answer:II/2. There are 90000 five-digit numbers (from 10000 to 99999). We can divide them into 900 columns, each containing 100 numbers, based on the value of k from 100 to 999:begin{array}{ccccc}100 cdot 100+0 & 100 cdot 101+0 & 100 cdot 102+0 & cdots & 100 cdot 999+0 100 cdot 100+1 & 100 cdot 101+1 & 100 cdot 102+1 & cdots & 100 cdot 999+1 vdots & vdots & vdots & vdots & vdots 100 cdot 100+99 & 100 cdot 101+99 & 100 cdot 102+99 & cdots & 100 cdot 999+99end{array}In each column where k is not divisible by 11, we find 9 values of the remainder o such that the sum k+o is divisible by 11. In columns where k is divisible by 11, there are 10 values of the remainder o such that the sum k+o is divisible by 11. In this case, the remainder o can be 0, 11, 22, ldots, 99.We then determine that there are 81 columns where the quotient k is divisible by 11. These are the columns where k has values 110=11 cdot 10, 121=11 cdot 11, 132=11 cdot 12, ldots, 990=11 cdot 90. Therefore, the number of columns where k is not divisible by 11 is 900-81=819. The sum k+o is divisible by 11 in 81 cdot 10 + 819 cdot 9 = 8181 five-digit numbers.Finding: For k that is not divisible by 11, there are 9 such values of o such that k+o is divisible by 11 ... 2 points Finding: For k that is divisible by 11, there are 10 such values of o such that k+o is divisible by 11 ... 2 points In 81 cases, k is divisible by 11 ..................................................................................................................... In 819 cases, k is not divisible by 11 .................................................................................................... The sum k+o is divisible by 11 in 8181 five-digit numbers .........................................................

answer:Solution: (1) According to the problem, we have begin{cases} frac{(sqrt{2})^2}{a^2} + frac{(frac{sqrt{2}}{2})^2}{b^2} = 1 frac{c}{a} = frac{sqrt{3}}{2} a^2 = b^2 + c^2 end{cases}, solving these equations yields a^2 = 4, b^2 = 1. Therefore, the equation of ellipse E is frac{x^2}{4} + y^2 = 1. (2) As k varies, m^2 is a constant value. Proof: From begin{cases} y = kx + m frac{x^2}{4} + y^2 = 1 end{cases}, we get (1 + 4k^2)x^2 + 8kmx + 4(m^2 - 1) = 0. Let P(x_1,y_1), Q(x_2,y_2), then x_1 + x_2 = -frac{8km}{1 + 4k^2}, x_1x_2 = frac{4(m^2 - 1)}{1 + 4k^2}, (*) Since the slopes of lines OP and OQ are k_1 and k_2 respectively, and 4k = k_1 + k_2, we have 4k = frac{y_1}{x_1} + frac{y_2}{x_2} = frac{kx_1 + m}{x_1} + frac{kx_2 + m}{x_2}, which leads to 2kx_1x_2 = m(x_1 + x_2), Substituting (*) into this equation yields m^2 = frac{1}{2}, which is verified to be true upon checking. Therefore, as k varies, m^2 is a constant value boxed{frac{1}{2}}.

answer:To analyze the monomial -frac{3x{y}^{2}}{5}, we break it down into its components:- Coefficient: The coefficient is the numerical part of the term, which in this case is -frac{3}{5}. This is because the coefficient includes the sign associated with the term, making it negative in this instance.- Degree: The degree of a monomial is the sum of the exponents of all its variables. Here, x has an exponent of 1 (since x = x^{1}), and y^{2} has an exponent of 2. Therefore, the degree is 1 + 2 = 3.Putting it all together, we have a coefficient of -frac{3}{5} and a degree of 3. Comparing this with the given options:A: Incorrect because the coefficient is not positive frac{3}{5}.B: Correct because it matches our findings: the coefficient is -frac{3}{5}, and the degree is 3.C: Incorrect because the degree is not 2.D: Incorrect because, although the coefficient is correctly identified as -frac{3}{5}, the degree is incorrectly stated as 2.Therefore, the correct option is boxed{B}.

