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1. [Group 5][Topic 2][May Exam Practice Question]

Perhaps there's an easier way to do this, but this is what I did: We can represent any number in base 10 by multiplying powers of 10 by numbers from 0-9 and adding them together. For example, 247=2(10)^2+4(10)^1+7(10)^0. Then we can write the number that is 2n 1s as the sum from k=0 to 2n-1 of 10^k. Notice that we have the upper bound as 2n-1 and not 2n because the amount of summands will actually be Upper bound - Lower bound +1=2n-1-0+1=2n, and the amount of summands must be equal to the amount of digits. We can write the number that is n 2s as 2 times the sum from k=0 to n-1 of 10^k. Thus we have reduced the problem to showing that [sum from k=0 to 2n-1 of 10^k]-2[sum from k=0 to n-1 of 10^k]. Using the formulas for finite geometric series (sum from k=0 to n-1 of r^k is [r^k-1]/[r-1]), we can reduce this to [10^(2n)-1]/[10-1]-2[10^n-1]/[10-1]. Factor out the 1/[10-1] and distribute the -2 to get: 1/9[10^2n-2*10^n+1]. You should recognize that [10^2n-2*10^n+1] is a perfect square, namely [10^n-1]^2. Thus we have 1/9[10^n-1]^2, which is [[10^n-1]/3]^2, which is a perfect square. I believe we might have to show that [10^n-1]/3 is an integer though. To do so, recall that a^n-b^n=[a-b][sum from k=0 to n-1 of a^k*b^(n-1-k)]. If we let a=10 and b=1 then we have that 10^n-1^n=[10-1][sum from k=0 to n-1 of 10^k]. This is evidently divisible by 3 and thus is an integer as it is the product of two integers. The 2nd term is in fact n 1s. So we have actually also shown that it is exactly equal to n 3's squared (this can also be seen by the two examples given). I did SL though and I've never encountered this before. Let me know if any of that was confusing. It's kind of hard to write out sums by hand
2. I mean, all the physics stuff is based on the math theory behind it. It's just a possible application, if that's what interested you. I like math theory, so the interesting part for me was proving all the theorems above. It helps to contextualize what you're doing. But I did my IA solely on an approximation method for a function that I created, which is all theory. I personally prefer the differential calculus part. It's a lot easier to grasp, especially if you have no experience with multivariable calculus. That should work out better for your IA, because you don't want to have a bunch of technical stuff that you can't explain [Criterion E].
3. Depends on how much you want to learn or what you're interested in. I just took the course so if you need any help with anything, let me know. Vector functions relate a lot to physics. For example, you can use a vector function to parametrize the trajectory of a particle (notably when the path is parabolic). You can use a line integral over a vector field to find the force done by a particle traversing the field. You can then go into conservative vector fields and the fundamental theorem for line integrals (the Gradient theorem). You can define flux (the amount of stuff passing through a surface) as a surface integral of a vector field. I believe flux integrals have an application in electromagnetism, but I'm not 100% sure. If you want, you can show how to evaluate these with the Gauss-Ostrogradsky theorem (the Divergence theorem). All of this is with integral calculus. On the differential calculus side, you can start defining tangent vectors, normal vectors, and binormal vectors to vector curves (curves in 3d space). From these you can derive the formulas for normal and tangent acceleration, if you wish. Two other useful properties of vector curves are curvature and torsion. Both of these uniquely define a vector curve up to rigid translation/rotation (this is called the fundamental theorem of curves). Also take a look at the Frenet-Serret equations, which show why this is the case. As an application, with curvature you can find the speed limit of car travelling along a path defined by a vector curve so that the car does not skid off. You can also find something called the osculating plane and the osculating circle, which give nice approximations to vector curves.
4. What was your Biology (or other science) IA topic?

5 trials for each of 5 concentrations. I totally forgot to do a control group lol, but I would just have it as testing enzyme activity w/o any enzyme.
5. What was your Biology (or other science) IA topic?

Ahh I can actually help you with this! I did my Bio IA (HL) on the effect of changing the concentration of enzyme on enzyme activity. I used yeast (catalase) and measured the decomposition of hydrogen peroxide using an O2 sensor. I did 5 different concentrations, measured the O2 concentration (in ppm) and tabulated all the data. I then standardized it (took each data point, subtracted the mean, divide by standard deviation. I gave the formulas for calculating the mean, standard deviation, and standardization and then gave sample calculations). I then calculated the rate of reaction by subtracting the initial O2 concentration from the final one, and dividing by the time interval [This is just how you calculate slope normally. y is O2 concentration, x is time]. Enzyme activity tends to be 0 order (or linear) at high substrate concentrations, so that's allowed. I then plotted these rates against the enzyme concentration, and that produced a graph that confirmed my hypothesis (rate increases w/increasing enzyme concentration). Other things I talked about: Things that went wrong (including how I wanted to vary pH at first, but that didn't work), sources of error and how they affected my data, uncertainty and how I had high uncertainty in measuring the concentrations of diluted enzyme, etc. Explaining what you're doing and why will add a lot of material to your paper. I got a 23/24. My IA was chosen by IB to be scored, and that was the grade they gave me. I hope that helped, and good luck!
6. Calculus- Optimization Problems

