Remarks on AP courses in high school

[moose]

Today I took my AP Calculus exam... The free response section was kind of challenging and fun.

EDIT: I'm dying in physics. In physics, we go over 40% of what will be on Monday's AP Physics exam. I haven't slept more than 5 hours in one night in the past two weeks because of that, biology, calculus, and some other things. I have almost everything for physics down though, I'm excited :)

 

[Pseudo Statistic]

Today I took my AP Calculus exam... The free response section was kind of challenging and fun.

EDIT: I'm dying in physics. In physics, we go over 40% of what will be on Monday's AP Physics exam. I haven't slept more than 5 hours in one night in the past two weeks because of that, biology, calculus, and some other things. I have almost everything for physics down though, I'm excited :)

Hmm... interesting..

A classmate of mine, who's a big math geek by the way, said that the AP Calculus exam screwed him over... makes me wonder what kind of questions came up. :D

What physics exam you doing BTW? Here I'm up for Physics B on Monday (Which I've spent quality time lubricating for...) and Physics C: Mechanics in like... 2 weeks. Physics C is what I'm not so worried about.

 

[mathwonk]

nice example curious. i think now i understand the whole problem with AP courses. The key issue is one raised earlier in posts by Hurkyl, motai and bomba, and also argued in a wordier way in my original essay.

Namely, AP courses have replaced honors courses. AP courses, which used to be meant foir honors level students, are now not challenging or
interesting enough to catch the imagination of the best students.


Maybe what we need to do is pose harder problems in our math couirses, even unsolved ones, to teach thinking and research habits, while it is still useful, i.e. while kids are young and bright and curious.

The worst thing we can do is take young kids and tell them their future depends on their performance on some standardized test, instead of their ability to solve hard problems.


Because then in grad school, we have students in their 20's asking us how to do research, since they have never done any in their studies.

 

[mathwonk]

and moose, I'm glad to hear there was a fun free response section on the calc exam, maybe they are getting better.

 

[EbolaPox]

I took the Calculus BC AP Exam yesterday, and I found the test to be pretty fun. Many of my classmates found it to be quite difficult and challenging, but it's a pretty good judge of computational skill. It is NOT meant to be a test to ask the student to prove anything, however.

Unfortunately, friends of mine in the IB program decided to take the AP Calculus AB (Calc I equivalent) test, and said it was "insanely difficult." Judging from tha tstatement, it appears that IB mathematics is even worse off than AP if the students that are in the top of the IB program where I live found the AB test extremely challenging.

However, what you say is true: many teachers focus purely on the student getting a minimum of a 3 on the test. In fact, there are only two students in my AP Calculus class (me included) that are actually interested in mathematics as a subject and do outside work. I've been reading Volume I of Calculus by Apostol and found it to be completely different from my textbook. Far more interesting and with problems that are actually enthralling, Apostol's seems to be better for a student that is interested in mathematics, as opposed to a student that merely want to exempt themselves from college calculus so they can get ahead in their college career or look better to admissions officers.

It would be nice if High School's offered both an AP Calculus AB/BC course and an "honors" level Calculus course that made the students do proofs and learn more than just the computational skills that are taught in the AP Course. However, I still support the AP program, it is probably the best thing that has happened to me in all of my education, besides my good math teachers.

(BTW: If you're curious, the CollegeBoard posts past AP Calculus BC and AB free response questions online. You can go see them and judge the test for yourself.)

 

[Jameson]

I just took AP Calc BC on Wednesday as well. To comment on a few things:

(1)My Calculus class will not do anything math related for the rest of the year now that the AP exam is over.

(2)The AP Calc test itself is not easy, but because it has to be so comprehensive, it's not rigorous. There are 45 multiple choice questions and 6 free response questions that are supposed to cover everything from limits to infinite series.

I'm attending the University of Florida this fall as a mathematics major and would like some advice. I'm confident I passed the AP Calc BC exam which would give me Calc I and II credit. Should I retake either course at UF?

 

[Cincinnatus]

You should not retake either course. There will probably be some sort of honors linear algebra or multivariable calculus course meant for first year math students. If its anything like where I go to school they will spend a good deal of time in these courses teaching things that you would have gotten in calc 1-2 had you taken them 30 years ago in a college but almost all high schools don't bother teaching.

