💡Okay, so I have this problem about a 5x5 calendar block of dates. It starts with 5 and increases by 1 each day horizontally. The task is to reverse the order of the numbers in the third and fifth rows, then calculate the sums of the two diagonals and find the positive difference between those sums. The answer choices are given from A to E, with numbers 6, 8, 10, 12, and 14.First, I need to visualize the original 5x5 grid. Since it starts with 5 and increases by 1 each day horizontally, the first row will be 5, 6, 7, 8, 9. The second row will start with 10, so it will be 10, 11, 12, 13, 14. The third row starts with 15, so 15, 16, 17, 18, 19. The fourth row is 20, 21, 22, 23, 24, and the fifth row is 25, 26, 27, 28, 29.So, the original grid looks like this:5 6 7 8 910 11 12 13 1415 16 17 18 1920 21 22 23 2425 26 27 28 29Now, the first step is to reverse the order of the numbers in the third and fifth rows. Let me do that.Reversing the third row: 15, 16, 17, 18, 19 becomes 19, 18, 17, 16, 15.Reversing the fifth row: 25, 26, 27, 28, 29 becomes 29, 28, 27, 26, 25.So, the modified grid after reversing the third and fifth rows is:5 6 7 8 910 11 12 13 1419 18 17 16 1520 21 22 23 2429 28 27 26 25Alright, now I need to calculate the sums of the two diagonals. Let me recall that in a matrix, the main diagonal runs from the top-left to the bottom-right, and the secondary diagonal runs from the top-right to the bottom-left.So, for the main diagonal, the elements are:First row, first column: 5Second row, second column: 11Third row, third column: 17Fourth row, fourth column: 23Fifth row, fifth column: 25So, adding these up: 5 + 11 + 17 + 23 + 25.Let me compute that step by step:5 + 11 = 1616 + 17 = 3333 + 23 = 5656 + 25 = 81So, the main diagonal sum is 81.Now, for the secondary diagonal, which runs from the top-right to the bottom-left. The elements are:First row, fifth column: 9Second row, fourth column: 13Third row, third column: 17Fourth row, second column: 21Fifth row, first column: 29Adding these up: 9 + 13 + 17 + 21 + 29.Let me compute that step by step:9 + 13 = 2222 + 17 = 3939 + 21 = 6060 + 29 = 89So, the secondary diagonal sum is 89.Now, the problem asks for the positive difference between the two diagonal sums. That means I need to subtract the smaller sum from the larger sum and take the absolute value, but since both sums are positive, just subtracting the smaller from the larger will give the positive difference.So, the two sums are 81 and 89.89 - 81 = 8Therefore, the positive difference is 8.Looking back at the answer choices: A)6, B)8, C)10, D)12, E)14.So, 8 is option B.Wait, let me double-check my calculations to make sure I didn't make any mistakes.Main diagonal: 5, 11, 17, 23, 25.5 + 11 is 16, plus 17 is 33, plus 23 is 56, plus 25 is 81. That seems correct.Secondary diagonal: 9, 13, 17, 21, 29.9 + 13 is 22, plus 17 is 39, plus 21 is 60, plus 29 is 89. That also seems correct.Difference: 89 - 81 = 8. So, yes, 8 is the correct answer, which is option B.I think that's solid. I don't see any errors in my reasoning or calculations.

