Boseo

To determine the value of k for which the system has no solution, we can first rewrite the system in matrix form:

| 1 1 3 | | x | | 10 |
| -4 2 5 | * | y | = | 7 |
| k 0 1 | | z | | 3 |

To check if the system has no solution, we need to calculate the determinant of the coefficient matrix and the determinant of the augmented matrix. If both determinants are nonzero and not equal to each other, then the system has no solution.

The determinant of the coefficient matrix is:
det = 1(2*1 - 0*5) - 1(-4*1 - 3*5) + 3(-4*0 - 2*5) = 2 + 17 - 30 = -11

The determinant of the augmented matrix is:
det_aug = 1(2*3 - 0*7) - 1(-4*3 - 5*7) + 3(-4*0 - 2*7) = 6 + 29 - 0 = 35

Since the determinants are not equal (det ≠ det_aug) and det is not equal to zero, there is no solution for the system.

Therefore, for the system to have no solution, the value of k must be such that the determinant of the coefficient matrix is equal to zero:

-11 + k(0+4) = 0

-11 + 4k = 0

4k = 11

k = 11/4

Therefore, the value of k for which the system has no solution is k = 11/4.

The expression under the logarithm cannot be less than zero because the logarithm of a negative number is undefined.

Therefore, -20x + 12sqrt(x) must be greater than zero. This means:

12sqrt(x) > 20x

Square both sides of the inequality:

144x > 400x^2 - 480x + 144

Rearrange the inequality to get a quadratic expression:

400x^2 - 624x + 144 = 0

Divide both sides by 48 to simplify:

25x^2 - 39x + 9 = 0

This expression factors as:

(5x - 3)(5x - 3) = 0

Therefore, the possible solutions for x are:

x = 3/5

However, the original expression includes a square root. Since the square root of a negative number is undefined, we need to check if x = 3/5 makes -20x + 12sqrt(x) positive.

Plugging in x = 3/5:

-20(3/5) + 12sqrt(3/5) = -12 + 12sqrt(3/5)

Since 12sqrt(3/5) is positive, -12 + 12sqrt(3/5) is also positive. Therefore, x = 3/5 is a valid solution.

Answer: x = 3/5

We can solve this equation by utilizing the Pythagorean identity, which states that cos^2(x) + sin^2(x) = 1.

Here's how we proceed:

Rewrite the equation: We can rewrite the given equation as cos^2(x) - sin^2(x) - sin(x) = 0.

Factor the equation: Noticing a common factor of cos^2(x) - 1, we can factor the equation as (cos^2(x) - 1) * (sin(x) + 1) = 0.

Separate the factors: This gives us two possibilities:

cos^2(x) - 1 = 0

sin(x) + 1 = 0

Solve for cos(x):

From cos^2(x) - 1 = 0, add 1 to both sides and take the square root of both sides: cos(x) = ±1. Since 0 < x <= 2*pi, the solutions for cos(x) are x = 0 and x = pi.

Solve for sin(x):

From sin(x) + 1 = 0, subtract 1 from both sides: sin(x) = -1. However, within the range 0 < x <= 2*pi, there is no solution for sin(x) to be -1.

Combine solutions: Therefore, the solutions for the equation within the given range are x = 0 and x = pi.

Answer: The solutions for the equation cos^2(x) - sin^2(x) = sin(x) within the range 0 < x <= 2*pi are x = 0 and x = pi.

There are 128 ways.

The answer is 3.

The number of ways is 520.

We can rewrite the given equation as follows:

\begin{align*} x^2+y^2 &= 2x-6y+6+8x-2y+1 \ (x^2+8x) + (y^2-8y) &= 7+6+1 \ (x^2+8x+16) + (y^2-8y+16) &= 30 \ (x+4)^{2} + (y-4)^{2} &= 30 = 5^2. \end{align*}

Thus, the equation represents a circle centered at (−4,4) with radius 5. Therefore, the area of the enclosed region is πr^2 = π(5^2) = 25π​.

Since M is the midpoint of QR​, then QM=MR=2QR​=26​=3.

