💡Okay, so I have this problem where I need to transform the graph of the function ( f(x) = sin 2x ). The transformations required are first compressing the x-coordinates to half their original length and then shifting it right by ( frac{pi}{6} ) units to get ( g(x) ). I need to figure out what ( g(x) ) is and choose the correct option from A to D.Let me start by recalling how transformations affect trigonometric functions, specifically sine functions. The general form of a sine function is ( y = sin(Bx + C) ), where ( B ) affects the period and ( C ) affects the phase shift. First, the function given is ( f(x) = sin 2x ). The coefficient 2 in front of ( x ) means that the period of the sine function is compressed. Normally, the period of ( sin x ) is ( 2pi ), but with ( sin 2x ), the period becomes ( pi ) because the period is ( frac{2pi}{B} ), so here ( B = 2 ), giving ( frac{2pi}{2} = pi ).Now, the first transformation is compressing the x-coordinates to half their original length. Hmm, compressing x-coordinates by half would mean that the graph is horizontally compressed. In terms of the function, horizontal compression by a factor of ( frac{1}{2} ) is achieved by multiplying the input ( x ) by 2. So, if we have ( f(x) = sin 2x ), compressing it further by half would mean replacing ( x ) with ( 2x ), resulting in ( sin(2*(2x)) = sin(4x) ).Wait, let me think about that again. If we have a function ( f(x) ) and we want to horizontally compress it by a factor of ( frac{1}{2} ), we replace ( x ) with ( 2x ). So, starting from ( f(x) = sin 2x ), after compression, it becomes ( sin(2*(2x)) = sin(4x) ). That seems right because compressing the x-axis by half would make the graph repeat twice as often, which is consistent with increasing the frequency from 2 to 4.So after the first transformation, the function becomes ( sin 4x ). Now, the next transformation is shifting it right by ( frac{pi}{6} ) units. To shift a function right by ( c ) units, we replace ( x ) with ( x - c ) in the function. So, starting from ( sin 4x ), shifting it right by ( frac{pi}{6} ) would give us ( sin(4(x - frac{pi}{6})) ).Let me compute that: ( sin(4(x - frac{pi}{6})) = sin(4x - 4*frac{pi}{6}) ). Simplifying ( 4*frac{pi}{6} ), that's ( frac{4pi}{6} ), which reduces to ( frac{2pi}{3} ). So, the transformed function is ( sin(4x - frac{2pi}{3}) ).Looking at the options, option C is ( sin(4x - frac{2pi}{3}) ), which matches what I just derived. So, that should be the correct answer.But wait, let me double-check my steps to make sure I didn't make a mistake. Starting with ( f(x) = sin 2x ). First transformation: compress x-coordinates to half their original length. That's a horizontal compression by factor 1/2, which is achieved by replacing ( x ) with ( 2x ), so ( sin(2*(2x)) = sin(4x) ). Then, shifting right by ( frac{pi}{6} ) is replacing ( x ) with ( x - frac{pi}{6} ), so ( sin(4(x - frac{pi}{6})) = sin(4x - frac{2pi}{3}) ). Yep, that seems consistent.Alternatively, sometimes people get confused between horizontal shifts and phase shifts. Let me recall that for a function ( sin(Bx - C) ), the phase shift is ( frac{C}{B} ). So, in this case, after the shift, the function is ( sin(4x - frac{2pi}{3}) ), which can be written as ( sin(4(x - frac{pi}{6})) ), confirming that the shift is indeed ( frac{pi}{6} ) to the right.So, all the steps check out. I think I can confidently say that the correct answer is option C.**Final Answer**boxed{C}

💡Let's break down the problem step by step to find the correct answer.1. **Define the variable**: Let the given number be ( y ).2. **Analyze Tom's incorrect execution**: Tom was supposed to subtract 5 and then divide by 7, but instead, he subtracted 14 and divided by 2, resulting in 50. This gives us the equation: [ frac{y - 14}{2} = 50 ]3. **Solve for ( y )**: Multiply both sides by 2 to isolate ( y - 14 ): [ y - 14 = 100 ] Then, add 14 to both sides to find ( y ): [ y = 114 ]4. **Determine the correct result**: Now, using the correct method of subtracting 5 and then dividing by 7: [ frac{114 - 5}{7} = frac{109}{7} approx 15.571 ] Since the answer choices are integers, we round down to the nearest whole number.5. **Conclusion**: The correct answer Tom should have obtained is ( 15 ).