💡Okay, so I have this problem here: Given that 180 degrees is less than alpha is less than 360 degrees, I need to find the value of the square root of (1 plus cosine alpha) and choose the correct option from A to D. The options are all expressions involving square roots and either cosine or sine of alpha over two, with some negative signs. First, I remember that there are some trigonometric identities that relate expressions like 1 plus cosine alpha to other trigonometric functions. Specifically, I think there's a double-angle identity that could help here. Let me recall: I believe it's something like cosine of alpha equals 2 cosine squared of alpha over two minus one. So, if I rewrite that, it would be cos(alpha) = 2 cos²(alpha/2) - 1. So, if I substitute that into the expression 1 + cos(alpha), it should become 1 + (2 cos²(alpha/2) - 1). Let me compute that: 1 plus 2 cos squared alpha over two minus one. The 1 and minus 1 cancel out, so I'm left with 2 cos squared alpha over two. So, 1 + cos(alpha) simplifies to 2 cos²(alpha/2). Now, the original expression is the square root of (1 + cos(alpha)), which is the square root of 2 cos squared alpha over two. The square root of a square is the absolute value, so this becomes the square root of 2 times the absolute value of cos(alpha/2). So, sqrt(1 + cos(alpha)) = sqrt(2) * |cos(alpha/2)|.But wait, I need to consider the sign of cos(alpha/2) because of the absolute value. The problem states that alpha is between 180 degrees and 360 degrees. So, if I divide that by two, alpha over two is between 90 degrees and 180 degrees. That means alpha over two is in the second quadrant. In the second quadrant, cosine is negative. So, cos(alpha/2) is negative. Therefore, the absolute value of cos(alpha/2) is equal to negative cos(alpha/2). So, putting it all together, sqrt(1 + cos(alpha)) equals sqrt(2) times negative cos(alpha/2). That simplifies to negative sqrt(2) times cos(alpha/2). Looking back at the options, option A is negative sqrt(2) cos(alpha/2), which matches what I just found. Option B is positive sqrt(2) cos(alpha/2), which would be incorrect because we determined that cos(alpha/2) is negative in this range. Option C is negative sqrt(2) sin(alpha/2), which doesn't match because we have cosine, not sine. Option D is positive sqrt(2) sin(alpha/2), which also doesn't match for the same reason. So, after going through the steps, I think the correct answer is option A: negative sqrt(2) cos(alpha/2). I should double-check my steps to make sure I didn't make any mistakes. Starting from the identity, substituting, simplifying, considering the quadrant for alpha over two, and determining the sign of cosine. Everything seems to add up. I don't see any errors in my reasoning, so I'm confident that option A is correct.

💡First, I need to determine how much fresh water the family uses in a day. Since each person requires 1/2 gallon of fresh water per day, I'll multiply the number of people by 1/2 to find the total daily water consumption.Next, I'll calculate the cost to purify this amount of water. Given that it costs 1 to purify one gallon of water, I'll multiply the total gallons of water used by 1 to find the total purification cost.Finally, I'll set up an equation where the total purification cost equals 3 and solve for the number of people in the family.

💡First, I recognize that the circumference of a circle is related to its diameter through the formula ( C = pi D ).Given that the circumference ( C ) is 36 cm, I can set up the equation:[ 36 = pi D ]To solve for the diameter ( D ), I divide both sides of the equation by ( pi ):[ D = frac{36}{pi} ]Therefore, the diameter of the circle is ( frac{36}{pi} ) centimeters.

💡Okay, so I have this problem where I need to find a matrix M such that when I multiply M by another given matrix, the result is the identity matrix I. The given matrix is a 3x3 matrix with some trigonometric functions and a 2 in the bottom right corner. The angle θ is given as π/4, which is 45 degrees. First, I should write down the given matrix to have a clear view. The matrix is:[begin{pmatrix}cos theta & -sin theta & 0 sin theta & cos theta & 0 0 & 0 & 2end{pmatrix}]Since θ is π/4, I can substitute the values of cos(π/4) and sin(π/4). I remember that cos(π/4) and sin(π/4) are both √2/2. So, substituting these values, the matrix becomes:[begin{pmatrix}frac{sqrt{2}}{2} & -frac{sqrt{2}}{2} & 0 frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & 2end{pmatrix}]Now, the problem states that M multiplied by this matrix equals the identity matrix I. So, mathematically, that is:[mathbf{M} times begin{pmatrix}frac{sqrt{2}}{2} & -frac{sqrt{2}}{2} & 0 frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & 2end{pmatrix} = mathbf{I}]I need to find matrix M. Since matrix multiplication is involved, I recall that if the product of two matrices is the identity matrix, then one matrix is the inverse of the other. So, M must be the inverse of the given matrix. Therefore, M = (given matrix)^{-1}.To find the inverse of a matrix, I can use the formula for the inverse of a 3x3 matrix, but that might be complicated. Alternatively, since the given matrix is block diagonal, meaning it can be divided into smaller blocks along the diagonal, I can find the inverse by inverting each block separately.Looking at the given matrix, it's composed of a 2x2 rotation matrix in the top-left corner and a 1x1 matrix [2] in the bottom-right corner. So, the inverse of the entire matrix will be the inverse of the 2x2 block combined with the inverse of the 1x1 block.First, let's find the inverse of the 2x2 rotation matrix:[begin{pmatrix}frac{sqrt{2}}{2} & -frac{sqrt{2}}{2} frac{sqrt{2}}{2} & frac{sqrt{2}}{2}end{pmatrix}]I remember that the inverse of a rotation matrix is its transpose because rotation matrices are orthogonal. The transpose of a matrix is obtained by flipping it over its diagonal, switching the row and column indices of each element. So, transposing the above matrix gives:[begin{pmatrix}frac{sqrt{2}}{2} & frac{sqrt{2}}{2} -frac{sqrt{2}}{2} & frac{sqrt{2}}{2}end{pmatrix}]Let me verify if this is indeed the inverse. Multiplying the original matrix by its transpose should give the identity matrix.Multiplying the original 2x2 matrix by its transpose:First row, first column: (√2/2)(√2/2) + (-√2/2)(-√2/2) = (2/4) + (2/4) = 1First row, second column: (√2/2)(√2/2) + (-√2/2)(√2/2) = (2/4) - (2/4) = 0Second row, first column: (√2/2)(√2/2) + (√2/2)(-√2/2) = (2/4) - (2/4) = 0Second row, second column: (√2/2)(√2/2) + (√2/2)(√2/2) = (2/4) + (2/4) = 1So, yes, the product is the identity matrix. Therefore, the inverse of the 2x2 block is indeed its transpose.Next, the 1x1 block is [2]. The inverse of a 1x1 matrix [a] is [1/a], so the inverse of [2] is [1/2].Putting it all together, the inverse of the entire 3x3 matrix is:[begin{pmatrix}frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 -frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & frac{1}{2}end{pmatrix}]Wait, hold on. Let me double-check. The inverse of the 2x2 block is the transpose, which we found to be:[begin{pmatrix}frac{sqrt{2}}{2} & frac{sqrt{2}}{2} -frac{sqrt{2}}{2} & frac{sqrt{2}}{2}end{pmatrix}]So, placing this in the top-left corner and the inverse of the 1x1 block in the bottom-right corner, the full inverse matrix M is:[begin{pmatrix}frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 -frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & frac{1}{2}end{pmatrix}]But wait, in the original problem, the multiplication is M times the given matrix equals I. So, is M the inverse on the left side or the right side? Because matrix multiplication is not commutative, so the inverse could be on the left or right.In this case, since M is multiplied on the left of the given matrix to get I, M must be the left inverse of the given matrix. However, for square matrices, if a matrix has a left inverse, it is also a right inverse, and they are equal. So, in this case, M is indeed the inverse of the given matrix.Alternatively, if I didn't recall that, I could have set up the equation and solved for M by writing M as a general 3x3 matrix and then performing the multiplication and setting each element equal to the corresponding element in the identity matrix. But since the given matrix is block diagonal, it's much simpler to invert each block separately.Just to be thorough, let me verify the multiplication. Let me denote the given matrix as A and M as the inverse we found. Then, M * A should equal I.Compute M * A:First row of M: [√2/2, √2/2, 0]Multiply by first column of A: (√2/2)(√2/2) + (√2/2)(√2/2) + 0*0 = (2/4) + (2/4) = 1First row of M times second column of A: (√2/2)(-√2/2) + (√2/2)(√2/2) + 0*0 = (-2/4) + (2/4) = 0First row of M times third column of A: 0*0 + 0*0 + 0*2 = 0Second row of M: [-√2/2, √2/2, 0]Multiply by first column of A: (-√2/2)(√2/2) + (√2/2)(√2/2) + 0*0 = (-2/4) + (2/4) = 0Second row of M times second column of A: (-√2/2)(-√2/2) + (√2/2)(√2/2) + 0*0 = (2/4) + (2/4) = 1Second row of M times third column of A: 0*0 + 0*0 + 0*2 = 0Third row of M: [0, 0, 