exercise 1.5.2. solve ,2yy′ 1=y2 x, with .

The trigonometric substitution to find the indefinite integral is x = 8sec(θ).

Explanation:

To find the trigonometric substitution for the given integral, follow these steps:

Step 1: we first notice that the expression inside the square root can be written as a difference of squares:

x^2 - 64 = (x^2 - 8^2)

Step 2: substitute x = 8sec(θ), which leads to the following substitutions:

x^2 = 64sec^2(θ)
x^2 - 64 = 64 tan^2(θ)

And
dx = 8sec(θ)tan(θ) dθ

Step 3: With these substitutions, the given integral can be rewritten as:

∫ x^2(x^2 - 64)^3/2 dx = ∫ (64sec^2(θ))(64tan^2(θ))^3/2 (8sec(θ)tan(θ)) dθ

Step 4: Simplifying this expression, we get:

∫ 2^18sec^3(θ)tan^4(θ) dθ

Therefore, the trigonometric substitution to find the indefinite integral is x = 8sec(θ).

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Answer:

Step-by-step explanation:

  A = [tex]\left[\begin{array}{cc}4&-4\\3&-2\end{array}\right][/tex]

3B = [tex]\left[\begin{array}{cc}12&12\\0&3\end{array}\right][/tex]

4 + 12 = 16 ; 12 + ( - 4) = 8

3 + 0 = 3  ; - 2 + 3 = 1

A + 3B = [tex]\left[\begin{array}{cc}16&8\\3&1\end{array}\right][/tex]

[tex](A+3B)^{-1}[/tex] = [tex]\left[\begin{array}{cc}-\frac{1}{8} &1\\\frac{3}{8} &-2\end{array}\right][/tex]

X = C ÷ ( A + 3B ) = C × [tex](A+3B)^{-1}[/tex]

X = [tex]\left[\begin{array}{cc}-1&0\\5&2\end{array}\right][/tex] × [tex]\left[\begin{array}{cc}-\frac{1}{8} &1\\\frac{3}{8} &-2\end{array}\right][/tex] = [tex]\left[\begin{array}{cc}\frac{1}{8} &-1\\\frac{1}{8} &1\end{array}\right][/tex]  

the final price will stay as $52

The P-value for a one-tailed z-test with Ha: p > 0.4 and z = 1.49 is 0.0675, indicating insufficient evidence to reject the null hypothesis at the 0.05 level of significance.

To find the P-value for a z-test with Ha: p > 0.4 and z = 1.49, we first calculate the corresponding area under the standard normal distribution curve.

Using a standard normal table or a calculator, we find that the area to the right of z = 1.49 is 0.0675.

Since the alternative hypothesis is one-tailed, the P-value is equal to the area in the tail to the right of z = 1.49.

Therefore, the P-value for this test is 0.0675 or 6.75% (rounded to four decimal places).

This means that if the null hypothesis is true, there is a 6.75% chance of observing a sample proportion as extreme as or more extreme than the one we obtained.

Since the P-value (6.75%) is greater than the significance level (α), we fail to reject the null hypothesis at the α = 0.05 level of significance. We do not have sufficient

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ANOVA is used when you have quantitative DV and IV with 3 or more levels, which means the correct answer is option A. True.


The One Way Repeated Measures ANOVA is a statistical test used to analyze the effects of an independent variable (IV) that has three or more levels on a dependent variable (DV) that is measured repeatedly on the same subjects over time. This test is appropriate when the IV is within-subjects in nature, meaning that each participant is exposed to all levels of the IV. Therefore, the statement is true as it accurately describes the use of this statistical test in relation to the IV and DV.
A. True

The One-Way Repeated Measures ANOVA is indeed used when you have a quantitative Dependent Variable (DV) and an Independent Variable (IV) with three or more levels that is within subjects in nature. In this case, the same subjects are exposed to different conditions or levels of the IV, allowing for the analysis of differences in the DV across those conditions.

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Answer:

7.81 miles

Step-by-step explanation:

pythagorean theorem, 6 units downwards, and 5 east, so we have to calculate the hypotenuse, or sqrt( 6^2 + 5^2) which is sqrt61 or 7.81 miles

When dealing with exponential functions given by y = (a + c)^x, where the constant 'c' is used to achieve horizontal shifts, there are particular effects on the domain, range, and asymptotes

The function's output values, or range, persist unchanged since it can assume any positive value for input from the vertical axis. Similarly, factorizing by adding constants does not impact the function's input values, otherwise known as the domain.

