Maths

Please view the attachments which have the question and mark scheme answer.

Can you please explain the following questions:
8b) I don't really understand the mark scheme answer - can you please draw or explain it in an easy way or with some sort of relevant analogy

9ai) Here the answers 77 and 76 are allowed but why is 77 preferred? Also, if you use the answer 76 for question 9b) then you would get 32, 30 as opposed to 33, 30.

Have you sketched the quadratic for 8b? The negative x^2 coefficient is the key thing. Without calculating k, the maximum must occur around the values stated.

The part of the mark scheme where it refers to 9ai)-44 ... seems to cover the part about whether its 76 or 77. The ans for 9ai) is ~76.8 and the 77 is the closest rounded value.

Sketch for Q8b) drawn - see the point you were making. Can we automatically assume it would be an upside down U-shape whenever we see -x^2? I thought we couldn't make this assumption as the kx or 0.4x part would make the graph such that it is not necessarily an upside down U-shape.

This is really gcse work, but a quadratic.
y(x) = kx - 0.001x^2
is a "n" shape which goes through the origin as y(0)=0, has a gradient of k at the origin so y'(0)=k and has a maximum at (k/0.002, k^2/0.004). In completed square form its
-0.001(x - k/0.002)^2 + k^2/0.004
The u or n shape is completely determined by the sign of the x^2 coefficient, as for large x, it dominates. 1*x^2 will be u as for large x, x^2 is postive large. -1*x^2 will be n as for large x, -x^2 is negative large.

So the number of primes, according to this model, initially increases as k>0 but decreases past the point x=k/0.002. Obviously its impossible as the number of primes <= x has to be non decreasing as you cant lose primes that have already occurred.

Have you sketched the quadratic for 8b? The negative x^2 coefficient is the key thing. Without calculating k, the maximum must occur around the values stated.

The part of the mark scheme where it refers to 9ai)-44 ... seems to cover the part about whether its 76 or 77. The ans for 9ai) is ~76.8 and the 77 is the closest rounded value.

For 9b)
200/(2log200) = 43.46 and yet this is rounded to 44. It seems regardless of whether it was 76.4 or 76.8 you round up either way. To me it doesn't make sense because we are making it seem as though there is an extra prime number when it may not be true.

Its an approximate formula and arguably it should be stated how you round, so either ceil (always round up) or floor (always round down) or the usual rounding to nearest integer. The mark scheme seems to allow any of them as its not stated/clear from the question.

Really the number of primes <200 is 46, so Id not worry about the first given number is 43 or 44. The primary answer in the mark scheme seems to be the usual rounding.