I'm going to give my input on this, and I too agree that #1 is a little tricky. First, @kw0573 is correct in that there are two ways to tackle this problem. The first is to consider the sides as having dimensions h by 2x and the front & back as having dimensions h by x. The second way is to just reverse the two dimensions. @FChaosi_ has done the method where the sides have dimensions h by 2x, although... This step is incorrect. The leading term should be 8/3x^2, not 2/3! Finding A'(x), setting A'(x)=0, and solving for x yields x=(49)^(1/3)/2, and h happens to be equal to this. The total cost function I wrote is: C(x)= x(2x)(3c)+x(h)(3c)+2x(h)(3c)(2)+x(h)(c)+x(2x)(c). This is equivalent to what @FChaosi_ has for the surface area multiplied by 3c, where c is the cost of the pine wood. I also multiplied by 3 to remove the fractions. Let c=0.6 (just a random test number) and find C(49^(1/3)/2)=48.2058658059. The method I used is to where the sides have dimensions h by x, which is also the method that @kw0573 seems to think is correct. This is implied (or was at least to me) as the "long" side would be used for the front (silly technicality, I know, but this is IB we're talking about). The cost function for this is similar to the one above and is given by C(x)= x(2x)(3c)+2x(h)(3c)+x(h)(3c)(2)+2x(h)(c)+x(2x)(c). We can make the same substitution for h, which is 12.25/(2x^2), and simplify C(x) to be: (12.25/x)(7c)+8(x^2)(c). C'(x)=c(16x-49(7)/(4x^2)). Solving for x gives 7/4, and h is 2. Much nicer numbers than the first! And if we let c=0.6 like before, and do C(7/4), we find that it is 44.1. This is lower than the first cost, so we should accept the dimensions x=7/4 dm and h=2 dm. For #2, we're going to write a bunch of smaller functions to model what's going on. First, let's write one for the total amount of apples after x months have passed (since September). This will be A(x)=10000-200x, as the amount decreases linearly, and 200 decay every month. Next let's write the price that we can sell them at after x months. In September, this is just 25, but it increases (linearly, again) by 5 cents per month. This gives P(x)=25+5x. Now let's look at the storage cost. This is going to be C(x)=350(100)x. I multiplied by 100 to convert to cents, because the cost is in dollars. The gross return will then be G(x)=A(x)(P(x))-C(x). In words, this tells us we have A(x) many apples selling for P(x) cents per apple, and we subtract the monthly storage cost. Let's simplify G(x) before we differentiate it. G(x)=(10000-200x)(25+5x)-350x=1000(50-x)(5+x)-35000x=1000(250+45x-x^2)-35000x. G'(x)=1000(45-2x)-35000. Setting G(x)=0 and solving for x, we get x=5 months, so next February.* *Disclaimer: I was never really that good at these cost problems, so something may be wrong here, but this is what makes sense to me!
7. To add onto this, even if you get a 20/20, there's still the chance of moderation. Last year, my teacher gave a few 20/20s and didn't get moderated. This year, she did the same (mine being one of them), and had them moderated down (mine to a 17/20). I still got the 7, which is all I really cared about. The point is that while you should still try your hardest to get as high a score as possible, there is no guarantee that you will get that score (it may even go up!).
8. Where can I find people who also want to be mathematicians?

Math (& Biochem) major right here! I want to do research in either (right now I'm leaning towards math), and end up being a professor. I really like teaching, but my hs math teacher is very adamant about me not becoming one, and I'd rather be a professor myself.
9. It seems to only be the topics: (https://ibpublishing.ibo.org/server2/rest/app/tsm.xql?doc=d_4_biolo_gui_1402_1_e&part=9&chapter=5) "The internal assessment requirements at SL and at HL are the same.", "Assessment criteria are the same for both SL and HL.", "The performance in internal assessment at both SL and HL is marked against common assessment criteria, with a total mark out of 24.", "The task will have the same assessment criteria for SL and HL. The five assessment criteria are personal engagement, exploration, analysis, evaluation, and communication." However, it does say that: "The range of practical carried out should reflect the breadth and depth of the subject syllabus at each level." So it seems that the criteria for grading are the same for SL and HL, and the only difference is the topics you can choose (HL, of course, has more). My personal guess is that HL students would be expected to produce higher quality IAs, but I have no evidence to support this.
10. Math IA Topic Help!