1/2]Multiply by first column of A: 0*√2/2 + 0*√2/2 + (1/2)*0 = 0Third row of M times second column of A: 0*(-√2/2) + 0*(√2/2) + (1/2)*0 = 0Third row of M times third column of A: 0*0 + 0*0 + (1/2)*2 = 1So, putting it all together, M * A is indeed the identity matrix:[begin{pmatrix}1 & 0 & 0 0 & 1 & 0 0 & 0 & 1end{pmatrix}]Therefore, my solution seems correct. So, the matrix M is:[begin{pmatrix}frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 -frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & frac{1}{2}end{pmatrix}]But wait, in the initial problem, the user wrote the given matrix with the third row as [0, 0, 2]. So, when inverting, the third element is 1/2. That matches with my inverse.Alternatively, if I had to write the inverse without using the block diagonal property, I could set up M as a general 3x3 matrix and solve the equations. Let me try that approach to confirm.Let M be:[begin{pmatrix}a & b & c d & e & f g & h & iend{pmatrix}]Then, M * A should equal I. So, let's compute each element of the product.First row of M times first column of A:a*(√2/2) + b*(√2/2) + c*0 = (a + b)*(√2/2)First row of M times second column of A:a*(-√2/2) + b*(√2/2) + c*0 = (-a + b)*(√2/2)First row of M times third column of A:a*0 + b*0 + c*2 = 2cSimilarly, second row of M times first column of A:d*(√2/2) + e*(√2/2) + f*0 = (d + e)*(√2/2)Second row of M times second column of A:d*(-√2/2) + e*(√2/2) + f*0 = (-d + e)*(√2/2)Second row of M times third column of A:d*0 + e*0 + f*2 = 2fThird row of M times first column of A:g*(√2/2) + h*(√2/2) + i*0 = (g + h)*(√2/2)Third row of M times second column of A:g*(-√2/2) + h*(√2/2) + i*0 = (-g + h)*(√2/2)Third row of M times third column of A:g*0 + h*0 + i*2 = 2iNow, setting each of these equal to the corresponding element in the identity matrix I, which is:[begin{pmatrix}1 & 0 & 0 0 & 1 & 0 0 & 0 & 1end{pmatrix}]So, we get the following equations:From first row, first column: (a + b)*(√2/2) = 1From first row, second column: (-a + b)*(√2/2) = 0From first row, third column: 2c = 0 => c = 0From second row, first column: (d + e)*(√2/2) = 0From second row, second column: (-d + e)*(√2/2) = 1From second row, third column: 2f = 0 => f = 0From third row, first column: (g + h)*(√2/2) = 0From third row, second column: (-g + h)*(√2/2) = 0From third row, third column: 2i = 1 => i = 1/2Now, let's solve these equations step by step.Starting with the first row:1. (a + b)*(√2/2) = 1 => a + b = 1 / (√2/2) = 2 / √2 = √22. (-a + b)*(√2/2) = 0 => -a + b = 0 => b = aFrom equation 2, b = a. Plugging into equation 1: a + a = √2 => 2a = √2 => a = √2 / 2Therefore, a = √2 / 2 and b = √2 / 2Next, second row:3. (d + e)*(√2/2) = 0 => d + e = 0 => e = -d4. (-d + e)*(√2/2) = 1 => (-d + e) = 1 / (√2/2) = 2 / √2 = √2But from equation 3, e = -d. Plugging into equation 4:(-d + (-d)) = √2 => -2d = √2 => d = -√2 / 2Therefore, d = -√2 / 2 and e = -d = √2 / 2Third row:5. (g + h)*(√2/2) = 0 => g + h = 0 => h = -g6. (-g + h)*(√2/2) = 0 => (-g + h) = 0 => h = gFrom equation 5, h = -g, and from equation 6, h = g. Therefore, -g = g => 2g = 0 => g = 0Thus, g = 0 and h = 0Finally, from third row, third column: i = 1/2So, compiling all the values:a = √2 / 2b = √2 / 2c = 0d = -√2 / 2e = √2 / 2f = 0g = 0h = 0i = 1/2Therefore, matrix M is:[begin{pmatrix}frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 -frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & frac{1}{2}end{pmatrix}]Which matches the inverse matrix I found earlier using the block diagonal approach. So, both methods lead to the same result, which gives me confidence that this is the correct matrix M.Just to recap, the key steps were recognizing that the given matrix is block diagonal, allowing me to invert each block separately. For the 2x2 rotation matrix, the inverse is its transpose, and for the 1x1 matrix [2], the inverse is [1/2]. Then, constructing M by combining these inverses. Alternatively, setting up M as a general matrix and solving the resulting system of equations also led to the same solution. Both methods confirmed that M is as above.I think I've covered all the necessary steps and verified the solution thoroughly. It's always good to check the result by performing the multiplication to ensure that M * A indeed equals the identity matrix, which I did and it worked out. So, I'm confident that this is the correct matrix M.**Final Answer**The matrix (mathbf{M}) is (boxed{begin{pmatrix} frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 -frac{sqrt{2}}{2} & frac{sqrt{2}}{2} & 0 0 & 0 & frac{1}{2} end{pmatrix}}).

