How hard does this test look to you?

[Mépris]

http://www.cmi.ac.in/admissions/sample-qp/ugmath2010.pdf

It's a sample entrance exam at a university for their Mathematics and Computer Science course. Three year long, so I suppose one should think of the level of difficulty and breadth/depth of material as comparable to second year college.

A lot of the syllabus content mirrors what I have in my Mathematics a-level syllabus, the catch is that, year in, year out, papers are ridiculously similar and over the years, the style in which we are taught (and self "study" as well - more like additional practice, I suppose) is ruled by that. It's really just plug and chug. By the time I get home, I'm way too tired to do much and right now, I'm trying to figure out a way around this, so I can work on my school work and on this entrance exam but that's beside the point.

tl;dr - what do you think of this entrance exam? (difficulty of questions/content wise)

 

[micromass]

I'd say this is actually a pretty difficult exam. You'll need quite a lot of experience with such questions to be able to solve them all. I mean, you have 6 minutes for each question, that looks like madness...

 

[Klockan3]

Content wise you got regular 1d calculus and an intro discrete maths course, so like a first year student. The biggest catch is the time, when you got 3 hours to do 19 problems with that much calculations you don't got the time to think. So the only way to have it like that is if the exams are quite similar so that the students can learn solution algorithms for most of the problems.

 

[Mépris]

I'd say this is actually a pretty difficult exam. You'll need quite a lot of experience with such questions to be able to solve them all. I mean, you have 6 minutes for each question, that looks like madness...

Content wise you got regular 1d calculus and an intro discrete maths course, so like a first year student. The biggest catch is the time, when you got 3 hours to do 19 problems with that much calculations you don't got the time to think. So the only way to have it like that is if the exams are quite similar so that the students can learn solution algorithms for most of the problems.

I managed to get hold of somebody studying that course and they said that they achieved ~70% in the test. This seems somewhat encouraging. Yes, it does seem insane. Preparing for that test though, is something I am willing to do because of the place itself. It's dead cheap. My whole course and living costs for the three years will be cheaper than the amount I'd have to pay in the US or UK for a year.

Also, from the descriptions I have read of MIT (mostly posts by two-fish quant), this school seems to resemble it a bit. It doesn't seem entirely focused on just Mathematics - hell, they have a compulsory English Lit course in the first year! Lots of its Mathematics grads have gone on to do their PhDs, many of which have went to the "big" universities. I'm not concerned about the "big" unis, I'm more concerned with how the students made their way there. It might be something to do with the school's culture or it might be that the students themselves found their own way to the US and A.

Also, can the thread title be changed to 'Chennai Mathematical Institute + Entrance Exam'? Or whatever a mod deems appropriate. This way, maybe students studying there might post - I remember about one or two on these boards.

 

[DrummingAtom]

I'm curious what classes are you supposed to have done before you take this? Some of those questions don't look too bad but others look completely foreign to me.

 

[Lavabug]

Definitely up there, I don't think I could answer all of those questions on the spot without some prior preparation, much less with those time constraints.

 

[flyingpig]

It was okay until I saw the second page...

 

[andyroo]

None of the questions seem like they are necessarily very hard to answer, but the time that you have to do each question would definitely be difficult since so many of the questions are proof based.

 

[mathwonk]

i expected to be able to do them in my head, but the only one that looked that easy was II7.

 

[Mépris]

I'm curious what classes are you supposed to have done before you take this? Some of those questions don't look too bad but others look completely foreign to me.

It's an entrance exam for a BSc course in Mathematics. It's aimed to students having just completed high school, generally. Although, iirc, additional learning has to be done for this exam. There's a specified list of books.

None of the questions seem like they are necessarily very hard to answer, but the time that you have to do each question would definitely be difficult since so many of the questions are proof based.

i expected to be able to do them in my head, but the only one that looked that easy was II7.

Looks like hard work. Should get done with current maths syllabus, then move on to either this or further maths.

 

[kramer733]

I can probably answer 2-4 questions but then I'm toast. I'm entering a double major in math and computer science. If I can't do this entrance exam, is it a bad indicator that i'll do bad in math and computer science?

 

[nlsherrill]

I can probably answer 2-4 questions but then I'm toast. I'm entering a double major in math and computer science. If I can't do this entrance exam, is it a bad indicator that i'll do bad in math and computer science?

No, I think not. Really, you should be able to figure out many of them, but the time constraint makes that test a nightmare. I have made an A in all math courses in college up to differential equations, and that test would probably murder me.

 

[mathwonk]

ok here's one i should have been able to do in my head but didn't right away. if f is a polynomial with integer coefficients and f(1) and f(0) are both odd then f has no integer roots. I tried to use the very elementary fact that an integer root would be a factor of the constant term f(0) but got no where fast. Then it dawned on me that an integer root would also be root mod 2, but the hypotheses say precisely that neither 0 nor 1 is a root mod 2, hence there are not roots mod 2, hence no roots.