While horizontally shifting the exponentially-decreasing function, its horizontal asymptote remains unaffected; however, the positional shift depends on the magnitude and direction of said diasporic events. Equivalently, rightward shifts append positively and leftward motions take away from the aforementioned translation distance.

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The radius of the circle is 17.78 cm.

The formula for calculating the perimeter of a circular sector with angle θ is given by

P = 2rθ

r = P / (2θ)

Substituting in the given values, we have:

r = 64 / (2 x 1.8)

r = 17.78

Therefore, the radius of the circle is 17.78 cm.

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The missing sides and angles of the triangle are

1. . A = 45 degrees.

2. B = 45 degrees.

3. BC = 3 and AB = 3 sqrt (2).

4. BC = 4 and AB = 4 sqrt (2).

5. BC = 9, then AB = 9 sqrt (2).

6. AB = 7V2, then BC = 7 .

7. If AB = 2√2, then AC = 2.​

An Isosceles Right Triangle is an angular design in the shape of a right triangle comprising two equal sides - forming congruent legs, and additionally, the third side (also known as the hypotenuse = c) being longer in length.

In this particular angle, the two legs are congruent to each other as well as proportional to the square root of two times one leg's length.

Mathematically, using Pythagoras' theorem

c^2 = a^2 + a^2

c^2 = 2a^2

Eventually, by taking the square root of both expressions, we obtain:

c = sqrt (2a^2)

c = a * sqrt (2)

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B) 30 - 75x

This is because of the distributive property.

In the expression 3(10-25x) you have to multiply 3 by the numbers inside.

3 x 10 = 30

3 x 25x = 75x

Then, we keep the subtraction sign. The final answer is 30 - 75x.

The average rate of change is increasing over this interval.

To find the average rate of change of the function f(x) = x^2 + 3 from x = 0 to x = 4, we can use the formula:

average rate of change = [f(4) - f(0)] / [4 - 0]

Substituting the values of x = 0 and x = 4 into the function f(x), we get:

f(0) = 0^2 + 3 = 3

f(4) = 4^2 + 3 = 19

So, the average rate of change of the function from x = 0 to x = 4 is:

average rate of change = [f(4) - f(0)] / [4 - 0] = (19 - 3) / 4 = 4

This means that the function increases at an average rate of 4 units per unit change in x from x = 0 to x = 4.

Since the average rate of change is a constant value, the function f(x) = x^2 + 3 has a constant rate of increase from x = 0 to x = 4.

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The statement "If f(x)≤g(x) and [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] diverges, then [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex]

also diverges" is true.

If f(x)≤g(x) for all x and [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] diverges, then we can conclude that

[tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] also diverges.

To see why, consider the integral [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex]. Since f(x) ≤ g(x) for all x,

we have:

[tex]\int\limits^{infinity}_0 {f(x)} \, dx[/tex] ≤ [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex]

If [tex]\int\limits^{infinity}_0 {g(x)} \, dx[/tex] diverges, then the integral on the right-hand side is

infinite. Since [tex]\int\limits^{infinity}_0 {f(x)} \, dx[/tex] is less than or equal to an infinite integral, it

must also be infinite. Therefore, [tex]\int\limits^{infinity}_0 {f(x)} \, dx[/tex] also diverges.

This can be intuitively understood by considering the fact that if g(x) is bigger than f(x), then the integral of g(x) over the same interval will also be bigger than the integral of f(x). Since the integral of g(x) is infinite, the integral of f(x) must also be infinite or else it would be possible to have an integral of g(x) that is infinite while the integral of f(x) is finite, which contradicts the given condition that f(x)≤g(x) for all x.

Therefore, the statement is true.

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The line of discontinuity is x = 0 or y = 0.

We have,

To identify the line of discontinuity in the function f(x, y) = ln|x y|, we need to determine the values of x and y for which the function becomes undefined or exhibits a discontinuity.

In this case, the natural logarithm function, ln, is undefined for non-positive values.

Therefore, we need to find the values of x and y that make the expression |x y| non-positive.

The absolute value of a real number is non-positive when the number itself is zero or negative.

So, we set the expression inside the absolute value, x y, to be zero or negative:

x y ≤ 0

This inequality indicates that either x ≤ 0 and y ≥ 0, or x ≥ 0 and y ≤ 0, for the expression to be non-positive.

Hence, the line of discontinuity occurs along the line where either x ≤ 0 and y ≥ 0, or x ≥ 0 and y ≤ 0.

The equation of this line can be written as:

x ≤ 0, y ≥ 0 or x ≥ 0, y ≤ 0

This line divides the plane into two regions:

one where x ≤ 0 and y ≥ 0, and the other where x ≥ 0 and y ≤ 0.

