💡Okay, so I have this problem about a triangle with sides 7, 24, and 25. It says the triangle has both an inscribed circle (incircle) and a circumscribed circle (circumcircle). I need to find the distance between the centers of these two circles, which are called the incenter and the circumcenter respectively.First, I remember that in any triangle, the incenter is the point where the angle bisectors meet, and it's also the center of the incircle. The circumcenter, on the other hand, is the point where the perpendicular bisectors of the sides meet, and it's the center of the circumcircle.I think the triangle with sides 7, 24, and 25 might be a right triangle because 7 squared is 49, 24 squared is 576, and 25 squared is 625. Let me check if 49 + 576 equals 625. Yes, 49 + 576 is 625, so this is a right triangle. That means the triangle has a right angle, and in a right triangle, the circumcenter is at the midpoint of the hypotenuse. So, the circumradius is half the length of the hypotenuse. Since the hypotenuse is 25, the circumradius should be 12.5.Now, for the inradius. I recall there's a formula for the inradius of a triangle: r = A/s, where A is the area of the triangle and s is the semi-perimeter. Let me calculate the semi-perimeter first. The semi-perimeter s is (7 + 24 + 25)/2, which is (56)/2 = 28.Next, the area A of the triangle. Since it's a right triangle, the area is (base * height)/2. The legs are 7 and 24, so the area is (7 * 24)/2 = 84.Now, plugging into the inradius formula: r = 84 / 28 = 3. So, the inradius is 3.Now, I need to find the distance between the incenter and the circumcenter. I remember there's a formula that relates the distance between the incenter and circumcenter with other elements of the triangle. Let me recall it. I think it's something like IO² = R(R - 2r), where R is the circumradius and r is the inradius.Let me verify that formula. Hmm, yes, I think that's correct for any triangle. So, if I use that, I can compute IO.Given that R is 12.5 and r is 3, plugging into the formula: IO² = 12.5(12.5 - 2*3) = 12.5*(12.5 - 6) = 12.5*6.5.Calculating 12.5 * 6.5: 12 * 6 is 72, 12 * 0.5 is 6, 0.5 * 6 is 3, and 0.5 * 0.5 is 0.25. Wait, maybe a better way is to compute 12.5 * 6.5 directly. 12.5 * 6 is 75, and 12.5 * 0.5 is 6.25, so total is 75 + 6.25 = 81.25.So, IO² = 81.25. Therefore, IO is the square root of 81.25. Let me compute that. 81.25 is equal to 325/4, because 81.25 * 4 = 325. So, sqrt(325/4) = sqrt(325)/2. Simplifying sqrt(325): 325 is 25*13, so sqrt(25*13) = 5*sqrt(13). Therefore, IO = (5*sqrt(13))/2.Wait, but earlier I thought the formula was IO² = R(R - 2r). Let me double-check if that's correct. Maybe I mixed up the formula. Alternatively, I remember another formula involving the sides of the triangle. Let me see.Alternatively, for a right triangle, maybe there's a simpler way to find the distance between the incenter and circumcenter. Since the triangle is right-angled, perhaps we can place it on a coordinate system to compute the coordinates of the incenter and circumcenter and then find the distance between them.Let me try that approach. Let's place the right-angled triangle with the right angle at the origin. Let me assign coordinates: Let’s say point B is at (0,0), point C is at (25,0), and point A is at (0,7). Wait, no, because the sides are 7, 24, 25. So, actually, the legs should be 7 and 24. So, let me correct that.Let me place point B at (0,0), point C at (24,0), and point A at (0,7). Then, the hypotenuse would be from (0,7) to (24,0), which should have length 25. Let me verify that distance: sqrt((24-0)^2 + (0-7)^2) = sqrt(576 + 49) = sqrt(625) = 25. Perfect.Now, the circumcenter O is the midpoint of the hypotenuse