2020 JAMB Mathematics Past Questions and Answers

Questions

1. 1.

   The acres for rice, pineapple, cassava, cocoa and palm oil in a certain district are given respectively as 2, 5, 3, 11 and 9. What is the angle of the sector of cassava in a pie chart?

   1. A.180$^o$
   2. B.36$^o$
   3. C.60$^o$
   4. D.108$^o$

   Show answer

   Answer: B. 36$^o$

   AI explanation

   1. Analyze the question

   We are given the distribution of land (in acres) for five different crops. We need to find the specific angle that the "cassava" portion would occupy if this data were represented in a pie chart.

   2. Explain the concept

   A pie chart represents the total data as a circle, which consists of $360^\circ$. To find the angle of a specific sector, we use the formula:

   $\text{Angle} = \left( \frac{\text{Value of the sector}}{\text{Total value of all sectors}} \right) \times 360^\circ$

   3. Show the solution

   Step 1: Calculate the total number of acres. Total = $2 (\text{rice}) + 5 (\text{pineapple}) + 3 (\text{cassava}) + 11 (\text{cocoa}) + 9 (\text{palm oil})$ Total = $30 \text{ acres}$

   Step 2: Identify the value for cassava. Value of cassava = $3 \text{ acres}$

   Step 3: Apply the formula.

   $\text{Angle of cassava} = \left( \frac{3}{30} \right) \times 360^\circ$

   Step 4: Simplify the calculation.

   $\text{Angle} = \left( \frac{1}{10} \right) \times 360^\circ$

   $\text{Angle} = 36^\circ$

   4. Identify the answer

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

   5. Exam Tip

   When dealing with pie chart questions, always calculate the total sum first. If the total sum is a factor of 360 (like 30, 40, 60, or 90), the calculation becomes much easier. Also, remember that the sum of all angles in your pie chart must equal $360^\circ$; if you have time, you can quickly check your work by adding the angles of all sectors together.

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2. 2.

   The sum of the interior angle of pentagon is 6x + 6y. Find y in terms of x.

   1. A.y = 6 - x
   2. B.y = 90 - x
   3. C.y = 120 - x
   4. D.y = 150 - x

   Show answer

   Answer: B. y = 90 - x

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3. 3.

   The goals scored by 40 football teams from three league  divisions are recorded below No of goals 0 1 2 3 4 5 6 Frequency 4 3 15 16 1 0 1 What is the total number of goals scored by all the teams?

   1. A.21
   2. B.40
   3. C.91
   4. D.96

   Show answer

   Answer: C. 91

   AI explanation

   Analysis

   To find the total number of goals scored by all the teams, we need to calculate the sum of the products of each number of goals and its corresponding frequency. This is essentially finding the sum ($\sum fx$) from a frequency distribution table.

   Concept

   When given a frequency table, the total value of a data set is calculated by multiplying each value ($x$) by its frequency ($f$) and then adding all those products together. Formula: $\text{Total} = \sum (x \times f)$

   Step-by-Step Solution

   Let's multiply each number of goals by its frequency:

   • 0 goals: $0 \times 4 = 0$
   • 1 goal: $1 \times 3 = 3$
   • 2 goals: $2 \times 15 = 30$
   • 3 goals: $3 \times 16 = 48$
   • 4 goals: $4 \times 1 = 4$
   • 5 goals: $5 \times 0 = 0$
   • 6 goals: $6 \times 1 = 6$

   Now, add these products together to get the total number of goals: $\text{Total} = 0 + 3 + 30 + 48 + 4 + 0 + 6$

   Calculation: $3 + 30 = 33$ $33 + 48 = 81$ $81 + 4 = 85$ $85 + 6 = 91$

   The total number of goals scored by all the teams is 91.

   Conclusion

   Comparing our result to the given options: A. 21 B. 40 C. 91 D. 96

   The correct option is C.

   Exam Tip

   In questions involving frequency tables, always double-check your multiplication before adding. A common mistake is to add the frequencies instead of the products, or to misread the table rows. Also, ensure the sum of your frequencies matches the total number of items mentioned in the question (in this case, $4 + 3 + 15 + 16 + 1 + 0 + 1 = 40$), which confirms you haven't missed any data points!

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4. 4.

   Reach each number to two significant figures and then evaluate $\frac{0.02174 \times 1.2047}{0.023789}$

   1. A.0
   2. B.0.9
   3. C.1.1
   4. D.1.2

   Show answer

   Answer: C. 1.1

   AI explanation

   1. Analyze the question

   The task requires two distinct steps:

   • Rounding: Convert each of the three numbers in the expression to two significant figures.
   • Evaluation: Perform the arithmetic operation (multiplication and division) using these rounded values.

   2. Explain the concept

   • Significant Figures: When rounding to two significant figures, we start counting from the first non-zero digit. For decimals, leading zeros (zeros before the first non-zero digit) are not significant.
   • Rounding Rules: If the digit after the second significant figure is 5 or greater, we round up the second digit. If it is less than 5, we keep the second digit as it is.

   3. Show the solution

   Step 1: Round each number to two significant figures

   • $0.02174$: The first two significant figures are $2$ and $1$. The next digit is $7$ (which is $\geq 5$), so we round up the $1$ to $2$. • Rounded value = $0.022$
   • $1.2047$: The first two significant figures are $1$ and $2$. The next digit is $0$ (which is $< 5$), so we keep the $2$ as it is. • Rounded value = $1.2$
   • $0.023789$: The first two significant figures are $2$ and $3$. The next digit is $7$ (which is $\geq 5$), so we round up the $3$ to $4$. • Rounded value = $0.024$

   Step 2: Evaluate the expression Substitute the rounded values into the expression:

   $\frac{0.022 \times 1.2}{0.024}$

   First, multiply the numerator:

   $0.022 \times 1.2 = 0.0264$

   Next, divide by the denominator:

   $\frac{0.0264}{0.024}$

   To make the division easier, multiply both numerator and denominator by $1000$:

   $\frac{26.4}{24} = 1.1$

   4. Identify the answer

   Comparing our result ($1.1$) to the given options: A. 0 B. 0.9 C. 1.1 D. 1.2

   The correct option is C.

   5. Exam Tip

   When dealing with significant figures in JAMB, always remember that leading zeros are never significant. If you are ever confused, write the number in standard form (scientific notation) first; the digits in the coefficient will clearly show you which ones are significant!

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5. 5.

   If N225.00 yields N27.00 in x years simple interest at the rate of 4% per annum, find x.