💡Okay, so Alice and Bob are playing this game on a circle with 18 equally spaced points, numbered 1 to 18 clockwise. They both start at point 18. Alice moves clockwise, and Bob moves counterclockwise. In each turn, Alice moves 7 points clockwise, and Bob moves 13 points counterclockwise. The game ends when they meet at the same point. I need to figure out how many turns this will take.First, let me visualize the circle. It's divided into 18 points, so each point is 20 degrees apart if I think in terms of a full circle being 360 degrees. But maybe I don't need to worry about the actual angles, just the positions.Both start at point 18. Alice moves clockwise, so her next position would be 18 + 7, but since there are only 18 points, I need to take modulo 18. Similarly, Bob is moving counterclockwise, which is the opposite direction. So moving 13 points counterclockwise from 18 would be like subtracting 13 from 18, but again, since it's a circle, I might need to add 18 to keep the number positive before taking modulo 18.Wait, maybe I should think of their movements in terms of modular arithmetic. Let's denote their positions after each turn.For Alice, starting at position 18, each turn she moves 7 points clockwise. So her position after k turns would be:( A(k) = (18 + 7k) mod 18 )Similarly, Bob is moving counterclockwise 13 points each turn. Moving counterclockwise can be thought of as negative movement in the clockwise direction. So, moving 13 points counterclockwise is equivalent to moving -13 points clockwise. Therefore, Bob's position after k turns would be:( B(k) = (18 - 13k) mod 18 )But I can also express this as:( B(k) = (18 + (-13k)) mod 18 )Since we're dealing with modulo 18, we can simplify -13k as (18 - 13)k = 5k, because -13 is congruent to 5 modulo 18 (since 18 - 13 = 5). So Bob's position can also be written as:( B(k) = (18 + 5k) mod 18 )Wait, that might not be accurate. Let me think again. If Bob moves 13 points counterclockwise, that's equivalent to moving 18 - 13 = 5 points clockwise. So each turn, Bob is effectively moving 5 points clockwise. So his position after k turns is:( B(k) = (18 + 5k) mod 18 )But Alice is moving 7 points clockwise each turn, so her position is:( A(k) = (18 + 7k) mod 18 )So, to find when they meet, we need to find the smallest k such that:( A(k) = B(k) )Which means:( (18 + 7k) mod 18 = (18 + 5k) mod 18 )Simplifying both sides, since 18 mod 18 is 0, we get:( 7k mod 18 = 5k mod 18 )Subtracting 5k from both sides:( 2k mod 18 = 0 )So, 2k must be a multiple of 18. Therefore, k must be a multiple of 9, because 2 and 18 share a common factor of 2. So, the smallest positive integer k that satisfies this is k = 9.Wait, let me double-check. If k = 9, then:Alice's position: ( 18 + 7*9 = 18 + 63 = 81 ). 81 mod 18 is 81 - 4*18 = 81 - 72 = 9.Bob's position: ( 18 + 5*9 = 18 + 45 = 63 ). 63 mod 18 is 63 - 3*18 = 63 - 54 = 9.Yes, both are at position 9 after 9 turns. So, the answer should be 9 turns.But just to make sure, let me check for smaller k. Let's see:k=1: Alice at (18+7)=25 mod18=7; Bob at (18-13)=5 mod18=5. Not same.k=2: Alice at 14; Bob at (5 +5)=10. Not same.k=3: Alice at 21 mod18=3; Bob at 15. Not same.k=4: Alice at 10; Bob at 20 mod18=2. Not same.k=5: Alice at 17; Bob at 7. Not same.k=6: Alice at 24 mod18=6; Bob at 12. Not same.k=7: Alice at 13; Bob at 17. Not same.k=8: Alice at 20 mod18=2; Bob at 22 mod18=4. Not same.k=9: Both at 9. So yes, it takes 9 turns.I think that's correct. So the answer is 9, which is option B.

💡First, identify the speeds of the two trains. The faster train has a speed of 24 m/s, and the slower train has half that speed, which is 12 m/s.Next, calculate the relative speed when the trains are moving in opposite directions. This is the sum of their individual speeds: 24 m/s + 12 m/s = 36 m/s.Determine the total distance that needs to be covered for the trains to completely pass each other. Since each train is 100 meters long, the total distance is 100 m + 100 m = 200 meters.Finally, use the formula for time, which is distance divided by speed, to find how long it takes for the trains to cross each other: 200 m ÷ 36 m/s ≈ 5.56 seconds.

💡First, calculate the total age of the cricket team by multiplying the average age by the number of members:[ text{Total age} = 11 times 27 = 297 text{ years} ]The wicket keeper is 3 years older than the average age, so:[ text{Wicket keeper's age} = 27 + 3 = 30 text{ years} ]Let ( P ) be the age of the other player. When excluding the wicket keeper and this player, the average age of the remaining 9 players is 26 years:[ frac{297 - 30 - P}{9} = 26 ]Solve for ( P ):[ 297 - 30 - P = 234 ][ 267 - P = 234 ][ P = 267 - 234 ][ P = 33 text{ years} ]Finally, calculate the new average age after excluding both the wicket keeper and the 33-year-old player:[ text{New average} = frac{297 - 30 - 33}{9} = frac{234}{9} = 26 text{ years} ]

💡First, determine the total number of ways to select three jellybeans from the 15 available. This is calculated using the combination formula:[{15 choose 3} = frac{15 times 14 times 13}{3 times 2 times 1} = 455]Next, calculate the number of ways to pick exactly one red jellybean and two blue jellybeans:- There are {5 choose 1} = 5 ways to choose one red jellybean.- There are {3 choose 2} = 3 ways to choose two blue jellybeans.Multiply these to find the total number of successful outcomes:[5 times 3 = 15]Finally, compute the probability by dividing the number of successful outcomes by the total number of possible outcomes:[frac{15}{455} = frac{3}{91}]