AC. So, coordinates of A are (0,7) and coordinates of C are (24,0). The midpoint O would be ((0 + 24)/2, (7 + 0)/2) = (12, 3.5).Next, the incenter I. The incenter can be found using the formula that weights the coordinates by the lengths of the sides. The formula is: I = (aA + bB + cC)/(a + b + c), where a, b, c are the lengths of the sides opposite to vertices A, B, C respectively.Wait, in triangle ABC, side a is opposite vertex A, which is BC. So, side a is 25, side b is opposite vertex B, which is AC = 7, and side c is opposite vertex C, which is AB = 24.Wait, actually, in standard notation, side a is opposite vertex A, side b opposite vertex B, and side c opposite vertex C. So, in our case, if we have triangle ABC with right angle at B, then side opposite A is BC, which is 25, side opposite B is AC, which is 7, and side opposite C is AB, which is 24.Wait, that seems inconsistent because in a right triangle, the hypotenuse is the longest side, so side opposite the right angle is the hypotenuse. So, if the right angle is at B, then side AC is the hypotenuse, which is 25. So, side a (opposite A) is BC, which is 24, side b (opposite B) is AC, which is 25, and side c (opposite C) is AB, which is 7.Wait, I'm getting confused. Let me clarify.In triangle ABC, with right angle at B, so angle B is 90 degrees. Then, side opposite A is BC, which is 24, side opposite B is AC, which is 25, and side opposite C is AB, which is 7.So, in the formula for incenter, the coordinates would be (aA + bB + cC)/(a + b + c). Wait, no, actually, the formula is (a*x_A + b*x_B + c*x_C)/(a + b + c) for the x-coordinate, and similarly for the y-coordinate.Wait, no, actually, the formula is weighted by the lengths of the sides. So, in barycentric coordinates, the incenter is at (a : b : c), but in Cartesian coordinates, it's ( (a*x_A + b*x_B + c*x_C)/(a + b + c), (a*y_A + b*y_B + c*y_C)/(a + b + c) ).Wait, but I think I might have it backwards. Let me check.Actually, the formula for the incenter in Cartesian coordinates is given by:I_x = (a*x_A + b*x_B + c*x_C)/(a + b + c)I_y = (a*y_A + b*y_B + c*y_C)/(a + b + c)where a, b, c are the lengths of the sides opposite to vertices A, B, C respectively.In our case, side a is opposite vertex A, which is BC = 24, side b is opposite vertex B, which is AC = 25, and side c is opposite vertex C, which is AB = 7.So, plugging in the coordinates:Point A is (0,7), so x_A = 0, y_A = 7Point B is (0,0), so x_B = 0, y_B = 0Point C is (24,0), so x_C = 24, y_C = 0Therefore,I_x = (a*x_A + b*x_B + c*x_C)/(a + b + c) = (24*0 + 25*0 + 7*24)/(24 + 25 + 7) = (0 + 0 + 168)/(56) = 168/56 = 3Similarly,I_y = (a*y_A + b*y_B + c*y_C)/(a + b + c) = (24*7 + 25*0 + 7*0)/(56) = (168 + 0 + 0)/56 = 168/56 = 3So, the incenter I is at (3,3).Earlier, the circumcenter O was at (12, 3.5).Now, to find the distance between I(3,3) and O(12, 3.5), we can use the distance formula:Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]Plugging in the values:Distance = sqrt[(12 - 3)^2 + (3.5 - 3)^2] = sqrt[(9)^2 + (0.5)^2] = sqrt[81 + 0.25] = sqrt[81.25]Now, 81.25 can be written as 325/4, because 81.25 * 4 = 325. So, sqrt(325/4) = sqrt(325)/2.Simplifying sqrt(325): 325 = 25 * 13, so sqrt(25*13) = 5*sqrt(13). Therefore, sqrt(325)/2 = (5*sqrt(13))/2.So, the distance between the incenter and circumcenter is (5*sqrt(13))/2.Wait, but earlier I thought the formula was IO² = R(R - 2r). Let me check if that gives the same result.Given R = 12.5, r = 3,IO² = 12.5*(12.5 - 2*3) = 12.5*(12.5 - 6) = 12.5*6.5 = 81.25Which is the same as before, so IO = sqrt(81.25) = (5*sqrt(13))/2.Yes, that matches. So, both methods give the same result.Therefore, the distance between the incenter and circumcenter is (5*sqrt(13))/2.

💡Okay, so I have this geometry problem here. Let me try to visualize it first. There's a figure with two right angles at EAB and ABC. So, point A is connected to E and B, and point B is connected to A and C. Both angles at EAB and ABC are right angles, which means they are 90 degrees each. Given lengths: AB is 5 units, BC is 7 units, and AE is 10 units. Also, lines AC and BE intersect at point D. The question is asking for the difference between the areas of triangles ADE and BDC. Hmm, okay. So, I need to find the areas of triangles ADE and BDC and then subtract them to find the difference. First, maybe I should sketch this figure to get a better idea. Let me describe it mentally. Since angle EAB is a right angle, triangle EAB is a right-angled triangle with legs AB and AE. Similarly, angle ABC is a right angle, so triangle ABC is also right-angled with legs AB and BC. So, point A is connected to E and B, and point B is connected to A and C. Then, lines AC and BE intersect at D. That makes sense. I think coordinate geometry might help here. If I can assign coordinates to points A, B, C, and E, I can find the equations of lines AC and BE, find their intersection point D, and then compute the areas of triangles ADE and BDC. Let me assign coordinates. Let's place point A at the origin (0,0) for simplicity. Since angle EAB is a right angle, and AE is 10 units, point E must be along the y-axis. So, point E is at (0,10). Similarly, since angle ABC is a right angle, and AB is 5 units, point B must be along the x-axis. So, point B is at (5,0). Now, point C is connected to B, and BC is 7 units. Since angle ABC is a right angle, point C must be along the line perpendicular to AB at point B. Since AB is along the x-axis from (0,0) to (5,0), the perpendicular would be along the y-axis. But wait, if ABC is a right angle, then BC is vertical or horizontal? Wait, AB is along the x-axis, so if ABC is a right angle, then BC must be vertical. So, point C is either above or below point B. Since lengths are positive, let's assume it's above. So, point B is at (5,0), and BC is 7 units upwards, so point C is at (5,7). Let me confirm: AB is from (0,0) to (5,0), so length AB is 5 units. BC is from (5,0) to (5,7), so length BC is 7 units. AE is from (0,0) to (0,10), so length AE is 10 units. That seems correct. Now, lines AC and BE intersect at D. Let me find the equations of lines AC and BE. First, line AC connects points A(0,0) and C(5,7). The slope of AC is (7-0)/(5-0) = 7/5. So, the equation of line AC is y = (7/5)x. Next, line BE connects points B(5,0) and E(0,10). The slope of BE is (10-0)/(0-5) = 10/(-5) = -2. So, the equation of line BE is y - 0 = -2(x - 5), which simplifies to y = -2x + 10. Now, to find point D, we need to solve the system of equations:y = (7/5)xy = -2x + 10Set them equal:(7/5)x = -2x + 10Multiply both sides by 5 to eliminate the denominator:7x = -10x + 50Add 10x to both sides:17x = 50So, x = 50/17 ≈ 2.941. Now, substitute x back into y = (7/5)x:y = (7/5)(50/17) = (350)/85 = 70/17 ≈ 4.118.So, point D is at (50/17, 70/17). Now, I need to find the areas of triangles ADE and BDC. Let's start with triangle ADE. Points A, D, and E. Point A is (0,0), point D is (50/17, 70/17), and point E is (0,10). To find the area of triangle ADE, I can