answer：First, I'll define the amounts that A and B have as ( a ) and ( b ) respectively.From the problem, I know that the total amount they have together is 1210, so:[ a + b = 1210 ]It's also given that ( frac{4}{5} ) of A's amount is equal to ( frac{2}{3} ) of B's amount:[ frac{4}{5}a = frac{2}{3}b ]To solve for one variable in terms of the other, I'll isolate ( a ) in the second equation:[ a = frac{5}{4} times frac{2}{3}b ][ a = frac{10}{12}b ][ a = frac{5}{6}b ]Now, I'll substitute ( a = frac{5}{6}b ) into the first equation:[ frac{5}{6}b + b = 1210 ][ frac{11}{6}b = 1210 ]To find ( b ), I'll multiply both sides by ( frac{6}{11} ):[ b = 1210 times frac{6}{11} ][ b = 660 ]Therefore, B has 660 rupees.

answer：First, I recognize that the train crosses a pole in 5 seconds, which means the time it takes to pass the pole is 5 seconds.Next, I note that the length of the train is 83.33333333333334 meters. To find the speed in kilometers per hour, I need to convert this distance from meters to kilometers.I convert 83.33333333333334 meters to kilometers by dividing by 1000, resulting in 0.08333333333333334 kilometers.Then, I convert the time from seconds to hours by dividing 5 seconds by 3600, which gives approximately 0.001388888888888889 hours.Finally, I calculate the speed by dividing the distance in kilometers by the time in hours:0.08333333333333334 kilometers / 0.001388888888888889 hours = 60 kilometers per hour.Therefore, the speed of the train is 60 km/hr.

answer：Okay, so I'm trying to figure out which of the statements A, B, C, or D about the function ( f(x) = sum_{i=1}^4 frac{sin[(2i-1)x]}{2i-1} ) are correct. Let me break this down step by step.First, let's write out the function explicitly. The sum from ( i=1 ) to ( 4 ) means we have four terms:[f(x) = frac{sin(1 cdot x)}{1} + frac{sin(3x)}{3} + frac{sin(5x)}{5} + frac{sin(7x)}{7}]So, simplifying, it's:[f(x) = sin x + frac{sin 3x}{3} + frac{sin 5x}{5} + frac{sin 7x}{7}]Now, let's look at each statement one by one.**Statement A: The graph of the function ( f(x) ) is symmetric about the line ( x = frac{pi}{2} ).**Hmm, symmetry about a vertical line ( x = a ) means that for any point ( (a + h, f(a + h)) ), there is a corresponding point ( (a - h, f(a - h)) ) such that ( f(a + h) = f(a - h) ). So, to check if ( f(x) ) is symmetric about ( x = frac{pi}{2} ), we can test if ( fleft(frac{pi}{2} + hright) = fleft(frac{pi}{2} - hright) ).Let me compute ( fleft(frac{pi}{2} + hright) ):[fleft(frac{pi}{2} + hright) = sinleft(frac{pi}{2} + hright) + frac{sinleft(3left(frac{pi}{2} + hright)right)}{3} + frac{sinleft(5left(frac{pi}{2} + hright)right)}{5} + frac{sinleft(7left(frac{pi}{2} + hright)right)}{7}]Using the sine addition formula ( sin(a + b) = sin a cos b + cos a sin b ):[sinleft(frac{pi}{2} + hright) = sinfrac{pi}{2}cos h + cosfrac{pi}{2}sin h = 1 cdot cos h + 0 cdot sin h = cos h]Similarly,[sinleft(3left(frac{pi}{2} + hright)right) = sinleft(frac{3pi}{2} + 3hright) = sinfrac{3pi}{2}cos 3h + cosfrac{3pi}{2}sin 3h = (-1)cos 3h + 0 cdot sin 3h = -cos 3h][sinleft(5left(frac{pi}{2} + hright)right) = sinleft(frac{5pi}{2} + 5hright) = sinfrac{5pi}{2}cos 5h + cosfrac{5pi}{2}sin 5h = 1 cdot cos 5h + 0 cdot sin 5h = cos 5h][sinleft(7left(frac{pi}{2} + hright)right) = sinleft(frac{7pi}{2} + 