Professors do understand that almost no students intending to major in mathematics today haven't taken these AP exams prior to coming to college and consequently are unlikely to take calc1-2 in college. So calc1-2 classes typically end up being "math for non-majors" type classes.

 

[G01]

I would say yes to be safe and for good measure to make sure you learned everything. My philosophy is to never test out of courses in your major. I think that can only hurt someone. Take the courses. You won't be behind anyone and it can only help your education. You don't want to chance going into higher level courses without enough prerequisit knowledge.

 

[omagdon7]

I go to UF both classes (calc 1 and 2) are a total waste of time if you know calculus. I retook the Honors version of the series after being in your position, just skip straight to calculus 3 and then start in on linear algebra. If you actually know calculus you will be fine.

 

[mathwonk]

I am not expert on UF, but it reminds me of the course I taught in high school a decade or so ago. The weakest student I had dropped my high school course in several variable calc because basically it was too hard for him.

He enrolled in UF and afterwards argued that he was right to drop my course because he "did fine" in calc at UF after only the basic AP course. So I got the impression that basic UF calc courses are pretty mickey mouse.

Of course we all have this problem in college because we HAVE to water down our courses to suit the weak preparation provided by most AP courses.

But usually, and I assume this is also true at UF, there are better courses available for people who want them, either honors versions, or some kind of higher level course.


(My stronger students, who stayed and worked in my course in high school, went to Harvard and Yale and Duke and UMass afterwards and did fine there also, in real calc courses, not joke ones that were spoiled by trying to please the AP crowd.)

 

[Jameson]

Mathwonk: I see your point. I'm definitely not trying to get by and only get credit. I want to learn mathematics as rigorously as possible. But it appears that UF Calc I and II don't offer the classes at a level where I would learn anything more than I already know on the subject.

omagdon7:Thanks for your advice. I think I'm going to go straight into multivariable and diff eq. Do you know anything else about the program? Anything about the Putnam competition?

 

[mathwonk]

well the fact is that there are college calc courses that are superfluous for someone who has done well in AP calc, I just don't teach too many of them myself.

We are forced to offer courses that our students can handle, and AP courses have forced us to lower the level of standard college courses to what average students can handle.

The point is that AP is a misnomer, i.e. they are not "Advanced placement" courses. One should not jump usually from AP to plain vanilla calc 3.

One should instead go from AP to a beginning course in calc at the elite honors level, i.e. a beginning course from Spivak or Apostol, if you can find one.

To go into ordinary calc 3 wil mean putting yourself in the same class with the weak students from UF or wherever that came up through the ordinary calc 1 and 2.

that way you never get above the non rigorous AL level of course.

 

[jbusc]

This is an interesting discussion for me as I did loads of AP's in high school (most of them self-study) including Calc BC (5) and both Physics C (5 & 5). All told I took 7 AP exams and got 6 fives and 1 four (in english) and got nearly a year's worth of elective credit.

Now that I've gone through 4 semesters of college though I'm definitely willing to say that "Advanced Placement" coursework is more of a "high-school plus" rather than true college level work. Given the option, I would have greatly preferred to take college classes rather than take more AP's - which is another consideration, since many high-schools and universities strongly insist on AP work rather than any regular college credits.

The point is that AP is a misnomer, i.e. they are not "Advanced placement" courses. One should not jump usually from AP to plain vanilla calc 3.

I did this semester, and I think I did well, getting an A (or at minimum an A-minus) in a class of 100 with a grade distribution centered around a B-minus average. *But* I'm not sure how much that's a function of the fact I learned calculus on my own and not through an AP course.

One should instead go from AP to a beginning course in calc at the elite honors level, i.e. a beginning course from Spivak or Apostol, if you can find one.

For physics I'm doing an honors sequence intended for people who did well on the Physics C exams, and it is not easy! I think even the regular 3-semester physics sequence here is more rigorous than the AP physics C, and the honors sequence is far, far more rigorous than anything I've seen before (the text is Ohanian, for those interested)

 

[Lucretius]

Well, on Monday I have my Physics B AP test. Wish me luck!

 

[Pseudo Statistic]

Good luck; I need it too. :P
I've decided not to go to school tomorrow (In the middle east our weekends are Thursday and Friday... don't ask) in order to study for the test... and to go to school at around 11AM on Monday so that I'm not too tired or anything. :)

 

[George Jones]

In the middle east our weekends are Thursday and Friday... don't ask

This makes perfect sense. The Saturday/Sunday weekend originated in the Christian world - the weekend is the day of worship and the day before. In the Islamic world, Friday is the day of worship (Khutbah/sermon and Jumu`ah prayers), so the weekend is Thursday/Friday.

