Watson

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question:A convex polyhedron Q has vertices V_1,V_2,ldots,V_n, and 120 edges. It is sliced by n planes P_1,P_2,ldots,P_n, such that each plane P_k cuts all the edges emanating from vertex V_k and each plane P_k adds 2 edges by forming new edges directly interconnecting any two newly created vertices from the cuts on any particular edge originating at V_k. No two planes intersect inside or on the surface of Q. After all slices, n pyramids and a new polyhedron R are formed. How many edges does R have? A) 720 B) 800 C) 840 D) 960 E) 1000

answer:1. **Edges from Original Polyhedron Q:** Each of the 120 edges of Q is cut, creating two new edge segments per edge at each vertex it is connected to. Thus, each original edge contributes two new edges, leading to 2 times 120 = 240 edge additions. 2. **Additional Edges from Planar Interconnections:** Each plane P_k adds 2 edges for every edge originally emitting from V_k. If we assume each vertex is approximately similar in connectivity, we can estimate each vertex connects to about 240 / n of these initial 240 new segments. Since each plane adds 2 edges per original edge emanating from V_k, 2 times 240 / n edges are added per plane, totaling 2 times 240 = 480. 3. **Total Edge Count:** Starting with 120 initial edges and with both new segment additions and plane-inserted edges: - Original and split edges: 120 + 240 = 360 - Additional edges from new planar connections: 480 - Total edges = 360 + 480 = 840. Conclusion: Thus, the new polyhedron R has 840 edges. The final answer is boxed{C) 840}

question:Among the following inequalities, the one that holds true for any x in mathbb{R} is (quadquad). A: frac{1}{x^2 + 1} < 1 B: x^2 + 1 geqslant 2|x| C: lg(x^2 + 1) geqslant lg 2x D: frac{4x}{x^2 + 4} geqslant 1

answer:For option A, when x = 0, the inequality does not hold. For option C, the domain is defined for x > 0, hence it does not hold for all real numbers. For option D, when x leqslant 0, the inequality does not hold. For option B, according to the fundamental inequality, we can derive that x^2 + 1 geqslant 2|x|. Therefore, the answer is boxed{B}. By taking special values, we can determine that options A, C, and D do not hold. The property of inequalities is used to determine that option B holds. This question tests the application of inequalities and is considered a basic question.

question:Given overrightarrow {a} = (2, 0), overrightarrow {b} = (1, 1), then the correct conclusion is ( ) A: overrightarrow {a} is parallel to overrightarrow {b} B: |overrightarrow {a}| = |overrightarrow {b}| C: overrightarrow {a} overrightarrow {b} is perpendicular to overrightarrow {b} D: The angle between overrightarrow {a} and overrightarrow {b} is frac {pi}{6}

answer:For option A, since 2 times 1 - 0 times 1 neq 0, it is incorrect. For option B, |overrightarrow {a}| = 2, |overrightarrow {b}| = sqrt {2}, so it is incorrect. For option C, since overrightarrow {a} = (2, 0), overrightarrow {b} = (1, 1), thus overrightarrow {a} - overrightarrow {b} = (1, -1), and (overrightarrow {a} - overrightarrow {b}) cdot overrightarrow {b} = 1 times 1 + (-1) times 1 = 0, therefore (overrightarrow {a} - overrightarrow {b}) is perpendicular to overrightarrow {b}, so it is correct. For option D, cos<overrightarrow {a}, overrightarrow {b}> = frac { overrightarrow {a} cdot overrightarrow {b}}{| overrightarrow {a}|| overrightarrow {b}|} = frac {2}{2 sqrt {2}} = frac { sqrt {2}}{2}, thus the angle is frac {pi}{4}, so it is incorrect. Therefore, the correct choice is: boxed{C}. This question examines the calculation of vector parallelism, perpendicularity, magnitude, and the angle between vectors, which is a basic question.

question:Among the following functions, the one that is an increasing even function on (0, frac{pi}{2}) is ( ). A: y=|sin x| B: y=|sin 2x| C: y=|cos x| D: y=tan x

answer:The function y=|sin x| is an increasing function on (0, frac{pi}{2}) and is also an even function, so option A satisfies the conditions. Since y=|sin 2x| does not have monotonicity on (0, frac{pi}{2}), option B is excluded. Since the function y=|cos x| is a decreasing function on (0, frac{pi}{2}), option C is excluded. Since y=tan x is an odd function, option D is excluded. Therefore, the correct choice is: boxed{A}. By using the graph and properties of the sine function, we can judge whether each option is correct, thus reaching a conclusion. This question mainly examines the graph and properties of the sine function and is considered a basic question.