Alan

As follows:

I'll leave you to get the last two values.

4*9 + x = 7*x + 0 

6*x = 36

x = 6

n = 7*6 = 42

As follows:

This has no solution.  If a is positive the left hand side is negative.  If a is negative the left hand side is complex.  In neither case do they match the right hand side.

Looking at a few sites it seems there is some variability in estimates of the radius of a single atom of gold.  However, 144pm seems like a reasonable value to use.  (1pm is 10^-12m).

a)  Add the components:   A + B = 3i - 2j + (-i -4j) = 2i +2j

b) Same as a) except you now subtract the components.

c) Use Pythagoras:  abs(A) = sqrt(3^2 + (-2)^2).  Do the same for B and add.

d) As c) but subtract the result for B from that for A.

e) The direction are gven by the angles, so direction of A + B is tan^-1(j component/i component) = tan^-1(2/2) = tan^-1(1) = 45º.  Do a similar calculation for the direction of A - B.

The three angles must sum to 180degrees, so

8x+5 + 6x-12 +4x+7 = 180

18x =180

x = 10

Now you can work out the measure of angle A.

The question is slightly ambiguous.  "The same scores" could mean identical scores for each individual problem, or it could mean the same total score.  I'll assume the latter.

With 22 problems and the three possible scores of 0, 1 and 3 per problem, the total score could be any of the numbers from 0 to 66.  Hence there are 67 possible totals.  Therefore we need a minimum of 68 candidates to ensure at least two have the same total score.

Cut it into three rectangles:

Find the area of each rectangle and add them up.

I get the following: