answer：Okay, so I have this problem about a sequence defined by a recurrence relation. Let me try to understand it step by step.The sequence is given by ( a_{n+2} = |a_{n+1}| - a_n ) for ( n = 1, 2, ldots ), with initial terms ( a_1 = a ) and ( a_2 = b ). The set ( M ) is defined as all the elements of the sequence, so ( M = {a_n | n in mathbb{N}^*} ).There are three parts to the problem:(I) If ( a = 1 ) and ( b = 2 ), list the elements of set ( M ) explicitly.(II) If ( a < 0 ) and ( b < 0 ), determine whether the sequence is periodic and explain why.(III) If ( a geq 0 ), ( b geq 0 ), and ( a + b neq 0 ), find the minimum number of elements in set ( M ).Let me tackle each part one by one.**Part I: ( a = 1 ) and ( b = 2 )**Alright, so I need to compute the terms of the sequence starting from ( a_1 = 1 ) and ( a_2 = 2 ), using the recurrence relation ( a_{n+2} = |a_{n+1}| - a_n ).Let me compute the first few terms:- ( a_1 = 1 )- ( a_2 = 2 )- ( a_3 = |a_2| - a_1 = |2| - 1 = 2 - 1 = 1 )- ( a_4 = |a_3| - a_2 = |1| - 2 = 1 - 2 = -1 )- ( a_5 = |a_4| - a_3 = |-1| - 1 = 1 - 1 = 0 )- ( a_6 = |a_5| - a_4 = |0| - (-1) = 0 + 1 = 1 )- ( a_7 = |a_6| - a_5 = |1| - 0 = 1 - 0 = 1 )- ( a_8 = |a_7| - a_6 = |1| - 1 = 1 - 1 = 0 )- ( a_9 = |a_8| - a_7 = |0| - 1 = 0 - 1 = -1 )- ( a_{10} = |a_9| - a_8 = |-1| - 0 = 1 - 0 = 1 )- ( a_{11} = |a_{10}| - a_9 = |1| - (-1) = 1 + 1 = 2 )- ( a_{12} = |a_{11}| - a_{10} = |2| - 1 = 2 - 1 = 1 )Hmm, I notice that after ( a_5 ), the sequence starts repeating some values. Let me list the terms I have so far:1, 2, 1, -1, 0, 1, 1, 0, -1, 1, 2, 1, ...So, the elements in set ( M ) are the distinct values from this sequence. Looking at the terms, the distinct values are 1, 2, -1, and 0. After ( a_5 ), the sequence cycles through 1, -1, 0, 1, 1, 0, -1, 1, 2, 1, and so on. So, the set ( M ) includes 1, 2, -1, and 0.Wait, let me check if there are any more distinct values beyond these. From ( a_1 ) to ( a_{12} ), I only see 1, 2, -1, and 0. So, I think that's all.**Conclusion for Part I:** The set ( M ) is ( {1, 2, -1, 0} ).**Part II: ( a < 0 ) and ( b < 0 )**I need to determine if the sequence is periodic when both initial terms are negative. Let me try to compute the sequence for some negative ( a ) and ( b ) to see if a pattern emerges.Let me choose ( a = -1 ) and ( b = -2 ) as an example.Compute the terms:- ( a_1 = -1 )- ( a_2 = -2 )- ( a_3 = |a_2| - a_1 = |-2| - (-1) = 2 + 1 = 3 )- ( a_4 = |a_3| - a_2 = |3| - (-2) = 3 + 2 = 5 )- ( a_5 = |a_4| - a_3 = |5| - 3 = 5 - 3 = 2 )- ( a_6 = |a_5| - a_4 = |2| - 5 = 2 - 5 = -3 )- ( a_7 = |a_6| - a_5 = |-3| - 2 = 3 - 2 = 1 )- ( a_8 = |a_7| - a_6 = |1| - (-3) = 1 + 3 = 4 )- ( a_9 = |a_8| - a_7 = |4| - 1 = 4 - 1 = 3 )- ( a_{10} = |a_9| - a_8 = |3| - 