use the formula for the area of a triangle given coordinates of its vertices. The formula is:Area = (1/2)| (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)) |Plugging in the coordinates:x1 = 0, y1 = 0x2 = 50/17, y2 = 70/17x3 = 0, y3 = 10So,Area = (1/2)| 0*(70/17 - 10) + (50/17)*(10 - 0) + 0*(0 - 70/17) |= (1/2)| 0 + (50/17)*10 + 0 |= (1/2)| 500/17 |= (1/2)*(500/17)= 250/17 ≈ 14.7059So, area of triangle ADE is 250/17.Now, let's find the area of triangle BDC. Points B, D, and C.Point B is (5,0), point D is (50/17, 70/17), and point C is (5,7).Again, using the area formula:Area = (1/2)| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |Plugging in the coordinates:x1 = 5, y1 = 0x2 = 50/17, y2 = 70/17x3 = 5, y3 = 7So,Area = (1/2)| 5*(70/17 - 7) + (50/17)*(7 - 0) + 5*(0 - 70/17) |Let me compute each term step by step.First term: 5*(70/17 - 7)= 5*(70/17 - 119/17)= 5*(-49/17)= -245/17Second term: (50/17)*(7 - 0)= (50/17)*7= 350/17Third term: 5*(0 - 70/17)= 5*(-70/17)= -350/17Now, add all three terms:-245/17 + 350/17 - 350/17= (-245 + 350 - 350)/17= (-245)/17Take the absolute value:| -245/17 | = 245/17Multiply by 1/2:Area = (1/2)*(245/17) = 245/34 ≈ 7.176So, area of triangle BDC is 245/34.Now, the question asks for the difference between the areas of triangle ADE and triangle BDC. So, compute 250/17 - 245/34.First, let me convert 250/17 to 500/34 so that both fractions have the same denominator.250/17 = (250*2)/(17*2) = 500/34So, 500/34 - 245/34 = (500 - 245)/34 = 255/34Simplify 255/34:Divide numerator and denominator by 17:255 ÷ 17 = 1534 ÷ 17 = 2So, 255/34 = 15/2 = 7.5Therefore, the difference between the areas is 7.5.Wait, let me double-check my calculations because 250/17 is approximately 14.7059 and 245/34 is approximately 7.176, so their difference is approximately 7.529, which is roughly 7.5. So, that seems consistent.Alternatively, maybe there's a smarter way without coordinate geometry. Let me think.Since lines AC and BE intersect at D, maybe we can use similar triangles or area ratios.Looking back at the initial problem, the user provided a solution using areas of triangles ABE and BAC, subtracting to get the difference. Let me see if that approach works.The area of triangle ABE is (1/2)*AB*AE = (1/2)*5*10 = 25.The area of triangle BAC is (1/2)*AB*BC = (1/2)*5*7 = 17.5.Then, the difference is 25 - 17.5 = 7.5.Wait, that's the same answer as before. So, that approach is much quicker. But how does that work? Let me understand.In the initial solution, they considered the areas of triangles ABE and BAC. Both triangles share the common triangle ABD (or maybe ABD and CBD? Wait, let me see.)Wait, actually, in triangle ABE, the area is split into triangle ADE and ABD. Similarly, in triangle BAC, the area is split into triangle BDC and ABD. So, subtracting the areas of BAC from ABE gives the difference between ADE and BDC.So, Area ABE = Area ADE + Area ABDArea BAC = Area BDC + Area ABDSubtracting, we get Area ABE - Area BAC = Area ADE - Area BDCWhich is 25 - 17.5 = 7.5.That's a much more straightforward method. So, I guess my coordinate geometry approach confirmed the answer, but the initial solution is more efficient.So, the difference between the areas is 7.5, which is 15/2.Therefore, the answer is 7.5, which can be written as 15/2 or 7.5. Since the question doesn't specify the form, both are correct, but in the box, it's usually preferred as a fraction or decimal. Since 7.5 is a terminating decimal, it's fine.**Final Answer**boxed{7.5}