   1. A.3
   2. B.4
   3. C.12
   4. D.17

   Show answer

   Answer: A. 3

   AI explanation

   Analysis

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

   Concept

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

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

   Where:

   • $I$ = Simple Interest (N27.00)
   • $P$ = Principal (N225.00)
   • $R$ = Rate per annum (4%)
   • $T$ = Time in years ($x$)

   Solution

   1. Substitute the given values into the formula: $$27 = \frac{225 \times 4 \times x}{100}$$

   2. Simplify the right side of the equation: First, multiply 225 by 4: $$225 \times 4 = 900$$

   Now, substitute this back:

   $27 = \frac{900 \times x}{100}$

   3. Divide 900 by 100: $$27 = 9 \times x$$ $$27 = 9x$$

   4. Solve for $x$: $$x = \frac{27}{9}$$ $$x = 3$$

   The time period is 3 years.

   Conclusion

   Comparing our result with the options provided, the correct value is 3. This matches option A.

   Exam Tip

   When dealing with simple interest questions in JAMB, always remember the acronym PRT (Principal, Rate, Time). If you are asked to find the time, you can quickly rearrange the formula to $T = \frac{100 \times I}{P \times R}$. This saves time during the exam!

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6. 6.

   Three consecutive terms of a geometric progression are give as n - 2, n and n + 3. Find the common ratio

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

   Show answer

   Answer: A. $\frac{3}{2}$

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7. 7.

   Find the value of x if $\frac{\sqrt{2}}{x + \sqrt{2}}$ = $\frac{1}{x - \sqrt{2}}$

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

   Show answer

   Answer: A. 3$\sqrt{2}$ + 4

   AI explanation

   Analysis

   To solve for $x$ in the equation $\frac{\sqrt{2}}{x + \sqrt{2}} = \frac{1}{x - \sqrt{2}}$, we need to eliminate the fractions by cross-multiplying and then isolate the variable $x$.

   Concept

   This is a linear equation involving surds. The key principle here is cross-multiplication: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$. Once the fractions are removed, we group the terms containing $x$ on one side and the constant terms on the other.

   Step-by-Step Solution

   Step 1: Cross-multiply the equation.

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

   Step 2: Expand the brackets. Multiply $\sqrt{2}$ by each term inside the parenthesis on the left side:

   $\sqrt{2} \cdot x - \sqrt{2} \cdot \sqrt{2} = x + \sqrt{2}$

   $x\sqrt{2} - 2 = x + \sqrt{2}$

   (Note: $\sqrt{2} \times \sqrt{2} = 2$)

   Step 3: Group the terms with $x$ on one side. Subtract $x$ from both sides and add $2$ to both sides:

   $x\sqrt{2} - x = \sqrt{2} + 2$

   Step 4: Factor out $x$ on the left side.

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

   Step 5: Solve for $x$ by dividing both sides by $(\sqrt{2} - 1)$.

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

   Step 6: Rationalize the denominator. To simplify, multiply the numerator and denominator by the conjugate of the denominator, which is $(\sqrt{2} + 1)$:

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

   Expand the numerator:

   $\sqrt{2} \cdot \sqrt{2} + \sqrt{2} \cdot 1 + 2 \cdot \sqrt{2} + 2 \cdot 1 = 2 + \sqrt{2} + 2\sqrt{2} + 2$

   $= 4 + 3\sqrt{2}$

   Expand the denominator (difference of two squares):

   $(\sqrt{2})^2 - (1)^2 = 2 - 1 = 1$

   So, $x = \frac{4 + 3\sqrt{2}}{1} = 3\sqrt{2} + 4$.

   Conclusion

   Comparing our result $3\sqrt{2} + 4$ with the given options, it matches option A.

   Exam Tip

   When you see a denominator with a surd (like $\sqrt{2} - 1$), always remember to rationalize it by multiplying by its conjugate (change the sign in the middle). This is a common requirement in JAMB Mathematics to simplify your final answer into the format provided in the options.

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8. 8.

   Simplify $\frac{324 - 4x^2}{2x + 18}$

   1. A.2(x - 9)
   2. B.2(9 + x)
   3. C.81 - x$^2$
   4. D.-2(x - 9)

   Show answer

   Answer: D. -2(x - 9)

   AI explanation

   To simplify the expression $\frac{324 - 4x^2}{2x + 18}$, we will use algebraic factorization techniques.

   1. Analyze the expression

   The numerator is $324 - 4x^2$. Notice that both terms are perfect squares:

   • $324 = 18^2$
   • $4x^2 = (2x)^2$

   This fits the Difference of Two Squares formula: $a^2 - b^2 = (a - b)(a + b)$.

   2. Factorize the numerator

   Applying the formula where $a = 18$ and $b = 2x$:

   $324 - 4x^2 = (18 - 2x)(18 + 2x)$

   Alternatively, we can factor out the common factor of $4$ first:

   $4(81 - x^2) = 4(9 - x)(9 + x)$

   3. Factorize the denominator

   The denominator is $2x + 18$. We can factor out the common factor of $2$:

   $2x + 18 = 2(x + 9)$

   Since addition is commutative, $2(x + 9)$ is the same as $2(9 + x)$.

   4. Simplify the fraction

   Now, substitute the factored forms back into the original expression:

   $\frac{4(9 - x)(9 + x)}{2(9 + x)}$

   Cancel out the common term $(9 + x)$ from the numerator and the denominator:

   $\frac{4(9 - x)}{2}$

   Divide $4$ by $2$:

   $2(9 - x)$

   5. Compare with the options

   We have $2(9 - x)$. Let's look at the options provided:

   • A. $2(x - 9)$
   • B. $2(9 + x)$
   • C. $81 - x^2$
   • D. $-2(x - 9)$

   Note that $2(9 - x)$ is mathematically equivalent to $-2(x - 9)$ because: $-2 \times x = -2x$ $-2 \times -9 = +18$ So, $-2(x - 9) = 18 - 2x$, which is the same as $2(9 - x)$.

   Exam Tip: When you see a negative sign inside a bracket, remember that $-(a - b)$ is the same as $(b - a)$. Always check if your result can be rearranged to match the options by factoring out a $-1$.

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9. 9.

   A sector of circle of radius 7.2cm which substends an angle of 300$^o$ at the centre is used to form a cone. What s the radius of the base of the cone?

   1. A.8cm
   2. B.9cm
   3. C.6cm
   4. D.7cm

   Show answer

   Answer: C. 6cm

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10. 10.

    A number is selected at random between 20 and 30, both numbers inclusive. Find the probability that the number is a prime.

    1. A.$\frac{2}{11}$
    2. B.$\frac{5}{11}$
    3. C.$\frac{6}{11}$
    4. D.$\frac{8}{11}$

    Show answer

    Answer: A. $\frac{2}{11}$

    AI explanation

    1. Analyze the question

    We need to find the probability of selecting a prime number from the set of integers between 20 and 30, inclusive.