4 = 3 - 4 = -1 )- ( a_{11} = |a_{10}| - a_9 = |-1| - 3 = 1 - 3 = -2 )- ( a_{12} = |a_{11}| - a_{10} = |-2| - (-1) = 2 + 1 = 3 )- ( a_{13} = |a_{12}| - a_{11} = |3| - (-2) = 3 + 2 = 5 )- ( a_{14} = |a_{13}| - a_{12} = |5| - 3 = 5 - 3 = 2 )- ( a_{15} = |a_{14}| - a_{13} = |2| - 5 = 2 - 5 = -3 )- ( a_{16} = |a_{15}| - a_{14} = |-3| - 2 = 3 - 2 = 1 )- ( a_{17} = |a_{16}| - a_{15} = |1| - (-3) = 1 + 3 = 4 )- ( a_{18} = |a_{17}| - a_{16} = |4| - 1 = 4 - 1 = 3 )- ( a_{19} = |a_{18}| - a_{17} = |3| - 4 = 3 - 4 = -1 )- ( a_{20} = |a_{19}| - a_{18} = |-1| - 3 = 1 - 3 = -2 )Looking at the terms from ( a_1 ) to ( a_{20} ):-1, -2, 3, 5, 2, -3, 1, 4, 3, -1, -2, 3, 5, 2, -3, 1, 4, 3, -1, -2,...I notice that starting from ( a_1 ), the sequence goes: -1, -2, 3, 5, 2, -3, 1, 4, 3, -1, -2, 3, 5, 2, -3, 1, 4, 3, -1, -2,...So, after ( a_{10} = -1 ) and ( a_{11} = -2 ), the sequence repeats the same pattern as from ( a_1 ) and ( a_2 ). That is, ( a_{10} = a_1 ) and ( a_{11} = a_2 ). Therefore, the sequence is periodic with a period of 9.Wait, let me check:From ( a_1 ) to ( a_{10} ): -1, -2, 3, 5, 2, -3, 1, 4, 3, -1From ( a_{11} ) to ( a_{20} ): -2, 3, 5, 2, -3, 1, 4, 3, -1, -2So, the sequence from ( a_{11} ) is the same as from ( a_2 ). Therefore, the period is 9 because ( a_{10} = a_1 ) and ( a_{11} = a_2 ), so the sequence repeats every 9 terms.But wait, is the period exactly 9 or could it be shorter? Let me see if the sequence repeats earlier.Looking at the terms:-1, -2, 3, 5, 2, -3, 1, 4, 3, -1, -2, 3, 5, 2, -3, 1, 4, 3, -1, -2,...From ( a_1 ) to ( a_{10} ): -1, -2, 3, 5, 2, -3, 1, 4, 3, -1From ( a_{11} ) to ( a_{20} ): -2, 3, 5, 2, -3, 1, 4, 3, -1, -2So, the sequence from ( a_{11} ) is the same as from ( a_2 ) to ( a_{10} ), but shifted by one. Therefore, the period is 9 because after 9 terms, the sequence repeats the initial terms.Wait, actually, the sequence from ( a_1 ) to ( a_{10} ) is 10 terms, but the period is 9 because ( a_{10} = a_1 ) and ( a_{11} = a_2 ). So, the period is 9.Let me confirm this by checking if ( a_{n+9} = a_n ) for all ( n ).From ( a_1 ) to ( a_{10} ): -1, -2, 3, 5, 2, -3, 1, 4, 3, -1From ( a_{10} ) to ( a_{19} ): -1, -2, 3, 5, 2, -3, 1, 4, 3, -1Yes, so ( a_{10} = a_1 ), ( a_{11} = a_2 ), and so on. Therefore, the sequence is periodic with period 9.But wait, is this specific to ( a = -1 ) and ( b = -2 ), or does it hold for any ( a < 0 ) and ( b < 0 )?Let me try another example with ( a = -2 ) and ( b = -3 ).Compute the terms:- ( a_1 = -2 )- ( a_2 = -3 )- ( a_3 = |a_2| - a_1 = |-3| - (-2) = 3 + 2 = 5 )- ( a_4 = |a_3| - a_2 = |5| - (-3) = 5 + 3 = 8 )- ( a_5 = |a_4| - a_3 = |8| - 5 = 8 - 5 = 3 )- ( a_6 = |a_5| - a_4 = |3| - 8 = 3 - 8 = -5 )- ( a_7 = |a_6| - a_5 = |-5| - 3 = 5 - 3 = 