7hright) = sinfrac{7pi}{2}cos 7h + cosfrac{7pi}{2}sin 7h = (-1)cos 7h + 0 cdot sin 7h = -cos 7h]Putting it all together:[fleft(frac{pi}{2} + hright) = cos h - frac{cos 3h}{3} + frac{cos 5h}{5} - frac{cos 7h}{7}]Now, let's compute ( fleft(frac{pi}{2} - hright) ):[fleft(frac{pi}{2} - hright) = sinleft(frac{pi}{2} - hright) + frac{sinleft(3left(frac{pi}{2} - hright)right)}{3} + frac{sinleft(5left(frac{pi}{2} - hright)right)}{5} + frac{sinleft(7left(frac{pi}{2} - hright)right)}{7}]Again, using the sine subtraction formula ( sin(a - b) = sin a cos b - cos a sin b ):[sinleft(frac{pi}{2} - hright) = sinfrac{pi}{2}cos h - cosfrac{pi}{2}sin h = 1 cdot cos h - 0 cdot sin h = cos h][sinleft(3left(frac{pi}{2} - hright)right) = sinleft(frac{3pi}{2} - 3hright) = sinfrac{3pi}{2}cos 3h - cosfrac{3pi}{2}sin 3h = (-1)cos 3h - 0 cdot sin 3h = -cos 3h][sinleft(5left(frac{pi}{2} - hright)right) = sinleft(frac{5pi}{2} - 5hright) = sinfrac{5pi}{2}cos 5h - cosfrac{5pi}{2}sin 5h = 1 cdot cos 5h - 0 cdot sin 5h = cos 5h][sinleft(7left(frac{pi}{2} - hright)right) = sinleft(frac{7pi}{2} - 7hright) = sinfrac{7pi}{2}cos 7h - cosfrac{7pi}{2}sin 7h = (-1)cos 7h - 0 cdot sin 7h = -cos 7h]So,[fleft(frac{pi}{2} - hright) = cos h - frac{cos 3h}{3} + frac{cos 5h}{5} - frac{cos 7h}{7}]Comparing ( fleft(frac{pi}{2} + hright) ) and ( fleft(frac{pi}{2} - hright) ), they are equal. Therefore, the function is symmetric about the line ( x = frac{pi}{2} ). So, statement A is correct.**Statement B: The graph of the function ( f(x) ) is symmetric about the point ( (0,0) ).**Symmetry about the origin means that ( f(-x) = -f(x) ) for all ( x ). Let's check this.Compute ( f(-x) ):[f(-x) = sin(-x) + frac{sin(-3x)}{3} + frac{sin(-5x)}{5} + frac{sin(-7x)}{7}]Since ( sin(-theta) = -sintheta ):[f(-x) = -sin x - frac{sin 3x}{3} - frac{sin 5x}{5} - frac{sin 7x}{7} = -left( sin x + frac{sin 3x}{3} + frac{sin 5x}{5} + frac{sin 7x}{7} right) = -f(x)]So, ( f(-x) = -f(x) ), which means the function is odd and symmetric about the origin. Therefore, statement B is correct.**Statement C: The function ( f(x) ) is a periodic function, and the smallest positive period is ( pi ).**First, let's recall that the period of ( sin(kx) ) is ( frac{2pi}{k} ). So, the periods of each term in ( f(x) ) are:- ( sin x ): period ( 2pi )- ( sin 3x ): period ( frac{2pi}{3} )- ( sin 5x ): period ( frac{2pi}{5} )- ( sin 7x ): period ( frac{2pi}{7} )The overall period of ( f(x) ) is the least common multiple (LCM) of these individual periods. Let's find the LCM of ( 2pi, frac{2pi}{3}, frac{2pi}{5}, frac{2pi}{7} ).First, factor out ( 2pi ):[2pi times text{LCM}left(1, frac{1}{3}, frac{1}{5}, frac{1}{7}right)]The LCM of ( 1, frac{1}{3}, frac{1}{5}, frac{1}{7} ) is the smallest number that is an integer multiple of each. Since these are reciprocals of integers, the LCM is the reciprocal of the greatest common divisor (GCD) of 1, 3, 5, 7. The GCD of 1, 3, 5, 7 is 1, so the LCM is 1.Wait, that doesn't seem right. Maybe I should think differently. The LCM of fractions can be found by taking the LCM of the numerators divided by the GCD of the denominators. But in this case, all fractions have numerator 1, so it's more about finding the LCM of the denominators.Actually, perhaps