    2. Explain the concept

    • Sample Space ($n(S)$): This is the total number of possible outcomes. Since the range is 20 to 30 inclusive, we count every integer from 20 up to 30.
    • Prime Number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
    • Probability Formula: $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

    3. Show the solution

    Step 1: Determine the total number of outcomes ($n(S)$) The numbers are: 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Counting these, we have: $n(S) = 11$

    Step 2: Identify the prime numbers in the set ($n(E)$) Let's check each number:

    • 20: Not prime (divisible by 2, 4, 5, 10)
    • 21: Not prime (divisible by 3, 7)
    • 22: Not prime (divisible by 2, 11)
    • 23: Prime
    • 24: Not prime (divisible by 2, 3, 4, 6, 8, 12)
    • 25: Not prime (divisible by 5)
    • 26: Not prime (divisible by 2, 13)
    • 27: Not prime (divisible by 3, 9)
    • 28: Not prime (divisible by 2, 4, 7, 14)
    • 29: Prime
    • 30: Not prime (divisible by 2, 3, 5, 6, 10, 15)

    The prime numbers are {23, 29}. So, $n(E) = 2$.

    Step 3: Calculate the probability $P(\text{Prime}) = \frac{n(E)}{n(S)} = \frac{2}{11}$

    4. Identify the answer

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

    5. Tip

    When counting numbers in an "inclusive" range (from $a$ to $b$), use the formula $(b - a) + 1$. For this question: $(30 - 20) + 1 = 10 + 1 = 11$. This ensures you don't accidentally miss one of the endpoints!

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11. 11.

    Find the non-zero positive value of x which satisfies the equation  $\begin{bmatrix} x & 1 & 0 \\ 1 & x & 1 \\ 0 & 1 & x \end{bmatrix}$ = 0

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

    Show answer

    Answer: C. $\sqrt{2}$

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12. 12.

    A crate of soft drinks contains 10 bottle of Coca-Cola 8 of Fanta and 6 of Sprite. If one bottle is selected at random, what is the probability that it is Not a Coca-Cola bottle?

    1. A.$\frac{5}{12}$
    2. B.$\frac{1}{3}$
    3. C.$\frac{3}{4}$
    4. D.$\frac{7}{12}$

    Show answer

    Answer: D. $\frac{7}{12}$

    AI explanation

    1. Analyze the question

    We are given a total collection of bottles and asked to find the probability of picking a bottle that is not Coca-Cola. To solve this, we need the total number of bottles and the number of bottles that are either Fanta or Sprite.

    2. Explain the concept

    Probability is defined as:

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

    When dealing with "Not" a specific item, you can use the Complementary Probability rule:

    $P(\text{Not A}) = 1 - P(\text{A})$

    Alternatively, you can simply add up all the items that satisfy the condition "Not Coca-Cola."

    3. Show the solution

    Step 1: Find the total number of bottles.

    • Coca-Cola = 10
    • Fanta = 8
    • Sprite = 6
    • Total = $10 + 8 + 6 = 24$

    Step 2: Find the number of bottles that are "Not Coca-Cola." These are the Fanta and Sprite bottles:

    • Not Coca-Cola = $8 (\text{Fanta}) + 6 (\text{Sprite}) = 14$

    Step 3: Calculate the probability.

    $P(\text{Not Coca-Cola}) = \frac{14}{24}$

    Step 4: Simplify the fraction. Divide both the numerator and the denominator by their greatest common divisor, which is 2:

    $\frac{14 \div 2}{24 \div 2} = \frac{7}{12}$

    (Verification using the Complementary Rule: $P(\text{Coca-Cola}) = \frac{10}{24} = \frac{5}{12}$. Therefore, $P(\text{Not Coca-Cola}) = 1 - \frac{5}{12} = \frac{7}{12}$.)

    4. Identify the answer

    The calculated probability is $\frac{7}{12}$, which matches option D.

    5. Tip

    In probability questions, always check if it is easier to calculate the probability of the event happening or the probability of it not happening. If the "not" category has fewer items, calculate that directly. If the "not" category is larger, subtract the probability of the event from 1 to save time!

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13. 13.

    Calculate the standard deviation of the following data: 7, 8, 9, 10, 11, 12, 13

    1. A.2
    2. B.4
    3. C.12
    4. D.17

    Show answer

    Answer: A. 2

    AI explanation

    To calculate the standard deviation of a set of data, we follow these systematic steps:

    1. Analyze the Question

    We are given a set of seven numbers: 7, 8, 9, 10, 11, 12, 13. We need to find the standard deviation ($\sigma$), which measures the amount of variation or dispersion in the set.

    2. Explain the Concept

    The formula for standard deviation is:

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

    Where:

    • $x$ = each value in the data set
    • $\bar{x}$ = the mean (average) of the data
    • $n$ = the number of values

    3. Show the Solution

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

    $\bar{x} = \frac{7 + 8 + 9 + 10 + 11 + 12 + 13}{7}$

    $\bar{x} = \frac{70}{7} = 10$

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

    • $(7 - 10)^2 = (-3)^2 = 9$
    • $(8 - 10)^2 = (-2)^2 = 4$
    • $(9 - 10)^2 = (-1)^2 = 1$
    • $(10 - 10)^2 = (0)^2 = 0$
    • $(11 - 10)^2 = (1)^2 = 1$
    • $(12 - 10)^2 = (2)^2 = 4$
    • $(13 - 10)^2 = (3)^2 = 9$

    Step 3: Find the sum of these squares

    $\sum(x - \bar{x})^2 = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28$

    Step 4: Divide by the number of values ($n$) and take the square root

    $\sigma = \sqrt{\frac{28}{7}}$

    $\sigma = \sqrt{4}$

    $\sigma = 2$

    4. Identify the Answer

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

    5. Exam Tip

    When you see a set of numbers that are in an arithmetic progression (like this one, where the difference between consecutive numbers is constant, i.e., 1), you can quickly calculate the variance using the formula $\frac{n^2 - 1}{12} \times d^2$, where $n$ is the number of terms and $d$ is the common difference. Here, $n=7$ and $d=1$: Variance $= \frac{7^2 - 1}{12} \times 1^2 = \frac{48}{12} = 4$. Standard Deviation $= \sqrt{4} = 2$. This is a great shortcut to save time in the exam hall!

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14. 14.

    Find the equation of the line through the points (5, 7) parallel to the line 7x + 5y = 12.

    1. A.5x + 7y = 120
    2. B.7x + 5y = 70
    3. C.x + y = 7
    4. D.15x + 17y = 90

    Show answer

    Answer: B. 7x + 5y = 70

    AI explanation

    1. Analyze the question

    We need to find the equation of a line that passes through the point $(5, 7)$ and is parallel to the given line $7x + 5y = 12$.

    2. Explain the concept

    • Parallel Lines: Two lines are parallel if and only if they have the same slope ($m$).
    • Slope-Intercept Form: The equation of a line can be written as $y = mx + c$, where $m$ is the slope.
    • General Form: For a line in the form $Ax + By = C$, the slope is given by $m = -\frac{A}{B}$.
    • Property: If two lines are parallel, their coefficients of $x$ and $y$ remain the same, but the constant term ($C$) changes. Therefore, a line parallel to $Ax + By = C$ will have the form $Ax + By = k$.

    3. Show the solution

    Step 1: Identify the slope of the given line. The given line is $7x + 5y = 12$. Using the formula $m = -\frac{A}{B}$, the slope is: $m = -\frac{7}{5}$

    Step 2: Use the parallel line property. Since the required line is parallel to $7x + 5y = 12$, it must also have the form: $7x + 5y = k$

    Step 3: Solve for $k$ using the given point $(5, 7)$. Substitute $x = 5$ and $y = 7$ into the equation $7x + 5y = k$: $7(5) + 5(7) = k$ $35 + 35 = k$ $70 = k$

    Step 4: Write the final equation. Substitute $k = 70$ back into the equation: $7x + 5y = 70$

    4. Identify the answer

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

    5. Exam Tip

    Whenever you see a question asking for a line parallel to $Ax + By = C$, simply write down $Ax + By = k$ and plug in the coordinates of the given point to find $k$ immediately. This saves you the time of calculating the slope and using the point-slope formula!

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15. 15.

    Find the simple interest rate percent annum at which N1000 accumulates to N1240 in 3 years.

    1. A.6%
    2. B.8%
    3. C.10%
    4. D.12%

    Show answer

    Answer: B. 8%

    AI explanation

    1. Analyze the question

    We are given:

    • Principal ($P$) = N1000
    • Amount ($A$) = N1240
    • Time ($T$) = 3 years
    • We need to find the rate ($R$) in percent per annum.

    2. Explain the concept

    Simple interest ($I$) is calculated using the formula:

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

    First, we must find the interest earned. The interest is the difference between the final amount and the principal:

    $I = A - P$

    Once we have $I$, we can rearrange the simple interest formula to solve for $R$:

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

    3. Show the solution

    Step 1: Calculate the interest ($I$)

    $I = 1240 - 1000$

    $I = 240$

    Step 2: Substitute the values into the rate formula

    $R = \frac{240 \times 100}{1000 \times 3}$

    Step 3: Simplify the calculation

    $R = \frac{24000}{3000}$

    $R = \frac{24}{3}$

    $R = 8\%$

    4. Identify the answer

    Comparing our result to the given options: A. 6% B. 8% C. 10% D. 12%

    The calculated rate is 8%, which corresponds to option B.

    5. Tip

    When dealing with simple interest questions in JAMB, always remember the acronym PRT. If you are asked for the rate, just remember: "Interest times 100, divided by the product of Principal and Time." Don't forget to subtract the Principal from the Amount first to get the Interest—many students mistakenly use the Amount ($1240) in the interest formula instead of the Interest ($240)!

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16. 16.

    What is the product of $\frac{27}{5}$, $(3)^{-3}$ and ($\frac{1}{5})^{-1}$?

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

    Show answer

    Answer: C. 1

    AI explanation

    Analysis

    To solve this problem, we need to find the product of three distinct terms:

    1. $\frac{27}{5}$
    2. $(3)^{-3}$
    3. $(\frac{1}{5})^{-1}$

    Concept

    We will apply the Laws of Indices:

    • Negative Index Rule: $a^{-n} = \frac{1}{a^n}$
    • Fractional Negative Index Rule: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$

    Step-by-Step Solution

    Step 1: Simplify each term.

    • The first term is already simplified: $\frac{27}{5}$
    • For the second term $(3)^{-3}$: Using the negative index rule, $(3)^{-3} = \frac{1}{3^3} = \frac{1}{27}$
    • For the third term $(\frac{1}{5})^{-1}$: Using the fractional negative index rule, $(\frac{1}{5})^{-1} = (\frac{5}{1})^1 = 5$

    Step 2: Multiply the simplified terms together. Product = $\frac{27}{5} \times \frac{1}{27} \times 5$

    Step 3: Perform the calculation. Product = $\left( \frac{27}{5} \right) \times \left( \frac{1}{27} \right) \times 5$

    Notice that the $27$ in the numerator and the $27$ in the denominator cancel each other out: Product = $\frac{1}{5} \times 5$

    Finally, the $5$ in the numerator and the $5$ in the denominator cancel out: Product = $1$

    Conclusion

    The product of the three terms is 1, which corresponds to option C.

    Exam Tip

    When dealing with products of fractions and indices, always look for opportunities to cancel out terms before multiplying the numerators and denominators. It saves time and reduces the risk of arithmetic errors!

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17. 17.

    The mean age group of some students is 15years. When the age of a teacher, 45 years old, is added to the ages of the students, the mean of their ages become 18 years. Find the number of students in the group.

    1. A.7
    2. B.9
    3. C.15
    4. D.42

    Show answer

    Answer: B. 9

    AI explanation

    1. Analyze the question

    We are given the mean age of a group of students. When a teacher (45 years old) is added to this group, the mean age increases. We need to find the original number of students.

    2. Explain the concept

    The formula for the mean is:

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

    From this, we can derive:

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

    3. Show the solution

    Let $n$ be the number of students. Let $S$ be the sum of the ages of the students.

    Step 1: Express the initial state The mean age of the students is 15.

    $\frac{S}{n} = 15 \implies S = 15n$

    Step 2: Express the state after adding the teacher When the teacher (45 years old) is added, the new sum of ages is $S + 45$, and the new number of people is $n + 1$. The new mean is 18.

    $\frac{S + 45}{n + 1} = 18$

    Step 3: Solve for $n$ Substitute $S = 15n$ into the equation:

    $\frac{15n + 45}{n + 1} = 18$

    Multiply both sides by $(n + 1)$:

    $15n + 45 = 18(n + 1)$

    $15n + 45 = 18n + 18$

    Rearrange the equation to group the $n$ terms:

    $45 - 18 = 18n - 15n$

    $27 = 3n$

    $n = \frac{27}{3}$

    $n = 9$

    4. Identify the answer

    The number of students in the group is 9. This matches option B.

    5. Tip

    When dealing with "mean" problems involving adding a new person, always use the "Sum = Mean × Count" approach. It prevents confusion when the number of people in the denominator changes. Also, always check if your answer makes sense: since the teacher is much older than the average, adding them should increase the mean, which is exactly what happened here (from 15 to 18).

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18. 18.

    A school boy lying on the ground 30m away from the foot of a water tank towel observes that the angle of elevation of the top of the tank is 60$^o$. Calculate the height of the tank.

    1. A.60$\sqrt{3m}$
    2. B.30$\sqrt{3m}$
    3. C.20$\sqrt{3m}$
    4. D.10$\sqrt{3m}$

    Show answer

    Answer: B. 30$\sqrt{3m}$

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19. 19.