2 )- ( a_8 = |a_7| - a_6 = |2| - (-5) = 2 + 5 = 7 )- ( a_9 = |a_8| - a_7 = |7| - 2 = 7 - 2 = 5 )- ( a_{10} = |a_9| - a_8 = |5| - 7 = 5 - 7 = -2 )- ( a_{11} = |a_{10}| - a_9 = |-2| - 5 = 2 - 5 = -3 )- ( a_{12} = |a_{11}| - a_{10} = |-3| - (-2) = 3 + 2 = 5 )- ( a_{13} = |a_{12}| - a_{11} = |5| - (-3) = 5 + 3 = 8 )- ( a_{14} = |a_{13}| - a_{12} = |8| - 5 = 8 - 5 = 3 )- ( a_{15} = |a_{14}| - a_{13} = |3| - 8 = 3 - 8 = -5 )- ( a_{16} = |a_{15}| - a_{14} = |-5| - 3 = 5 - 3 = 2 )- ( a_{17} = |a_{16}| - a_{15} = |2| - (-5) = 2 + 5 = 7 )- ( a_{18} = |a_{17}| - a_{16} = |7| - 2 = 7 - 2 = 5 )- ( a_{19} = |a_{18}| - a_{17} = |5| - 7 = 5 - 7 = -2 )- ( a_{20} = |a_{19}| - a_{18} = |-2| - 5 = 2 - 5 = -3 )So, the sequence is:-2, -3, 5, 8, 3, -5, 2, 7, 5, -2, -3, 5, 8, 3, -5, 2, 7, 5, -2, -3,...Again, we see that ( a_{10} = a_1 = -2 ) and ( a_{11} = a_2 = -3 ), so the sequence repeats every 9 terms.This suggests that regardless of the initial negative values ( a ) and ( b ), the sequence becomes periodic with period 9.But why is that? Let me think about the structure of the recurrence relation.The recurrence is ( a_{n+2} = |a_{n+1}| - a_n ). When both ( a ) and ( b ) are negative, the first term ( a_3 = |b| - a ). Since ( b ) is negative, ( |b| = -b ), so ( a_3 = -b - a ). Similarly, ( a_4 = |a_3| - b ). Depending on the sign of ( a_3 ), this can lead to positive or negative terms.But in both examples I tried, the sequence eventually cycles back to the initial terms after 9 steps. This suggests that the period is 9 for any negative initial terms.**Conclusion for Part II:** The sequence is periodic with period 9 when both ( a ) and ( b ) are negative.**Part III: ( a geq 0 ), ( b geq 0 ), and ( a + b neq 0 ). Find the minimum number of elements in set ( M ).**So, we need to find the smallest possible size of ( M ) given that ( a ) and ( b ) are non-negative and not both zero.Let me consider different cases for ( a ) and ( b ).**Case 1: ( a = 0 ) and ( b > 0 )**Let me set ( a = 0 ) and ( b = c ) where ( c > 0 ).Compute the sequence:- ( a_1 = 0 )- ( a_2 = c )- ( a_3 = |c| - 0 = c )- ( a_4 = |c| - c = 0 )- ( a_5 = |0| - c = -c )- ( a_6 = |-c| - 0 = c )- ( a_7 = |c| - (-c) = c + c = 2c )- ( a_8 = |2c| - c = 2c - c = c )- ( a_9 = |c| - 2c = c - 2c = -c )- ( a_{10} = |-c| - c = c - c = 0 )- ( a_{11} = |0| - (-c) = 0 + c = c )- ( a_{12} = |c| - 0 = c )- ( a_{13} = |c| - c = 0 )- ...So, the sequence cycles through 0, c, c, 0, -c, c, 2c, c, -c, 0, c, c, 0, ...The distinct elements are 0, c, -c, 2c. So, set ( M ) has 4 elements.