💡Okay, so I have this problem with three parts. It involves functions f(x) = ln x and g(x) = (1/2)x² + m x + 7/2, where m is negative. There's a line l that's tangent to both f(x) and g(x), and the tangent point on f(x) has an abscissa (which is the x-coordinate) of 1. Starting with part 1: I need to find the equation of line l and the value of m. Hmm, since the line is tangent to f(x) at x=1, I should find the equation of the tangent line at that point. First, let me find f(1). Since f(x) = ln x, f(1) = ln 1 = 0. So the point is (1, 0). Now, to find the slope of the tangent line, I need the derivative of f(x). The derivative of ln x is 1/x, so f’(1) = 1/1 = 1. So the slope of the tangent line is 1. Therefore, the equation of the tangent line l is y = 1*(x - 1) + 0, which simplifies to y = x - 1. So that's the equation of line l. Now, this same line is also tangent to g(x). So, I need to find m such that the line y = x - 1 is tangent to the parabola g(x). For a line to be tangent to a parabola, the system of equations should have exactly one solution. So, setting g(x) equal to l(x):(1/2)x² + m x + 7/2 = x - 1Let me rearrange this equation:(1/2)x² + m x + 7/2 - x + 1 = 0Simplify the terms:(1/2)x² + (m - 1)x + (7/2 + 1) = 0Which becomes:(1/2)x² + (m - 1)x + 9/2 = 0To make it easier, multiply both sides by 2 to eliminate the fractions:x² + 2(m - 1)x + 9 = 0Now, for this quadratic equation to have exactly one solution, the discriminant must be zero. The discriminant D is [2(m - 1)]² - 4*1*9. Calculating D:D = 4(m - 1)² - 36Set D = 0:4(m - 1)² - 36 = 0Divide both sides by 4:(m - 1)² - 9 = 0So, (m - 1)² = 9Take square roots:m - 1 = ±3Therefore, m = 1 + 3 = 4 or m = 1 - 3 = -2But the problem states that m < 0, so m = -2.Alright, so part 1 is done: the equation of line l is y = x - 1, and m is -2.Moving on to part 2: h(x) = f(x + 1) - g’(x). I need to find the maximum value of h(x).First, let's write down what h(x) is. f(x + 1) is ln(x + 1). g’(x) is the derivative of g(x). Since g(x) = (1/2)x² + m x + 7/2, its derivative is g’(x) = x + m. But from part 1, m = -2, so g’(x) = x - 2.Therefore, h(x) = ln(x + 1) - (x - 2) = ln(x + 1) - x + 2.Now, to find the maximum value of h(x), I need to find its critical points by taking the derivative and setting it equal to zero.Compute h’(x):h’(x) = derivative of ln(x + 1) is 1/(x + 1), derivative of -x is -1, and derivative of 2 is 0. So,h’(x) = 1/(x + 1) - 1Set h’(x) = 0:1/(x + 1) - 1 = 01/(x + 1) = 1Multiply both sides by (x + 1):1 = x + 1So, x = 0Now, check if this is a maximum. Let's compute the second derivative or analyze the sign changes of h’(x).Compute h''(x):h''(x) = derivative of 1/(x + 1) is -1/(x + 1)², derivative of -1 is 0. So,h''(x) = -1/(x + 1)²Since (x + 1)² is always positive for x ≠ -1, h''(x) is always negative where it's defined. Therefore, the function h(x) is concave down everywhere, so x = 0 is a point of local maximum, which is also the global maximum.Now, compute h(0):h(0) = ln(0 + 1) - 0 + 2 = ln(1) + 2 = 0 + 2 = 2So, the maximum value of h(x) is 2.Alright, part 2 is done.Part 