JAMB UTME - Mathematics - 2023

Intersecting Chords Theorem states that If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal.

⇒ AE * EB = CE * ED

⇒ 6 * x $\mathrm{<br>}$ = 4 * (x $\mathrm{<br>}$ + 5)

⇒ 6x $\mathrm{<br>}$ = 4x $\mathrm{<br>}$ + 20

⇒ 6x $\mathrm{<br>}$ - 4x $\mathrm{<br>}$ = 20

⇒ 2x $\mathrm{<br>}$ = 20

∴ x=202 $\mathrm{<br>}=\frac{20}{2}$ = 10 units

Total possible outcome = 6 x 6 = 36

Required outcome = 15

∴ Pr(E) = 1536=512

A binary operation '*' on a set A is said to be commutative if the order in which the elements are combined does not affect the result. In other words, for any two elements a and b in A, the operation a * b is equal to b * a.
Let's say we have a set A with elements {a, b, c}. For the operation '*' to be commutative, it should hold true for all possible combinations of elements in A.
1. We need to check if a * b is equal to b * a. If a * b = b * a for all a and b in A, then the operation is commutative. 2. We also need to check if (a * b) * c is equal to a * (b * c). If this equation holds true for all a, b, and c in A, then the operation is associative, but it does not necessarily guarantee commutativity. 3. Finally, we need to check if (b ο c) * a is equal to (b * a) ο (c * a). If this equation holds true for all a, b, and c in A, then the operation is commutative.
In summary: - If the operation a * b = b * a holds true for all a and b in A, then the operation is commutative. - If (a * b) * c = a * (b * c) holds true for all a, b, and c in A, the operation is associative, but not necessarily commutative. - If (b ο c) * a = (b * a) ο (c * a) holds true for all a, b, and c in A, the operation is commutative.
Therefore, the option that satisfies the condition of commutativity is **a * b = b * a**.

To estimate the median, we need to find the midpoint of the data set. In other words, we need to find the value that separates the data into two equal parts.
To do this, we can use the cumulative frequency. The cumulative frequency is the sum of the frequencies up to a certain point. In this case, we add up the frequencies starting from the youngest age group (5-6 years) and continue until we reach a cumulative frequency that is greater than the total number of students divided by 2.
Let's calculate the cumulative frequency:
5-6 years: 29 students 5-6 years + 7-8 years = 29 + 40 = 69 students 5-6 years + 7-8 years + 9-10 years = 69 + 38 = 107 students
Since the total number of students is 107 and we want to find the midpoint, which is the median, we divide 107 by 2 to get 53.5. This means that the median falls between the second and third age groups.
To estimate the median, we can use the cumulative frequency of the second age group (69) as a reference. The median lies within this age group. To find the exact estimated median, we can use the formula:
Median = Lower boundary of median group + ((Total/2) - Cumulative frequency of previous group) / Frequency of median group * Width of median group
The lower boundary of the median group is 7, the cumulative frequency of the previous group is 29, the frequency of the median group is 40, and the width of the median group is 2 (since the age group is 7-8 years).
Now, let's plug in these values into the formula:
Median = 7 + ((53.5 - 29) / 40) * 2
Calculating this, we get:
Median ≈ 7.725
Therefore, the estimated median of the ages of the students is approximately 7.725.

$\frac{\mathrm{<br>}}{}$

To find the value of y in the equation 402y = 102ten, we need to solve for y.
First, let's simplify the equation. We can start by dividing both sides of the equation by 402 to isolate y on one side.
402y / 402 = 102ten / 402
Simplifying further, we get:
y = 102ten / 402
To evaluate 102ten / 402, we can break it down into two steps.
Step 1: Evaluate 102ten To evaluate 102ten, we need to convert the base of the number to the base 10. Since "ten" is another way of saying base 10, we can rewrite 102ten as 10210.
So, 102ten = 10210
Step 2: Divide 10210 by 402 Now, we can divide 10210 by 402:
10210 ÷ 402 = 25
Therefore, y = 25.
The correct answer is 5.

The inequality that describes the graph above is 5y + 4x <= 20.