**Case 2: ( a > 0 ) and ( b = 0 )**Similarly, set ( a = c ) and ( b = 0 ).Compute the sequence:- ( a_1 = c )- ( a_2 = 0 )- ( a_3 = |0| - c = -c )- ( a_4 = |-c| - 0 = c )- ( a_5 = |c| - (-c) = c + c = 2c )- ( a_6 = |2c| - c = 2c - c = c )- ( a_7 = |c| - 2c = c - 2c = -c )- ( a_8 = |-c| - c = c - c = 0 )- ( a_9 = |0| - (-c) = 0 + c = c )- ( a_{10} = |c| - 0 = c )- ( a_{11} = |c| - c = 0 )- ( a_{12} = |0| - c = -c )- ...The sequence cycles through c, 0, -c, c, 2c, c, -c, 0, c, c, 0, -c,...Again, the distinct elements are c, 0, -c, 2c. So, set ( M ) has 4 elements.**Case 3: ( a > 0 ) and ( b > 0 )**Now, both ( a ) and ( b ) are positive. Let me consider different subcases based on the relationship between ( a ) and ( b ).**Subcase 3.1: ( a = b )**Let ( a = b = c ).Compute the sequence:- ( a_1 = c )- ( a_2 = c )- ( a_3 = |c| - c = 0 )- ( a_4 = |0| - c = -c )- ( a_5 = |-c| - 0 = c )- ( a_6 = |c| - (-c) = c + c = 2c )- ( a_7 = |2c| - c = 2c - c = c )- ( a_8 = |c| - 2c = c - 2c = -c )- ( a_9 = |-c| - c = c - c = 0 )- ( a_{10} = |0| - (-c) = 0 + c = c )- ( a_{11} = |c| - 0 = c )- ( a_{12} = |c| - c = 0 )- ...The sequence cycles through c, c, 0, -c, c, 2c, c, -c, 0, c, c, 0,...Distinct elements: c, 0, -c, 2c. So, set ( M ) has 4 elements.**Subcase 3.2: ( a neq b )**Let me consider two scenarios: ( a > b ) and ( a < b ).**Subsubcase 3.2.1: ( a > b )**Let me choose ( a = 2 ) and ( b = 1 ).Compute the sequence:- ( a_1 = 2 )- ( a_2 = 1 )- ( a_3 = |1| - 2 = 1 - 2 = -1 )- ( a_4 = |-1| - 1 = 1 - 1 = 0 )- ( a_5 = |0| - (-1) = 0 + 1 = 1 )- ( a_6 = |1| - 0 = 1 - 0 = 1 )- ( a_7 = |1| - 1 = 1 - 1 = 0 )- ( a_8 = |0| - 1 = 0 - 1 = -1 )- ( a_9 = |-1| - 0 = 1 - 0 = 1 )- ( a_{10} = |1| - (-1) = 1 + 1 = 2 )- ( a_{11} = |2| - 1 = 2 - 1 = 1 )- ( a_{12} = |1| - 2 = 1 - 2 = -1 )- ...The sequence cycles through 2, 1, -1, 0, 1, 1, 0, -1, 1, 2, 1, -1,...Distinct elements: 2, 1, -1, 0. So, set ( M ) has 4 elements.**Subsubcase 3.2.2: ( a < b )**Let me choose ( a = 1 ) and ( b = 2 ) (as in Part I).Compute the sequence:- ( a_1 = 1 )- ( a_2 = 2 )- ( a_3 = |2| - 1 = 2 - 1 = 1 )- ( a_4 = |1| - 2 = 1 - 2 = -1 )- ( a_5 = |-1| - 1 = 1 - 1 = 0 )- ( a_6 = |0| - (-1) = 0 + 1 = 1 )- ( a_7 = |1| - 0 = 1 - 0 = 1 )- ( a_8 = |1| - 1 = 1 - 1 = 0 )- ( a_9 = |0| - 1 = 0 - 1 = -1 )- ( a_{10} = |-1| - 0 = 1 - 0 = 1 )- ( a_{11} = |1| - (-1) = 1 + 1 = 2 )- ( a_{12} = |2| - 1 = 2 - 1 = 1 )- ...The sequence cycles through 1, 2, 1, -1, 0, 1, 1, 0, -1, 1, 2, 1,...Distinct elements: 1, 2, -1, 0. So, set ( M ) has 4 elements.**Conclusion for Part III:** In all cases where ( a geq 0 ), ( b geq 0 ), and ( a + b neq 0 ), the set ( M ) has at least 4 distinct elements. From the examples above, we see that 4 elements are achievable, so the minimum number of elements in ( M ) is 4.**Final Answer**(I) The set ( M ) is ( boxed{{1, 2, -1, 0}} ).(II) The sequence is periodic with period 9.(III) The minimum number of elements in set ( M ) is ( boxed{4} ).