Good luck to everyone!

Regards,
George

 

[Astronuc]

Interesting thread, Mathwonk! Thanks! And you are right on the mark. I could have said the same thing 30+ years ago, and the situation is not any better now than then - which is a sad indictment of general state of education in the US.

I took AP Calculus (BC), Chem, and Physics as a senior, and got 5, 5, 4. I placed out of the fresheman introductory courses, 1st year Math (Calc), Chemistry (but had to the Lab course), and one semester of Physics (I had to take the modern physics part which was intro QM). I did OK in Chem and Physics, and since I was a Physics major, I did not take any more courses in Chem afterward.

In Math, I jumped into an Honors Sophomore course in Linear Algebra and promptly got my a** kicked, so I dropped back to the general sophomore course in multivariable calculus and did OK. What I realized was that as rigourous as our AP Calc (BC) course was, there were certain areas we did not touch, which was unfortunate.

In our high school, the teacher for AP Calc BC was the department head and qualified math teacher with a degree in mathematics. She used a college level textbook, so we actually had the equivalent of a freshman intro calculus course, including ordinary diff EQ and integral calculus. When we learned about the derivative and differential calculus, we did the epsilon-delta proofs, and we did various theorems and proofs in differential and integral calculus. But that seems to be the exception, not the rule.

Similar, our high school chemistry course, was taught by the head of the department, and she had an MS in Chemistry. Again, that seems to be an exception, not the rule.

I would recommend that any student, who takes an AP class, simply go and talk to the appropriate professors and find out about the courses directly. Alternatively, toward the end of the semester, find out what texts are being used at the university one will attend, and then try to compare the college level texts with the one being used in high school.

 

[mathwonk]

jbusc, i did not mean you would never succeed by jumping from AP calc to ordinary calc III.

There are two scenarios likely there: 1) either someone is not too strong and then they will not succeed in calc III (this is the most common one), or else 2) the person is strong and then they are in too easy a course in calc III.

i.e. the weakly prepared AP student is out of their depth in even the ordinary calc II, and the well prepared AP student is in the shallow end of the pool in ordinary calc III.

the real honors student does not belong in the regular calc sequence at all, no matter whether at the I level or the III. Thus an AP course designed to let them skip calc I and go to calc II or II is missing the point of accurate placement for good stduents, and denying them the college course best suited to their needs.

In fact the ideal college course for strong students, the beginning spivak course, does not even exist at Harvard or many other schools today, precisely because so many strong students went the AP route, that the demand for this superior course disappeared.

A few schools, in cluding UGA in Athens, still offer this cousre to a handful of students. E.g. when I teach ordinary calc I or II, I try to identify misplaced strong students in the first few days.

As much as it pains me to lose them, if I have a strong stduent, I immediately advise them to get out of calc II and transfer to the beginning spivak course.

sadly, often they decline, because mommy and daddy want them to save the tuition from the one course that they have earned with AP credit and push on in the non honors track instead.

we could counter this by denying college credit for AP courses, and some coleges are beginning to do this, but we would lose money and talent as many students would go where they are offered that financial incentive.I did nolt always feel this way. In the past i have overheard professors in my department advising students not to take their ordinary calc I course if they have a 4 or 5 on AP but to go to honors. I was surprized as I used to try to teach even ordinary calc I at a higher level than that. I never advised anyone to skip my course. But over the years my course too has slipped down to the level that almost anyone can survive.

When a student comes to me for advice and says they have had AP calc, I may ask them to state the fundamental theorem of calculus, or the mean value theorem, with hypotheses. (it seldom gets as far as asking for the proof.) usually they cannot. If not, then i tell them they will probably learn something in my course, even the first one.

 

[mathwonk]

Last semester I had a very strong student in my ordinary calc II. I tried unsuccessfully to get him to transfer. Since there were also 4 or 5 other good students, I ratcheted up the course level slightly, as it seemed appropriate. I enjoyed it immensely for the talented group of students, but 19 out of 35 dropped out at midpoint, and 5 others got D's at the end.

I am not talking about an impossibly diffcult course since the top student averaged about 107/100. I.e., I set his grade at A+, not A. E.g. no proofs were required at all.