answer:2850

answer:(I) Since the least common multiple of 2, 3, and 5 is 30, a cube with an edge length of 30 formed by 2, 3, and 5 will have its diagonal passing through an integer number of small cuboids. Therefore, the number of small cuboids that a diagonal of the cube with an edge length of 90, formed by 2, 3, and 5, passes through should be a multiple of 3. Among the four options, only B meets this requirement. Hence, the choice is boxed{B}.(II) Let the coordinates of the midpoint of the line segment AB be (x, y). By the problem statement, we have 2x = x_A + x_B, 2y = y_A + y_B. Therefore, 4x^2 = (x_A + x_B)^2 and 4y^2 = (y_A + y_B)^2. Given x^2+y^2-24x-28y-36=0, we have (x_A)^2+(y_A)^2-24x_A-28y_A-36=0 and (x_B)^2+(y_B)^2-24x_B-28y_B-36=0. Adding the above two equations, we get (x_A)^2+(y_A)^2-24x_A-28y_A-36+(x_B)^2+(y_B)^2-24x_B-28y_B-36=0. Therefore, (x_A)^2+(x_B)^2+(y_A)^2+(y_B)^2-24(x_A+x_B)-28(y_A+y_B)-72=0. Thus, (x_A+x_B)^2-2x_A cdot x_B+(y_A+y_B)^2-2y_A cdot y_B-24 cdot 2x-28 cdot 2y-72=0. Therefore, 4x^2+4y^2-48x-56y-72=2(x_A cdot x_B+y_A cdot y_B). Since PA perp PB, we have frac{(y_A-2)}{(x_A-4)} cdot frac{(y_B-2)}{(x_B-4)}=-1. Therefore, (x_A-4) cdot (x_B-4)+(y_A-2) cdot (y_B-2)=0. Thus, x_A cdot x_B+y_A cdot y_B=4(x_A+x_B)+2(y_A+y_B)-20. Therefore, x_A cdot x_B+y_A cdot y_B=4 cdot 2x+2 cdot 2y-20. Thus, 16x+8y-40=2(x_A cdot x_B+y_A cdot y_B). Therefore, 4x^2+4y^2-48x-56y-72=16x+8y-40. After simplification, we get x^2+y^2-16x-16y-8=0. Hence, the trajectory equation of the midpoint Q of AB is a circle: (x-8)^2+(y-8)^2=136 (located inside the circle). boxed{(x-8)^2+(y-8)^2=136}.

answer:To analyze the function f(x)=sum_{i=1}^4{frac{{sin[(2i-1)x]}}{{2i-1}}} and its properties, we will look into each statement given in the options:For Option A: Symmetry about the line x=frac{pi}{2}Given f(x)=sin x + frac{sin 3x}{3} + frac{sin 5x}{5} + frac{sin 7x}{7}, let's check the symmetry about the line x=frac{pi}{2} by evaluating f(pi + x):[begin{align*}f(pi + x) &= sin(pi + x) + frac{sin(3(pi + x))}{3} + frac{sin(5(pi + x))}{5} + frac{sin(7(pi + x))}{7} &= -sin x - frac{sin 3x}{3} - frac{sin 5x}{5} - frac{sin 7x}{7} &= -(sin x + frac{sin 3x}{3} + frac{sin 5x}{5} + frac{sin 7x}{7}) &= -f(x)end{align*}]This shows that the function is symmetric about the line x=frac{pi}{2}, making statement A correct.For Option B: Symmetry about the point (0,0)To check if the graph is symmetric about the origin, we evaluate f(-x):[begin{align*}f(-x) &= sin(-x) + frac{sin(-3x)}{3} + frac{sin(-5x)}{5} + frac{sin(-7x)}{7} &= -sin x - frac{sin 3x}{3} - frac{sin 5x}{5} - frac{sin 7x}{7} &= -f(x)end{align*}]This confirms that the function f(x) is odd, and the graph is symmetric about the point (0,0), making statement B correct.For Option C: Periodicity and smallest positive periodGiven f(x+pi)=-f(x) neq f(x), this indicates that the function does not repeat its values every pi units in a manner that f(x+pi) = f(x). Therefore, the statement C is incorrect because the function does not have a smallest positive period of pi in the context of being a periodic function with the same values.For Option D: Maximum value of the derivative f'(x)The derivative of f(x) is given by f'(x)=cos x + cos 3x + cos 5x + cos 7x. Since the maximum value of cos function is 1, the maximum value of f'(x) can be calculated as:[f'(x) leqslant 1 + 1 + 1 + 1 = 4]Thus, the maximum value of f'(x) is 4, making statement D correct.Therefore, the correct choices are boxed{ABD}.