    Find the gradient of the line passing through the points (-2, 0) and (0, -4)

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

    Show answer

    Answer: C. -2

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20. 20.

    A surveyor walks 500m up a hill which slopes at an angle of 30$^o$. Calculate the vertical height through which he rises

    1. A.252m
    2. B.500m
    3. C.250m
    4. D.255m

    Show answer

    Answer: C. 250m

    AI explanation

    1. Analyze the question

    We are given a scenario involving a surveyor walking up a hill. We have:

    • The distance walked along the slope (the hypotenuse of a right-angled triangle) = 500m
    • The angle of inclination (the angle the slope makes with the horizontal) = 30°
    • We need to find the vertical height (the side opposite the angle).

    2. Explain the concept

    This is a trigonometry problem. In a right-angled triangle, the relationship between the angle, the opposite side (height), and the hypotenuse (slope) is defined by the Sine function:

    $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

    Where:

    • $\theta = 30^\circ$
    • Hypotenuse = 500m
    • Opposite = Vertical height ($h$)

    3. Show the solution

    Using the formula:

    $\sin(30^\circ) = \frac{h}{500}$

    We know from trigonometric ratios that $\sin(30^\circ) = 0.5$ (or $\frac{1}{2}$).

    Substitute the value:

    $0.5 = \frac{h}{500}$

    To find $h$, multiply both sides by 500:

    $h = 500 \times 0.5$

    $h = 250\text{m}$

    4. Identify the answer

    The vertical height is 250m. Comparing this to the given options: A. 252m B. 500m C. 250m D. 255m

    The correct option is C.

    5. Tip

    Remember the mnemonic SOH CAH TOA:

    • SOH: Sine = Opposite / Hypotenuse
    • CAH: Cosine = Adjacent / Hypotenuse
    • TOA: Tangent = Opposite / Adjacent

    Whenever you see "vertical height" in a slope problem, you are almost always looking for the "Opposite" side, so SOH is your best friend!

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21. 21.

    The chord ST of a circle is equal to the radius r of the circle. Find the length of arc ST

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

    Show answer

    Answer: A. $\frac{\pi r}{3}$

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22. 22.

    A cylindrical tank has a capacity of 3080m$^3$. What is the depth of the tank if the diameter of its base is 14m?

    1. A.25m
    2. B.23m
    3. C.22m
    4. D.20m

    Show answer

    Answer: D. 20m

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23. 23.

    In a class of 40 students, 32 offer mathematics, 24 offer physics and 4 offer neither mathematics nor physics. How many offer both mathematics and physics?

    1. A.4
    2. B.8
    3. C.16
    4. D.20

    Show answer

    Answer: D. 20

    AI explanation

    1. Analyze the question

    We are given the total number of students in a class and the number of students who offer Mathematics, Physics, or neither. We need to find the number of students who offer both subjects. This is a classic Set Theory problem involving two sets.

    2. Explain the concept

    We use the principle of sets. Let:

    • $n(U)$ = Total number of students = 40
    • $n(M)$ = Students who offer Mathematics = 32
    • $n(P)$ = Students who offer Physics = 24
    • $n(M \cup P)'$ = Students who offer neither = 4
    • $n(M \cap P)$ = Students who offer both (this is what we need to find)

    The formula for the union of two sets is: $n(M \cup P) = n(M) + n(P) - n(M \cap P)$

    Also, the total number of students is the sum of those who offer at least one subject and those who offer none: $n(U) = n(M \cup P) + n(M \cup P)'$

    3. Show the solution

    Step 1: Find the number of students who offer at least one subject. Since 4 students offer neither, the number of students who offer either Mathematics or Physics (or both) is: $n(M \cup P) = n(U) - n(M \cup P)'$ $n(M \cup P) = 40 - 4 = 36$

    Step 2: Use the set formula to find the intersection. Substitute the values into the formula: $n(M \cup P) = n(M) + n(P) - n(M \cap P)$ $36 = 32 + 24 - n(M \cap P)$ $36 = 56 - n(M \cap P)$

    Step 3: Solve for $n(M \cap P)$. $n(M \cap P) = 56 - 36$ $n(M \cap P) = 20$

    4. Identify the answer

    The number of students who offer both Mathematics and Physics is 20. This corresponds to option D.

    5. Tip

    When solving set problems, always subtract the "neither" group from the total first. A common mistake is to forget that the 4 students who offer neither must be excluded from the calculation of the union of the two sets. You can also visualize this using a Venn diagram to ensure your numbers add up correctly!

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24. 24.

    What is the circumference of latitude 0$^o$S if R is the radius of the earth?

    1. A.cos $\theta$
    2. B.2$\pi$R cos $\theta$
    3. C.R sin $\theta$
    4. D.2 $\pi$ r sin $\theta$

    Show answer

    Answer: B. 2$\pi$R cos $\theta$

    AI explanation

    1. Analyze the Question

    The question asks for the circumference of a specific latitude on the Earth's surface. You are given the latitude as $0^\circ$S and the radius of the Earth as $R$.

    2. Explain the Concept

    • Latitude $0^\circ$: This is the Equator. The Equator is a "Great Circle," meaning it is the largest circle that can be drawn on the Earth's surface.
    • General Formula: The radius ($r$) of a circle of latitude at an angle $\theta$ is given by $r = R \cos \theta$, where $R$ is the radius of the Earth.
    • Circumference Formula: The circumference of any circle is $2\pi r$. Substituting the radius of the latitude, the formula becomes: $$\text{Circumference} = 2\pi R \cos \theta$$

    3. Show the Solution

    • Step 1: Identify the latitude angle ($\theta$). Here, $\theta = 0^\circ$.
    • Step 2: Recall the trigonometric value for $\cos 0^\circ$. We know that $\cos 0^\circ = 1$.
    • Step 3: Apply the formula for the circumference of a latitude: $$\text{Circumference} = 2\pi R \cos \theta$$ $$\text{Circumference} = 2\pi R \cos 0^\circ$$ $$\text{Circumference} = 2\pi R (1)$$ $$\text{Circumference} = 2\pi R$$

    4. Identify the Answer

    Looking at the options provided: A. $\cos \theta$ B. $2\pi R \cos \theta$ C. $R \sin \theta$ D. $2\pi r \sin \theta$

    While the specific value for $0^\circ$ is $2\pi R$, the general formula for the circumference of any latitude $\theta$ is represented by option B. In JAMB questions of this type, they often ask for the general formula expression. Therefore, option B is the correct representation of the circumference of a latitude.

    5. Exam Tip

    Always remember the "Cos" rule for latitudes: "Latitude uses Cosine, Longitude uses Sine." If you are asked for the radius of a parallel of latitude, always multiply the Earth's radius ($R$) by the cosine of the latitude angle.

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25. 25.