answer：Alright, so I have this problem here about a sequence. The general term is given as ( a_n = 2(n + 1) + 3 ). I need to figure out if this sequence is arithmetic and, if so, what the common difference is. The options are (A) with a difference of 2, (B) with 3, (C) with 5, or (D) not an arithmetic sequence at all.Okay, first things first. Let me recall what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference. So, if I can show that ( a_{n+1} - a_n ) is the same for any ( n ), then it's arithmetic, and that difference will be the common difference.Given ( a_n = 2(n + 1) + 3 ), let me simplify this expression first to make it easier to work with. Expanding the terms, I get:( a_n = 2n + 2 + 3 )( a_n = 2n + 5 )Hmm, that looks simpler. So, the general term is ( a_n = 2n + 5 ). Now, to check if it's arithmetic, I can compute the difference between consecutive terms.Let me compute ( a_{n+1} ):( a_{n+1} = 2(n + 1) + 5 )( a_{n+1} = 2n + 2 + 5 )( a_{n+1} = 2n + 7 )Now, subtract ( a_n ) from ( a_{n+1} ):( a_{n+1} - a_n = (2n + 7) - (2n + 5) )( a_{n+1} - a_n = 2n + 7 - 2n - 5 )( a_{n+1} - a_n = 2 )So, the difference between consecutive terms is 2, which is constant. That means this sequence is indeed arithmetic with a common difference of 2.Wait, let me double-check to make sure I didn't make a mistake. Maybe I should compute the first few terms to see if they follow an arithmetic progression.Let's compute ( a_1 ):( a_1 = 2(1) + 5 = 2 + 5 = 7 )( a_2 = 2(2) + 5 = 4 + 5 = 9 )( a_3 = 2(3) + 5 = 6 + 5 = 11 )( a_4 = 2(4) + 5 = 8 + 5 = 13 )So, the sequence starts as 7, 9, 11, 13, and so on. The differences between terms are 2 each time (9 - 7 = 2, 11 - 9 = 2, etc.). That confirms it's an arithmetic sequence with a common difference of 2.Just to ensure I'm thorough, let me think about the original expression ( a_n = 2(n + 1) + 3 ). If I hadn't simplified it, would I have gotten the same result? Let's see:Compute ( a_{n+1} ) using the original expression:( a_{n+1} = 2((n + 1) + 1) + 3 )( a_{n+1} = 2(n + 2) + 3 )( a_{n+1} = 2n + 4 + 3 )( a_{n+1} = 2n + 7 )Then, subtract ( a_n ):( a_{n+1} - a_n = (2n + 7) - (2(n + 1) + 3) )( a_{n+1} - a_n = 2n + 7 - 2n - 2 - 3 )( a_{n+1} - a_n = 2n + 7 - 2n - 5 )( a_{n+1} - a_n = 2 )Same result. So, regardless of whether I simplify the general term first or compute ( a_{n+1} ) directly, the common difference is 2. Therefore, the sequence is arithmetic with a common difference of 2.I think I've covered all bases here. I simplified the expression, checked the difference between consecutive terms, computed the first few terms to verify, and even double-checked using the original expression. Everything points to option (A) being correct.

answer：Alright, so I have this problem about translating the graph of ( f(x) = sin 2x ) to the right by ( varphi ) units, where ( 0 < varphi < frac{pi}{2} ). This translation gives me a new function ( g(x) ). The problem states two conditions: first, that ( g(x) ) is monotonically increasing on the interval ( left[ 0, frac{pi}{3} right] ), and second, that the largest negative zero of ( g(x) ) lies within the interval ( left( -frac{pi}{3}, -frac{pi}{12} right) ). I need to find the range of ( varphi ) that satisfies both conditions.Okay, let's break this down step by step. First, translating ( f(x) = sin 2x ) to the right by ( varphi ) units means that ( g(x) = sin(2(x - varphi)) ). Simplifying that, ( g(x) = sin(2x - 2varphi) ). So, ( g(x) ) is a sine function with a phase shift of ( 2varphi ).Next, I need to consider the monotonicity of ( g(x) ). The sine function ( sin(theta) ) is increasing on intervals where its derivative ( cos(theta) ) is positive. The