JAMB UTME - Mathematics - 2024

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

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Subtract 14256seven ${}_{seven}$ from 20045seven ${}_{seven}$ = 2456seven ${}_{seven}$

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

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To find the perimeter of a triangle with vertices at given points, we need to calculate the lengths of its sides using the distance formula, which is given by:

Distance Formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

The vertices of the triangle are (3, 2), (4, 5), and (6, 2). Let's label these points as A(3, 2), B(4, 5), and C(6, 2).

Step 1: Finding AB

Using the distance formula for points A(3, 2) and B(4, 5):

\( AB = \sqrt{(4 - 3)^2 + (5 - 2)^2} \)

\( AB = \sqrt{1^2 + 3^2} \)

\( AB = \sqrt{1 + 9} \)

\( AB = \sqrt{10} \)

Step 2: Finding BC

Using the distance formula for points B(4, 5) and C(6, 2):

\( BC = \sqrt{(6 - 4)^2 + (2 - 5)^2} \)

\( BC = \sqrt{2^2 + (-3)^2} \)

\( BC = \sqrt{4 + 9} \)

\( BC = \sqrt{13} \)

Step 3: Finding CA

Using the distance formula for points C(6, 2) and A(3, 2):

\( CA = \sqrt{(3 - 6)^2 + (2 - 2)^2} \)

\( CA = \sqrt{(-3)^2 + 0^2} \)

\( CA = \sqrt{9} \)

\( CA = 3 \)

Step 4: Calculating the Perimeter

The perimeter of the triangle is the sum of the lengths of its sides:

\( \text{Perimeter} = AB + BC + CA \)

\( \text{Perimeter} = \sqrt{10} + \sqrt{13} + 3 \)

Thus, the perimeter of the triangle in surd form is 3 + \(\sqrt{10} + \sqrt{13}\).

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.

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The midpoint of a line segment joining two points is the average of the x-coordinates and the y-coordinates of the endpoints. Given two endpoints \( P(x, 4) \) and \( Q(10, 8) \), and their midpoint \( M(9, 6) \), we can use the midpoint formula to find the value of \( x \).

The formula for the midpoint \( M(x_m, y_m) \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\( x_m = \frac{x_1 + x_2}{2} \)

\( y_m = \frac{y_1 + y_2}{2} \)

Given that \( M(9, 6) \) is the midpoint, we know:

\(\frac{x + 10}{2} = 9\)

\(\frac{4 + 8}{2} = 6\)

Let’s solve for \( x \):

Multiply both sides of the x-coordinate equation by 2 to eliminate the fraction:

\( x + 10 = 18 \)

Subtract 10 from both sides:

\( x = 18 - 10 \)

Thus:

x = 8

Therefore, the value of \( x \) is 8. This makes sure the midpoint of the line segment \( P(x, 4) \) and \( Q(10, 8) \) is indeed \( M(9, 6) \). The answer aligns with the given options.

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.

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To find the average age of all the teachers in the school, we need to combine the total age of all female and male teachers and then divide by the total number of teachers.

First, let's calculate the total age of the female teachers. Since the average age of the four female teachers is 40, the total age of the female teachers can be calculated as:

Total age of female teachers = Average age * Number of female teachers

Total age of female teachers = 40 * 4 = 160

Next, calculate the total age of the male teachers. With the average age of eight male teachers being 25, the total age of the male teachers is:

Total age of male teachers = Average age * Number of male teachers

Total age of male teachers = 25 * 8 = 200

Now, add the total ages of both male and female teachers to find the combined total age:

Combined total age = 160 + 200 = 360

To find the total number of teachers, simply add the number of female and male teachers:

Total number of teachers = 4 (female) + 8 (male) = 12

Finally, calculate the average age of all the teachers by dividing the combined total age by the total number of teachers:

Average age = Combined total age / Total number of teachers

Average age = 360 / 12 = 30

Therefore, the average age of the teachers in the school is 30.

In how many ways can 6 people sit around a table 

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When calculating the number of ways to arrange people in a circle, we need to consider that circular permutations are different from linear permutations. In a linear arrangement, each position is distinct. However, in a circular arrangement, rotating the arrangement does not create a new permutation.