    Express the product of 0.0014 and 0.011 in standard form

    1. A.1.54 x 10$^{-2}$
    2. B.1.54 x 10$^{-3}$
    3. C.1.54 x 10$^{-2}$
    4. D.1.54 x 10$^{-5}$

    Show answer

    Answer: D. 1.54 x 10$^{-5}$

    AI explanation

    Analysis

    To solve this problem, we need to multiply two decimal numbers and then express the result in standard form (also known as scientific notation). Standard form follows the format $a \times 10^n$, where $1 \le a < 10$ and $n$ is an integer.

    Concept

    1. Multiplication of decimals: You can multiply the numbers as if they were whole numbers and then place the decimal point by counting the total number of decimal places in the original factors.
    2. Standard Form: To convert a number to standard form, move the decimal point until you have a number between 1 and 10, then multiply by $10$ raised to the power of how many places you moved the decimal.

    Step-by-Step Solution

    Step 1: Multiply the numbers

    • $0.0014$ has 4 decimal places.
    • $0.011$ has 3 decimal places.
    • Multiply the significant digits: $14 \times 11 = 154$.
    • Total decimal places required = $4 + 3 = 7$.
    • Starting from the right of $154$, move the decimal point 7 places to the left: $0.0000154$

    Step 2: Convert to standard form

    • We need the decimal point to be after the first non-zero digit (which is 1).
    • Move the decimal point from $0.0000154$ to $1.54$.
    • Counting the jumps: • From $0.0000154$ to $0.000154$ (1) • To $0.00154$ (2) • To $0.0154$ (3) • To $0.154$ (4) • To $1.54$ (5)
    • Since we moved the decimal point 5 places to the right, the exponent is $-5$.
    • Result: $1.54 \times 10^{-5}$

    Conclusion

    Comparing our result ($1.54 \times 10^{-5}$) with the given options, it matches option D.

    Exam Tip: When multiplying decimals, always count the total number of digits after the decimal points in the question. Your final answer must have that exact number of decimal places before you convert it to standard form. This helps you avoid common errors with zeros!

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26. 26.

    Four boys and ten girls can cut a field first in 5 hours. If the boys work at $\frac{5}{4}$ the rate at which the girls work, how many boys will be needed to cut the field in 3 hours?

    1. A.180
    2. B.60
    3. C.25
    4. D.20

    Show answer

    Answer: D. 20

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27. 27.

    A binary operation x is defined by a x b = a$^b$. If a x 2 = 2 - a, find the possible values of a?

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

    Show answer

    Answer: A. 1, -2

    AI explanation

    1. Analyze the question

    We are given a binary operation defined as $a * b = a^b$. We are also given a specific condition: $a * 2 = 2 - a$. We need to find the values of $a$ that satisfy this equation.

    2. Explain the concept

    A binary operation is simply a rule for combining two numbers. Here, the rule is "raise the first number to the power of the second number."

    • Given: $a * b = a^b$
    • Condition: $a * 2 = 2 - a$

    We need to substitute $b = 2$ into the definition and solve the resulting algebraic equation.

    3. Show the solution

    Step 1: Apply the definition to the condition. Since $a * b = a^b$, then $a * 2$ must be $a^2$.

    Step 2: Set up the equation. Equate the definition to the given condition: $a^2 = 2 - a$

    Step 3: Rearrange into a standard quadratic equation. Move all terms to one side to set the equation to zero: $a^2 + a - 2 = 0$

    Step 4: Factorize the quadratic equation. We look for two numbers that multiply to $-2$ and add up to $+1$. These numbers are $+2$ and $-1$. $(a + 2)(a - 1) = 0$

    Step 5: Solve for $a$. Set each factor to zero:

    • $a + 2 = 0 \implies a = -2$
    • $a - 1 = 0 \implies a = 1$

    The possible values for $a$ are $1$ and $-2$.

    4. Identify the answer

    Comparing our results ($1, -2$) with the given options, we see that they match Option A.

    5. Tip

    When solving quadratic equations in JAMB, always remember the "sum and product" rule. For an equation $ax^2 + bx + c = 0$, the roots must add up to $-b/a$ and multiply to $c/a$. Here, $1 + (-2) = -1$ (which is $-b$) and $1 \times (-2) = -2$ (which is $c$). This is a great way to double-check your work quickly!

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28. 28.

    Evaluate (212)$_3$ - (121)$_3$ + (222)$_3$

    1. A.(313)$_3$
    2. B.(1000)$_3$
    3. C.(1020)$_3$
    4. D.(1222)$_3$

    Show answer

    Answer: C. (1020)$_3$

    AI explanation

    To solve this problem, we need to perform arithmetic operations in Base 3. Remember that in base 3, the digits allowed are only 0, 1, and 2. When you add or subtract, any value that reaches 3 is carried over or borrowed as "10" in base 3.

    Step 1: Perform the subtraction (212)$_3$ - (121)$_3$

    Align the numbers:

      2 1 2
    - 1 2 1
    -------

    • Units column: 2 - 1 = 1
    • Threes column: 1 - 2. We cannot do this, so we borrow 1 from the "nines" column. The 2 becomes 1, and the 1 becomes (1 + 3) = 4. Now, 4 - 2 = 2.
    • Nines column: 1 - 1 = 0.

    Result of subtraction: (021)$_3$

    Step 2: Perform the addition (021)$_3$ + (222)$_3$

    Align the numbers:

      0 2 1
    + 2 2 2
    -------

    • Units column: 1 + 2 = 3. In base 3, 3 is written as 10. Write down 0 and carry over 1.
    • Threes column: 2 + 2 + 1 (carried) = 5. In base 3, 5 is (1 × 3) + 2, which is written as 12. Write down 2 and carry over 1.
    • Nines column: 0 + 2 + 1 (carried) = 3. In base 3, 3 is written as 10. Write down 10.

    Result of addition: (1020)$_3$

    Summary of Calculation

    1. (212)$_3$ - (121)$_3$ = (021)$_3$
    2. (021)$_3$ + (222)$_3$ = (1020)$_3$

    Exam Tip: When working in any base $n$, if your sum or difference involves a number equal to or greater than $n$, always divide by $n$ to find the remainder (the digit to write) and the quotient (the carry-over). For example, in base 3, if you get 5, divide 5 by 3: $5 \div 3 = 1$ remainder $2$. So, you write 2 and carry 1.

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29. 29.

    The angle of a sector of a circle radius 10.5 cm is 48$^o$. Calculate the perimeter of the sector.

    1. A.25.4cm
    2. B.25.4cm
    3. C.25.6cm
    4. D.29.8cm

    Show answer

    Answer: D. 29.8cm

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30. 30.

    In this fiqure, PQ = PR = PS and SRT = 68$^o$. Find QPS

    1. A.136$^o$
    2. B.124$^o$
    3. C.112$^o$
    4. D.68$^o$

    Show answer

    Answer: B. 124$^o$

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31. 31.