It is not at all unusual for us to have many students in calc II who do not know the product rule or the chain rule, who cannot write an equation for a line, do not know the derivative of sin or tan, cannot graph y = e^x, and even students who factor the 3 out of expressions like cos(3x) "=" 3cos(x) ! Seriously. And not being able to add fractions is very common. I have even had students who literally could not multiply 2 - digit numbers without a calculator.

It turned out my top student had been admitted to a top school but came to UGA for financial reasons. I believe such students belong at better schools for the camaraderie of other fine scholars, but if they choose to do so, they can also challenge themselves at UGA by following our advice.

 

[Astronuc]

It is not at all unusual for us to have many students in calc II who do not know the product rule or the chain rule, who cannot write an equation for a line, do not know the derivative of sin or tan, cannot graph y = e^x, and even students who factor the 3 out of expressions like cos(3x) "=" 3cos(x) ! Seriously. And not being able to add fractions is very common. I have even had students who literally could not multiply 2 - digit numbers without a calculator.

That's worrisome. We necessarily had to master the chain rule in my AP calc class and then use it to prove various identities. So we learned it very well. In fact, IIRC, we had to use the definition of the derivative to prove the chain rule.

The first part of our Calc class was elementary analysis and analytical geometry, and we had to know equations for line, plane, min distance between lines, distance between planes, closest distance between surfaces, conic sections, . . . . Somewhere in there, we did series (infinite series), continuity/discontinuity, then jumped into limits, then into the definition of the derivative.

The Calculus class was developed on the basis of our honor (major works) Algebra II class in which one year of advanced algebra and one year of trigonometry was crammed into one year - i.e. we did the equivalent of the one of algebra in one semester and one year of trig in the other semester during my junior year.

What I didn't see was coordination between Calculus and Physics, and it was not until university that I began to understand better the relationship between Physics and Math.

 

[Astronuc]

. . . . the generalized fundamental theorem of calculus often mentioned here, that a lipschitz continuous function which is differentiable almost everywhere, with derivative equal to the value of a riemann integrable function at points where that function is continuous, does equal the indefinite integral of that function.

in particular we studied the contrast between continuity, uniform continuity, and lipschitz continuity.

I think this would blow most high school math teachers away.

 

[mathwonk]

well i only learned it recently myself, but it is amazing how simple you can make something seem, once you understand it yourself.

the benefit of my courses, i hope, is that I learn something new every year and put it in my courses. i am still infected with the bug i caught at harvard as a student, i.e. we expected our professors to teach us things that they knew that others did not. we did not expect our courses to be as easy as possible, but to be as valuable as possible. we wanted to learn things that students at easier schools did not learn.

i am a beginner at diff eq and have never liked or understood it before, so i worked hard this semester and came to love the stuff. I consulted extensively in v.i.arnol'd's works, and when he recommended the 100 year old treatise of edouard goursat i bought a used copy of that and consulted that too. I also used as references: braun, guterman - nitecki, boyce - di prima, edwards - penney, coddington, waltman, hurewicz, tenenbaum - pollard, henry helson, jerry kazdan's harvard notes, that's all i can remember.

 

[mathwonk]

here is the list of topics from my course:
math 2700 sp2006 Topics

first order equations, review of integration from 2210
the inverse function theorem and separable variables.
linear equations, exactness, integrating factors. why not every vector field is a gradient, hence exactness cannot always be achieved.

exponential functions, problems involving exponential growth,
population problems, mortgage problems, radioactive dating of paintings

systems, vector fields, predator prey models, equilibrium points, intro to linearization and stability.

second order equations, linearity, characteristic polynomial of a second order equation. factoring second order constant coefficient equations using linear constant coefficient operators, and uniqueness of solutions.

method of annihilators, method of power series, for finding formal inverse of operators, partial fractions.

method of judicious guessing, when the non homogeneous function is simple, e^x or sin or cos (i.e. has a known annihilator).

power series methods for solving homogeneous linear equations with anlytic coefficients, especially simple ones, like constants or polynomials.