💡Okay, so I have this function ( f(x) = log_{2}(x^{2} - 2x - 3) ), and I need to figure out which interval E from the options A to D makes the function monotonically decreasing. That means, for any two points ( x_1 ) and ( x_2 ) in E where ( x_1 < x_2 ), it should always be true that ( f(x_1) > f(x_2) ).First, I remember that logarithmic functions are only defined when their arguments are positive. So, I need to find where ( x^{2} - 2x - 3 > 0 ). Let me solve that inequality.The quadratic equation ( x^{2} - 2x - 3 = 0 ) can be factored. Let me see, looking for two numbers that multiply to -3 and add to -2. Hmm, that would be -3 and +1. So, factoring gives ( (x - 3)(x + 1) = 0 ). Therefore, the roots are at ( x = 3 ) and ( x = -1 ).To determine where the quadratic is positive, I can use a sign chart. The critical points are at ( x = -1 ) and ( x = 3 ). So, the number line is divided into three intervals:1. ( x < -1 )2. ( -1 < x < 3 )3. ( x > 3 )Testing each interval:1. For ( x < -1 ), say ( x = -2 ): ( (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 > 0 ).2. For ( -1 < x < 3 ), say ( x = 0 ): ( 0 - 0 - 3 = -3 < 0 ).3. For ( x > 3 ), say ( x = 4 ): ( 16 - 8 - 3 = 5 > 0 ).So, the quadratic is positive when ( x < -1 ) or ( x > 3 ). Therefore, the domain of ( f(x) ) is ( (-infty, -1) cup (3, +infty) ).Next, I need to determine where ( f(x) ) is decreasing. Since ( f(x) ) is a logarithmic function, its behavior depends on the inside function ( t(x) = x^{2} - 2x - 3 ). The logarithm base 2 is an increasing function, so the behavior of ( f(x) ) will depend on whether ( t(x) ) is increasing or decreasing.Let me find the derivative of ( t(x) ) to determine its monotonicity. The derivative ( t'(x) = 2x - 2 ).Setting ( t'(x) = 0 ) gives ( 2x - 2 = 0 ) → ( x = 1 ). So, the critical point is at ( x = 1 ).Analyzing the intervals around ( x = 1 ):1. For ( x < 1 ), say ( x = 0 ): ( t'(0) = -2 < 0 ), so ( t(x) ) is decreasing.2. For ( x > 1 ), say ( x = 2 ): ( t'(2) = 4 - 2 = 2 > 0 ), so ( t(x) ) is increasing.But wait, the domain of ( f(x) ) is only ( (-infty, -1) ) and ( (3, +infty) ). So, within the domain, how does ( t(x) ) behave?In the interval ( (-infty, -1) ), since ( x < 1 ), ( t(x) ) is decreasing. Therefore, ( t(x) ) is decreasing on ( (-infty, -1) ).In the interval ( (3, +infty) ), since ( x > 1 ), ( t(x) ) is increasing. Therefore, ( t(x) ) is increasing on ( (3, +infty) ).Since ( f(x) = log_{2}(t(x)) ) and ( log_{2} ) is an increasing function, the monotonicity of ( f(x) ) will follow the monotonicity of ( t(x) ). So:- On ( (-infty, -1) ), ( t(x) ) is decreasing, so ( f(x) ) is decreasing.- On ( (3, +infty) ), ( t(x) ) is increasing, so ( f(x) ) is increasing.Therefore, ( f(x) ) is decreasing on ( (-infty, -1) ) and increasing on ( (3, +infty) ).Now, looking at the options:A: ( (-3, -1) ) – This is a subset of ( (-infty, -1) ), where ( f(x) ) is decreasing.B: ( (-1, 0) ) – This interval is not in the domain of ( f(x) ) because ( x^{2} - 2x - 3 ) is negative here.C: ( (1, 2) ) – Also not in the domain, since ( x^{2} - 2x - 3 ) is negative here.D: ( (3, 6) ) – This is a subset of ( (3, +infty) ), where ( f(x) ) is increasing, not decreasing.So, only option A is a valid interval where ( f(x) ) is decreasing.Wait, just to double-check, let me pick two points in interval A, say ( x_1 = -2 ) and ( x_2 = -1.5 ). Compute ( f(x_1) ) and ( f(x_2) ).Compute ( t(x_1) = (-2)^2 - 2*(-2) - 3 = 4 + 4 - 3 = 5 ). So, ( f(x_1) = log_{2}(5) approx 2.32 ).Compute ( t(x_2) = (-1.5)^2 - 2*(-1.5) - 3 = 2.25 + 3 - 3 = 2.25 ). So, ( f(x_2) = log_{2}(2.25) approx 1.17 ).Since ( x_1 < x_2 ) and ( f(x_1) > f(x_2) ), it works as expected.Similarly, in interval D, pick ( x_1 = 4 ) and ( x_2 = 5 ).Compute ( t(4) = 16 - 8 - 3 = 5 ), so ( f(4) = log_{2}(5) approx 2.32 ).Compute ( t(5) = 25 - 10 - 3 = 12 ), so ( f(5) = log_{2}(12) approx 3.58 ).Here, ( x_1 < x_2 ) but ( f(x_1) < f(x_2) ), which is increasing, so D is not suitable.Therefore, I'm confident that A is the correct answer.**Final Answer**boxed{A}