    If x is positive real number, find the range of values for which $\frac{1}{3x}$ + $\frac{1}{2}$ > $\frac{1}{4x}$

    1. A.x > -$\frac{1}{6}$
    2. B.x > 0
    3. C.0 < x < 6
    4. D.0 < x <$\frac{1}{6}$

    Show answer

    Answer: B. x > 0

    AI explanation

    Analysis

    We are given the inequality $\frac{1}{3x} + \frac{1}{2} > \frac{1}{4x}$ where $x$ is a positive real number. Our goal is to isolate $x$ to find the range of values that satisfy this condition.

    Concept

    To solve inequalities involving fractions, we need to:

    1. Find a common denominator to combine the terms.
    2. Simplify the inequality.
    3. Apply the properties of inequalities (remembering that multiplying or dividing by a negative number flips the inequality sign).

    Step-by-Step Solution

    Step 1: Move all terms to one side. $\frac{1}{3x} - \frac{1}{4x} + \frac{1}{2} > 0$

    Step 2: Find a common denominator for the $x$ terms. The common denominator for $3x$ and $4x$ is $12x$. $\frac{4 - 3}{12x} + \frac{1}{2} > 0$ $\frac{1}{12x} + \frac{1}{2} > 0$

    Step 3: Combine the fractions. To add $\frac{1}{12x}$ and $\frac{1}{2}$, the common denominator is $12x$: $\frac{1 + 6x}{12x} > 0$

    Step 4: Analyze the inequality. We are given that $x$ is a positive real number ($x > 0$). If $x > 0$, then $12x$ is definitely positive. For the fraction $\frac{1 + 6x}{12x}$ to be greater than zero (positive), the numerator $(1 + 6x)$ must also be positive because a positive divided by a positive is positive.

    $1 + 6x > 0$ $6x > -1$ $x > -\frac{1}{6}$

    Step 5: Consider the constraint. The problem explicitly states that $x$ is a positive real number ($x > 0$). Since our calculated range $x > -\frac{1}{6}$ includes negative numbers, we must intersect this with the given condition $x > 0$. The intersection of $x > -\frac{1}{6}$ and $x > 0$ is simply $x > 0$.

    Conclusion

    The inequality holds true for all positive values of $x$. Therefore, the range is $x > 0$.

    Exam Tip

    When dealing with inequalities involving variables in the denominator, always check if the variable is restricted (e.g., $x > 0$). If you multiply both sides by $x$ to clear the fraction, ensure you know the sign of $x$ so you know whether to flip the inequality sign!

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32. 32.

    Factorise (4a + 3) $^2$ - (3a - 2)$^2$

    1. A.(a + 1)(a + 5)
    2. B.(a - 5)(7a - 1)
    3. C.(a + 5)(7a + 1)
    4. D.a(7a + 1)

    Show answer

    Answer: C. (a + 5)(7a + 1)

    AI explanation

    Analysis

    This problem requires the application of the Difference of Two Squares algebraic identity. The expression is in the form $x^2 - y^2$, where $x = (4a + 3)$ and $y = (3a - 2)$.

    Concept

    The difference of two squares formula is:

    $x^2 - y^2 = (x - y)(x + y)$

    Using this identity allows us to factorise the expression without having to expand the squares and simplify a long polynomial, which saves time during your JAMB exam.

    Solution

    Given: $(4a + 3)^2 - (3a - 2)^2$

    Step 1: Identify $x$ and $y$

    • $x = (4a + 3)$
    • $y = (3a - 2)$

    Step 2: Apply the formula $(x - y)(x + y)$

    • First bracket $(x - y)$: $(4a + 3) - (3a - 2)$ $= 4a + 3 - 3a + 2$ $= (4a - 3a) + (3 + 2)$ $= a + 5$

    • Second bracket $(x + y)$: $(4a + 3) + (3a - 2)$ $= 4a + 3 + 3a - 2$ $= (4a + 3a) + (3 - 2)$ $= 7a + 1$

    Step 3: Combine the results $(a + 5)(7a + 1)$

    Conclusion

    Comparing our result $(a + 5)(7a + 1)$ with the given options, it matches option C.

    Exam Tip

    Whenever you see a question in the format $A^2 - B^2$, always use the difference of two squares formula $(A-B)(A+B)$ instead of expanding the brackets. Expanding $(4a+3)^2$ and $(3a-2)^2$ separately is prone to sign errors (especially with the negative sign in the middle) and takes much longer to solve.

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33. 33.

    A room is 12m long, 9m wide and 8m high. Find the cosine of the angle which a diagonal of the room makes with the floor of the room

    1. A.$\frac{15}{17}$
    2. B.$\frac{8}{17}$
    3. C.$\frac{8}{15}$
    4. D.$\frac{12}{17}$

    Show answer

    Answer: A. $\frac{15}{17}$

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34. 34.

    In a class of 150 students, the sector in a pie chart representing the students offering physics has angle 12$^o$. How many students are offering physics?

    1. A.18
    2. B.15
    3. C.10
    4. D.5

    Show answer

    Answer: D. 5

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35. 35.

    The pie chart shows the income of a civil servant in month. If his monthly income is N6,000. Find his monthly basic salary.

    1. A.N2,050
    2. B.N2600
    3. C.N3,100
    4. D.N3,450

    Show answer

    Answer: A. N2,050

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36. 36.

    If log$_{10}$2 = 0.3010 and log$_{10}$3 = 0.4771, eventually without using the logarithm tables, log$_{10}$4.5

    1. A.0.3010
    2. B.0.4771
    3. C.0.6532
    4. D.0.9542

    Show answer

    Answer: C. 0.6532

    AI explanation

    1. Analyze the question

    We are given the values of $\log_{10}2$ and $\log_{10}3$. We need to find the value of $\log_{10}4.5$ by expressing $4.5$ in terms of $2$ and $3$.

    2. Explain the concept

    To solve this, we use the laws of logarithms:

    • Quotient Rule: $\log_b(\frac{x}{y}) = \log_b x - \log_b y$
    • Product Rule: $\log_b(xy) = \log_b x + \log_b y$
    • Power Rule: $\log_b(x^n) = n \log_b x$

    First, convert the decimal $4.5$ into a fraction: $4.5 = \frac{45}{10} = \frac{9}{2}$

    3. Show the solution

    Step 1: Express $\log_{10}4.5$ using the quotient rule.

    $\log_{10}4.5 = \log_{10}\left(\frac{9}{2}\right)$

    $\log_{10}4.5 = \log_{10}9 - \log_{10}2$

    Step 2: Express $9$ as a power of $3$. Since $9 = 3^2$, we have:

    $\log_{10}9 = \log_{10}(3^2) = 2 \log_{10}3$

    Step 3: Substitute the known values into the equation.