derivative of ( g(x) ) is ( g'(x) = 2cos(2x - 2varphi) ). So, ( g(x) ) is increasing when ( cos(2x - 2varphi) > 0 ).The cosine function is positive in the intervals ( (-frac{pi}{2} + 2kpi, frac{pi}{2} + 2kpi) ) for any integer ( k ). Therefore, ( 2x - 2varphi ) must lie within one of these intervals for ( g(x) ) to be increasing. Let's solve for ( x ):[-frac{pi}{2} + 2kpi < 2x - 2varphi < frac{pi}{2} + 2kpi]Dividing through by 2:[-frac{pi}{4} + kpi < x - varphi < frac{pi}{4} + kpi]Adding ( varphi ) to all parts:[varphi - frac{pi}{4} + kpi < x < varphi + frac{pi}{4} + kpi]So, the intervals where ( g(x) ) is increasing are ( left( varphi - frac{pi}{4} + kpi, varphi + frac{pi}{4} + kpi right) ) for any integer ( k ).Given that ( g(x) ) is monotonically increasing on ( left[ 0, frac{pi}{3} right] ), this interval must lie entirely within one of the increasing intervals of ( g(x) ). Since ( varphi ) is between 0 and ( frac{pi}{2} ), let's consider ( k = 0 ) first:[varphi - frac{pi}{4} < 0 quad text{and} quad varphi + frac{pi}{4} > frac{pi}{3}]These inequalities ensure that the interval ( left[ 0, frac{pi}{3} right] ) is contained within ( left( varphi - frac{pi}{4}, varphi + frac{pi}{4} right) ).Solving the first inequality:[varphi - frac{pi}{4} < 0 implies varphi < frac{pi}{4}]Solving the second inequality:[varphi + frac{pi}{4} > frac{pi}{3} implies varphi > frac{pi}{3} - frac{pi}{4} = frac{4pi - 3pi}{12} = frac{pi}{12}]So, from the monotonicity condition, ( varphi ) must satisfy:[frac{pi}{12} < varphi < frac{pi}{4}]Alright, that's the first condition. Now, moving on to the second condition: the largest negative zero of ( g(x) ) is in ( left( -frac{pi}{3}, -frac{pi}{12} right) ).First, let's find the zeros of ( g(x) = sin(2x - 2varphi) ). The zeros occur when:[2x - 2varphi = kpi implies x = varphi + frac{kpi}{2}]for any integer ( k ).We are interested in the negative zeros, so let's consider negative values of ( k ). The largest negative zero would correspond to the smallest negative ( k ) such that ( x ) is still negative.Let's take ( k = -1 ):[x = varphi - frac{pi}{2}]Is this the largest negative zero? Let's check ( k = -2 ):[x = varphi - pi]This is more negative than ( varphi - frac{pi}{2} ), so indeed, the largest negative zero is at ( x = varphi - frac{pi}{2} ).We are told that this zero lies in ( left( -frac{pi}{3}, -frac{pi}{12} right) ). Therefore:[-frac{pi}{3} < varphi - frac{pi}{2} < -frac{pi}{12}]Let's solve these inequalities for ( varphi ).First inequality:[-frac{pi}{3} < varphi - frac{pi}{2}]Adding ( frac{pi}{2} ) to both sides:[-frac{pi}{3} + frac{pi}{2} < varphi]Calculating ( -frac{pi}{3} + frac{pi}{2} ):[-frac{2pi}{6} + frac{3pi}{6} = frac{pi}{6}]So, ( varphi > frac{pi}{6} ).Second inequality:[varphi - frac{pi}{2} < -frac{pi}{12}]Adding ( frac{pi}{2} ) to both sides:[varphi < -frac{pi}{12} + frac{pi}{2}]Calculating ( -frac{pi}{12} + frac{pi}{2} ):[-frac{pi}{12} + frac{6pi}{12} = frac{5pi}{12}]So, ( varphi < frac{5pi}{12} ).Putting these together, from the zero condition, we have:[frac{pi}{6} < varphi < frac{5pi}{12}]Now, combining this with the earlier result from the monotonicity condition:From monotonicity: ( frac{pi}{12} < varphi < frac{pi}{4} )From zero condition: ( frac{pi}{6} < varphi < frac{5pi}{12} )So, the overlap of these two intervals is:[frac{pi}{6} < varphi < frac{pi}{4}]But wait, we also need to consider the endpoints. Let's check if ( varphi = frac{pi}{6} ) or ( varphi = frac{pi}{4} ) are included.From the monotonicity condition, ( varphi ) must be greater than ( frac{pi}{12} ) and less than ( frac{pi}{4} ). So, ( varphi = frac{pi}{4} ) is allowed, but ( varphi = frac{pi}{6} ) is