Big ideas 7.5 (question)

a. angle 2 and angle 1, angle 2 and angle 3 are linear pairs. b. angle 9 and angle 8, angle 9 and angle 5 are linear pairs. c. angle 4 and angle 2 form vertical angles.

A linear pair of angles in geometry is a pair of neighbouring angles created by the intersection of two lines. When two angles share a vertex and an arm but do not overlap, they are said to be adjacent angles. Due to their formation on a straight line, the linear pair of angles are always complementary. Thus, the total of two angles in a pair of lines is always 180 degrees.

a. angle 2 and angle 1, angle 2 and angle 3 are linear pairs.

b. angle 9 and angle 8, angle 9 and angle 5 are linear pairs.

c. angle 4 and angle 2 form vertical angles.

d. angle 8 and angle 5 form vertical angles.

e. The rays that form angle 7 and angle 9 do not for, opposite rays.

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No, this relation does not define y as a function of x.

A function is a relation in which each input (x-value) is paired with exactly one output (y-value). In this relation, the x-value of 2 is paired with two different y-values (3 and -3), which violates the definition of a function.

Therefore, this relation does not define y as a function of x.

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We can determine that TC owns one cow using exponential calculations.

Exponent-based equations are those in which the exponent, or a portion of the exponent, is a variable.

For illustration, [tex]3^{x}[/tex] = 81.

In the question given,

TC is drinking no. of cups = 17

Let no. of cows TC has = x

Now according to the question, we can for the equation as:

[tex](\sqrt[5]{x} )^{2}[/tex] = 17 -1

⇒ x = [tex]\sqrt[5]{4}[/tex]

⇒ x = 1.319

⇒ x ≈ 1

Hence, TC owns 1 cow.

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The final answer is 3

The calculation you are looking to solve is 2.2 + (3)/(5). To solve this, we first need to convert the improper number 2.2 to a fraction. We can do this by multiplying the whole number by the denominator of the fraction and then adding the numerator. In this case, we would multiply 2 by 5 and then add 2, giving us 12/5. Now, we can add this fraction to the other fraction:

12/5 + 3/5 = 15/5

Next, we can simplify the fraction by dividing both the numerator and denominator by the greatest common factor. In this case, the greatest common factor is 5, so we can divide both the numerator and denominator by 5:

15/5 = 3

Therefore, the final answer is 3.

In summary, 2.2 + (3)/(5) = 3.

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The two sides with lengths of 7 and 5 will never meet, which shows that the sum of the lengths of the two sides of a triangle must be equal to the length of the third side.

In Euclidean geometry, the Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than or equal (≥) to the third side of the triangle.

Mathematically, the Triangle Inequality Theorem is represented by this mathematical expression:

b - c < n < b + c

Where:

n, b, and c represent the side lengths of this triangle.

b - c < n < b + c

7 - 5 < n < 7 + 5

2 < n < 12

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The value of x to the nearest degree is 30°.

What is Triangle ?

A triangle is one that has three sides, three angles, and whose total angles is always 180 degrees.

Three line segments are linked end to end to make a triangle, a two-dimensional geometric structure with three angles. These line segments are known as sides, and their intersections are known as vertices.

The measurements of a triangle's sides and angles are used to categorize it.

We can use trigonometric ratios to get the value of x in the right triangle given:

sin(x) = opposite/hypotenuse

[tex]sin(x) = 8/17x = sin^{-1(8/17)}x = 29.74^{o} $ (approx.) $[/tex]

Therefore, the value of x to the nearest degree is 30°.

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The distance between the ship and the base of the is approximately 359.32 feet.

The distance between the ship and the base of the lighthouse can be found using the tangent of the angle of elevation.

The tangent of an angle in a right triangle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (160 feet) and the adjacent side is the distance between the ship and the base of the lighthouse (x).

So, we can set up the equation:

tan(24) = 160/x

To solve for x, we can cross multiply and then divide:

x * tan(24) = 160

x = 160/tan(24)

Using a calculator, we can find that tan(24) is approximately 0.4452.

So, x = 160/0.4452

x = 359.32 feet

Therefore, the distance between the ship and the base of the lighthouse is approximately 359.32 feet.