So that suggests to me these problems are indeed doable, or some of them, but take a little alertness. I.e. they seem to use basic information, but you have to be a bit clever to see what to use. Try another one or two. Maybe you'll get faster.

 

[wisvuze]

I thought the test looked pretty hard (especially for an entrance exam, but then is it an acceptance exam? or just an assessment test? ), but I've always been lousy at "doing math";
For 2) , the f(0 ) and f( 1 ) being odd thing, I used an inductive approach; that factor of f(0) thing worked out for me in the base case, then I examined x^n( a_n x^n-1 + ... +.. + a_0/x^n ) = 0

 

[wisvuze]

edit: nevermind, I thought a "repeated" root was something else for a second

 

[mathwonk]

#13 is just the binomial thm, #5 is just fermat little thm, #11 is just repeated squaring, #8 seems tricky to prove but clearly the max number of points occurs for the vertices and center of a regular hexagon.

moral: a test should be taken for what we can learn, not what it says about us.

for #9, by the derivative test the only possible repeated root is 0 which isn't a root.

 

[jbunniii]

A lot of these look hard but turn out not to be after you think about them for a few minutes.

#1 in the second section is a good example. It seems hard until you restate it in terms of sums from 1 to n and take everything modulo 100. Then the question becomes, is it possible for the sum from 1 to n to be different (and nonzero) for each n? A simple application of the pigeonhole principle shows that it's not.

The problem is that you need to have a similar insight on a lot of the problems, and you need to do it quickly. It's the time constraint that makes this test hard.

Another example is #1 in the first part. It's very easy if you use the trig identity

cos(a) cos(b) = (1/2) (cos(a+b) + cos(a-b))

but if you don't recognize that quickly, you will probably be spinning your wheels for a while.

 

[Vanadium 50]

For 2) , the f(0 ) and f( 1 ) being odd thing, I used an inductive approach

Too much work. Think about the parity of the problem - i.e. what it looks like mod 2. In Z2, if P(1) = 1 (odd), and P(0) = 1 (odd), then P(n) = 1 for all n. Therefore P(n) is never 0. QED.

 

[kramer733]

what do you guys mean by "mod x"? I've never been exposed to that. Also how would you guys do number 3b)?

 

[jbunniii]

what do you guys mean by "mod x"? I've never been exposed to that. Also how would you guys do number 3b)?

I don't think the limit in 3b exists. If x->0 from the negative side, the limit is -infinity, but from the positive side it's zero.

 

[lisab]

http://www.cmi.ac.in/admissions/sample-qp/ugmath2010.pdf

It's a sample entrance exam at a university for their Mathematics and Computer Science course. Three year long, so I suppose one should think of the level of difficulty and breadth/depth of material as comparable to second year college.

A lot of the syllabus content mirrors what I have in my Mathematics a-level syllabus, the catch is that, year in, year out, papers are ridiculously similar and over the years, the style in which we are taught (and self "study" as well - more like additional practice, I suppose) is ruled by that. It's really just plug and chug. By the time I get home, I'm way too tired to do much and right now, I'm trying to figure out a way around this, so I can work on my school work and on this entrance exam but that's beside the point.

tl;dr - what do you think of this entrance exam? (difficulty of questions/content wise)

It looks pretty challenging, compared to http://libraries.mit.edu/archives/exhibits/exam/algebra.html" .

 

[General_Sax]

Well, let's just say that I probably wouldn't be accepted :/

 

[mathwonk]

take it as one set of advised knowledge. it is saying one should know about modular arithmetic, basic euclidean geometry, the fact that a repeated root of a polynomial is one that is also a root of the derivative,...

 

[mathwonk]

good observation on 3b, to look at both sided limits. otherwise this would look like a l'
hopital problem, so another rule is everyone should know l'hopital's =rule, (which my university does not always teach).

problem #4 yields to a little geometric reasoning too after some brief thought. I.e. squares get further apart as you go up, so no infinite sequence of them can be the same distance apart. I.e. for every n, there exists k such that all squares after k^2 are further apart than n. since (k+1)^2 = k^2 + 2k + 1, just take k = n. this occurs to one while looking at a graph of the parabola y= x^2, and realizing the terms of the series have to lie on it.I recommend working out this test. It look like one of the most imaginative set of problems to measure ones ability to use of elementary college math that I have seen.

 

[jbunniii]

It looks pretty challenging, compared to http://libraries.mit.edu/archives/exhibits/exam/algebra.html" .

Ha, those were certainly simpler times. A freshman in high school today could answer those without breaking a sweat, but I wonder how many Americans in 1869 could do so.

 

[QuarkCharmer]

Ha, those were certainly simpler times. A freshman in high school today could answer those without breaking a sweat, but I wonder how many Americans in 1869 could do so.

I would like to know that as well. I would have guessed it to be a bit harder than todays entrance exams.

