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The errors in the proposed statement to prove by contradiction that 3·√2 - 7 is an irrational number, is the option;

Proving by contradiction is an indirect method of proving a fact or a reductio ad absurdum, which is a method of proving a statement by the assumption that the opposite of the statement is true, then showing that a contradiction is obtained from the assumption.

The definition of rational numbers are numbers that can be expressed in the form a/b, where a and b are integers

The assumption that 3·√2 - 7 is a rational number indicates that we get;

3·√2 - 7 = a/b, where a and b are integers

Therefore, the error in the method used to prove that 3·√2 - 7 is an irrational number is the option; To apply the definition of rational, a and b must be integers. This is so as the value 3·√2 - 7 is a real number, which is also an irrational number, thereby contradicting the supposition.

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To draw the 6th term, represent it visually within the context of the pattern or sequence from which it is derived.

To draw the 6th term, we need to understand the context or pattern from which the term is derived.

Drawing the term usually involves representing the elements or characteristics of the pattern in a visual form.

Without specific information about the pattern, we can provide a general approach to drawing the 6th term.

Identify the Pattern:

Determine the sequence or pattern from which the 6th term is derived.

It could be a numerical sequence, a geometric pattern, or any other pattern.

For example, if the pattern is a number sequence of multiples of 3, the first few terms would be 3, 6, 9, 12, 15, and so on.

Visualize the Pattern: Based on the identified pattern, visualize how the elements change or progress from term to term.

This could involve drawing a diagram, a graph, or any visual representation that captures the pattern.

Consider using a coordinate grid, a number line, or any other suitable visual aid.

Locate the 6th Term:

Use the information from the pattern and the visualization to determine the specific position or value of the 6th term.

In our example of multiples of 3, the 6th term would be 18.

Draw the 6th Term: Finally, represent the 6th term in your chosen visual form.

This could mean marking the position on a number line, plotting a point on a graph, or incorporating the value into a diagram.

Note that the specific method of drawing the 6th term will depend on the nature of the pattern and the context in which it is given.

Providing more details about the pattern would allow for a more accurate and specific visual representation of the 6th term.

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The inverse of the given matrix [1 0 0; 1 1 0; 1 1 1] is:

[1 0 0]

[-1 1 0]

[0 -1 1]

To find the inverse of a matrix, we can follow these steps:

Step 1: Write the given matrix and the identity matrix side by side.

[1 0 0 | 1 0 0]

[1 1 0 | 0 1 0]

[1 1 1 | 0 0 1]

Step 2: Apply row operations to transform the given matrix into the identity matrix on the left side.

Subtract the first row from the second row: R2 = R2 - R1

[1 0 0 | 1 0 0]

[0 1 0 | -1 1 0]

[1 1 1 | 0 0 1]

Subtract the first row from the third row: R3 = R3 - R1

[1 0 0 | 1 0 0]

[0 1 0 | -1 1 0]

[0 1 1 | -1 0 1]

Subtract the second row from the third row: R3 = R3 - R2

[1 0 0 | 1 0 0]

[0 1 0 | -1 1 0]

[0 0 1 | 0 -1 1]

Step 3: The matrix on the right side is now the inverse of the given matrix. Therefore, the inverse of the given matrix is:

[1 0 0]

[-1 1 0]

[0 -1 1]

The inverse of the given matrix [1 0 0; 1 1 0; 1 1 1] is:

[1 0 0]

[-1 1 0]

[0 -1 1]

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The values into the slope-intercept form, we have y = -5x - 6

The slope-intercept form of a linear equation is given by:

y = mx + b

where 'm' represents the slope of the line, and 'b' represents the y-intercept.

In this case, the y-intercept is (0, -6), which means that the line crosses the y-axis at the point (0, -6).

The slope is given as -5.

Therefore, substituting the values into the slope-intercept form, we have:

y = -5x - 6

This equation represents the line with a y-intercept of (0, -6) and a slope of -5.

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The correct statement about rectangles is:

1. All rectangles are parallelograms.

2. Parallelograms have exactly 1 pair of parallel sides.

3. So, all rectangles have exactly 1 pair of parallel sides.

A rectangle is a type of parallelogram that has additional properties. By definition, a rectangle is a quadrilateral with four right angles. This means that opposite sides of a rectangle are parallel. Since all four sides of a rectangle are right angles, it follows that a rectangle has exactly 1 pair of parallel sides.

Option 1 states that only some rectangles are parallelograms, which is incorrect. All rectangles are parallelograms because they have opposite sides that are parallel.

Option 2 states that parallelograms have 2 pairs of parallel sides, which is also incorrect. Parallelograms have exactly 2 pairs of parallel sides, not 4. A rectangle is a special type of parallelogram that has additional properties such as all angles being right angles.