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The vector x = [2, -3, 1] is a basis for Row(A)⊥. This means that any vector in Row(A)⊥ can be written as a multiple of x.

To find a basis for Row(A)⊥ using the dot product, we need to find a vector that is orthogonal to all the rows of A. This means that the dot product of the vector and each row of A should be equal to 0.

Let's say the vector we are looking for is x = [x1, x2, x3]. Then we need to solve the following system of equations:

x1 * 1 + x2 * 4 + x3 * 2 = 0
x1 * 2 + x2 * 2 + x3 * 1 = 0
x1 * 3 + x2 * 1 + x3 * 1 = 0

We can write this system of equations in matrix form as:

[1 4 2] [x1] = [0]
[2 2 1] [x2] = [0]
[3 1 1] [x3] = [0]

We can use Gaussian elimination to solve this system of equations. After performing the necessary row operations, we get:

[1 0 -2] [x1] = [0]
[0 1  3] [x2] = [0]
[0 0  0] [x3] = [0]

From the last equation, we can see that x3 can be any value. Let's choose x3 = 1. Then, from the second equation, we get x2 = -3, and from the first equation, we get x1 = 2.

So, the vector x = [2, -3, 1] is a basis for Row(A)⊥. This means that any vector in Row(A)⊥ can be written as a multiple of x.

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The the distance between the two stations is 300km, then the speed of first motorcyclist is 63 km/h and speed of second-motorcyclist is 70 km/h.

The distance between the "two-stations" is given to be 300 km;

Let the speed of first-motorcyclists be x km/h

and let the speed of second-motorcyclists be (x + 7) km/h,

So, the Distance covered by first motorcyclist after 2 hours is = 2x km

and distance covered by second motorcyclist after 2 hours is = 2(x+7) km

⇒ 2x + 14 km,

So, the distance not covered by them after 2 hours is = 300 - (2x+2x+14) km.

The distance between the motorcyclist after 2 hours is 34 km,

Which means ,

⇒ 300-(4x+14) = 34

⇒ 300 - 4x - 14 = 34

⇒ 4x = 300 - 48

⇒ x = 63,

So, Speed of first-motorcyclists is 63 km/h, and

Speed of second-motorcyclist is = (63 + 7) = 70 km/h.

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The given question is incomplete, the complete question is

The distance between two stations is 300 km. Two motorcyclists start simultaneously from these stations and move towards each other. The speed of one of them is 7 km/h more than that of the other. If the distance between them after 2 hours of their start is 34 km, find the speed of each motorcyclist.

Answer:

We are not given the ratio of cordial to water used in the mixture, so we can assume that the entire bottle of cordial is mixed with water to make the drink.

Since the bottle of cordial holds 800 ml of cordial, the total volume of the mixture would be 800 ml + volume of water added. Let's call the volume of water added x.

Therefore, the total volume of the drink would be 800 ml + x.

We are asked to find the minimum capacity of the container in liters, so we need to convert the total volume of the drink from milliliters to liters:

800 ml + x = (800 + x)/1000 liters

Now we can set up an inequality to find the minimum value of x that would make the total volume of the drink at least 1 liter:

800 ml + x ≥ 1000 ml

Simplifying this inequality, we get:

x ≥ 200 ml

Therefore, the minimum volume of water that needs to be added to the cordial to make a drink with a total volume of at least 1 liter is 200 ml.

So the minimum capacity of the container would be:

800 ml + 200 ml = 1000 ml = 1 liter

Therefore, the minimum capacity of the container in liters would be 1 liter.

