

    Given:

    • $S_\infty = 100$
    • $r = -\frac{1}{2}$

    Step 1: Substitute the values into the formula

    $100 = \frac{a}{1 - (-\frac{1}{2})}$

    Step 2: Simplify the denominator

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

    $1 + 0.5 = 1.5 \text{ (or } \frac{3}{2} \text{)}$

    Step 3: Solve for $a$

    $100 = \frac{a}{1.5}$

    Multiply both sides by 1.5:

    $a = 100 \times 1.5$

    $a = 150$

    4. Identify the answer

    The calculated first term is 150, which corresponds to option A.

    5. Tip

    When dealing with a negative common ratio in the sum to infinity formula, always be careful with the signs in the denominator ($1 - r$). Many students mistakenly add the ratio directly instead of subtracting a negative, which leads to $1 - 0.5 = 0.5$ instead of the correct $1 + 0.5 = 1.5$. Always write out the signs clearly to avoid this common JAMB trap!

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