not included in the zero condition since it's strictly greater than ( frac{pi}{6} ).Wait, actually, looking back at the zero condition, the inequality was ( varphi > frac{pi}{6} ), so ( varphi = frac{pi}{6} ) is not included. Similarly, the upper bound from the zero condition is ( varphi < frac{5pi}{12} ), but from the monotonicity, ( varphi < frac{pi}{4} approx 0.785 ), while ( frac{5pi}{12} approx 1.308 ), so the stricter upper bound is ( frac{pi}{4} ).Therefore, combining both conditions, the range of ( varphi ) is:[frac{pi}{6} < varphi leq frac{pi}{4}]But wait, let me double-check. The zero condition gives ( varphi < frac{5pi}{12} ), but since ( frac{pi}{4} ) is approximately 0.785 and ( frac{5pi}{12} ) is approximately 1.308, the upper bound is actually ( frac{pi}{4} ) because of the monotonicity condition.So, the overlapping interval is ( frac{pi}{6} < varphi leq frac{pi}{4} ).Looking back at the options:A: ( left[ frac{pi}{12}, frac{pi}{4} right] )B: ( left[ frac{pi}{6}, frac{5pi}{12} right) )C: ( left[ frac{pi}{6}, frac{pi}{3} right] )D: ( left( frac{pi}{6}, frac{pi}{4} right] )So, my conclusion is ( left( frac{pi}{6}, frac{pi}{4} right] ), which is option D.But wait, let me make sure I didn't make a mistake in the zero condition. The largest negative zero is ( varphi - frac{pi}{2} ), and it's in ( left( -frac{pi}{3}, -frac{pi}{12} right) ). So:[-frac{pi}{3} < varphi - frac{pi}{2} < -frac{pi}{12}]Adding ( frac{pi}{2} ):[-frac{pi}{3} + frac{pi}{2} < varphi < -frac{pi}{12} + frac{pi}{2}]Calculating:Left side: ( -frac{2pi}{6} + frac{3pi}{6} = frac{pi}{6} )Right side: ( -frac{pi}{12} + frac{6pi}{12} = frac{5pi}{12} )So, ( frac{pi}{6} < varphi < frac{5pi}{12} )But from the monotonicity, ( frac{pi}{12} < varphi < frac{pi}{4} )So, overlapping these, ( frac{pi}{6} < varphi < frac{pi}{4} )But since ( frac{pi}{4} ) is approximately 0.785 and ( frac{5pi}{12} ) is approximately 1.308, the upper limit is ( frac{pi}{4} ).Now, considering whether ( varphi = frac{pi}{4} ) is allowed. Let's check the zero condition:If ( varphi = frac{pi}{4} ), then the largest negative zero is ( frac{pi}{4} - frac{pi}{2} = -frac{pi}{4} ). Is ( -frac{pi}{4} ) in ( left( -frac{pi}{3}, -frac{pi}{12} right) )?Calculating:( -frac{pi}{3} approx -1.047 ), ( -frac{pi}{4} approx -0.785 ), ( -frac{pi}{12} approx -0.2618 )So, ( -frac{pi}{4} ) is approximately -0.785, which is greater than ( -frac{pi}{3} ) (-1.047) and less than ( -frac{pi}{12} ) (-0.2618)? Wait, no, -0.785 is less than -0.2618? No, -0.785 is less than -0.2618 in terms of magnitude, but in terms of actual value, -0.785 is less than -0.2618 because it's further to the left on the number line.Wait, actually, ( -frac{pi}{4} approx -0.785 ) is greater than ( -frac{pi}{3} approx -1.047 ) because -0.785 is to the right of -1.047. So, ( -frac{pi}{4} ) is indeed in ( left( -frac{pi}{3}, -frac{pi}{12} right) ) because ( -1.047 < -0.785 < -0.2618 ). Wait, no, ( -0.785 ) is greater than ( -1.047 ) but less than ( -0.2618 ). So, yes, ( -frac{pi}{4} ) is within ( left( -frac{pi}{3}, -frac{pi}{12} right) ).Therefore, ( varphi = frac{pi}{4} ) is allowed because the zero is still within the required interval.Similarly, checking ( varphi = frac{pi}{6} ):The largest negative zero would be ( frac{pi}{6} - frac{pi}{2} = -frac{pi}{3} ). But the interval is ( left( -frac{pi}{3}, -frac{pi}{12} right) ), which is open at ( -frac{pi}{3} ). So, ( -frac{pi}{3} ) is not included, meaning ( varphi = frac{pi}{6} ) is not allowed because the zero would be exactly at ( -frac{pi}{3} ), which is excluded.Therefore, the interval for ( varphi ) is ( left( frac{pi}{6}, frac{pi}{4} right] ), which is option D.

