answer:According to the problem, f(x)=x^{2}+ax+4 always has two different fixed points in [1,3], which means x=x^{2}+ax+4 has two real roots in [1,3], i.e., x^{2}+(a-1)x+4=0 has two different real roots in [1,3], Let g(x)=x^{2}+(a-1)x+4. It has two different intersections in [1,3], therefore begin{cases} g(1)geqslant 0 g(3)geqslant 0 1 < dfrac {1-a}{2} < 3 (a-1)^{2}-16 > 0end{cases}, which means begin{cases} a+4geqslant 0 3a+10geqslant 0 1 < dfrac {1-a}{2} < 3 (a-1)^{2}-16 > 0end{cases}, Solving these, we get: a in left[- dfrac {10}{3},-3right); Therefore, the answer is: boxed{left[- dfrac {10}{3},-3right)}. A fixed point is essentially the real root of the equation f(x_{0})=x_{0}. The quadratic function f(x)=x^{2}+ax+4 having a fixed point means the equation x=x^{2}+ax+4 has real roots. That is, the equation x=x^{2}+ax+4 has two different real roots, and then solve the inequalities based on the roots. This problem examines the coordinate characteristics of points on the graph of a quadratic function and the comprehensive application of functions and equations. In solving this problem, the knowledge point of the discriminant and roots of a quadratic equation is used.

answer:A. If three lines intersect pairwise, forming an equilateral triangle, this condition is satisfied, but m is not parallel to n, so A is incorrect. B. According to the property of parallel planes, if two planes are parallel, then any line in one plane is parallel to the other plane, so B is correct. C. If the angles formed by m, n with alpha are equal, there is no specific relationship between line m and n, so C is incorrect. D. If the angles formed by gamma with planes alpha, beta are equal, then alpha and beta may intersect or be parallel, so D is incorrect. Therefore, the correct choice is boxed{text{B}}.

answer:Since the function f(x)=sin frac{x+phi}{3} is even, we have frac{phi}{3}=kpi + frac{pi}{2}, where kin mathbb{Z}. Considering the range of phi, we take k=0, which gives us phi = frac{3pi}{2} in [0,2pi]. Hence, the answer is (C): boxed{frac{3pi}{2}}. To solve this problem, we first find an expression for phi using the even function property and then compute the value of phi accordingly. This question tests your understanding of the even-odd property of sine functions, application of trigonometric function formulas, and computational skills.

answer:By Vieta's formulas for a cubic equation of the form (ax^3 + bx^2 + cx + d = 0), the product of the roots (including multiplicities) is given by (-frac{d}{a}). For the given equation (2x^3 - 3x^2 - 8x + 10 = 0), the coefficients (a) and (d) are 2 and 10, respectively. Therefore, the product of the roots is: [ -frac{10}{2} = -5 ] Thus, the product of the roots of the equation (2x^3 - 3x^2 - 8x + 10 = 0) is (boxed{-5}).