 

[cdotter]

Ha, those were certainly simpler times. A freshman in high school today could answer those without breaking a sweat, but I wonder how many Americans in 1869 could do so.

You are severely overestimating the mathematical aptitude of American high school students...

 

[jbunniii]

You are severely overestimating the mathematical aptitude of American high school students...

Maybe so, but the questions look like they could have been taken from a basic non-honors Algebra I exam, which all students have to pass if they plan to graduate. Unless Algebra I has been severely watered down since I was in high school (mid '80s).

I was under the impression that today's students are, if anything, a bit more advanced, with many students now taking algebra in middle school. Certainly that's true of the district where I live.

 

[mathwonk]

nice observation jbunniii on that question mod 100. I did not notice until your answer how to look at it, adding one additional term each time, so that if you ever got the same answer, you would know you had added zero mod 100.

anyway that gives us aniother basic principle one should know, pigeonhole.

by the way, this is a little out of date but a friend of mine documented quite convincingly that high school preparation in the US went down significantly for 100 years between 1890 and 1990.take a look at a high school math book from around 1900, by david eugene smith?, at amazon. there are algebra problems in there few high school teachers today could do. its not the topics, its the difficulty level of the treatment that has diminished.

 

[fluidistic]

take a look at a high school math book from around 1900, by david eugene smith?, at amazon. there are algebra problems in there few high school teachers today could do. its not the topics, its the difficulty level of the treatment that has diminished.

Interesting. I'm curious to know if 100 years ago high schoolers had to take as many courses as now. If they had less courses and could specialize themselves say in mathematics as early as they wanted, this might explain the level of difficulty of their book.

 

[gb7nash]

I did number 2 a little differently. Without much rigor, supposing that f(x) = a_nxⁿ + ... + a₀:

f(0): a₀ = odd
f(1): a_n + ... + a₀ = odd

Case 1: the root (call it b) is even

So a_nbⁿ + ... a₁b is even since b is even, causing arbitrary a_kb^k (k>0) to be even. a₀ is odd, so we have an odd number. -> <-

Case 2: the root is odd

Since for any a_k, if a_k is even, a_kb^k is still even. If a_k is odd, a_kb^k is still odd. So since by (2) , a_nbⁿ + ... a₁b is odd. -> <-

Too much work. Think about the parity of the problem - i.e. what it looks like mod 2. In Z2, if P(1) = 1 (odd), and P(0) = 1 (odd), then P(n) = 1 for all n. Therefore P(n) is never 0. QED.

That's certainly much cleaner than my approach.

 

[Shackleford]

Interesting. I'm curious to know if 100 years ago high schoolers had to take as many courses as now. If they had less courses and could specialize themselves say in mathematics as early as they wanted, this might explain the level of difficulty of their book.

That's a good question. One of my university friends from Germany said in high school they chose two subjects on which to focus. He said he had already covered multi-variable calculus in high school. I certainly did not. I took the entire Cal I-III sequence in college.

 

[Ryker]

Too much work. Think about the parity of the problem - i.e. what it looks like mod 2. In Z2, if P(1) = 1 (odd), and P(0) = 1 (odd), then P(n) = 1 for all n. Therefore P(n) is never 0. QED.

I assume you go to "then P(n) = 1 for all n" by induction, since one of n or n + 1 is 0 and the other 1, right? But why would you be able to generalize this from Z2 to R?

 

[mathwonk]

The difference apparently was the selection of students. 100-120 years ago only about 10% of all students even went to high school, so they were what would be called gifted or at least privileged today. My father had only a high school diploma, graduating in 1906, but read Archimedes, Aristotle, Shakespeare, Dante,... In his father's home they had the complete works of Washington Irving, Charles Dickens, ... I still have some of them.

I also have his, or his brother's, copy of Alexander Dumas' unabridged (1400 pages) Count of Monte Cristo, apparently read when he was a middle school student, inscribed in pencil "this the best book I ever read".

All my childhood I recall he read both the New York Times and the Chicago Tribune every Sunday, to get the major news. And we lived in Nashville, Tennessee. Recall that high school students used to learn Euclid, not the bowdlerized trivia we call geometry today. Get a hold of a copy of Euclid and compare it to the contents of even a good geometry book of today.

The best high school geometry book of today I know of is that of Harold Jacobs and it is nowhere near the quality of Euclid. Euclid was written as if the reader could be assumed to be intelligent, whereas todays books are written as if the reader is a moron, or at best a child of limited attention span.

Take a look at Euler's elementary algebra book, "Elements of algebra", written the publishers says for the benefit of a young servant knowing only arithmetic, and "not above mediocrity" in mathematical ability.

http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ElementsAlgebra.html

This only seems to be the first few parts...

 

[jbunniii]

I assume you go to "then P(n) = 1 for all n" by induction, since one of n or n + 1 is 0 and the other 1, right? But why would you be able to generalize this from Z2 to R?

If P(n) were zero for some n in R, then the same would have to be true mod 2. But that possibility has been excluded.