1) Line L intersects the x axis going towards the top right hand corner of the grid. It forms a 60 degree angle to the right of the point of intersection. What is the slope of the line?
If you know the angle of a line, one way to figure out the slope is to use results from trigonometry. You can draw a right triangle whose hypotenuse lies on the line: pick a point on the line and draw a vertical line to the x-axis then take the point where the line intersects the x-axis and draw a horizontal line along the axis over to the point where the vertical line hits the x-axis.
We can say that the hypotenuse of this triangle has a length of h (its pretty cool that it won't actually matter how long it is). Then the base has a length
h*cos(60), and the height has a length of h*sin(60) and the slope is the height divided by the length:
slope = h*sin(60)/h*cos(60) = sin(60)/cos(60) = sqrt(3) (see how the h drops out).
If you haven't learned about sin and cos functions but you know properties of a 60-30 triangle you should know that a 30-60 triangle has side length ratios 1,2,sqrt(3). So the slope of the line is again the length of the opposite side (opposite 60) divided by the adjacent side (rise over run) which is sqrt(3)/1 = sqrt(3).
For problem 2)
remember that the side opposite the angle-60 has a length twice the side adjacent to the angle-60, and the hypotenuse has a length sqrt(3) times the length of the side adjacent to the angle-60.