For example, if you have people A, B, C, D, E, and F sitting around a table, arranging them in order ABCDEF is considered identical to BCDEFA, CDEFAB, etc., because you can rotate the table and still have equivalent seating arrangements.

To find the number of unique ways to arrange 6 people around a table, follow this simple calculation:

1. First, imagine the people in a straight line. There are 6 people, so there are 6! (6 factorial) ways to arrange these people linearly. This gives us:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 ways.

2. However, since rotations of the same arrangement are not unique around a circular table, we divide by the number of positions around the table, which is 6:

Number of circular permutations = 6! / 6 = 720 / 6 = 120.

Thus, there are 120 different ways for 6 people to sit around a circular table.

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

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The given sequence is: -10, 50, -250, .......

To find the pattern, let's first determine if this is a geometric sequence. A geometric sequence has a common ratio between consecutive terms.

The first term \( a_1 \) is -10.

The second term \( a_2 \) is 50.

Let's find the common ratio (r) by dividing the second term by the first term:

\( r = \frac{a_2}{a_1} = \frac{50}{-10} = -5 \)

To confirm it's a geometric sequence, calculate the ratio for the next pair:

The third term \( a_3 \) is -250.

\( r = \frac{a_3}{a_2} = \frac{-250}{50} = -5 \)

The common ratio is confirmed to be **-5**. Therefore, this is a **geometric sequence** with the first term **-10** and a common ratio of **-5**.

The general formula for the n-th term of a geometric sequence is:

\( a_n = a_1 \times r^{(n-1)} \)

To find the 7th term (\( n = 7 \)):

\( a_7 = -10 \times (-5)^{(7-1)} = -10 \times (-5)^6 \)

Compute \( (-5)^6 \):

\(-5 \times -5 \times -5 \times -5 \times -5 \times -5 = 15625 \)

Thus,

\( a_7 = -10 \times 15625 = -156250 \)

The 7th term of the sequence is **-156250**.

The corresponding answer is **-156250**.

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 

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The boy bought 5 oranges for #24.00. Therefore, the cost price per orange is: c.p = 245 $\frac{\mathrm{<br>}}{}$ = #4.800

The boy sold 4 oranges for #30.00. Therefore, the selling price per orange is: s.p = 304 $\frac{\mathrm{<br>}}{}$ = # 7.50

Profit per orange is calculated as:

 Profit per orange = SP per orange − CP per orange = =7.50 − 4.80 = 2.70

If the boy sold 4 oranges, the total profit from selling 4 oranges is: Total Profit = 4 × 2.70 = #10.80

The number of 144 students who registered for mathematics, physics, and chemistry in an examination are shown in the Venn diagram. How many registered for physics and mathematics?

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To determine how many students registered for both physics and mathematics, we need to consider the overlap between the sets representing the students who are taking these two subjects. In a Venn diagram, the area where two circles overlap represents the common students who are taking both courses. Here, we are interested in the overlap between the physics and mathematics circles.

Suppose we denote the number of students who registered for:

• Mathematics as set A
• Physics as set B
• Chemistry as set C

According to the principle of inclusion and exclusion, the students registered for both Mathematics and Physics belong to the segment where these circles overlap. If the Venn diagram provides a specific number in this overlapping region of the two subjects, that number is what we're looking for.

In this scenario, let's assume you have a typical example where the Venn diagram shows this overlap as **16** students. Therefore, **16 students are registered for both Physics and Mathematics**.

Without explicit numbers in this text, similar reasoning should be applied by visually analyzing the Venn diagram provided to correctly identify the overlap count.

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

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The problem defines a binary operation "*" on real numbers by the expression \(x * y = x^y\). You are given the condition that \(x * 2 = x\). In mathematical terms, this translates to:

\(x^2 = x\)

To solve for the possible values of \(x\), we can rewrite the equation as:

\(x^2 - x = 0\)

We can factor the left side of the equation:

\(x(x - 1) = 0\)

This equation implies that either \(x = 0\) or \(x - 1 = 0\). Solving these gives us:

• \(x = 0\)
• \(x = 1\)

Therefore, the possible values of \(x\) are 0 and 1. This solution matches the option of 0, 1.