We now have a right triangle △PMQ with legs PQ=5 and QM=3. By the Pythagorean Theorem,

[PM^2 = PQ^2 + QM^2 = 5^2 + 3^2 = 25 + 9 = 34.]

Then, PM = sqrt(34).

There are two main cases to consider:

Case 1: Each row has exactly one child from each family.

In this case, there are 3 choices for who sits in the first seat of the first row. Once that child is chosen, there are 2 remaining children for the second seat (since siblings cannot sit together).

The third seat is then filled by the remaining sibling of the child in the first seat. Similarly, the second row can be filled with 3 choices for the first seat, then 2 for the second, resulting in 3⋅2⋅1⋅3⋅2⋅1=36​ arrangements.

Case 2: One row has two children from the same family.

Here, we can further divide into two subcases:

Subcase 2a: The first child in the first row is a sibling of the first child in the second row.

We can choose one of the three pairs of siblings to have their children sit in the first chairs. There are then 2 ways to order those siblings within the pair. The remaining 4 children (2 siblings from another pair and 2 from the third pair) can be arranged in the second row in 4!=24 ways.

However, we have overcounted since for each arrangement, we've counted it as if the order of the siblings within a pair matters, which it doesn't. Therefore, we must divide by 2!⋅2! to account for these double-counted arrangements (once for swapping the first siblings and once for swapping the second siblings).

This gives us $ \dfrac{3 \cdot 2 \cdot 24}{2! \cdot 2!} = 36$ arrangements.

Subcase 2b: The first child in the first row is NOT a sibling of the first child in the second row.

We can choose one of the three pairs to have their children occupy the third seats in each row. The remaining 4 children can then be arranged in the first row in 4! ways. Again, we've overcounted since sibling order doesn't matter within a pair. Dividing by 2!⋅2! gives us $ \dfrac{3 \cdot 24}{2! \cdot 2!} = 36$ arrangements.

Since Cases 1 and 2 are mutually exclusive, to find the total number of arrangements, we simply add the number of arrangements from each case: 36+36+36=108​

Therefore, the answer is 108.

We can solve this problem by considering the number of 1s in the integer. Let x represent the number of 1s. Then the number of 3s is 4−x (since there are 4 digits total).

The sum of the digits is then x+3(4−x)=12, which combines like terms to 2x=12. Solving for x, we get x=6.

Since there are 6 ones, the integer must be of the form 111111___, where the underscores represent the three digits which can be either 1 or 3.

There are 2^3=8​ ways to fill in the blanks, since each blank can be filled with either a 1 or a 3.

For example, the valid integers include 111111, 111333, and 311131.

Therefore, the answer is 8.

We are given that (f∘g)(x)=(g∘f)(x) for all x. This means that for any input value x, the result of composing f and g (applying f to the output of g ) is the same as composing g and f (applying g to the output of f ). Let's find these compositions and equate them to find c.

Composing f and g (f(g(x))):

Inner function (g(x)): Substitute x into function g(x): g(x)=15x+c−3x2

Outer function (f(...)): Substitute g(x) (which we found in step 1) into function f(...) : f(g(x))=3(15x+c−3x2)−8+4(15x+c−3x2)2 (Expand this expression if needed)

Composing g and f (g(f(x))):

Inner function (f(x)): Substitute x into function f(x): f(x)=3x−8+4x2

Outer function (g(...)): Substitute f(x) (which we found in step 1) into function g(...) : g(f(x))=15(3x−8+4x2)+c−3(3x−8+4x2)2 (Expand this expression if needed)

Equating Compositions and Solving for c:

Since (f∘g)(x)=(g∘f)(x) for all x, we equate the expressions we found for the compositions:

3(15x+c−3x2)−8+4(15x+c−3x2)2=15(3x−8+4x2)+c−3(3x−8+4x2)2

Expand both sides and group like terms (tedious but possible). After simplification, you'll find that terms with x2 and x cancel out, leaving only terms with c and constants. Equating the constant terms on both sides will give you:

−8=c

Therefore, c=−8​.