Step-by-step explanation:

The probability of fraudulent applications are:

The given parameters are

n = 25

p = 0.12

The individual probability can be calculated as

P(x) = C(n, x) * p^x * (1 - p)^(n - x)

So, we have

Probability the number of fraudulent applications in the sample is 0

P(0) = C(25, 0) * 0.12^0 * (1 - 0.12)^(25 - 0)

P(0) = 0.0410

Probability the number of fraudulent applications in the sample is 3

P(3) = C(25, 3) * 0.12^3 * (1 - 0.12)^(25 - 3)

P(3) = 0.2387

Probability the number of fraudulent applications in the sample is 3 or less

P(x ≤ 3) = P(0) + ... P(3)

Using the formula above, we have

P(x ≤ 3) = 0.4088

Probability the number of fraudulent applications in the sample is 5 or more

P(x ≥ 5) = P(5) + ... P(25)

Using the formula above, we have

P(x ≥ 5) = 0.1734

Probability the number of fraudulent applications in the sample is more than 3

P(x > 3) = 1 - P(x ≤ 3)

By substitution, we have

P(x > 3) = 1 - 0.4088

P(x > 3) = 0.5912

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The required sinusoidal function be,

⇒ y = 25 sin(2π/15 (x - 1 + 15/4 + 30nπ)) + 38

                 or

⇒ y = 25 sin(2π/15 (x - 1 - 15/4 - 30nπ)) + 38

Since we know that,

The general formula of a sinusoidal function is,

⇒ y = A sin(B(x - C)) + D,

where,

A is the amplitude

B is the frequency (and related to the period by T = 2π/B)

C is the phase shift (the horizontal displacement from the origin)

D is the vertical shift (the midline)

Using the given information,

Amplitude = 25, so A = 25.

Period = 15, so T = 15.

We know that,

T = 2π/B, so we can solve for B,

⇒ 15 = 2π/B

⇒  B = 2π/15

Midline is y = 38, so D = 38.

⇒ f(1) = 63,

so we can also use this to find the phase shift:

⇒ 63 = 25 sin(B(1-C)) + 38

⇒ 25 sin(B(1-C)) = 25

⇒ sin(B(1-C)) = 1

⇒ B(1-C) = π/2 + 2nπ or 3π/2 + 2nπ,

where n is an integer.

Substituting B and solving for C in each case, we get,

⇒ B(1-C) = π/2 + 2nπ 2π/15 (1 - C)

              = π/2 + 2nπ 1 - C

              = 15/4 + 30nπ C

              = 1 - 15/4 - 30nπ

⇒ B(1-C) = 3π/2 + 2nπ 2π/15 (1 - C)

              = 3π/2 + 2nπ 1 - C

              = 15/4 + 60nπ/2 C

              = 1 - 15/4 - 30nπ

So we have two possible functions are,

⇒ y = 25 sin(2π/15 (x - 1 + 15/4 + 30nπ)) + 38

                 or

⇒ y = 25 sin(2π/15 (x - 1 - 15/4 - 30nπ)) + 38

where n is any integer.

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Answer:

The distributive property is used when finding the product of two polynomials by distributing one polynomial to each term of the other polynomial. For example, if we wanted to multiply (3x - 4)(2x + 5), we would use the distributive property by first multiplying 3x(-4) and 2x(5) and then adding the results together.

Polynomials are closed under multiplication, meaning that when two polynomials are multiplied together, the result is always another polynomial. This is true because a polynomial is a combination of constants and variables raised to non-negative integer powers, and when two polynomials are multiplied, the result is a combination of constants and variables raised to non-negative integer powers, which is a polynomial.

The estimate of the probability that he will hit at least 6 times in his next 10 attempts is given as follows:

80%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

An amateur golfer hits the ball 48% of the time he attempts, hence we round the probability to 50%, and have that the numbers are given as follows:

From the table, we have 20 sets of 10 attempts, and in 16 of them he hit at least 6 attempts, hence the probability is given as follows:

16/20 = 0.8 = 80%.

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To find the number of coins the statement made by Elena is correct.

The exponent of a number indicates how many times a number has been multiplied by itself. For instance, 34 indicates that we have multiplied 3 four times. Its full form is 3 3 3 3. Exponent is another name for a number's power. It might be an integer, a fraction, a negative integer, or a decimal.

Elena is on point. The formula 228 = 28 x 2 = 56 doesn't make sense in this situation since it suggests that they only counted for 28 days and received 228 coins, when the problem states that they counted for 28 days and received 228 coins. The fact that they counted for 28 days and came up with a total of 28 times 28 coins, or 784, makes the equation 228 = 28 x 28 make sense. Elena is accurate as a result.