a better approach is to consider the periods as ( 2pi, frac{2pi}{3}, frac{2pi}{5}, frac{2pi}{7} ). The LCM of these periods is the smallest ( T ) such that ( T ) is a multiple of each period.So, ( T ) must satisfy:[T = 2pi times k_1 = frac{2pi}{3} times k_2 = frac{2pi}{5} times k_3 = frac{2pi}{7} times k_4]where ( k_1, k_2, k_3, k_4 ) are integers.Dividing both sides by ( 2pi ):[1 times k_1 = frac{1}{3} times k_2 = frac{1}{5} times k_3 = frac{1}{7} times k_4]Let ( c = k_1 = frac{k_2}{3} = frac{k_3}{5} = frac{k_4}{7} ). So, ( k_2 = 3c ), ( k_3 = 5c ), ( k_4 = 7c ). To make all ( k_i ) integers, ( c ) must be an integer. The smallest such ( c ) is 1, so:[T = 2pi times 1 = 2pi]Therefore, the fundamental period of ( f(x) ) is ( 2pi ), not ( pi ). So, statement C is incorrect.**Statement D: The maximum value of the derivative of the function ( f(x) ), denoted as ( f'(x) ), is 4.**First, let's find the derivative ( f'(x) ):[f'(x) = frac{d}{dx}left( sin x + frac{sin 3x}{3} + frac{sin 5x}{5} + frac{sin 7x}{7} right) = cos x + cos 3x + cos 5x + cos 7x]To find the maximum value of ( f'(x) ), we need to find the maximum of the sum of these cosine functions. The maximum value of each ( cos kx ) is 1, so the maximum possible value of ( f'(x) ) is when all cosines are equal to 1 simultaneously.Is there an ( x ) such that ( cos x = cos 3x = cos 5x = cos 7x = 1 )?Let's see. For ( cos x = 1 ), ( x = 2pi n ), where ( n ) is an integer.Check ( cos 3x ) at ( x = 2pi n ):[cos(3 times 2pi n) = cos(6pi n) = 1]Similarly, ( cos 5x = cos(10pi n) = 1 ), and ( cos 7x = cos(14pi n) = 1 ).So, at ( x = 2pi n ), all cosines are 1, and thus:[f'(x) = 1 + 1 + 1 + 1 = 4]Therefore, the maximum value of ( f'(x) ) is indeed 4. So, statement D is correct.**Summary:**- Statement A: Correct- Statement B: Correct- Statement C: Incorrect- Statement D: CorrectSo, the correct statements are A, B, and D.

answer：First, I need to verify that the point (1, -1) lies on the curve by evaluating the function at x = 1.Next, I'll find the derivative of the function f(x) to determine the slope of the tangent line at the given point.After obtaining the derivative, I'll evaluate it at x = 1 to find the specific slope at the point (1, -1).Finally, I'll use the point-slope form of a linear equation to write the equation of the tangent line and simplify it to the standard form.

answer：First, I need to evaluate the expression ((90 θ 33) θ 17) - (99 θ (33 θ 17)).I'll start by calculating 90 θ 33. This represents the remainder when 90 is divided by 33. Dividing 90 by 33 gives a quotient of 2 with a remainder of 24. So, 90 θ 33 equals 24.Next, I'll find 33 θ 17, which is the remainder when 33 is divided by 17. Dividing 33 by 17 gives a quotient of 1 with a remainder of 16. Therefore, 33 θ 17 equals 16.Now, I'll calculate 99 θ (33 θ 17). Since 33 θ 17 is 16, this becomes 99 θ 16. Dividing 99 by 16 gives a quotient of 6 with a remainder of 3. So, 99 θ 16 equals 3.With these values, the expression simplifies to (24 θ 17) - 3. I'll now find 24 θ 17, which is the remainder when 24 is divided by 17. Dividing 24 by 17 gives a quotient of 1 with a remainder of 7. Therefore, 24 θ 17 equals 7.Finally, I'll subtract 3 from 7 to get the result: 7 - 3 equals 4.

