Alan

If 5x² - 10x + k has remainder 3 when divided by x - 1 then we must have:

5x² - 10x + k = (x - 1)*m + 3  where m is an integer.

When x = 1 the right-hand side is just 3 and m disappears so put x = 1 in the left-hand side as well to get

5 - 10 + k = 3

or k = 8

(Chris beat me to it!)

Incidentally, you can't upload images as long as you stay Anonymous.  Why not register, then you will have access to the upload images function.

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See Nauseated's solution below.

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1. Perimeter

Yes, though it can be expressed more simply.  If s is arc length (i.e. perimeter of sector), r is radius and θ is angle of sector then:  s = r*θ where θ is in radians  (s = r*θ*pi/180 if θ is in degrees: this is the equivalent of your expression:  s = r*θ*2pi/360)

2. Area

Area of sector = (θ/360)*pi*r^2   (if θ is in degrees).

(I've assumed you are talking about the sector of a circle)

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I think it's more complicated than this Chris.  Your result can be written as x² + y² = constant.

This gives 2x + 2ydy/dx = 0  or dy/dx = -x/y.

Here's my attempt:

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$$d_\infty= \lim_{p \to \infty} \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}$$

Without the limit it is a p-norm.  With the infinite limit I think it's just called an infinite p-norm or a maximum p-norm.

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This problem appears in the movie Die Hard 3.  Bruce Willis and Samuel L Jackson had 30 seconds to solve it in order to prevent an explosion which would have killed them!

Luckily, the penalty for getting problems wrong here is not so severe!

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If we assume that the number of cars arriving follows a Poisson distribution then the probability of exactly 5 cars appearing in the next hour is:

p = λ⁵e^-λ/5!  where λ is the mean arrival rate (= 7 here)

$${\mathtt{p}} = {\frac{{{\mathtt{7}}}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{{\mathtt{e}}}^{\left(-{\mathtt{7}}\right)}}{{\mathtt{5}}{!}}} \Rightarrow {\mathtt{p}} = {\mathtt{0.127\: \!716\: \!668\: \!292\: \!289\: \!6}}$$

(For exactly k cars arriving the probability is λ^ke^-λ/k!)

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1 litre is 0.001 cubic metres, so 512 litres is 0.512 cubic metres.

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We should note that Chris's expansion is still correct.  It's just that the resulting grand sum evaluates to zero!

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I was deriving thejamesmachine's method for determining if a number is divisible by 7.  Here's how he expressed it

For seven:

Take the last digit, double it, and subtract it from the rest of the number. If the answer is divisible by 7 (including 0), then the number is also divisible by 7:

861:86-1x2=84

84/7=12

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