Along this line, the function f(x, y) = ln|x y| becomes undefined or discontinuous.

Note that when x = 0 or y = 0, the function f(x, y) = ln|x y| is also undefined, but these points do not form a continuous line.

Thus,

The line of discontinuity is x = 0 or y = 0.

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For the given equation there are 3 real solutions they are -4/3, -3, 1 , under the condition that the given equation is  x² = 1/(x+3)

The equation x²= 1/(x+3) can be restructured as
x³ + 3x² - 1 = 0.
This is a cubic equation and could be evaluated applying the cubic formula. Then, we can also apply the rational root theorem to search the rational roots of the equation.
The rational root theorem projects that if a polynomial equation has integer coefficients, then any rational root of the equation should be of the form p/q
Here,
p = factor of the constant term and q is a factor of the leading coefficient.
For the given case,
the constant term is -1 and the leading coefficient is 1.
Hence, any rational root of the equation should be of the form p/q
Here, p is a factor of -1 and q is a factor of 1.
The possible rational roots are ±1 and ±1/3.
Applying the principle of testing these values, we evaluate that
x = -1/3 is a root of the equation.
Then, we can factorize
x³ + 3x² - 1 as (x + 1/3)(x² + 2x - 3).
The quadratic factor can be simplified further as
(x + 3)(x - 1),
Then, the solutions to the original equation are
x = -4/3, x = -3, and x = 1.
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The value of the function is dy/dx = f(x) = 6x²-10x-14

Differentiation is an element of personalized learning which involves changing the instructional approach to meet the diverse needs of students. It can involve designing and delivering instruction using an assortment of teaching styles and giving students options for taking in information and making sense of ideas.

the given function f(x) 2x³ -5x² -14x + 8

F(x) =dy/dx = 2*3(x)³⁻¹ -5*2(x²⁻¹) -14(x¹⁻¹)

Therefore the derivative of the function is f(x) = 6x²-10x-14

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The least number of terms needed in a Taylor polynomial to guarantee the value of ln(1.08) has an accuracy of 10⁻¹⁰ is 30. Option a is correct.

The Taylor series expansion of ln(1+x) is given by:

For ln(1.08), we have x = 0.08. Therefore, the nth term of the series is given by:

To guarantee the accuracy of ln(1.08) to 10⁻¹⁰, we need to ensure that the absolute value of the remainder term (i.e., the difference between the actual value and the value obtained using the Taylor polynomial approximation) is less than 10⁻¹⁰.

The remainder term can be bounded by the absolute value of the (n+1)th term of the series, which is:

Therefore, we need to find the smallest value of n such that:

Solving this inequality numerically, we get n > 29.82. Therefore, we need at least 30 terms in the Taylor polynomial to guarantee the accuracy of ln(1.08) to 10⁻¹⁰. Hence Option a is correct.

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The complete question is:

Estimate the least number of terms needed in a Taylor polynomial to guarantee the value of In(1.08)has accuracy of 10⁻¹⁰.

The probability that a randomly selected person will have an IQ score of less than 80 is approximately 0.0918, or 9.18%

To find the probability that a randomly selected person will have an IQ score of less than 80, we need to consider the properties of the normal distribution, as IQ scores typically follow a normal distribution with a mean (μ) of 100 and a standard deviation (σ) of 15.

1. Calculate the z-score: The z-score represents the number of standard deviations a data point is from the mean. Use the formula:

z = (X - μ) / σ

where X is the IQ score, μ is the mean, and σ is the standard deviation.

z = (80 - 100) / 15
z = -20 / 15
z = -1.3333

2. Look up the z-score in a standard normal distribution table or use a calculator to find the corresponding probability. In this case, the probability is 0.0918.

Therefore, the probability that a randomly selected person will have an IQ score of less than 80 is approximately 0.0918, or 9.18% when rounded to four decimal places.

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If the function f(2)=25 and f' (2) = -2.5, then f(2.5) is approximately 23.75

The first-order Taylor's approximation formula, also known as the linear approximation formula, is a mathematical formula that provides an approximate value of a differentiable function f(x) near a point a. The formula is given as

f(x) ≈ f(a) + f'(a)(x - a)

where f'(a) is the derivative of f(x) at the point a. This formula is based on the tangent line to the graph of f(x) at the point (a, f(a)). The approximation becomes more accurate as x gets closer to a.

