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question:Compute the value of binom{21075}{21074}.

answer:The binomial coefficient binom{n}{k} represents the number of ways to choose k items from a set of n items without regard to order. When k = n - 1, the binomial coefficient simplifies to binom{n}{n-1} = n. Therefore, binom{21075}{21074} = 21075.

question:Find the value of the infinite continued fraction frac{1}{x + frac{1}{x + frac{1}{x + ddots}}} when x = frac{26761}{13953}.

answer:The given infinite continued fraction represents the reciprocal of the golden ratio for the given x. Therefore, we need to find the geometric mean of x and frac{1}{x}: frac{1}{x+frac{1}{x+frac{1}{x+ddots}}} = sqrt{x cdot frac{1}{x}} = sqrt{1} However, since the question asks for the specific value when x = frac{26761}{13953}, we can also represent it algebraically. Let y = frac{1}{x+frac{1}{x+frac{1}{x+ddots}}}. Then we have: y = frac{1}{x + y} Solving for y: xy + y^2 = 1 y^2 + xy - 1 = 0 This is a quadratic equation in y. Using the quadratic formula: y = frac{-b pm sqrt{b^2 - 4ac}}{2a} y = frac{-x pm sqrt{x^2 + 4}}{2} Since x = frac{26761}{13953}, we have: y = frac{-frac{26761}{13953} pm sqrt{left(frac{26761}{13953}right)^2 + 4}}{2} Calculating the square root and simplifying, we get: y = frac{sqrt{1494895957} - 26761}{27906} So, the value of the infinite continued fraction is: frac{sqrt{1494895957} - 26761}{27906}

question:Find the roots of the quadratic polynomial -6 sqrt{5} x^2 + 4 sqrt{5} x.

answer:[ begin{array}{l} text{Solve the equation: } -6 sqrt{5} x^2 + 4 sqrt{5} x = 0 text{Factor out } sqrt{5} x: sqrt{5} x (-6x + 4) = 0 text{Set each factor equal to zero: } sqrt{5} x = 0 quad text{or} quad -6x + 4 = 0 text{Solve for } x: x = 0 quad text{or} quad 6x = 4 text{Divide both sides by 6: } x = 0 quad text{or} quad x = frac{4}{6} text{Simplify the fraction: } x = 0 quad text{or} quad x = frac{2}{3} end{array} ] [ text{Answer: } x = 0 quad text{or} quad x = frac{2}{3} ]

question:Find the curl of the vector field mathbf{F} = f(x,y,z)mathbf{i} + g(x,y,z)mathbf{j} + h(x,y,z)mathbf{k} where f(x,y,z) = z^2, g(x,y,z) = ln (y), and h(x,y,z) = sin (y-z^2).

answer:The curl of mathbf{F} is given by nabla times mathbf{F} = left(frac{partial h}{partial y} - frac{partial g}{partial z}right)mathbf{i} + left(frac{partial f}{partial z} - frac{partial h}{partial x}right)mathbf{j} + left(frac{partial g}{partial x} - frac{partial f}{partial y}right)mathbf{k}. Plugging in the given functions, we get begin{split}nabla times mathbf{F} &= left(cos (y-z^2) - 0right)mathbf{i} + left(2z - 0right)mathbf{j} + left(0 - frac{1}{y}right)mathbf{k} &= cos (y-z^2)mathbf{i} + 2zmathbf{j} - frac{1}{y}mathbf{k}. end{split} Therefore, the curl of mathbf{F} is boxed{cos (y-z^2)mathbf{i} + 2zmathbf{j} - frac{1}{y}mathbf{k}}.