3: If ln(x + 1) < x + c holds true for any x, determine the range of values for the real number c.So, the inequality is ln(x + 1) < x + c for all x. I need to find the range of c such that this inequality holds for any x.First, let's consider the domain of ln(x + 1). The argument of ln must be positive, so x + 1 > 0 => x > -1. So, the inequality must hold for all x > -1.We can rearrange the inequality:ln(x + 1) - x < cSo, c must be greater than ln(x + 1) - x for all x > -1.Therefore, c must be greater than the maximum value of ln(x + 1) - x.Wait, that's interesting. From part 2, h(x) = ln(x + 1) - x + 2. So, ln(x + 1) - x = h(x) - 2.Therefore, the maximum of ln(x + 1) - x is the maximum of h(x) - 2. Since the maximum of h(x) is 2, then the maximum of ln(x + 1) - x is 2 - 2 = 0.Therefore, ln(x + 1) - x ≤ 0 for all x > -1, with equality at x = 0.Hence, to have ln(x + 1) < x + c for all x > -1, c must be greater than the maximum of ln(x + 1) - x, which is 0. So, c must be greater than or equal to 0.Wait, but the inequality is strict: ln(x + 1) < x + c. Since the maximum of ln(x + 1) - x is 0, we have ln(x + 1) - x ≤ 0. So, to have ln(x + 1) < x + c for all x, c must be greater than 0. Because if c = 0, then ln(x + 1) < x + 0 would not hold at x = 0, since ln(1) = 0 = 0 + 0. So, the inequality is strict, so c must be greater than 0.Wait, but in part 2, h(x) = ln(x + 1) - x + 2, which had a maximum at 2. So, ln(x + 1) - x = h(x) - 2, which has a maximum at 0. So, ln(x + 1) - x ≤ 0, meaning that ln(x + 1) ≤ x. Therefore, ln(x + 1) - x ≤ 0, so to have ln(x + 1) < x + c, we need c > 0. Because if c = 0, then ln(x + 1) ≤ x, but the inequality is strict, so c must be greater than 0.Wait, but actually, the maximum of ln(x + 1) - x is 0, so ln(x + 1) - x ≤ 0 for all x > -1. Therefore, ln(x + 1) < x + c is equivalent to c > ln(x + 1) - x. Since ln(x + 1) - x ≤ 0, c must be greater than the maximum of ln(x + 1) - x, which is 0. Therefore, c must be greater than or equal to 0. But since the inequality is strict, ln(x + 1) < x + c, and at x = 0, ln(1) = 0, so 0 < 0 + c, which implies c > 0.Therefore, c must be greater than 0.Wait, but in part 2, h(x) = ln(x + 1) - x + 2, which has a maximum at x = 0, h(0) = 2. So, ln(x + 1) - x = h(x) - 2, which has a maximum at 0. So, ln(x + 1) - x ≤ 0 for all x > -1. Therefore, ln(x + 1) - x is always less than or equal to 0. So, to have ln(x + 1) < x + c, we need c > ln(x + 1) - x for all x. Since the maximum of ln(x + 1) - x is 0, c must be greater than 0.Therefore, the range of c is c > 0.Wait, but in part 2, h(x) = ln(x + 1) - x + 2, which has a maximum of 2. So, ln(x + 1) - x = h(x) - 2, which has a maximum of 0. So, ln(x + 1) - x ≤ 0. Therefore, ln(x + 1) < x + c is equivalent to c > ln(x + 1) - x. Since the maximum of ln(x + 1) - x is 0, c must be greater than 0. So, c > 0.Yes, that makes sense. So, the range of c is c > 0.Wait, but let me double-check. If c = 0, then ln(x + 1) < x. But at x = 0, ln(1) = 0, which is equal to x, so the inequality is not strict. Therefore, c must be greater than 0 to ensure that ln(x + 1) < x + c for all x > -1.Yes, that seems correct.So, summarizing:1. The equation of line l is y = x - 1, and m = -2.2. The maximum value of h(x) is 2.3. The range of c is c > 0.I think that's it.