My teacher said the exact opposite (granted, I didn't take her advice...). So long as it's a part in the curriculum, you're good. To quote IB (https://ibpublishing.ibo.org/live-exist/rest/app/tsm.xql?doc=d_5_matsl_tsm_1205_1_e&part=1&chapter=8), Criterion E: "Students are expected to produce work that is commensurate with the level of the course. The mathematics explored should either be part of the syllabus, or at a similar level or beyond. It should not be completely based on mathematics listed in the prior learning. If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion. " Statistics is fully within the SL syllabus. According to the syllabus (which can be found here: http://highered.colorado.gov/academics/admissions/PLA docs/Mathematics (GT-MA1)/IB – Mathematics SL Curriculum Guide.pdf), under Topic 5, you can see that it covers: Data, data processing (mean, median, mode, etc.), correlation, and more. Keep in mind that if you do correlation, you must talk about the correlation coefficient and its meaning. A personal friend of mine wrote their IA solely on the probability of winning different lotteries, and got 20/20. My IA was a proof far beyond the SL scope, and I got 20/20 too. So long as you're in the SL curriculum, it doesn't matter what you choose. What matters is how well you explain the math, how sound/logical it is, and that you reflect on it well. An example IA that my teacher gave us (that our previous IB teacher graded) was on Euler's Column formula and measuring load, and it got something like 11/20. Of course this was far beyond the SL level, but they barely explained anything and it didn't really make sense overall. For an example of how to NOT do an IA on BMI, see: https://ibpublishing.ibo.org/live-exist/rest/app/tsm.xql?doc=d_5_matsl_tsm_1205_1_e&part=2&chapter=21 Also, understand that you can still score well in Criterions A-D (Communication, Presentation, Personal Engagement, and Reflection) even if you do choose something below SL, as you'll only be penalized once (i.e., the max score of 2 in Criterion E). Recall that even in Criterion E, you're being graded on how well YOU understand the topic -- it's worse to choose something hard and not understand it well than to choose something easy and understand it spectacularly.
11. Viewing Topic: Math Studies Paper 2

As @IB_taking_over mentioned, it would be a lot easier to explain with the full question. With what you've given, though, I may be able to answer, so I'll try my best. When the question says 30 people use Airplanes and Trains, they mean that 30 use solely Airplanes and Trains AND Airplanes and Trains and Buses (i.e, the entire overlapping part between the Airplane and Train circles if you remove the Bus circle). This is because the people who use all 3 still use Airplanes and Trains. When the question says 35 use Airplanes and Buses, they mean that 35 use solely Airplanes and Buses AND Airplanes and Trains and Buses. When the question says 20 use all 3, they mean exactly that: 20 use Airplanes and Trains and Buses. Here it is visually: Now, the 30 people who use Airplanes and Trains is represented by the red (R) and purple (P) areas, the 35 people who use Airplanes and Buses is represented by the blue (Bl) and purple (P) areas, and the 20 people who use all 3 is represented by the purple area. The purple area, P, is 20. We can deduce then that the red area, R , is = 30-20 =10, because R+P=30. The blue area, Bl, is 35-20=15, because B+P=35. Next, the total amount of people who use Airplanes is given by the white part of Circle A, W, the red area, R, the purple area, P, and the blue area, Bl. We have that W+R+P+B=47, with R=10, P=20, and B=15. Therefore, W+10+20+15=47 => W+45=47 => W=2. This makes sense, because it is positive (and it means that 2 people solely used Airplanes). In what you did, you said that R=30, P=20, and Bl=35, which would mean that W+30+20+35=47 => W+85=47 => W=-48, which doesn't make sense because it's negative. You said that the 35 and 30 amounts were apart from the 20 who use all, but if this were true, then (as you said) it wouldn't make sense. Did the original question specify this, or was it a misinterpretation? I know when I was doing these Venn Diagram problems, I had trouble discerning from the two as well.
12. I find that it depends on you, your school, and the subject. For instance, I usually had 1-2 hours of homework every night, but more like 7 per week. Yet, that's because I did all of my Math homework during other classes, didn't have any Bio homework (except for studying for weekly tests), and left all reading-based homework to actually do at home [Which is what took up the most time for me, being a slow reader]. I was able to maintain a very good sleep schedule (I went to bed around 8:30-9:00 every night), but I know others in my school had TONS of homework to do because they didn't manage their time efficiently. Lucky for me, I tend to readily understand concepts & information, so studying wasn't something I really had to do. I didn't do MYP, so I can't help you there unfortunately!
13. 3 sciences in sophomore year?

