answer:To solve the problem, we will use the given information and substitute specific values for x to find the sum of a_{2}+a_{4}+a_{6}. First, let's substitute x=1 into the given polynomial expansion of (2x+1)^{6}. This gives us: [a_{0}+a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6} = (2cdot1+1)^{6} = 3^{6} = 729.] Next, we substitute x=-1 into the same polynomial expansion. This results in: [a_{0}-a_{1}+a_{2}-a_{3}+a_{4}-a_{5}+a_{6} = (2cdot(-1)+1)^{6} = 1^{6} = 1.] By adding these two equations together, we eliminate the terms with odd powers of x (since they are subtracted in one equation and added in the other), leading to: [2(a_{0}+a_{2}+a_{4}+a_{6}) = 730.] Dividing both sides by 2 gives us: [a_{0}+a_{2}+a_{4}+a_{6} = 365.] To find a_{0}, we substitute x=0 into the original polynomial, which simplifies to: [a_{0} = (2cdot0+1)^{6} = 1^{6} = 1.] Subtracting a_{0} from the sum a_{0}+a_{2}+a_{4}+a_{6}, we get: [a_{2}+a_{4}+a_{6} = 365 - 1 = 364.] Therefore, the sum of a_{2}+a_{4}+a_{6} is boxed{364}, which corresponds to choice boxed{C}.

answer:First, compute the squares of 20 and 15 respectively. - (20^2 = 400) - (15^2 = 225) Next, calculate the difference of these squares: - (400 - 225 = 175) Now, take the absolute value (though not necessary since the difference is positive): - (|175| = 175) Then, calculate half of this value: - (frac{175}{2} = 87.5) The final answer is: (boxed{87.5})

answer:Given tan alpha = 4, we aim to find the value of sin alpha cos alpha. We start by expressing sin alpha cos alpha in terms of tan alpha and using the Pythagorean identity sin^2 alpha + cos^2 alpha = 1: [ sin alpha cos alpha = frac{sin alpha cos alpha}{sin^2 alpha + cos^2 alpha} = frac{tan alpha}{tan^2 alpha + 1} ] Substituting tan alpha = 4 into the equation: [ frac{tan alpha}{tan^2 alpha + 1} = frac{4}{4^2 + 1} = frac{4}{16 + 1} = frac{4}{17} ] Therefore, the value of sin alpha cos alpha is frac{4}{17}, which corresponds to choice: [ boxed{B} ]

answer:Consider the function [ f(z) = frac{sin^4 z}{z^2} ] 1. **Analyticity and Isolated Singular Points**: The function ( f(z) ) is analytic for all ( z neq 0 ). However, it has an isolated singularity at ( z = 0 ). To find the zeros of the function, we need to solve ( f(z) = 0 ). 2. **Finding the Zeros**: We need to determine the points where ( sin^4 z = 0 ), excluding ( z = 0 ). The zeros of ( sin z ) occur at ( z = pi k ), where ( k ) is an integer. Thus, the zeros of ( sin^4 z ), and consequently ( f(z) ), are: [ z_k = pi k, quad k = pm 1, pm 2, pm 3, ldots ] 3. **Order of Zeros**: Since ( sin z ) has simple zeros at ( z = pi k ), the function ( sin^4 z ) will have zeros of order 4 (since ( sin z ) raised to the power of 4). The factor ( z^2 ) in the denominator is non-zero at ( z_k = pi k ). This does not affect the order of the zeros because it remains non-zero at these points. We can express the function ( f(z) ) near each zero as: [ f(z) = frac{(sin z)^4}{z^2} ] Near ( z_k ): [ f(z) approx (z - z_k)^4 cdot varphi(z) ] where ( varphi(z) ) is some analytic function satisfying ( varphi(z_k) neq 0 ). 4. **Conclusion**: From the above analysis, it follows that ( z_k = pi k ) (for ( k = pm 1, pm 2, ldots )) are zeros of the function ( f(z) ), and each zero is of order 4. [ boxed{z_k = pi k, quad k = pm 1, pm 2, pm 3, ldots text{ are zeros of order 4}.} ]