    $\log_{10}4.5 = 2(\log_{10}3) - \log_{10}2$

    $\log_{10}4.5 = 2(0.4771) - 0.3010$

    Step 4: Perform the arithmetic.

    $2 \times 0.4771 = 0.9542$

    $0.9542 - 0.3010 = 0.6532$

    4. Identify the answer

    Comparing our result to the given options: A. 0.3010 B. 0.4771 C. 0.6532 D. 0.9542

    The correct option is C.

    5. Tip

    Whenever you see a decimal in a logarithm question (like 4.5 or 0.75), always convert it to a fraction first. It makes it much easier to break the number down into the prime factors (2, 3, 5, etc.) provided in the question!

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37. 37.

    Find all real number x which satisfy the inequality $\frac{1}{3}$ (x + 1) - 1 > $\frac{1}{5}$(x + 4)

    1. A.x < 11
    2. B.x < -1
    3. C.x > 6
    4. D.x > 11

    Show answer

    Answer: D. x > 11

    AI explanation

    To solve the inequality $\frac{1}{3}(x + 1) - 1 > \frac{1}{5}(x + 4)$, we will follow these steps:

    1. Analyze the inequality

    We have a linear inequality involving fractions. To make the calculation easier, we should eliminate the fractions by multiplying the entire inequality by the Least Common Multiple (LCM) of the denominators (3 and 5).

    The LCM of 3 and 5 is 15.

    2. Step-by-step solution

    Step 1: Multiply every term by 15 to clear the fractions: $15 \times [\frac{1}{3}(x + 1)] - 15 \times [1] > 15 \times [\frac{1}{5}(x + 4)]$

    Step 2: Simplify the terms: $5(x + 1) - 15 > 3(x + 4)$

    Step 3: Expand the brackets: $5x + 5 - 15 > 3x + 12$

    Step 4: Combine like terms on the left side: $5x - 10 > 3x + 12$

    Step 5: Group the $x$ terms on one side and the constants on the other: Subtract $3x$ from both sides: $5x - 3x - 10 > 12$ $2x - 10 > 12$

    Add 10 to both sides: $2x > 12 + 10$ $2x > 22$

    Step 6: Solve for $x$: Divide both sides by 2: $x > \frac{22}{2}$ $x > 11$

    3. Conclusion

    Comparing our result ($x > 11$) with the given options, we find that it matches option D.

    Exam Tip

    When dealing with inequalities involving fractions, always multiply by the LCM of the denominators first. This turns the problem into a simple linear inequality and helps you avoid errors with fraction arithmetic. Also, remember that if you multiply or divide an inequality by a negative number, you must reverse the inequality sign (e.g., $>$ becomes $<$). In this problem, we only multiplied by a positive number (15), so the sign remained the same.

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38. 38.

    Find the median of the numbers 89, 141, 130, 161, 120, 131, 131, 100, 108 and 119

    1. A.131
    2. B.125
    3. C.123
    4. D.120

    Show answer

    Answer: B. 125

    AI explanation

    To find the median of a set of numbers, follow these steps:

    1. Analyze the question

    The median is the middle value of a data set when the numbers are arranged in order of magnitude (either ascending or descending). Since there are 10 numbers in this set (an even number), the median will be the average of the two middle terms.

    2. Explain the concept

    • Step 1: Arrange the numbers in ascending order (from smallest to largest).
    • Step 2: Identify the middle position. For $n$ numbers, if $n$ is even, the median is the average of the $(\frac{n}{2})^{th}$ and the $(\frac{n}{2} + 1)^{th}$ terms.
    • Step 3: Calculate the average of those two middle terms.

    3. Show the solution

    Step 1: Arrange the numbers in ascending order: Original list: 89, 141, 130, 161, 120, 131, 131, 100, 108, 119

    Ordered list:

    1. 89
    2. 100
    3. 108
    4. 119
    5. 120
    6. 130
    7. 131
    8. 131
    9. 141
    10. 161

    Step 2: Identify the middle terms: Since there are 10 numbers ($n=10$), the middle terms are the 5th and 6th positions:

    • 5th term = 120
    • 6th term = 130

    Step 3: Calculate the average:

    $\text{Median} = \frac{120 + 130}{2}$

    $\text{Median} = \frac{250}{2}$

    $\text{Median} = 125$

    4. Identify the answer

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

    5. Exam Tip

    Always count the number of items in your list before you start. If you have an odd number of items, the median is just the single middle number. If you have an even number of items, never forget to divide the sum of the two middle numbers by 2! A common mistake is to just pick one of the two middle numbers.

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39. 39.

    In preparing rice cutlets, a cook used 75g of rice, 40g of margarine, 105g of meat and 20g of bread crumbs. Find the angle of the sector which represent meat in a pie chart?

    1. A.30$^o$
    2. B.60$^o$
    3. C.112.5$^o$
    4. D.157.5$^o$

    Show answer

    Answer: D. 157.5$^o$

    AI explanation

    To solve this problem, we need to determine the proportion of the total weight that the meat represents and then convert that proportion into degrees for a pie chart.

    1. Analyze the question

    We are given the weights of four ingredients used to make rice cutlets. We need to find the sector angle for the meat in a pie chart. A full circle (pie chart) represents $360^\circ$.

    2. Explain the concept

    To find the angle of a sector for a specific item, use the formula:

    $\text{Sector Angle} = \left( \frac{\text{Weight of the item}}{\text{Total weight of all items}} \right) \times 360^\circ$

    3. Show the solution

    Step 1: Calculate the total weight of all ingredients.

    $\text{Total weight} = 75\text{g (rice)} + 40\text{g (margarine)} + 105\text{g (meat)} + 20\text{g (bread crumbs)}$

    $\text{Total weight} = 240\text{g}$

    Step 2: Calculate the fraction represented by the meat.

    $\text{Fraction of meat} = \frac{\text{Weight of meat}}{\text{Total weight}}$

    $\text{Fraction of meat} = \frac{105}{240}$

    Step 3: Convert the fraction to degrees.

    $\text{Angle} = \left( \frac{105}{240} \right) \times 360^\circ$

    Simplify the calculation:

    $\text{Angle} = 105 \times \left( \frac{360}{240} \right)$

    $\text{Angle} = 105 \times 1.5$

    $\text{Angle} = 157.5^\circ$

    4. Identify the answer

    Comparing our result to the given options: A. $30^\circ$ B. $60^\circ$ C. $112.5^\circ$ D. $157.5^\circ$

    The calculated value matches option D.

    5. Tip

    When dealing with pie chart questions, always check if your total angle adds up to $360^\circ$. If you are ever unsure of your calculation, remember that the sector angle is simply the "share" of the total. If the meat is roughly half of the total weight (105 is slightly less than half of 240), your answer should be slightly less than $180^\circ$ (half of $360^\circ$). This helps you quickly eliminate impossible options!

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