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I'm back from my exams! Here are some questions.

If f(x)=\(arcsin(\sin(\pi x))\).

Find

  1. f(2.7)

  2. f'(2.7)

  3. \(\int^{2.5}_0 {f(x)} {dx}\)

If 1000 people are sitting around a circular table and each person makes a donation of the average of the two people sitting adjacent to him/her and it is known that one person made a contribution of 500 dollars, then find, if you can, the total amount donated and the maximum amount of money donated.

If \(p(x)=(x+a_1)(x+a_2)...(x+a_{10})\) is a polynomial such that \(\forall a_i \in R\) and all the 11 coefficients are positive, then answer true or false for the following questions:

  1. p(x) has a global minimum.

  2. \(a_i\) are all positive.

  3. p'(x) has all negative real roots.

  4. p'(x) has all real roots.

Note by Vishnu C
2 years, 7 months ago

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The second one was actually modified a little bit, but the answer is still the same.

Let A be the guy who donated 500 dollars. C must have donated 500+x and G must have donated 500-x, where x is any real number.With a little bit of simplification, you can see that D must have given 500+2x and F must have given 500-2x, and so on. The last guy, E, also gives 500 dollars. Because of symmetry, you can see that if you add it all up, you get 500*1000=500000.

Note that the maximum amount of money donated can never be found out because everyone could have given the same amount, i.e, x=0 is also a possible scenario.

Vishnu C - 2 years, 7 months ago

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@Brian Charlesworth , I could some help with the third one.

Vishnu C - 2 years, 7 months ago

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Okay, here's how you do the first one:

First draw the graph of the function. Although mathematically inaccurate, it's going to be a series of triangles with base on the x-axis.

2.7=2.5+0.2. When the arcsin function is applied, it is going to give you (0.5-0.2)pi=0.3pi

As the function has a slope = -PI at x=2.7.

The integral is pretty simple. It's the area under the curve. Height =PI/2 and base = 1/2 and there is one triangle like that because the positive and negative areas cancel out. And you get PI/8.

Vishnu C - 2 years, 7 months ago

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