answer：Okay, so I have this problem here about Sam and Lee sharing lunch. Let me try to understand it step by step. First, it says that Lee's portion is 150% the size of Sam's. Hmm, okay, so if Sam has a portion, let's say, of size Y, then Lee has 1.5 times that, right? So Lee's portion would be 1.5Y. That makes sense because 150% is the same as 1.5 in decimal form.Now, both of them initially eat 2/3 of their respective portions. So, Sam eats 2/3 of Y, and Lee eats 2/3 of 1.5Y. Let me write that down:Sam eats: (2/3)YLee eats: (2/3)(1.5Y)Let me calculate what Lee eats. 2/3 of 1.5Y. Well, 1.5 is the same as 3/2, so 2/3 times 3/2 is... 1. So Lee eats Y. That's interesting. So Lee eats Y, and Sam eats (2/3)Y.Now, after eating 2/3 of their portions, they both have some left. Let's figure out how much is left for each.Sam has Y - (2/3)Y = (1/3)Y left.Lee has 1.5Y - Y = 0.5Y left.So, Sam has (1/3)Y left, and Lee has 0.5Y left.Then, the problem says Lee gives Sam half of what he has left. So Lee gives half of his remaining 0.5Y to Sam. Let me compute that.Half of 0.5Y is 0.25Y. So Lee gives 0.25Y to Sam.Now, let's see how much each has after this transfer.Sam had (1/3)Y left and received 0.25Y from Lee. So Sam now has:(1/3)Y + 0.25YLet me convert 0.25 to a fraction to make it easier. 0.25 is 1/4. So:(1/3)Y + (1/4)YTo add these, I need a common denominator, which is 12.(4/12)Y + (3/12)Y = (7/12)YSo Sam now has (7/12)Y left.Lee, on the other hand, gave away 0.25Y, so he has:0.5Y - 0.25Y = 0.25Y left.Now, the problem says that when they finish eating, they realize they both ate exactly the same amount. Hmm, okay. So let's figure out how much each of them ate in total.Sam initially ate (2/3)Y and then ate the amount he received from Lee, which is 0.25Y. Wait, no. Wait, when Lee gives Sam 0.25Y, does Sam eat that immediately, or does he eat it later? The problem says they both finish eating, so I think the transfer happens before they finish eating. So Sam has (7/12)Y left, and Lee has 0.25Y left. Then they finish eating, meaning they eat the remaining portions.So Sam eats his remaining (7/12)Y, and Lee eats his remaining 0.25Y.So total amount Sam ate is:(2/3)Y + (7/12)YLet me compute that. Convert (2/3)Y to twelfths: (8/12)Y. So:(8/12)Y + (7/12)Y = (15/12)Y = (5/4)Y = 1.25YSimilarly, Lee ate initially Y, and then he ate his remaining 0.25Y. So total Lee ate:Y + 0.25Y = 1.25YOkay, so both Sam and Lee ate 1.25Y each. That checks out with the problem statement that they both ate exactly the same amount.Now, the question is asking how much lunch did they eat together. So together, they ate 1.25Y + 1.25Y = 2.5Y.Wait, but let me double-check. The total lunch they had was Sam's Y and Lee's 1.5Y, so total was 2.5Y. And they both ate 1.25Y each, so together 2.5Y. That makes sense.So the answer should be 2.5Y, which is option B.But let me just go through the steps again to make sure I didn't make any mistakes.1. Sam has Y, Lee has 1.5Y.2. Both eat 2/3 of their portions: - Sam eats (2/3)Y, leaves (1/3)Y. - Lee eats (2/3)(1.5Y) = Y, leaves 0.5Y.3. Lee gives half of his remaining 0.5Y to Sam, which is 0.25Y.4. Now, Sam has (1/3)Y + 0.25Y = (7/12)Y left. - Lee has 0.25Y left.5. They finish eating, so Sam eats (7/12)Y and Lee eats 0.25Y.6. Total eaten by Sam: (2/3)Y + (7/12)Y = 1.25Y - Total eaten by Lee: Y + 0.25Y = 1.25Y7. Together: 1.25Y + 1.25Y = 2.5YYes, that seems correct. So the answer is 2.5Y, which is option B.