 

[mathwonk]

Euler’s equation - factoring it using the linear operators (xD-a).

variation of parameters, for solving non homogeneous equations when general solution of homogweneous equation is known.

linear systems, matrix exponentials, eigenvalues, eigenvectors, determinants for 2x2 matrices, diagonalizing matrices (2x2 case), dealing with non diagonalizable matrices as diagonalizable + nilpotent ones. how to compute matrix exponentials in that case (2x2 “jordan form”).
application to electrical circuits, harmonic motion, simple or complex pendulum. stability in terms of eigenvalues, attractors, repellers, centers, hyperbolic points.

non - linear systems, linearization, Hartman Grobman theorem on when stability agrees with linearization.

approximation by Euler’s method related to exponentiation of vector fields. Connection with existence and uniqueness theorem.

 

[courtrigrad]

I hated my AP Calculus BC class. The teacher was inexperienced, and the course was not rigorous. It seemed that all we did was memorize formulas, and 'plug and chug' to get ready for the "exam of our life." I honestly feel that I did not learn anything in this class. Right now, I am starting from scratch, reading Spivak's book. As I will be a math major, I want to learn the subject properly.

 

[mgiddy911]

I am currently a High School Senior (woohoo only 5 more days!) from a private Jesuit high school.
My junior year I opted to take regular Physics B, the non-AP course, however taught by the same teacher as the AP class who was also my water polo coach.
I regret not taking the AP course because it would have been more rigorous and I would have learned more problem solving techniques. However as for the AP test, I feel I was still prepared to take the AP test even though I didn't take the class.(note I actually decided to not take the test, but prepared for it for a while before i made the decision, I opted not to take it only because I plan to major in maths/physics and decided that AP credit in algebra based physics would not really mean i should skip out on any college physics.)
I bring this up for two reasons. One is to show that I certainly agree with previous posters who mention that the merit of one AP class over another at various schools often comes down to the teachers. I felt prepared for the AP test having not taken the AP class because my teacher was amazing. Yet I do not think that this means in any way I was prepared to be placed in advanced college physics classes.
Another point I bring up, and I apologize if i skimmed the 7 pages of this topic and didn't see anyone else address this. My chemistry teacher, and a few members of my class, previously this year, discussed the old way our school did certain AP subjects. My school used to teach Accelerated Chemistry, where the teacher taught how and what he felt necessary. And student took the AP exam if they wanted to if they felt prepared. However now that our school offers AP Chemistry, our teacher is forced to teach in a specific manner, because now students expect to be able to do well on the AP exam, so he has to make sure he covers what The College Board wants him to cover including lab work.
I also recently took the AP AB Calculus exam. I think our class may be a little more in depth than those of some posters. I feel confident going into Calc 2 for college (though if i receive a 5 , my college could grant me credit for calc 1 and 2 for some reason). The only exception i have is proofs, and some definitions. My AB class did almost no proofs, and we skipped or glanced over some definitions.

I also took the AP Computers AB exam ( the computers AB is the harder of the two exams A and AB, possibly equivalent to the relation of BC to AB in calculus, as the computers AB would count as credit in you intro computers class as well as Data Structures)
I am wondering if anyone else has taken the exam and gone into college with advanced placement form it? Do you feel like you missed out on anythign big? were you prepared to take whatever the third computer class would generally be at your university?
I guess I should note that the AP Computers exam/class is now completely Java Based.

I am not certain where I stand on the AP issue as a whole, as it seems to differ form school to school and particularly from teachers. One thing I do think is still good is that Students who have some knowledge of what they want to major in are able to get elective credit out of the way. So someone who wants to major in math can get history credit out of the way, giving them more time to broaden their scope of mathematics. And depending on their school, they may get some value as far as writing or problem solving or study habits go, from taking an AP course.
I don't see AP as a good way for students who want to pursue maths to challenege themselves unless the AP class happens to be the highest math they can take at their high school. I wish my high school had offered me dual enrollment for things like math. I would have rather dual enrolled Calc 1 at a local university than taken the AP Calc exam where The College Board determines what I should know to be on par with college level.

I'd liek to hear back on anyone's comments, especially on the AP Computer topic, or my poor grammar / typing ability.
thanks

 

[Hurkyl]

I do think is still good is that Students who have some knowledge of what they want to major in are able to get elective credit out of the way.

Only to some extent.

I got a 3 on the music theory AP exam, but still had to take a music general ed class. (Actually, it wasn't so bad, but still!)

 

[jbusc]

It is very difficult to say whether a computer science AB exam will substitute well for a "intro to programming" or "data structures" class at a university. For example, both classes at my university are waived for the AB exam, and it is recommended to skip "intro to programming" if you passed the AP. If you are good at programming "data structures" can be skipped, but some do not, opting for better preparation for later classes.