Option 3 states that only some rectangles have 2 pairs of parallel sides, which is incorrect. All rectangles have exactly 1 pair of parallel sides, not 2. Having 2 pairs of parallel sides would make a shape a parallelogram, not a rectangle.

Therefore, the correct statement is that all rectangles are parallelograms and have exactly 1 pair of parallel sides. 1,2,3 are correct.

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We can conclude that point A is located at the origin (0, 0).

There is only one possible location for point A is at the origin.

Let's revisit the given information to determine the possible location of point A.

Point C has the same y-coordinate as point B.

This means that the y-coordinate of point C is equal to the y-coordinate of point B.

The distance between point B and point C is equal to the distance between point C and the y-axis.

Let's assume the distance between point B and point C is represented by "d".

According to the information given, the distance between point C and the y-axis is also "d".

Point A has the same x-coordinate as point C.

This implies that the x-coordinate of point A is equal to the x-coordinate of point C.

The distance between point A and point C is twice the distance between point B and point C.

Let's assume the distance between point B and point C is represented by "d".

According to the information given, the distance between point A and point C is 2d.

Based on this information, we can analyze the relationships between the points:

Since the distance between point B and point C is equal to the distance between point C and the y-axis, we can infer that point B lies on the y-axis.

The x-coordinate of point B is 0.

As point C has the same y-coordinate as point B, the y-coordinate of point C is also determined to be the same as the y-coordinate of point B.

Since point A has the same x-coordinate as point C, the x-coordinate of point A will also be 0.

The distance between point A and point C is twice the distance between point B and point C.

As the distance between point B and point C is "d", the distance between point A and point C is 2d.

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Answer:

[tex]\textsf{23)} \quad y=3\left(\dfrac{5}{3}\right)^x[/tex]

[tex]\textsf{24)} \quady=2\left(\dfrac{1}{2}\right)^x[/tex]

Step-by-step explanation:

The general formula for an exponential function is:

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]

From inspection of the given graph, the exponential curve passes through the points (0, 3) and (1, 5).

The y-intercept is the value of y when x = 0. Therefore, a = 3.

To find the value of b, substitute point (1, 5) and the found value of a into the exponential function formula:

[tex]\begin{aligned}y&=ab^x\\\\\implies 5&=3b^1\\\\5&=3b\\\\b&=\dfrac{5}{3}\end{aligned}[/tex]

To write an equation for the graphed exponential function, substitute the found values of a and b into the formula:

[tex]\boxed{y=3\left(\dfrac{5}{3}\right)^x}[/tex]

[tex]\hrulefill[/tex]

From inspection of the given graph, the exponential curve passes through points (-1, 4) and (0, 2).

The y-intercept is the value of y when x = 0. Therefore, a = 2.

To find the value of b, substitute point (-1, 4) and the found value of a into the exponential function formula:

[tex]\begin{aligned}y&=ab^x\\\\\implies 4&=2b^{-1}\\\\2&=b^{-1}\\\\2&=\dfrac{1}{b}\\\\b&=\dfrac{1}{2}\end{aligned}[/tex]

To write an equation for the graphed exponential function, substitute the found values of a and b into the formula:

[tex]\boxed{y=2\left(\dfrac{1}{2}\right)^x}[/tex]

Answer:

Step-by-step explanation:

Surface area of a cone = surface area of a circle = pi r^2

250 = pi r^2

[tex]r = \sqrt{ \frac{250}{2} } = 5 \sqrt{5} \: cm[/tex]

Because the height (h) of the cone is double the length of its radius

Then

h = 2r

[tex]h \: = 2 \times 5 \sqrt{5} = 10 \sqrt{5} = 22.36 \: cm[/tex]

Angle "a" in the given isosceles triangle is 40 degrees.

To find the size of angle "a" in the isosceles triangle with a 40-degree angle, we can use the properties of isosceles triangles. In an isosceles triangle, the two equal sides are opposite the two equal angles.

Since the given angle is 40 degrees, we know that the other two angles in the triangle are also equal. Let's call these angles "b" and "c." Therefore, we have:

b = c

Since the sum of the angles in a triangle is always 180 degrees, we can write the equation:

40 + b + c = 180

Since b = c, we can rewrite the equation as:

40 + b + b = 180

Combining like terms, we have:

2b + 40 = 180

Subtracting 40 from both sides, we get:

2b = 140

Dividing both sides by 2, we find:

b = 70

Therefore, both angles "b" and "c" are 70 degrees.

Now, we can find angle "a" by subtracting the sum of angles "b" and "c" from 180 degrees:

a = 180 - (b + c)

= 180 - (70 + 70)

= 180 - 140

= 40

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Answer:

Angle ACB = 44°

There are two ways to solve it. Both are right

Solution number 1

From triangle ABC

angle BAC = 180°-(102° +44°) = 36°

Because BG is parallel with AC

Another solution

The sum of angles in the shape AGBC = 360°

Answer:

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