Alan

Let f(x)=12x+7 and g(x) = 5x+2 whenever x is a positive integer. Define h(x) to be the greatest common divisor of f(x) and g(x). What is the sum of all possible values of h(x)?

The gcd of f(x) and g(x) when x is a positive integer, is 1 for most values of x and 11 when x is 4, 15, 26, ...etc.

So the sum is 12.

Oops!  Quite right.  I messed up that detail.  The method still holds but Firebolt will need to go through it correcting the numerics.

Like so:

Make sure you check this carefully to see if I've made any silly mistakes!

Suppose that we win $2 if we flip a heads on a coin toss, but lose $1 if we flip tails. What is the expected value, in dollars, of our winnings after one flip?

Expected value = prob_win*$2 - prob_lose*$1  =  $0.5  (assuming a fair coin!)

This question was answered here yesterday: https://web2.0calc.com/questions/help_62999#r1

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In the expansion of \(ax+b)^{2000}, where a and b are relatively prime positive integers, the coefficients of x^2 and x^3 are equal.   Find a and b. 

Consider the set S=1,2,3,4,5,6,7,8,9,12,13,14,15,16,17,18,19,23,24,. . . . ,123456789\}, which consists of all positive integers whose digits strictly increase from left to right. This set is finite. What is the median of the set?

The median here is given by (123456789 + 1)/2  

Daniel had 80 more stickers than Elle. He gave 1/4 of his stickers to Elle. She then gave 3/5 of her stickers to Daniel. In the end, Daniel had 92 more stickers than Elle. How many stickers did Daniel have at first?

Since you are only being given information about the horizontal distance, I suspect the 1m refers to the point (-1, 0).

So you would have \(D(0) = a*cos(b*0)+d \text{ or }D(0)=a+d\)

The pi/6 refers to a time, not a distance, so \(D(\frac{\pi}{6})=a*cos(b*\frac{\pi}{6})+d\) and because this is the centre where D = 0 this becomes \(0=a*cos(b*\frac{\pi}{6})+d\)

Does this answer your questions?