💡Okay, so I need to find the determinant of a transformation matrix T. This matrix T does two things: first, it scales all vectors by a factor of 3, and then it rotates them counter-clockwise by 45 degrees. Hmm, determinants... I remember that determinants have something to do with the scaling factor of the area when you apply a linear transformation. So, if I can figure out how scaling and rotation affect the determinant, I can find det(T).Let me start by recalling what the scaling matrix looks like. If we scale by a factor of 3, the scaling matrix S should be:S = [3 0] [0 3]Right? Because scaling each component by 3. Now, the determinant of a scaling matrix is just the product of the scaling factors along the diagonal. So, det(S) should be 3 * 3 = 9. Got that down.Next, the rotation matrix. Rotating counter-clockwise by 45 degrees. I remember the general form of a rotation matrix is:R = [cosθ -sinθ] [sinθ cosθ]Where θ is the angle of rotation. So, plugging in 45 degrees, which is π/4 radians. I need to compute cos(45°) and sin(45°). Both of these are √2/2. So, the rotation matrix R becomes:R = [√2/2 -√2/2] [√2/2 √2/2]Now, what's the determinant of R? The determinant of a 2x2 matrix [a b; c d] is ad - bc. So, plugging in the values:det(R) = (√2/2)(√2/2) - (-√2/2)(√2/2)Let me compute that step by step. First, (√2/2)(√2/2) is (√2 * √2)/(2*2) = 2/4 = 1/2.Then, the second term is (-√2/2)(√2/2). Wait, no, the formula is ad - bc, so it's (√2/2)(√2/2) - (-√2/2)(√2/2). So, that's 1/2 - (-1/2). Which is 1/2 + 1/2 = 1.So, det(R) = 1. That makes sense because rotation doesn't change the area, it just moves things around, so the determinant should be 1.Now, the transformation matrix T is the combination of scaling and then rotating. So, in matrix terms, that means T = R * S, right? Because we first apply S and then R. Wait, actually, no. Wait, in linear transformations, the order matters. If we first scale and then rotate, the matrix multiplication should be R * S, because when you multiply matrices, the order is from right to left. So, T = R * S.But wait, actually, I might be getting confused here. Let me think. If we have a vector v, and we first scale it by S, then rotate it by R, the transformation is R * (S * v) = (R * S) * v. So, yes, T = R * S.But does the order matter for the determinant? Hmm, determinants are multiplicative, so det(T) = det(R * S) = det(R) * det(S). Since determinant is multiplicative regardless of the order, right? Wait, no, actually, determinant is multiplicative, so det(R * S) = det(R) * det(S). So, regardless of the order, the determinant is the product of the determinants.Wait, but actually, in this case, since both S and R are 2x2 matrices, det(R * S) = det(R) * det(S) = 1 * 9 = 9. So, regardless of the order, the determinant is 9.But just to be thorough, let me compute T explicitly. Let's compute R * S.R = [√2/2 -√2/2] [√2/2 √2/2]S = [3 0] [0 3]So, R * S is:First row: [√2/2 * 3 + (-√2/2) * 0, √2/2 * 0 + (-√2/2) * 3] = [3√2/2, -3√2/2]Second row: [√2/2 * 3 + √2/2 * 0, √2/2 * 0 + √2/2 * 3] = [3√2/2, 3√2/2]So, T = R * S = [3√2/2 -3√2/2] [3√2/2 3√2/2]Now, let's compute det(T). Using the formula for a 2x2 matrix:det(T) = (3√2/2)(3√2/2) - (-3√2/2)(3√2/2)Compute each term:First term: (3√2/2)(3√2/2) = (9 * 2)/4 = 18/4 = 9/2Second term: (-3√2/2)(3√2/2) = (-9 * 2)/4 = -18/4 = -9/2But since it's subtracting this term, it becomes:det(T) = 9/2 - (-9/2) = 9/2 + 9/2 = 18/2 = 9So, det(T) = 9.Wait, that's consistent with the earlier result where det(T) = det(R) * det(S) = 1 * 9 = 9.So, both methods give the same answer, which is reassuring.I think that's solid. So, the determinant of the transformation matrix T is 9.