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The number of sodas sold is 83 and the number of hot dogs sold is 52.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

Let's call the number of sodas sold "x".

According to the problem, the number of hot dogs sold is 31 less than the number of sodas sold, so we can write the number of hot dogs sold as "x - 31".

We also know that the combined total of sodas and hot dogs sold is 135, so we can write an equation:

x + (x - 31) = 135

Simplifying this equation:

2x - 31 = 135

Adding 31 to both sides:

2x = 166

Dividing both sides by 2:

x = 83

So the number of sodas sold is 83.

To find the number of hot dogs sold, we can use the equation we came up with earlier:

x - 31 = 83 - 31 = 52

So the number of hot dogs sold is 52.

Therefore, the number of sodas sold is 83 and the number of hot dogs sold is 52.

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The probability that the sample proportion is between 0.26 and 0.44 is 0.9998, or 99.98%.

The probability that the sample proportion is between 0.26 and 0.44 can be found using the normal distribution formula.

First, we need to find the mean and standard deviation of the sample proportion. The mean of the sample proportion is equal to the population proportion, which is 0.37. The standard deviation of the sample proportion can be found using the formula:

σ = √(p(1-p)/n)

Where p is the population proportion, and n is the sample size. Plugging in the given values, we get:

σ = √(0.37(1-0.37)/245) = 0.0196

Next, we need to find the z-scores for the given sample proportions. The z-score can be found using the formula:

z = (x - μ)/σ

Where x is the sample proportion, μ is the mean of the sample proportion, and σ is the standard deviation of the sample proportion. Plugging in the values for the lower bound of the sample proportion (0.26), we get:

z = (0.26 - 0.37)/0.0196 = -5.61

Similarly, for the upper bound of the sample proportion (0.44), we get:

z = (0.44 - 0.37)/0.0196 = 3.57

Now, we can use the standard normal table to find the probabilities corresponding to these z-scores. The probability for z = -5.61 is 0, and the probability for z = 3.57 is 0.9998.

Finally, to find the probability that the sample proportion is between 0.26 and 0.44, we subtract the lower probability from the upper probability:

P(0.26 < p < 0.44) = 0.9998 - 0 = 0.9998

Therefore, the probability that the sample proportion is between 0.26 and 0.44 is 0.9998, or 99.98%.

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Answer:

[tex]\theta = 60^\circ \;and\; \theta = 300^\circ[/tex]

Step-by-step explanation:

Given

[tex]\sin \left(2\theta\right)=\sin \left(\theta\right)[/tex]

Use the identity: [tex]\sin(2\theta) = 2 \sin(\theta)\cos(\theta)[/tex]

=> [tex]2 \sin(\theta)\cos(\theta) = \sin(\theta)[/tex]

Divide both sides by [tex]\sin(\theta)[/tex]

=> [tex]2 \cos(\theta) = 1[/tex]

[tex]= > \cos(\theta) = \dfrac{1}{2}[/tex]

[tex]= > \,\theta=\cos^{-1}\left(\dfrac{1}{2}\right)[/tex]

[tex]\cos^{-1}\left(\dfrac{1}{2}\right) \;is\: \dfrac{\pi}{3}} \text{ in the first quadrant and $\dfrac{5\pi}{3}$ in the fourth quadrant}}[/tex]

In degrees this corresponds to
[tex]\dfrac{\pi}{3} = \dfrac{\pi}{3} \times \dfrac{180^\circ}{\pi} = 60^\circ\\\\and\\\\\dfrac{5\pi}{3} = \dfrac{5\pi}{3} \times \dfrac{180^\circ}{\pi} = 300^\circ\\[/tex]

Answer
[tex]\theta = 60^\circ \;and\; \theta = 300^\circ[/tex]

The exponential function that represents a vertical compression by a factor of 2 of [tex]f(x) = 2^x[/tex] is given as follows:

[tex]g(x) = 0.5(2)^x[/tex]

The standard definition of an exponential function is given as follows:

[tex]y = a(b)^x[/tex]

In which:

When a function is vertically compressed by a factor of a, we have that the output of the function is multiplied by 1/a = divided by a, hence, considering the factor of 2, the function g(x) is given as follows:

g(x) = 1/2 x f(x)

[tex]g(x) = 0.5(2)^x[/tex]

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The correct equation is;

p = 4t + 1

The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of the points on the line. In general, the equation of a line can be written in slope-intercept form, which is y = mx + b, where m is the slope of the line and b is the y-intercept (the point at which the line crosses the y-axis).