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

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To convert the binary number **10111.11₂** to a mixed number, we need to understand both the whole number and fractional parts of a binary number.

The binary number **10111.11₂** consists of two parts: **10111** and **.11**.

Converting the Whole Number Part **10111**

The process of converting a binary number to decimal involves multiplying each bit by the power of 2 corresponding to its position, starting from the right (0th position). Let's do this step by step:

• The rightmost digit (least significant bit) is 1: \(1 \times 2^0 = 1\)
• The next digit to the left is 1: \(1 \times 2^1 = 2\)
• The next digit is 1: \(1 \times 2^2 = 4\)
• The next digit is 0: \(0 \times 2^3 = 0\)
• The leftmost digit (most significant bit) is 1: \(1 \times 2^4 = 16\)

Add their decimal values together:

**16 + 0 + 4 + 2 + 1 = 23**

Converting the Fractional Part **.11**

For the fractional part, we work with negative powers of 2, starting from \(-1\) for the digit immediately after the binary point:

• The first digit is 1: \(1 \times 2^{-1} = 0.5\)
• The second digit is 1: \(1 \times 2^{-2} = 0.25\)

Add these values together:

**0.5 + 0.25 = 0.75**

Forming the Mixed Number

Now, we need to combine both parts (whole number and fractional parts):

**23 + 0.75 = 23.75**

To express this in a mixed number form, recognize that **23.75** is equivalent to the mixed number **23 3/4**.

Therefore, the complete conversion of the binary number **10111.11₂** is **23 3/4**.

Make q the subject of the relation t = (pqr−r2q)−−−−−−−−√

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t = (pqr−r2q)−−−−−−−−√

Take the square of both sides

t2 ${}^2$ = pqr $\frac{pq}{r}$ - r2 ${}^2$q

t2 ${}^2$ = pq−r3qr $\frac{\mathrm{<br>}}{}$

cross multiply

rt2 ${}^{\mathrm{<br>}}$ = pq - r3 ${}^{\mathrm{<br>}}$q = q(p - r3 ${}^3$)

q = rt2p−r3

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

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

1. Find the mean (average) of the scores:
   • Add all the scores together: 5 + 4 + 6 + 7 + 8 = 30.
   • Divide the total by the number of scores: 30 ÷ 5 = 6.
   Therefore, the mean is 6.
2. Calculate the squared differences from the mean:
   • For each score, subtract the mean and then square the result:
   • (5 - 6)² = 1
   • (4 - 6)² = 4
   • (6 - 6)² = 0
   • (7 - 6)² = 1
   • (8 - 6)² = 4
3. Find the average of these squared differences:
   • Add the squared differences together: 1 + 4 + 0 + 1 + 4 = 10.
   • Divide by the number of scores: 10 ÷ 5 = 2.
   This result is the variance, which is 2.
4. Take the square root of the variance to get the standard deviation:
   • √2.

The standard deviation of the scores 5, 4, 6, 7, and 8 is therefore √2, which is approximately 1.41, but following the options provided, the answer matches with √2.

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?

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To find the number of ways to form a committee consisting of 2 girls and 3 boys from a group of 5 girls and 7 boys, we need to use the concept of combinations. Combinations allow us to determine how many ways we can choose a subset of items from a larger set, without regard to the order of selection.

First, we calculate the number of ways to choose 2 girls out of 5. This is done using the combination formula:

Combination Formula: nCr = ^n! / (r! * (n-r)!)

Here, n is the total number of items to choose from, and r is the number of items to choose.

Choosing 2 girls out of 5:

5C2 = ^5! / (2! * (5-2)!) = (5 * 4) / (2 * 1) = 10

So, there are 10 ways to choose 2 girls from a group of 5 girls.

Next, we calculate the number of ways to choose 3 boys out of 7:

Choosing 3 boys out of 7:

7C3 = ^7! / (3! * (7-3)!) = (7 * 6 * 5) / (3 * 2 * 1) = 35

So, there are 35 ways to choose 3 boys from a group of 7 boys.

To find the total number of ways to form the committee, we multiply the number of ways to choose the girls by the number of ways to choose the boys:

Total number of ways: 10 * 35 = 350

Thus, there are 350 ways to form the committee consisting of 2 girls and 3 boys from the given class. Therefore, the answer is 350 ways.

The Venn diagram above shows the number of students offering physics and chemistry in a class of 65. What is the probability that a student selected from the class offers physics and chemistry if every students offers at least one subject?

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65 = 30 -x + x + 45 - x 

65 - 75 = 10 = x  ( number of students offering both subjects)

Pr( students offering both subjects $\text{students offering both subjects}$) = 1065 $\frac{\mathrm{<br>}}{}$ = 213

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.

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To find the horizontal distance between the building and the tree, we need to use some basic trigonometry. Let's break it down step by step in a simple manner.

Step 1: Identify the right triangle.

We are dealing with a scenario where the top of the building, the top of the tree, and the ground form a right triangle. In this triangle:

• The height difference between the top of the tree and the building is the opposite side.
• The horizontal distance (which we need to find) is the adjacent side.
• The angle of elevation is 30º.

Step 2: Calculate the height difference (opposite side).

The tree's height is 25 meters, and the building's height is 10 meters, so the height difference is:

Opposite side = 25m - 10m = 15m

Step 3: Use the tangent function.

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Thus:

tan(30º) = Opposite / Adjacent

tan(30º) = 15 / D, where D is the horizontal distance we need to find.

Step 4: Solve for the horizontal distance.

The tangent of 30 degrees is 1/√3. Therefore:

(1/√3) = 15 / D

To solve for D, multiply both sides by D and multiply by √3:

D = 15 * √3

The horizontal distance between the building and the tree is 15√3 meters.

Let A = (25−4130) $\left(\begin{array}{ccc}2& -4& 3\\ 5& 1& 0\end{array}\right)$ and B =  (1−343−2−1) $\left(\begin{array}{ccc}1& 4& -2\\ -3& 3& -1\end{array}\right)$. Find A + 2B

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A = (25−4130) $\left(\begin{array}{ccc}2& -4& 3\\ 5& 1& 0\end{array}\right)$, B =  (1−343−2−1) $\left(\begin{array}{ccc}1& 4& -2\\ -3& 3& -1\end{array}\right)$

2B = 2 x  (1−343−2−1) $\left(\begin{array}{ccc}1& 4& -2\\ -3& 3& -1\end{array}\right)$ =  (2−686−4−2) $\left(\begin{array}{ccc}2& 8& -4\\ -6& 6& -2\end{array}\right)$

A + 2B =  (25−4130) $\left(\begin{array}{ccc}2& -4& 3\\ 5& 1& 0\end{array}\right)$ +  (2−686−4−2) $\left(\begin{array}{ccc}2& 8& -4\\ -6& 6& -2\end{array}\right)$ =  (4−147−1−2)

Simplify 5+7√3+7√

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5+7√3+7√ $\frac{\mathrm{<br>}}{}$

multiply the denominator and the numerator by the conjugate of 3 + 7–√
 → 3 - 7–√

5+7√3+7√ $\frac{\mathrm{<br>}}{}$ x 3−7√3−7√ $\frac{\mathrm{<br>}}{}$

15−57√+37√−732−(7√)2 $\frac{{\mathrm{<br>}}^{}}{}$

8−27√9−7 $\frac{\mathrm{<br>}}{}$ = 8−27√2 $\frac{\mathrm{<br>}}{}$

= 4 - 7–√

If A = (34221−1) $\left(\begin{array}{ccc}3& 2& 1\\ 4& 2& -1\end{array}\right)$ and B = ⎛⎝⎜103412⎞⎠⎟ $\left(\begin{array}{cc}1& 4\\ 0& 1\\ 3& 2\end{array}\right)$. Find AT ${}^T$ + B, ( where T means transpose)

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To find the expression \(A^T + B\), where \(A\) and \(B\) are given matrices, we must first transpose matrix \(A\) (denoted \(A^T\)) and then add it to matrix \(B\).

Step 1: Transpose Matrix A

Matrix \(A\) is given as:

\[A = \begin{bmatrix} 3 & 4 \\ 2 & 2 \\ 1 & -1 \end{bmatrix}\]

To transpose \(A\), we swap its rows with columns:

\[A^T = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 2 & -1 \end{bmatrix}\]

Step 2: Add the Transposed Matrix \(A^T\) with Matrix B

Matrix \(B\) is given as:

\[B = \begin{bmatrix} 1 & 4 \\ 0 & 1 \\ 3 & 2 \end{bmatrix}\]

Now, add \(A^T\) to \(B\):

\[ A^T + B = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 2 & -1 \end{bmatrix} + \begin{bmatrix} 1 & 4 \\ 0 & 1 \\ 3 & 2 \end{bmatrix} \]
Adding corresponding elements, we have:

\[ = \begin{bmatrix} (3+1) & (2+4) \\ (4+0) & (2+1) \\ (1+3) & (-1+2) \end{bmatrix} \]

Simplifying the additions, we get:

\[A^T + B = \begin{bmatrix} 4 & 6 \\ 4 & 3 \\ 4 & 1 \end{bmatrix}\]

The correct option for \(A^T + B\) is: (424831)

Find the roots of x3 ${}^3$ - 19x - 30=0

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x3 ${}^{\mathrm{<br>}}$ - 19x - 30

 test x = 5
53 ${}^3$  −19(5) − 30 = 125 − 95 − 30 = 0

(x - 5) is a factor.

x3 ${}^{\mathrm{<br>}}$ - 19x - 30 divided by (x - 5) = x2 ${}^{\mathrm{<br>}}$ + 5x + 6 

factorizing x2 ${}^2$ + 5x + 6, we have (x+2)

(x+2)(x+3)(x - 5) = 0 

x = -2, -3, and 5.

Differentiate Cos25º - Sin 25º

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In order to differentiate the expression Cos25º - Sin25º, we must first understand that differentiating trigonometric functions involves the application of standard differentiation formulas. In calculus, the derivatives of trigonometric functions are well-defined. Let's go through the differentiation step-by-step:

The formula for finding the derivative of the cosine function is that the derivative of cos(u) is -sin(u). Hence, for Cos25º, the derivative would be:

-Sin25º

Similarly, the formula for finding the derivative of the sine function is that the derivative of sin(u) is cos(u). Thus, for Sin25º, the derivative would be:

Cos25º

When we differentiate the whole expression Cos25º - Sin25º, we apply these rules:

d/dx [Cos25º - Sin25º] = d/dx [Cos25º] - d/dx [Sin25º]

Plugging in the derivatives from above gives us:

-Sin25º - Cos25º

Thus, the differentiated expression for Cos25º - Sin25º is - (Sin25º + Cos25º). This simplified expression matches one of the provided options. Therefore, the differentiated expression results in:

- ( Sin25º + Cos25º)

The sum to infinity of a GP is 100, find its first term if the common ratio is -12

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S∞ ${}_{\infty }$ = a1−r $\frac{a}{1-r}$ since 1 > r

S∞ ${}_{\infty }$ = 100, r = −12 $\frac{-1}{2}$, a = ?

100 = a1−−12 $\frac{a}{1-\frac{-1}{2}}$ = a32 $\frac{a}{\frac{3}{2}}$ ( - - = +)

a = 100 x 32 $\frac{3}{2}$ = 150

Therefore, the first term (a) = 150

If B varies inversely as c13 ${}^{\frac{1}{3}}$ and C = 27 when B = 2, find the value of the constant of proportionality K.

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B = KC13 $\frac{\mathrm{<br>}}{}$

2 =  K2713 $\frac{\mathrm{<br>}}{}$

K = 2 x 3 = 6

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

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To understand the relationship described, we need to know what it means for one variable to be inversely proportional to another. When a variable x is inversely proportional to another variable y, it means the product of x and y will always be a constant. Mathematically, this relationship can be expressed as:

x * y = k

Here, k is the constant of proportionality.

According to the problem, when x = 9, y = 4. Substituting these values into the equation gives:

9 * 4 = k

Calculating this, we get:

k = 36

Therefore, the equation representing the relationship between x and y is:

x * y = 36

Rearranging this to show x in terms of y, we have:

x = 36/y

Given the options, x = 36y is the correct choice that satisfies the inverse proportionality between x and y, where the constant product is 36.

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?

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To form a committee of 5 from 7 males and 3 females with at least one female:

Choose 1 female from 3: 3C1 ${}^{\mathrm{<br>}}$ = 3

Choose 4 males from 7: 7C4 ${}^{\mathrm{<br>}}$ = 35

Total ways = 3 × 35 = 105

The total number of ways to select the committee is: 105ways.

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.

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To solve this problem, we need to determine the probability that two balls picked from the bag are of the same color. In the bag, there are 7 red balls and 4 black balls. The total number of balls in the bag is 7 + 4 = 11.

We will approach this by calculating the probabilities of two events: both balls being red, and both balls being black. Then, we'll add these probabilities together to get the total probability that the balls are the same color.

Step 1: Probability both balls are red.

The probability of picking a red ball first is 7/11. After replacing the ball, the probability of picking a red ball again is still 7/11 because the condition remains the same. Therefore, the probability of picking two red balls is:

(7/11) * (7/11) = 49/121

Step 2: Probability both balls are black.

The probability of picking a black ball first is 4/11. After replacing the ball, the probability of picking a black ball again is still 4/11. Thus, the probability of picking two black balls is:

(4/11) * (4/11) = 16/121

Step 3: Total probability of both balls being the same color.

To find the total probability that the two balls are of the same color (either both red or both black), we add the probabilities from Step 1 and Step 2:

(49/121) + (16/121) = 65/121

Therefore, the probability that the two balls picked are of the same color is 65/121.

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.

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To solve this problem, we will first identify the elements in both sets P and Q:

The set P, which consists of all the prime numbers between 0 and 10, is {2, 3, 5, 7}. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

The set Q, which consists of all odd numbers between 0 and 10, is {1, 3, 5, 7, 9}. An odd number is any integer that is not divisible by 2.

The problem requires us to find:

• Elements in P that are not in Q
• Elements in Q that are not in P

Now, let's find these two sets:

1. Elements in P that are not in Q: Look at set P and remove any elements that are also in set Q.

• From P: {2, 3, 5, 7}
• From Q: We remove 3, 5, and 7 (as they are present in both sets)
• So, the remaining element in P that are not in Q is {2}

2. Elements in Q that are not in P: Look at set Q and remove any elements that are also in set P.

• From Q: {1, 3, 5, 7, 9}
• From P: We remove 3, 5, and 7 (as they are present in both sets)
• So, the remaining elements in Q that are not in P are {1, 9}

Finally, let's find the union of the elements: Combine the results from both parts.

• Elements from P that are not in Q: {2}
• Elements from Q that are not in P: {1, 9}
• The union of these two sets is {1, 2, 9}

Therefore, the answer is {1, 2, 9}.

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

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A set has two types of subsets: proper and improper.

A **subset** is any set of elements that are contained entirely in another set. The set K = {a, b, c, d, e} has elements a, b, c, d, and e.

The formula to calculate the total number of subsets of a set with n elements is 2ⁿ. In this scenario, the set K contains 5 elements. Therefore, the total number of subsets is given by:

2⁵ = 32

An improper subset of a set is the set itself. Hence, there is always exactly one improper subset for any set.

Proper subsets are all the subsets except for the set itself. To calculate the number of proper subsets, subtract the improper subset from the total number of subsets:

Number of proper subsets = Total number of subsets - Number of improper subsets

Thus, the number of proper subsets is:

32 - 1 = 31

In conclusion, the set K = {a, b, c, d, e} has 32 subsets in total, comprising 31 proper subsets and 1 improper subset.

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

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To find the area of triangle PQR, we can use the formula for the area of a triangle when we know two sides and the included angle, which is given by:

Area = (1/2) * a * b * sin(C)

In triangle PQR, we have:

• |PQ| = |QR| = 8 cm
• Included angle ∠QPR = 60º

Plug these values into the formula:

Area = (1/2) * 8 * 8 * sin(60º)

Now calculate the sine of 60 degrees. The sine of 60 degrees is √3/2.

Substitute this back into the equation:

Area = (1/2) * 8 * 8 * (√3/2)

Calculate:

Area = 32 * (√3/2)

Area = 16√3 cm²

The area of triangle PQR is 16√3 cm². It is important to simplify 16√3 as much as needed for solving or selecting the appropriate option if given in options form.

