Ok, this is the final TNDOT on the 1911 pass Algebra Matric exam, consisting of the final three questions. The previous parts are TNDOT 10 and TNDOT 11, both of which have now been updated with solutions.
These last three questions don’t seem to me so interesting, except for pondering the way they were asked and what was expected. In any case, I’ll be interested in people’s thoughts.
9. Find the value of 
without assuming the value of 
.
In how many ways can n different letters be arranged in a row, 3 specified letters not being permitted to stand at either end of the row ?
10. Prove the Binomial Theorem for a positive integral exponent.
Shew that the coefficient of the 
term of 
is equal to the sum of the coefficient of the 
term and terms of 
.
Give a reason why this is necessarily the case.
11. Define a logarithm ; and prove that
Given 
, and 
, find the 540th root of .00007.
In Q11 the logarithms are in base ten if that helps anyone…
OK, no one is answering, so I will offer a suggestion for Q10.
An induction proof for the binomial theorem would likely have been a standard proof to know in 1911.
Explaining the second part is easy for anyone sufficiently familiar with Pascal’s triangle.
Thanks, RF. I’m not expecting too many responses on this one. I doubt that induction was expected for Q110. What makes you think so? Also, I think the second part is probably intended to be done directly by enumerating the number of ways of choosing by two different methods.
I suppose induction just came to mind as the “natural” way to prove the binomial theorem.
As to the second part, writing each out as a binomial coefficient and then simplifying the algebra would be my first thought – but I’m not sure if it is the only approach that would have been taught in 1911.
I agree though that these are not the most interesting questions on the paper.
No big deal. I decided it was worth doing full papers, in chunks like I’m doing it. So I’ll complete the papers already begun. I don’t expect people to work on questions of little interest. I’ll still try to supply good solutions, with some thought on how the questions were expected to be tackled at the time.
All good. Now you have used trigonometry and algebra papers, will calculus be next?
Yeah, possibly. It’ll take a while to complete the tests already started, so I’m not thinking about it now.
For Q11, I have seen textbooks from the 1940s that define a logarithm as the area under a
graph from 
to 
. I assume this is what they wanted.
For the second part, writing
as 
etc gives 
which then becomes 
Thanks, RF. I doubt if the 1911 answer to logarithm is an area thing.
Unless you have any textbooks from 1911, we may never know.
I think it is more likely to NOT be what I suggested given this was the pass exam and they probably wanted the logarithm-exponential relationship definition.
If we go back to 1202, in Liber Abaci, Fibonacci expected the reader to be able to multiply 8 digit numbers by 8 digit numbers essentially as an exercise in mental arithmetic – and this was only chapter 2.
Arithmetic in 1911 was a paper on its own so such skills would not be tested on the algebra paper.
For the second part of Q11, it helps to imagine a draft version of the question being simply phrased without the “hints”: Find the 540th root of .00007. How do you answer that with 1911 technology? Logarithm tables.
Take the log of .00007, divide it by 540, then take the antilog.
The log of .00007 won’t be in your table (which runs from log 1 = 0 to log 10 = 1 in increments of .0001 or so), but the log of 7 will be, so you look that up and add the log of .00001 which is obviously -5, yielding -4.154902 as your estimate of the log of .00007.
Next you divide by 540 (the most tedious part, I think, using plain old long division). Now you have the log of the 540th root of .00007, and you want to take the antilog, which means finding your number in the “log” column of the log table. But it isn’t in range so you have to add an integer to make something between 0 and 1, look up the antilog of that, and divide by a power of 10 to cancel the effect of the integer you added.
But during an exam, you don’t want people referencing huge tables, so you include in the question the exact 2 log table entries they’re going to need, in a similar form to how they would be found in a book, requiring the same amount of “compensate for adding an integer to the log by shifting a decimal point” logic. That’s an important calculator skill, from when people were the calculators, and testing that skill seems to be the main point of the question.
Giving just those 2 log table entries, with no added decoys, made the question a little easier. If you know the outline of the solution, you can skip a lot of steps, guess that the answer is 9.824394 times some power of 10, and only one choice of power of 10 makes the answer remotely plausible.
Thanks very much, Alan.