Depending on which Group 4 [Science] IB Class you're going to take, you might want to drop either AP Physics or AP Bio. I know that, especially with Bio, there's a lot of overlap between IB Bio and AP Bio. With that said, in both Sophomore and Junior year, I took two science classes (Honors Physics & Honors Chemistry in 10th, IB Chemistry and IB Biology in 11th) -- so I know what it's like to take multiple science classes at once. I can assure you that it's not as difficult as it is daunting. I should mention that I was an A student throughout highschool. 2 is definitely manageable -- 3, however, I'm unsure. If you're worried due to the fact that they're all AP classes, I wouldn't worry. Many of my friends took AP Gov & AP Macro in 10th grade, along with a variety of other AP classes. If available, they took AP Spanish Lang/Lit., and AP Bio/Chem. All of us were required to take AP English Lang and AP Euro. For one in particular, the amount of AP classes was staggering; They had AP Gov, Macro, Eng Lang, Spanish Lang, and Euro. They also happened to be in my Honors Physics class, and seemed to be doing fine (actually, pre-cal seemed to be what hurt them. This is only because of the teacher we had though). At my school, we were not allowed to take AP Physics/Biology without having taken Chemistry (which I thought was a silly rule). Due to this, I had to take Honors Physics over AP Physics, and cannot afford you guidance as to how hard it is or what the content is like. I can help you with AP Biology, though, given that it is similar to IB Biology, which I took. You will not need much chem knowledge for AP Biology. The only time you really deal with chem is in Biochem (clearly) and with metabolic pathways (Krebs/Calvin Cycle). And even then, the chem knowledge would only serve to better your understanding, but was not imperative. I would like to point out that, while the strength of your schedule is important, universities do not want to see you overburden yourself and fail. They want to see you excel while challenging yourself both in and outside of school. I believe that you should be fine dropping an AP, given that you're also focusing on your extracurriculars. IB will make up for what you might have "lost" in 10th grade. And, in the event that you have spare time, you can always take the class online, DE, or self-study for the exam.
14. IB Spanish is more about the culture surrounding Spanish-speaking countries. You'll learn things about the celebrations, festivals, traditions, etc. as well as certain topics dealing with the environment, technology, personal health, etc. You will also learn how to write in various styles, like in a diary, blog, essay, etc. While you will review the grammar and other aspects of the language (all while learning definitions to words concerning the aforementioned topics), you will also supplement your current knowledge with a variety of new tenses (if you haven't already, perfect tenses, past and future subjunctive, commands), rules, and more. I'm not sure if your school does anything differently, but mine went up to Spanish 4 before we were supposed to enter IB Spanish 4, and then take IB Spanish 5 in our senior year. However, I decided to skip IB Spanish 4 and transition from Spanish 4 to IB Spanish 5 from sophomore to junior year, and I took the test last year. I will assume that my Spanish 4 is equivalent to your Spanish 3 given that you said you have IB Spanish 3 & 4. Because of this, and from my own experience skipping a year of Spanish, you should be fine. I really didn't know a lot of Spanish before IB (In terms of tenses, I probably knew the present, preterite, imperfect, future, conditional, and subjunctive) and my vocabulary was extremely limited. So I wasn't the strongest, and I also didn't want to skip a year of Spanish -- I needed to take 2 tests my junior year -- because I was very uncomfortable with it. A majority of the students who skipped a year got 7s [Including me], a handful got 6s, and one that I know got a 5.
15. For those currently in post-secondary or starting it soon, how did you decide on your degree?

My plan is to double-major in Mathematics and Biochemistry! I've always excelled in math and science. I genuinely love them to death, ever since a young age. In highschool, I had the opportunity to entirely skip Calc I and move into Calc II (AP Calc BC), which really influenced me a lot because I was finally "exposed" to higher mathematics. I was also one of two students to take two IB sciences (which isn't uncommon here lol). Although, I switch from liking one more than the other often, hence why I'm majoring in both ! In math, I really enjoy calculus and complex analysis (although I've never had the opportunity to study the latter, the US education system neglects it heavily. Everything is done under the reals); In science, I really like Microbiology & Virology. Other factors? My senior year, as aforementioned, I took AP Calculus BC. The math teacher happened to also be my IB SL math teacher, so I saw her quite often. She's really one of my closer friends, and had a huge impact on me. She's the reason why math became more than rote memorization of steps and processes, and actually explained the underlying principles. Also, it didn't really have a role in my decision, but I was generally known as the "Math kid" in school. Peers would always start a conversation with asking if I've been up to any 'recreational mathematics' (an inside joke among us IBers) Did my parents play a role in my decision? Not really, I kind of had my mind up for a loooong time! They did support me, however, which is nice. Science kind of runs in my family -- My father's a marine biologist, my aunt is a botanist, etc. I might change my mind in the future, though, so who knows? Right now I'm really into learning math, so in grad school I'll probably drop the science and focus on that. The goal is to become a professor (Though not a teacher. Literally everyone I talk to dissuades me from that path. Even my own math teachers!)
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