💡I need to calculate the value of the expression 8 + 15 ÷ 3 - 4 × 2.First, I should remember the order of operations, often abbreviated as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).Looking at the expression, there are no parentheses or exponents, so I start with multiplication and division.I see two operations to perform: 15 ÷ 3 and 4 × 2.Calculating 15 ÷ 3 gives me 5.Calculating 4 × 2 gives me 8.Now, I substitute these results back into the original expression: 8 + 5 - 8.Next, I perform the addition and subtraction from left to right.Adding 8 and 5 gives me 13.Then, subtracting 8 from 13 results in 5.Therefore, the value of the expression is 5.

💡Alright, so I've got this problem with four propositions, and I need to figure out how many of them are true. Let me take them one by one and think through each carefully.Starting with proposition ①: "The negation of the proposition 'There exists (x in mathbb{R}) such that (x^{2}+1 > 3x)' is 'For all (x in mathbb{R}), (x^{2}+1 < 3x)'." Hmm, okay. I remember that the negation of an existential quantifier ("there exists") is a universal quantifier ("for all"). So, the original statement is saying that there is at least one real number x where (x^2 + 1) is greater than (3x). The negation should then be that for every real number x, (x^2 + 1) is not greater than (3x). But wait, does that mean it's less than, or less than or equal to? The negation of (>) is (leq), right? So, the negation should be "For all (x in mathbb{R}), (x^{2}+1 leq 3x)." But the proposition says it's (x^{2}+1 < 3x). So, that's not exactly correct because it's using a strict inequality instead of a non-strict one. Therefore, proposition ① is false.Moving on to proposition ②: "'If (a > 2)' is a sufficient but not necessary condition for '(a > 5)'." Okay, let's unpack this. A sufficient condition means that if (a > 2), then it's enough to ensure (a > 5). But wait, is (a > 2) sufficient for (a > 5)? No, because (a) could be 3, which is greater than 2 but not greater than 5. So, (a > 2) doesn't guarantee (a > 5). Therefore, it's not a sufficient condition. Also, is it a necessary condition? If (a > 5), then (a) must be greater than 2, so (a > 2) is a necessary condition for (a > 5). But the proposition says it's sufficient but not necessary, which is incorrect because it's actually necessary but not sufficient. So, proposition ② is false.Next, proposition ③: "The contrapositive of the proposition 'If (xy=0), then (x=0) and (y=0)' is true." Let me recall that the contrapositive of "If P, then Q" is "If not Q, then not P," and they have the same truth value. So, the original statement is "If (xy=0), then (x=0) and (y=0)." The contrapositive would be "If (x neq 0) or (y neq 0), then (xy neq 0)." Wait, is that true? If either (x) or (y) is not zero, does that mean their product isn't zero? No, because if one is zero and the other isn't, the product is still zero. So, the contrapositive is actually false because the original statement is false. For example, if (x=0) and (y=5), then (xy=0), but (y) isn't zero. So, the original implication is false, which means its contrapositive is also false. Therefore, proposition ③ is false.Finally, proposition ④: "Given (p) and (q) are two propositions, if '(p lor q)' is false, then '(neg p land neg q)' is true." Let me think about this. The disjunction (p lor q) is false only when both (p) and (q) are false. So, if (p lor q) is false, that means both (p) is false and (q) is false. Therefore, (neg p) is true and (neg q) is true, so their conjunction (neg p land neg q) is true. That makes sense. So, proposition ④ is true.So, after evaluating all four propositions, only proposition ④ is true. Therefore, the number of true propositions is 1.**Final Answer**boxed{C}