answer：Okay, so I have this probability problem here about Xiao Ming waiting for a shuttle bus. Let me try to understand it step by step.First, the shuttle buses depart at 7:30, 8:00, and 8:30. Xiao Ming arrives at the bus station somewhere between 7:50 and 8:30. His arrival time is random, meaning it's equally likely for him to arrive at any time within that window. The question is asking for the probability that he waits for less than 10 minutes for the bus.Alright, so let's break this down. The total time window during which Xiao Ming can arrive is from 7:50 to 8:30. That's a span of 40 minutes. So, the total possible arrival time is 40 minutes.Now, we need to figure out during which parts of this 40-minute window Xiao Ming would have to wait less than 10 minutes for the bus. To do that, let's look at the bus departure times and see what arrival times would result in a wait time of less than 10 minutes.The buses leave at 7:30, 8:00, and 8:30. Since Xiao Ming arrives between 7:50 and 8:30, the 7:30 bus is already gone by the time he arrives. So, the relevant buses he can catch are the ones at 8:00 and 8:30.Let's consider the 8:00 bus first. If Xiao Ming arrives just before 8:00, he might have to wait a little while. To find out when he would wait less than 10 minutes, we look at the time window from 7:50 to 8:00. If he arrives between 7:50 and 8:00, the wait time until 8:00 would be less than 10 minutes. Specifically, if he arrives at 7:50, he has to wait 10 minutes, and if he arrives just before 8:00, he waits almost 0 minutes. So, the entire 10-minute window before 8:00 is when he would wait less than 10 minutes for the 8:00 bus.Similarly, for the 8:30 bus, if Xiao Ming arrives just before 8:30, he would have a short wait. So, the time window from 8:20 to 8:30 would result in a wait time of less than 10 minutes. If he arrives at 8:20, he waits exactly 10 minutes, and if he arrives just before 8:30, he waits almost 0 minutes.So, putting this together, the total time during which Xiao Ming would wait less than 10 minutes is from 7:50 to 8:00 (10 minutes) and from 8:20 to 8:30 (another 10 minutes). That gives us a total of 20 minutes where he waits less than 10 minutes.Now, the total time window for his arrival is 40 minutes (from 7:50 to 8:30). So, the probability that he waits less than 10 minutes is the ratio of the favorable time (20 minutes) to the total time (40 minutes). Calculating that, we get 20/40, which simplifies to 1/2. So, the probability is 1/2.Wait, let me double-check that. Is there any overlap or something I missed? Hmm, no, the two time windows (7:50-8:00 and 8:20-8:30) don't overlap, and they each contribute 10 minutes. So, 10 + 10 is 20 minutes. The total possible arrival time is 40 minutes. So, 20/40 is indeed 1/2.But hold on, the answer options are A: 1/3, B: 3/4, C: 2/3, D: 1/2. So, according to my calculation, the answer should be D: 1/2.Wait a second, let me think again. Maybe I'm missing something. Is there another bus that he could catch? The buses are at 7:30, 8:00, and 8:30. He arrives between 7:50 and 8:30. So, the 7:30 bus is already gone when he arrives at 7:50. So, the next buses are at 8:00 and 8:30. So, if he arrives between 7:50 and 8:00, he can catch the 8:00 bus with a wait time less than 10 minutes. If he arrives between 8:00 and 8:20, he would have to wait longer than 10 minutes for the next bus at 8:30. And if he arrives between 8:20 and 8:30, he can catch the 8:30 bus with a wait time less than 10 minutes.So, the favorable time is 7:50-8:00 (10 minutes) and 8:20-8:30 (another 10 minutes), totaling 20 minutes. The total possible arrival time is 40 minutes. So, 20/40 is 1/2. So, yeah, D: 1/2 seems correct.But wait, another thought: is the arrival time uniformly distributed? The problem says his arrival time is random, which I assume means uniform distribution. So, the probability is just the ratio of the favorable time to the total time. So, 20/40 is 1/2.Hmm, I think that's right. So, the answer is D: 1/2.

answer：To find the cube root of 6880, I start by examining the given values: the cube root of 68.8 is 4.098, and the cube root of 6.88 is 1.902.I notice that 6880 is significantly larger than both 68.8 and 6.88. Specifically, 6880 is 1000 times larger than 6.88. Since 1000 is a perfect cube (10 cubed), I can use this relationship to simplify the problem.By expressing 6880 as 6.88 multiplied by 1000, I can apply the property of cube roots that allows me to separate the cube root of a product into the product of the cube roots. This means that the cube root of 6880 is equal to the cube root of 6.88 multiplied by the cube root of 1000.Given that the cube root of 6.88 is 1.902 and the cube root of 1000 is 10, I can multiply these two results together to find the cube root of 6880.Finally, multiplying 1.902 by 10 gives me the cube root of 6880, which is 19.02.

answer：First, I need to translate the given percentages into mathematical equations. The statement "45% of z is 39% of y" can be written as:0.45z = 0.39yNext, the statement "y is 75% of x" can be written as:y = 0.75xNow, I want to find what percent of x is z. To do this, I'll express z in terms of x.Starting with the first equation:0.45z = 0.39yI can solve for y:y = (0.45z) / 0.39Substituting this expression for y into the second equation:(0.45z) / 0.39 = 0.75xNow, solve for z:0.45z = 0.39 * 0.75x0.45z = 0.2925xDivide both sides by x to find the ratio of z to x:z / x = 0.2925 / 0.45z / x = 0.65To express this as a percentage, multiply by 100:(z / x) * 100 = 65%Therefore, z is 65% of x.