We can use the first-order Taylor's approximation formula to estimate the value of f(2.5) based on the information given

f(x) ≈ f(a) + f'(a)(x - a)

where a = 2 and x = 2.5. Plugging in the values, we get

f(2.5) ≈ f(2) + f'(2)(2.5 - 2)

f(2.5) ≈ 25 + (-2.5)(0.5)

f(2.5) ≈ 23.75

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The requried value of x between lines m and n is 59.5.

When two lines intersect at a point, they form four angles around the intersection point. The pairs of opposite angles and sides are similar, meaning they have the proportionate measure.

As shown in the figure,
lines l,m, and n are parallel and cut by two transversal lines,
following the property of proportion of transversal line on a parallel line,
12/51 = 14/x

Simplifying the above expression,
x = 51 * [14/12]
x = 59.5

Thus, the requried value of x between lines m and n is 59.5.

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Answer: no

Step-by-step explanation: if we'd substitute the numbers, it'd look like this 10≤-10+7  which isn't true  as "≤" this symbol means more than or equals to but -10 plus 7 is equal to 3 so it doesn't fit the inequality

Step-by-step explanation:

a quadratic equation has 2 zeros.

luckily we got 2 points with y = 0, so these define the zero points.

a quadratic function is usually looking like

ax² + bx + c = 0

and with the zeros being the factors, we get

y = a(x - z1)(x - z2) = a(x + 4)(x - 2) =

= a(x² - 2x + 4x - 8) = a(x² + 2x - 8)

to get "a" we use the third point.

-2 = a((-2)² + 2×-2 - 8) = a(4 - 4 - 8) = -8a

a = -2/-8 = 1/4

and the equation is

y = (1/4)x² + (1/2)x - 8/4 = (1/4)x² + (1/2)x - 2

When each point on a plane of a two-dimensional coordinate system is decided by a distance from a reference point and an angle is taken from a reference direction, it is known as the polar coordinate system.

Polar coordinates are advantageous when the region being integrated over has a circular or symmetric shape. This is because polar coordinates use angles and radii to describe points in a two-dimensional plane, which aligns well with circular and symmetric shapes. Additionally, polar coordinates can simplify the integrand, as some functions are more easily expressed in terms of angles and radii rather than Cartesian coordinates.

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To graph the quadratic function y=-2x^2 using the given values of x, one can create a table with two columns: one for x and the other for y. Starting with x=-2, we can substitute this value into the equation to find the corresponding value of y, which is y=-8. Similarly, by substituting -1, 0, 1, and 2 into the equation, we can find corresponding values of y as 2, 0, -2, and -8, respectively. By plotting these points on a graph and connecting them, we get a downward facing parabola with its vertex at (0,0).

The sequence converges to: lim n→[infinity] an = ln(3) = 1.0986. So the sequence converges to 1.0986.

To determine whether the sequence converges or diverges and find the limit, we'll use the properties of logarithms and the concept of limits at infinity.

Given sequence: a_n = ln(3n² + 4) - ln(n² + 4)

Using the logarithm property, ln(a) - ln(b) = ln(a/b), we can rewrite the sequence as:

a_n = ln[(3n² + 4)/(n² + 4)]

Now, we'll find the limit as n approaches infinity:

lim (n→∞) a_n = lim (n→∞) ln[(3n² + 4)/(n² + 4)]

To evaluate this limit, we can divide both the numerator and the denominator by the highest power of n, which is n^2 in this case:

lim (n→∞) ln[(3 + 4/n²)/(1 + 4/n²)]

As n approaches infinity, the terms with n² in the denominator will approach 0:

lim (n→∞) ln[(3 + 0)/(1 + 0)] = ln(3)

So, the sequence converges, and the limit is ln(3).

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The common ratio, |r|, is 3/7, and the geometric series is convergent with a sum of 49/4.

The given geometric series is Σ(−3)ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity. To find the common ratio, |r|, let's simplify the series.



1. Rewrite the series: Σ(−3ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity.
2. Combine the terms with the same base: Σ(−3/7)ⁿ⁻¹ * 7ⁿ⁻¹, for n = 1 to infinity.
3. Now, the common ratio, |r| = |-3/7| = 3/7.

Since |r| < 1, the geometric series is convergent.

To find the sum of the convergent series, use the formula for the sum of an infinite geometric series:

S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

4. Find the first term (n=1): a = (−3)¹⁻¹ * 7^1 = 1 * 7 = 7.
5. Use the formula: S = 7 / (1 - (3/7)) = 7 / (4/7) = 7 * (7/4) = 49/4.