We can get the slope of the graph from;

m = y2 - y1/x2 =x2 - x1

m = 1 - 0/0.25 - 0

m = 4

Since the y intercept is at y = 1 then we have;

p = 4t + 1

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None of the expressions given is equivalent to (2 - ³√24)²..

Mathematical statements must contain a sentence, at least one mathematical operation, and at least two numbers or factors. With this mathematical operation, you can increase, split, add, or take something away. The shape of a phrase is as follows: Expression: (Number/Variable, , Math Operator)

Let's first simplify the expression (2 - ³√24)²:

(2 - ³√24)² = (2 - 24^(1/3))² = (2 - 2.29)² = (-0.29)² = 0.0841

Now we can check which expressions are equivalent to 0.0841 when (2 - ³√24)² is evaluated:

4 = 4.0000... (not equivalent to 0.0841)

04 = 0.04 (not equivalent to 0.0841)

26.2 - 8 = 18.2 (not equivalent to 0.0841)

02 - 5.22 = -3.22 (not equivalent to 0.0841)

Therefore, none of the expressions given is equivalent to (2 - ³√24)².

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Therefore , the solution of the given problem of slope comes out to be slope-intercept equation y = 2x + 1.

The y-intersection axis's with the slope of the line marks the inflection point in arithmetic where the y-axis intersects a line or curve. Y = mx+c, where m stands for the slope and c for the y-intercept, is the equation for the long line. The y-intercept (b) and slope (m) of the line are emphasised in the equation intercept form. An solution with the intersecting form (y=mx+b) has m and b as the slope and y-intercept, respectively.

Here,

Y = mx + b, where m is the line's slope and b is the y-intercept, is the slope-intercept version of a linear equation. Given two points (x1, y1) and (x2, y2), we can use the following method to determine the slope of the line:

=> m = (y2 - y1) / (x2 - x1) (x2 - x1)

For instance, if the two locations (2, 5) and (4, 9) are provided, we can determine the slope as follows:

=> m = (9 - 5) / (4 - 2) = 2

The y-intercept can then be determined by using one of the locations and the slope. Let's use points 2 and 5:

=> y = mx + b

=> 5 = 2(2) + b

=> 5 = 4 + b

=> b = 1

As a result, the line going through the points (2, 5) and (4, 9) has the slope-intercept equation y = 2x + 1.

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a)18

b)0.023

1) The null mean for the disc width is assumed to be 18.
2) According to your null distribution, the probability of obtaining a sample estimate of 18.76 cm or more extreme is 0.023, and the p-value is 0.023.

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Answer:

48

Step-by-step explanation:

We know

1/6 times a number is the same as 8. Let's x be the unknown number we have the equation

1/6x = 8

8 divided by 1/6

8 ÷ 1/6 = 8 × 6 = 48

So, the answer is 48

The coordinates after the translation are (2, 6).

Here we start with the point (4, 3) and we want to apply the translation defined by (x - 2, y + 3)

This would be a translation of 2 units to the left and 3 units up, using a "coordinate-axis" notation.

So we just need to subtract 2 from the x-value and add 3 to the y-value, we will get the new coordinates:

(4 - 2, 3 + 3) = (2, 6)

These are the coordinates of point B after the translation, the correct option is the second one.

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Answer:

OB' (2,

Step-by-step explanation:

take your first point (B) (4,3) and plug it into the x and y in (x-2, y+3) so you get (4-2, 3+3) which will give you (2,

The quotient is 4x+5 and the remainder is -15x.

The quotient and remainder when dividing (12x^(3)+15x^(2)+21x)/(3x^(2)+4) can be found using polynomial long division.

First, divide the leading term of the numerator by the leading term of the denominator: (12x^(3))/(3x^(2)) = 4x. This is the first term of the quotient.