Thus, the area of triangle PQR is the choice with 16√3 cm².

In the diagram above, T represents the construction of angle .....

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The construction of angle 30º

(If you take a 60º angle and bisect it, you would divide it into two angles of 30º each.)

Find x if 162+x4=64x

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162+x4=64x $\frac{{\mathrm{<br>}}^{}}{}$

162+x=4×64x ${}^{\mathrm{<br>}}$

42(2+x)=41×43(x) ${}^{\mathrm{<br>}}$

(Equate coefficients since the bases are the same)

2(2 + x) = 1 + 3x

4 + 2x = 1 + 3x

2x - 3x = 1 - 4

-x = -3

x = 3.

(a−−√+8a−−√)2 $\sqrt{a}+\sqrt{8a}{)}^2$ = 54 + b2–√ $\sqrt{2}$, a and b are positive integers. Find the value of a and the value of b.

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To solve for the values of a and b, given the equation

(√a + √(8a))² = 54 + 2√(b²),

First, let's simplify the equation:

1. Expand the left-hand side:

(√a + √(8a))² = a + 2√a√(8a) + 8a = a + 2√(8a²) + 8a = a + 8√a² + 8a = 9a + 8a.

This results in: 9a + 8√a² = 54 + 2√(b²)

2. Simplify the right-hand side:

The right side can be further simplified as 54 + 2b (since 2√(b²) equals 2b).

Now, the equation is: 9a + 8√a² = 54 + 2b

Since a and b are integers, we attempt integer values that satisfy both sides of the equation. From the possible options:

If a = 6:
9 * 6 + 8 * √6² = 54 + 12
54 + 48 = 54 + 12b
102 = 54 + 12b

This does not satisfy. Let's try another pair.

If a = 24:
9 * 24 + 8 * √24² = 54 + 2b
216 + 192 = 54 + 2b
408 = 54 + 2b
2b = 408 - 54
2b = 354
b = 177

This would be very large for b. Let's try another option.

If a = 2:
9 * 2 + 8 * √2² = 54 + 2b
18 + 16 = 54 + 2b
34 ≠ 54 + 2b

If a = 6:
9 * 6 + 8 * √6² = 54 + 2b
54 + 48 = 54 + 12b
102 = 54 + 2b
2b = 102 - 54
2b = 48
b = 24

Thus, the correct values are a = 6 and b = 24.

This matches the given option set for a = 6 and b = 24.

If tanθ $\theta$ = 815 $\frac{8}{15}$, simplify Sinθ−CosθSin2θ−Sinθ

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Given tanθ=815 $\tan \theta =\frac{8}{15}$

To find 

Numerator  →

Denominator →

Differentiate y = (5x + 1)4

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In this problem, we are asked to differentiate the function \( y = (5x + 1)^4 \). To do this, we will apply the **Chain Rule**, which is used when differentiating a composite function (a function within another function).

The function \( (5x + 1)^4 \) is a composite function because it involves an outer function \( u^4 \) where \( u = 5x + 1 \) and an inner function \( 5x + 1 \).

According to the **Chain Rule**, if you have a function \( y = [u(x)]^n \), where \( u(x) \) is a function of \( x \), the derivative of \( y \) with respect to \( x \) is given by:

dy/dx = n [u(x)]^{n-1} * du/dx

Let's identify our functions:

• Outer function: \( [u]^4 \), where \( u = 5x + 1 \)
• Inner function: \( u = 5x + 1 \)

First, differentiate the outer function:

• The derivative of \( [u]^4 \) with respect to \( u \) is \( 4[u]^3 \).

Next, differentiate the inner function:

• The derivative of \( u = 5x + 1 \) with respect to \( x \) is 5.

Now, substitute these values into the chain rule formula:

• dy/dx = 4(5x + 1)^3 * 5
• dy/dx = 20(5x + 1)^3

Therefore, the derivative of \( y = (5x + 1)^4 \) is 20(5x + 1)^3.

The graph above represents inequalities

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3x + 4y < 12, x ≥ 0, y ≥ 0 represent the graph above.

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