Hey! I worked out the question - I'm not really sure how to explain this properly, as I am only doing 3/4 methods this year, but I tried to take it step by step through the working out Sorry its not in much detail, but I do hope it helps you!

Essentially, the best thing is to sketch a graph of the hybrid function first, and then remember that the sum of the area under the function must always equal to 1.

Well done for having a go at the question. Since you're taking Methods 3&4, I'm going to give you some feedback.

Sketching the graph of the function in this question wouldn't get you any marks. While it might help with your understanding, just be wary that in an exam situation, you're pressed for time, so just be careful spending extra time on things that don't get you any marks.

Please do not write the terminals of integration on the left side of your integral sign. This is an abuse of notation. The question also already defines the critical value as \(k\) but you've used \(m\). Stick to what the question has already defined for you.

In part b, since \[\Pr(X<5)=\int_0^5 \frac{x}{50}\,\text{d}x=\frac14,\] we know that \(5\leq k\leq 20\), and so one can just solve \[\int_k^{20} \left(\frac{-x}{150}+\frac{2}{15}\right)\text{d}x=\frac15\implies \left[\frac{-x^2}{300}+\frac{2x}{15}\right]_k^{20}=\frac15\implies \cdots\] which is a little nicer. Guessing that \(5\leq k\leq 20\) is a bit risky.

For part c, you should define the binomial variable \(N\sim\text{Bi}(5,\,1/4)\) for clarity. Then it becomes much clearer that \[\Pr(N=4)=\binom{5}{4}\left(\frac14\right)^{\!4}\!\left(\frac34\right)=\frac{15}{1024}.\] The notation \(^n\text{C}_k\) is correct, but using the binomial coefficient notation, \(\displaystyle \binom{n}{k}\), is preferred.