Let's break down the reasoning behind this answer:

In the given inequality options, we can see that the coefficients in front of x and y are 5 and 4 respectively. In the graph, we can observe that the line is steeper in the y-direction compared to the x-direction. This means that the slope of the line is greater in the y-direction compared to the x-direction.

The inequality sign for the equation is less than or equal to in the selected option of 5y + 4x <= 20. This suggests that the shaded region below or on the line represents the solution. This is because any point below or on the line will satisfy the condition.

Therefore, the graph above corresponds to the inequality 5y + 4x <= 20.

To find the profit or loss percentage, we need to compare the selling price to the cost price. Let's call the cost price of the article C.

We know that when the article is sold for ₦230.00, it makes a 15% profit. Profit is calculated by subtracting the cost price from the selling price. So, we can write the equation:

Selling Price - Cost Price = Profit

₦230.00 - C = 15% of C

Since we want to find the profit or loss percentage when the article is sold for ₦180.00, we can use the same equation:

₦180.00 - C = x% of C

where x is the profit or loss percentage we want to find.

Now, let's solve these equations to find the value of C and x.

From the first equation:

₦230.00 - C = 0.15C

₦230.00 = 0.15C + C

₦230.00 = 1.15C

To solve for C:

C = ₦230.00 / 1.15

C = ₦200.00

Now, let's substitute the value of C into the second equation:

₦180.00 - ₦200.00 = x% of ₦200.00

-₦20.00 = x% of ₦200.00

To solve for x:

x% = (-₦20.00 / ₦200.00) * 100

x% = -10%

Therefore, when the article is sold for ₦180.00, it results in a 10% loss.

To solve the logarithmic equation log2(6-x) = 3 - log2x, we can use the properties of logarithms.
First, let's simplify the equation by combining the logarithms on the right side:
log2(6-x) + log2x = 3
Next, we can use the logarithmic product rule, which states that logb(M * N) = logb(M) + logb(N), to combine the logarithms on the left side:
log2[(6-x) * x] = 3
To solve for x, we can rewrite the equation using exponential form. Since the base of the logarithm is 2, we can rewrite the equation as:
2^3 = (6-x) * x
Simplifying the left side gives us:
8 = (6-x) * x
Now, we have a quadratic equation. Let's expand the right side:
8 = 6x - x^2
Re-arranging the equation gives us:
x^2 - 6x + 8 = 0
To solve this quadratic equation, we can factor or use the quadratic formula. Factoring the equation gives us:
(x - 4)(x - 2) = 0
Setting each factor equal to zero gives us two possible solutions:
x - 4 = 0 or x - 2 = 0
Solving these equations gives us:
x = 4 or x = 2
Therefore, the solutions to the logarithmic equation log2(6-x) = 3 - log2x are:
x = 4 or x = 2

To calculate the mean deviation of the first five prime numbers, we need to follow these steps:
Step 1: Find the mean/average of the five prime numbers. The mean is calculated by adding up all the numbers and dividing the sum by the total number of values. In this case, we have 2, 3, 5, 7, and 11 as our prime numbers. So we add them up: 2 + 3 + 5 + 7 + 11 = 28. Then, we divide this sum by 5 (the total number of values): 28 ÷ 5 = 5.6.
Step 2: Find the deviation of each number from the mean. To do this, we subtract the mean from each number. For the given prime numbers, the deviations are as follows: - Deviation of 2 from the mean: 2 - 5.6 = -3.6 - Deviation of 3 from the mean: 3 - 5.6 = -2.6 - Deviation of 5 from the mean: 5 - 5.6 = -0.6 - Deviation of 7 from the mean: 7 - 5.6 = 1.4 - Deviation of 11 from the mean: 11 - 5.6 = 5.4
Step 3: Find the absolute value of each deviation. Absolute value means removing the negative sign, if any, and considering only the magnitude of the deviation. The absolute values of the deviations in this case are: - Absolute value of -3.6: 3.6 - Absolute value of -2.6: 2.6 - Absolute value of -0.6: 0.6 - Absolute value of 1.4: 1.4 - Absolute value of 5.4: 5.4
Step 4: Find the average of these absolute deviations, which is the mean deviation. To do this, we add up all the absolute deviations and divide by the total number of values. In this case, we have 3.6 + 2.6 + 0.6 + 1.4 + 5.4 = 13.6. Then, we divide this sum by 5 (the total number of values): 13.6 ÷ 5 = 2.72.
Therefore, the mean deviation of the first five prime numbers is 2.72.

To find the value of y in the equation log(y + 8) + log(y - 8) = 2log3 + 2log5, we can use some logarithmic properties.
First, let's simplify the right side of the equation. Using the property log(a) + log(b) = log(ab), we can rewrite 2log3 + 2log5 as log(3^2) + log(5^2), which becomes: log(9) + log(25)
Next, the left side of the equation can be simplified using the property log(a) + log(b) = log(ab): log((y + 8)(y - 8))
Now, our equation becomes: log((y + 8)(y - 8)) = log(9) + log(25)
To further simplify, we can apply the exponential function to both sides of the equation, which allows us to remove the logarithm function: (y + 8)(y - 8) = 9 * 25
Expanding the equation on the left side, we get: y^2 - 64 = 225
Rearranging terms, we have: y^2 = 289
Taking the square root of both sides, we get: y = ±17
Therefore, the correct answer is y = ±17.

To find the set (A - B) ⋃ (B - A) or the union of the set difference between A and B and the set difference between B and A, we can follow a few simple steps.
1. Find the set difference between A and B (A - B): - A - B includes all elements that are in set A but not in set B. - In this case, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. - Subtracting the common elements from set A, we have A - B = {1, 3, 5}.
2. Find the set difference between B and A (B - A): - B - A includes all elements that are in set B but not in set A. - In this case, A is still {1, 2, 3, 4, 5, 6} and B is still {2, 4, 6, 8}. - Subtracting the common elements from set B, we have B - A = {8}.
3. Take the union (⋃) of the two sets: - The union of two sets includes all elements that are in either set. - In this case, the union of (A - B) and (B - A) is {1, 3, 5} ⋃ {8}. - Combining these two sets, we have (A - B) ⋃ (B - A) = {1, 3, 5, 8}.
Therefore, the correct answer is {1, 3, 5, 8}.

To solve the equation 3(x – 1) ≤ 2(x – 3), we need to find the range of values for x that satisfy the inequality.
Let's simplify the inequality step by step:
1. Distribute the multiplication: 3x - 3 ≤ 2x - 6
2. Combine like terms by subtracting 2x from both sides: 3x - 2x - 3 ≤ -6
3. Simplify further by combining like terms on the left side: x - 3 ≤ -6
4. Add 3 to both sides of the inequality to isolate x: x - 3 + 3 ≤ -6 + 3 x ≤ -3
Therefore, the solution to the inequality 3(x – 1) ≤ 2(x – 3) is x ≤ -3. This means that any value of x that is less than or equal to -3 will satisfy the inequality.
So, the correct option is x ≤ -3.

To solve this problem, let's first calculate the total work that needs to be done. We can do this by multiplying the number of men, the number of hours they work per day, and the number of days it takes them to complete the work.
For the first scenario, we have: - 12 men working together for 8 hours a day - It takes them 4 days to finish the work
So, the total work done by these 12 men can be calculated as: work = (12 men) * (8 hours/day) * (4 days) = 384 man-hours
Now, let's find out how long it would take 4 men working 16 hours a day to complete the same piece of work.
If 12 men can do the work in 4 days, it means that the total man-hours required to complete the work is the same for both scenarios.
Let's use the same formula to calculate the total work for the second scenario: work = (4 men) * (16 hours/day) * (x days) = 384 man-hours
We need to solve for x, which represents the number of days required by 4 men to complete the work.
Dividing both sides of the equation by (4 men) and (16 hours/day), we get: x = 384 man-hours / (4 men * 16 hours/day) x = 384 / 64 x = 6
So, it will take 4 men working 16 hours a day approximately 6 days to complete the same piece of work.
Therefore, the correct answer is 6 days.

tan θ = oppadj=RQQP=68 $\frac{\mathrm{<br>}}{}$

tan θ = 0.75

θ = tan−1(0.75)=36.87o ${}^{-1}$

${\mathrm{<br>}}^{}$
∴ The bearing of R from P = 360o ${}^{\mathrm{<br>}}$ - 36.87o ${}^{\mathrm{<br>}}$ = 323o ${}^{\mathrm{<br>}}$ (to the nearest degree)

To find the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually, we can use the formula for compound interest:
CI = P(1 + r/n)^(nt) - P
Where: - CI is the compound interest - P is the principal amount (₦15,700 in this case) - r is the annual interest rate (8% or 0.08 as a decimal) - n is the number of times the interest is compounded per year (since it's compounded annually, n is 1) - t is the number of years (2 in this case)
Now we can substitute the values into the formula:
CI = ₦15,700(1 + 0.08/1)^(1*2) - ₦15,700
Simplifying the equation:
CI = ₦15,700(1.08)^2 - ₦15,700
CI = ₦15,700(1.1664) - ₦15,700
CI = ₦18,312.48 - ₦15,700
CI = ₦2,612.48
Therefore, the compound interest (CI) on ₦15,700 for 2 years at 8% per annum compounded annually is ₦2,612.48.

To find the value of k, we can use the given information that the line passes through points A(-2, k) and B(4, 8).
The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.
First, let's find the slope (m) of the line using the coordinates of the two points. The formula for slope is (y2 - y1) / (x2 - x1).
Substituting the coordinates of point A and B into the formula, we have: m = (8 - k) / (4 - (-2)) m = (8 - k) / 6
The line also passes through point A(-2, k), so we can substitute these values into the slope-intercept form equation: k = m(-2) + b
Now we have two equations: k = m(-2) + b m = (8 - k) / 6
To simplify the situation, we need to eliminate the variable b by isolating it in the first equation. Let's solve the first equation for b: b = k + 2m
Now we have: m = (8 - k) / 6 b = k + 2m
Next, substitute the expression for b into the second equation: m = (8 - k) / 6 k + 2m = (8 - k) / 6
To solve this equation for k, we will multiply both sides by 6 to eliminate the denominator: 6k + 12m = 8 - k
To simplify the equation, we bring like terms together: 6k + k = 8 - 12m 7k = 8 - 12m 7k + 12m = 8
Now, we have a linear equation in two variables (k and m). To solve for k, we need to know the value of m.
Assuming we know the value of m, we can substitute it into the equation 7k + 12m = 8 and solve for k.
Based on the given options, we can assume a value for k and calculate the corresponding value of m using the equation m = (8 - k) / 6.
Let's try k = 20: m = (8 - 20) / 6 m = -12/6 m = -2
Now we substitute m = -2 into the equation 7k + 12m = 8: 7k + 12(-2) = 8 7k - 24 = 8 7k = 32 k = 32/7
Therefore, when k = 20, the equation satisfies the given line equation and passes through points A(-2,20) and B(4,8).
Hence, the value of k is 20 when the line passes through points A(-2, k) and B(4, 8).

To find the probability of obtaining at least one head when a coin is thrown 3 times, we need to consider the possible outcomes.
First, let's determine the total number of possible outcomes when a coin is thrown 3 times. For each coin toss, there are 2 possibilities: either a head or a tail. Since we are tossing the coin 3 times, the total number of outcomes is 2 * 2 * 2 = 8.
Now, let's calculate the number of outcomes in which at least one head is obtained. To have at least one head, we can have the following outcomes: - 1 head and 2 tails: HHT, HTH, or THH - 2 heads and 1 tail: HHT, HTH, or THH - 3 heads: HHH
So, there are a total of 7 outcomes in which at least one head is obtained.
Therefore, the probability of obtaining at least one head when a coin is thrown 3 times is 7/8.
The correct answer is None of the above (since none of the options match 7/8).

If the majority are women implies that:

3 Women and 2 Men Or 4 Women and 1 Man

= 4C3×6C2+4C4×6C1 ${}^{\mathrm{<br>}}$

= 4×15+1×6 $4\times 15+1\times 6$

= 60 + 6 = 66