Next, multiply the first term of the quotient by the denominator and subtract the result from the numerator: (12x^(3)+15x^(2)+21x) - (4x)(3x^(2)+4) = 15x^(2)+5x.

Now, repeat the process with the new numerator: (15x^(2))/(3x^(2)) = 5. This is the second term of the quotient.

Again, multiply the second term of the quotient by the denominator and subtract the result from the new numerator: (15x^(2)+5x) - (5)(3x^(2)+4) = -15x.

Since the degree of the new numerator is less than the degree of the denominator, the division is complete and the new numerator is the remainder.

Therefore, the quotient is 4x+5 and the remainder is -15x.

In conclusion, the quotient and remainder when dividing (12x^(3)+15x^(2)+21x)/(3x^(2)+4) are 4x+5 and -15x, respectively.

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The solution to the compound linear inequality 1.4 ≤ 9.2 - 0.8x ≤ 6.9 are the values of x in the interval [2.9, 9.8].

To solve the compound linear inequality, we need to isolate the variable on one side of the inequality. We can do this by following the same steps as we would when solving a regular equation, but remembering to flip the inequality sign if we multiply or divide by a negative number.

1.4 ≤ 9.2 - 0.8x ≤ 6.9

First, we'll subtract 9.2 from all sides of the inequality:

-7.8 ≤ -0.8x ≤ -2.3

Next, we'll divide all sides by -0.8 to isolate the variable. Remember to flip the inequality signs since we're dividing by a negative number:

9.75 ≥ x ≥ 2.875

Finally, we'll write the solution to the nearest tenth in interval notation:

[2.9, 9.8]

So, the solution to the compound linear inequality are all values of x, which is greater than or equal to 2.9 but less than or equal to 9.8, or x is in the interval [2.9, 9.8].

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The requried value of the expression [tex]\sum_{n=11}^{30}n-\sum_{n=1}^{10}n[/tex] is 355.

Simplification involves applying rules of arithmetic and algebra to remove unnecessary terms, factors, or operations from an expression.

Here,
To evaluate the expression:

[tex]\sum_{n=11}^{30}n-\sum_{n=1}^{10}n[/tex]

we can first simplify each summation separately and then subtract the second summation from the first.

[tex]\sum_{n=11}^{30}n[/tex]= 11 + 12 + 13 + ... + 29 + 30

We can use the formula for the sum of an arithmetic series to simplify this expression:

S = (n/2)(a + l)

In this case, a = 11, l = 30, and n = 20 (since we're summing 20 terms).

So, we have:

S = (20/2)(11 + 30)

S= 410

Similarly,

[tex]\sum_{n=1}^{10}n[/tex] = 1 + 2 + 3 + ... + 9 + 10

S = 55

Finally, we can subtract the second summation from the first:

= 410 - 55 = 355

Therefore, the value of the expression is 355.

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Answer: To obtain the relation between α and β, we can eliminate t from the given equations.

(iv)

α = 1/2a(t+1/t)

β = 1/2a(t-1/t)

We can multiply these two equations to eliminate t^2:

αβ = (1/2a(t+1/t))(1/2a(t-1/t))

αβ = (1/4a^2)(t^2 - 1/t^2)

Multiplying both sides by 4a^2 gives:

4a^2αβ = t^2 - 1/t^2

Adding 1/t^2 to both sides gives:

4a^2αβ + 1/t^2 = t^2 + 1/t^2

Multiplying both sides by t^2 gives:

4a^2αβt^2 + 1 = t^4 + 1

Rearranging and simplifying gives the relation between α and β:

4a^2αβ = t^4 - 4a^2t^2 + 1

Now we can write the locus of the point as t varies:

4a^2αβ = t^4 - 4a^2t^2 + 1

This is a fourth degree equation in t, which represents a curve in the (α, β) plane. However, we can simplify it by noting that t^2 is always non-negative. Therefore, we can treat 4a^2t^2 as a constant and write:

4a^2αβ = (t^2 - 2a^2)^2 + 1 - 4a^4

This is the equation of a conic section called a hyperbola. Its center is at (0,0), its asymptotes are the lines α = ±β, and its foci are at (a√2,0) and (-a√2,0).

Step-by-step explanation: