answer:(1) From the given function, we can simplify it using trigonometric identities as follows: f(x)=2sin frac{3}{2}x cdot (cos frac{3}{2}x+sqrt{3}sin frac{3}{2}x)-sqrt{3} =2sin(frac{3}{2}x+frac{pi}{3})-sqrt{3} (2) To find the interval where f(x) is monotonically increasing, we need to find the range of x where sin(frac{3}{2}x+frac{pi}{3}) is increasing. This occurs when frac{3}{2}x+frac{pi}{3} is in the first and second quadrants. -frac{pi }{2}+2kpi leqslant frac{3}{2}x+frac{pi}{3}leqslant frac{pi }{2}+2kpi, for any integer k. Solving for x, we get -frac{pi }{18}+frac{2kpi }{3}leqslant xleqslant frac{5pi }{18}+frac{2kpi }{3}. Thus, the function f(x) is monotonically increasing in the intervals [frac{-π}{18}+frac{2kπ}{3}, frac{5π}{18}+frac{2kπ}{3}] for all integers k. (3) When xin [0,frac{pi }{6}], the range of f(x) is [-√3, 1]. Since 2+fleft( x right) > 0, we can rewrite the inequality as follows: mfleft( x right)+2mgeqslant fleft( x right) Rightarrow mgeqslant frac{fleft( x right)}{2+fleft( x right)}=1-frac{2}{2+fleft( x right)}. The maximum value of frac{fleft( x right)}{2+fleft( x right)} is frac{1}{3} given the range of f(x). Therefore, the range of values for the real number m is boxed{mgeqslant frac{1}{3}}.

answer:To find the probability that the sum of two dice is either 7 or 11, we need to consider all the possible outcomes when two dice are rolled and then identify the outcomes that result in a sum of 7 or 11. There are a total of 6 sides on each die, so when two dice are rolled, there are 6 x 6 = 36 possible outcomes. Now, let's find the outcomes that result in a sum of 7 or 11: For a sum of 7, the possible combinations are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) There are 6 outcomes that result in a sum of 7. For a sum of 11, the possible combinations are: (5, 6), (6, 5) There are 2 outcomes that result in a sum of 11. In total, there are 6 + 2 = 8 outcomes that result in a sum of either 7 or 11. To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes: Probability = Number of favorable outcomes / Total number of possible outcomes Probability = 8 / 36 Probability = 2 / 9 So, the probability that the sum of two dice is either 7 or 11 is boxed{2/9} .

answer:(1) Let the coordinates of point C be C(x, y). Since M is the midpoint of AC, we have: begin{cases} frac{x + 5}{2} = 0 frac{y - 3}{2} = 0 end{cases} Solving these equations, we get: begin{cases} x = -5 y = -3 end{cases} Thus, the coordinates of point C are C(-5, -3). (2) Given A(5,-2), B(7,3), and C(-5, -3), we can find the midpoints M and N: M = (0, -frac{5}{2}) quad text{and} quad N = (1, 0) The equation of line MN is given by: frac{x - 0}{1 - 0} = frac{y + frac{5}{2}}{0 + frac{5}{2}} Simplifying and rearranging, we get: boxed{5x - 2y - 5 = 0} (3) First, find the slope of line AB: k_{AB} = frac{-2 - 3}{5 - 7} = frac{5}{2} Now, the equation of line AB is: y - (-2) = frac{5}{2}(x - 5) Simplifying, we get: boxed{5x - 2y - 29 = 0} To find the area of the triangle formed by line AB and the coordinate axes, first find the x- and y-intercepts of the line AB: Let x = 0, then y = -frac{29}{2}; Let y = 0, then x = frac{29}{5}. The area of the triangle is given by: S = frac{1}{2} times frac{29}{2} times frac{29}{5} = boxed{frac{841}{20}}

answer:Let ( n ) be the number of sides of the polygon. The sum of the interior angles of any ( n )-sided polygon is given by ( 180(n-2) ) degrees. If each interior angle measures ( 156^circ ), then the sum of the interior angles is ( 156n ). Setting these equal gives: [ 180(n-2) = 156n ] Expanding and simplifying: [ 180n - 360 = 156n ] [ 24n = 360 ] [ n = frac{360}{24} ] [ n = boxed{15} ] We can confirm this by considering the exterior angles, which are ( 180^circ - 156^circ = 24^circ ) each. Since the sum of the exterior angles of any polygon is ( 360^circ ), the number of sides is ( frac{360^circ}{24^circ} = 15 ). Conclusion: The regular polygon with each interior angle measuring ( 156^circ ) has ( boxed{15} ) sides.