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question:6. Find the sum of the integers that belong to the set of values of the function f(x)=log _{2}(5 cos 2 x+11) for x in[1,25(operatorname{arctg}(1 / 3)) cos (pi+arcsin (-0.6)) ; operatorname{arctg} 2] quad (10 points)
answer:# Solution:Since quad cos (pi+arcsin (-0.6))=cos (pi-arcsin 0.6)=-cos (arcsin 0.6)=-0.8, quad then x in[1.25(operatorname{arctg}(1 / 3)) cos (pi+arcsin (-0.6)) ; operatorname{arctg} 2]=[-operatorname{arctg}(1 / 3) ; operatorname{arctg} 2] Therefore, 2 x in[-2 operatorname{arctg}(1 / 3) ; 2 operatorname{arctg} 2] Since quad 0<operatorname{arctg}(1 / 3)<operatorname{arctg} 1=pi / 4, quad 0<2 operatorname{arctg}(1 / 3)<pi / 2, -pi / 2<-2 operatorname{arctg}(1 / 3)<0, quad and quad pi / 4<operatorname{arctg} 2<pi / 2, quad pi / 2<2 operatorname{arctg} 2<pi, quad then cos 2 x in[cos (2 operatorname{arctg} 2) ; 1] . quad Using the formula cos 2 alpha=frac{1-operatorname{tg}^{2} alpha}{1+operatorname{tg}^{2} alpha}, quad we get cos (2 operatorname{arctg} 2)=-0.6, quad and quad cos 2 x in[-0.6 ; 1]. From this, we have 5 cos 2 x+11 in[8 ; 16], and f(x)=log _{2}(5 cos 2 x+11) inleft[log _{2} 8 ; log _{2} 16right]=[3 ; 4].The interval [3 ; 4] is the range of the function f(x)=log _{2}(5 cos 2 x+11) for x in[1.25(operatorname{arctg}(1 / 3)) cos (pi+arcsin (-0.6)) ; operatorname{arctg} 2].The sum of the integers in the interval [3 ; 4] is 7.Answer: E_{f}=[3 ; 4], the sum of the integers is 7.
question:Given the set {x|x^2-3x+2=0} is a superset of {x|ax-2=0}, find the set of all possible values of a.
answer:From the given condition, we have {1, 2} is a superset of {x|ax-2=0}.Since the subsets of {1, 2} are emptyset, {1}, {2}, and {1, 2}, we analyze each case:- If {x|ax-2=0} = emptyset, then a=0.- If {x|ax-2=0} = {1}, then solving a-2=0 gives a=2.- If {x|ax-2=0} = {2}, then solving 2a-2=0 gives a=1.Therefore, the possible values of a are 0, 1, and 2.Thus, the set of all possible values of a is boxed{{0, 1, 2}}.
question:Given that the vertex of angle (alpha) coincides with the origin of the Cartesian coordinate system, its initial side is on the non-negative half-axis of (x), and its terminal side passes through point (P(-1,2)), find the value of (sin (2alpha+ dfrac {2}{3}pi)).
answer:From the given information and the definition of trigonometric functions, we can obtain (sin alpha= dfrac {2}{ sqrt {5}}) and (cos alpha= dfrac {-1}{ sqrt {5}}).Therefore, (sin 2alpha=2sin alphacos alpha=- dfrac {4}{5}) and (cos 2alpha=cos ^{2}alpha-sin ^{2}alpha=- dfrac {3}{5}).Thus, (sin (2alpha+ dfrac {2}{3}pi)=sin 2alphacos dfrac {2pi}{3}+cos 2alphasin dfrac {2pi}{3})(=- dfrac {4}{5}times(- dfrac {1}{2})+(- dfrac {3}{5})times dfrac { sqrt {3}}{2}= dfrac {4-3 sqrt {3}}{10}).So, the final answer is boxed{dfrac {4-3 sqrt {3}}{10}}.
question:7. If p, q are prime numbers, and 5p+3q=91, then p=ldots, q= qquad
answer:ニ、7.17,2.Since 5p + 3q = 91 is an odd number, one of p or q must be even, and the only even prime number is 2.Upon inspection, we find that p = 17, q = 2.
question:A7. A straight angle is divided into four angles by three rays, the sizes of which are in the ratio 1: 2: 5: 1. What is the measure of the largest angle?(A) 20^{circ}(B) 50^{circ}(C) 100^{circ}(D) 200^{circ}(E) 270^{circ}
answer:A7. The sizes of the angles, which in the ratio 1: 2: 5: 1 form a straight angle, are 1 x, 2 x, 5 x, and 1 x. It holds that: 1 x+2 x+5 x+1 x=180^{circ} or x=20^{circ}. The largest angle measures 5 x=100^{circ}.
question:During a certain period, the probability of no rain in location A is 0.3, and the probability of no rain in location B is 0.4. Assuming that the rain status in these two locations is independent of each other, the probability of it raining in both locations during this period is ( )A: 0.12B: 0.88C: 0.28D: 0.42
answer:Given the problem, this is a case of the probability of two independent events occurring simultaneously.∵ The probability of no rain in location A is 0.3, and the probability of no rain in location B is 0.4,∴ The probability of it raining in both location A and location B during this period isP = (1-0.3)(1-0.4) = 0.42.Therefore, the correct choice is boxed{D}.