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Complete question:

consider the following geometric series. [infinity] Σ(−3)ⁿ⁻¹ * 7ⁿ = 1 Find the common ratio. |r| = Determine whether the geometric series is convergent or divergent. convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

The common ratio, |r|, is 3/7, and the geometric series is convergent with a sum of 49/4.

The given geometric series is Σ(−3)ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity. To find the common ratio, |r|, let's simplify the series.



1. Rewrite the series: Σ(−3ⁿ⁻¹ * 7ⁿ, for n = 1 to infinity.
2. Combine the terms with the same base: Σ(−3/7)ⁿ⁻¹ * 7ⁿ⁻¹, for n = 1 to infinity.
3. Now, the common ratio, |r| = |-3/7| = 3/7.

Since |r| < 1, the geometric series is convergent.

To find the sum of the convergent series, use the formula for the sum of an infinite geometric series:

S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

4. Find the first term (n=1): a = (−3)¹⁻¹ * 7^1 = 1 * 7 = 7.
5. Use the formula: S = 7 / (1 - (3/7)) = 7 / (4/7) = 7 * (7/4) = 49/4.

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Complete question:

consider the following geometric series. [infinity] Σ(−3)ⁿ⁻¹ * 7ⁿ = 1 Find the common ratio. |r| = Determine whether the geometric series is convergent or divergent. convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Answer:

y(x)=7x^2+56x+115

y(x)=7(x^2+8x+115/7) ( Factor out )

y(x)=7(x^2+8x+(4)^2-1(4)^2+115/7) ( Complete the square )

y(x)=7((x+4)^2-1(4)^2+115/7) ( Use the binomial formula )

y(x)=7((x+4)^2+3/7) ( simplify )

y(x)=7*(x+4)^2+3 done!

Step-by-step explanation:

hope helps:)

Answer:

log₂ (4x)= log₂4 + log₂x

Step-by-step explanation:

log₂ (4x)= log₂4 + log₂x illustrates the product rule for logarithmic equations.

The product rule states that logb (mn) = logb m + logb n. In this case, b is 2, m is 4, and n is x. So,

log₂ (4x) = log₂ 4 + log₂ x.

Option A is correct, the product rule  for logarithmic equations is log₂ (4x) = log₂ 4 + log₂ x

Two or more expressions with an Equal sign is called as Equation.

The logarithm is the inverse function to exponentiation.

The product rule for logarithmic equations states that the logarithm of a product of two numbers is equal to the sum of the logarithms of the individual numbers.

logab=loga + logb

log₂ (4x) = log₂ 4 + log₂ x

Therefore, the correct illustration of the product rule  for logarithmic equations is log₂ (4x) = log₂ 4 + log₂ x

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The average cost per month of owning an iPhone decreases as the number of months of ownership increases. After 8 months, the average cost per month is $145.

Assuming a constant monthly service fee of $70, the total cost (C) of owning an iPhone for n months can be calculated as:

C = 600 + 70n

where n is the number of months of ownership.

Using this formula, we can calculate the total cost of owning an iPhone after:

i. 2 months:

C = 600 + 70(2) = 740

ii. 4 months:

C = 600 + 70(4) = 880

iii. 6 months:

C = 600 + 70(6) = 1020

iv. 8 months:

C = 600 + 70(8) = 1160

To find the average cost per month, we can divide the total cost by the number of months:

i. Average cost per month after 2 months: 740 / 2 = 370

ii. Average cost per month after 4 months: 880 / 4 = 220

iii. Average cost per month after 6 months: 1020 / 6 = 170

iv. Average cost per month after 8 months: 1160 / 8 = 145

Therefore, the average cost per month of owning an iPhone decreases as the number of months of ownership increases. After 8 months, the average cost per month is $145.

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Answer: 1/63

Step-by-step explanation:

1/9 ÷ 7 can be rewritten as 1/9 x 1/7

= 1/63

Answer:

To divide a fraction by a whole number, we can flip the whole number upside down and multiply. So, 1/9 ÷ 7 is the same as 1/9 * (1/7).

To multiply fractions, we multiply the numerators and the denominators. So, 1/9 * (1/7) = (1 * 1) / (9 * 7) = 1/63.

Therefore, 1/9 ÷ 7 = 1/63.

Step-by-step explanation:

The coordinates of N by the 270 degree rotation clockwise rule is (-7, 3)

From the question, we have the following parameters that can be used in our computation:

N = (-3, 7)

The transfomation rule is given as

270 degree rotation rule clockwise

Mathematically, this is represented as

(x, y) = (-y, x)

Substitute the known values in the above equation, so, we have the following representation

N' = (-7, 3)

Hence, the coordinates of N after the rotation is (-7, 3)

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