However, at some universities Computer Science is taught from the beginning in a higher, more theoretical fashion and "intro to programming" classes are taught in functional languages such as Lisp or Scheme. Students should _not_ skip those classes unless they are familiar with Lisp or Scheme and the AP computer science test provides little preparation for those languages and functional programming.

 

[Pseudo Statistic]

I took the AP Physics B exam earlier on today. (I figure people in the US are taking it as we speak
After taking it, I have to say, I was pretty surprised at it..
I mean, I expected a harder test-- not to say that I'm 100% certain I got a 5. (I guessed a lot of the multiple choice questions and, from what I predict I got in the free-response (71/80), I have to get somewhere in the region of 23-27 wrong to pull off a 5 (I skipped 7 questions), according to my predictions)
However, the test, from my perspective, didn't do a good job of properly measuring people's thinking skills, in my opinion.
Everything was so plug-and-chug... with the exception of one part of one of the questions...
And the multiple choice was so... well, it wasn't that bad, but it would have been VERY straightforward had I shifted my focus in the topics I studied.
Overall it was pretty OK. However, I don't think I'll be able to say how I did for certain, lest I get <5 meaning I would have spoken too soon. :)

 

[0rthodontist]

I think that the AP exams when I took them were far too easy. A 60% or higher raw score on the CompSci AB was a 5. Maybe if the test scores were given as a straight percentile, the AP tests would be held in higher regard.

 

[mgiddy911]

On the note of the scores possibly being reported as a straight percentile. Could anyone explain to me why they are scored the way they are? besdies the fact that I guess it spreads out the scores neatly on most tests. And on some tests give you a lot of leway with your mistakes. I think for some tests a percentage grade would make more sense. For example tomorrow I take the AP chemistry exam. Now in theory tht class, and exam tests your knowledge over that would generally be covered in what most colleges call General Chemistry I and II plus their respective labs. Now that is a lot of material. It seems like it would be easier for a school to determine whetehr you are qualified to skip their class based on seing your overall percent score and the test itself. I don't know if they have access to those materials

 

[jbusc]

I think, it's for two reasons:

1. The actual grades are supposedly determined(in part) by giving the same questions in college classes, and comparing the raw scores from those students with their final grade in the class.
2. Free response changes in difficulty from year to year, so percentiles change also. Standardised grading makes scores comparable year to year.

 

[0rthodontist]

I said percentile, not percentage. A percentile is a number that indicates what percent of other test-takers scored lower or higher than you, such as "80th percentile."

Obviously not all courses at all colleges are equal in difficulty or have students who are equal in aptitude. Therefore normalizing the test to a "standard college course" is not legitimate if indeed that is what they do. There is no reason to deliberately toss away data.

 

[Lucretius]

Well, I found the Physics B test somewhat difficult. Let me clarify:

The multiple choice I breezed through — left only about 5 blank, and I was fairly sure of the ones I did fill out, so that wasn't a problem.

The free response questions, a few of them at least, I had to leave totally blank. I cannot discuss the nature of these questions yet, or else the AP guys will hunt me down and burn my test — suffice to say, I had a fair amount of trouble with some of them. Still, I'm fairly sure my score will be high enough to wave a physics 101 class in college next year.

 

[Pseudo Statistic]

Well, I found the Physics B test somewhat difficult. Let me clarify:

The multiple choice I breezed through — left only about 5 blank, and I was fairly sure of the ones I did fill out, so that wasn't a problem.

The free response questions, a few of them at least, I had to leave totally blank. I cannot discuss the nature of these questions yet, or else the AP guys will hunt me down and burn my test — suffice to say, I had a fair amount of trouble with some of them. Still, I'm fairly sure my score will be high enough to wave a physics 101 class in college next year.

48 hours have passed for me.
So DUDE, wait a sec.
The free-response... the ones I got were like super easy. The first one was like, measuring g and you had a table of d vs. t. The second one was on electricity I think. The third was on reflection/refraction and slit stuff. The fourth was on Thermodynamics. The mechanics question was somewhere in there and one on an electron and a positron and stuff.
If that's the paper you had, I seriously think you were mistaken if you thought it was difficult and the multiple choice was easy... 'cause... well... the multiple choice pretty much raped me. :(