Consider + and – first. Let f(x)=sin(x), g(x)=1 iff x is rational and g(x)=0 iff x is irrational, then both f(x) and g(x) are periodic. Let F(x)=f(x)-g(x). We claim that F(x) is not periodical. Otherwise, Let T>0 be a period of F(x). Then we have F(0)=0=F(T)=sin(T)-g(T), So sin(T)=g(T). If sin(T)=1, then T=2k*pi+pi/2 is irrational. Contradiction. If sin(T)=0, then T=2k*pi. However, F(1)=sin(1)-g(1)=sin(1)-1, F(1+2k*pi)=sin(1+2k*pi)-g(1+2k*pi)=sin(1). Contradiction. So F(x) is not periodical.
Now let’s consider * and /. Let f(x)=x-[x], g(x)=sin(x), F(x)=f(x)*g(x). We claim that F(x) is not periodical. Otherwise let T>0 be a period of F(x). We have F(0)=0=F(T)=(T-[T])*sin(T). If T-[T]=0, then T is integral. So F(2*pi)=2*pi-[2*pi]=F(2*pi+T)=(2*pi-[2*pi])*sin(T), so sin(T)=1. Impossible. If sin(T)=0, then T=2k*pi. So F(1)=0=F(1+2k*pi)=(2k*pi-[2k*pi])*sin(1). Impossible. So F(x) is not periodical.
Step 1. Let’s start with the approach. You have probably encountered people in your daily life who snidely demean the SAT or at least the experience of taking the SAT. However, you must approach this important experience with a fundamentally different mindset. Okay, perhaps the SAT is a test full of tricks – a test purely to be gamed. If so however, learning to work within a system is a very valuable skill to have in life. Furthermore, I believe that the fundamental basis of the SAT is not its tricks, but its call for a rapid comprehension of certain situations, a supple maneuverability, and a positive approach to the material. After all, a multiple-choice test with any semblance of difficulty can be said to contain tricks. How good is your knowledge if you can’t manipulate it to a small challenge? Don’t demean your opponent – that’s a recipe for disaster. The last quality, a positive approach to the material, is the most important and the one you can control the easiest. However, it does not come naturally (as can be seen with scores of grumbling teens) and takes reinforcing.
Step 2. A second word about approach: You didn’t pay CollegeBoard 45 bucks so that you could be nice. When you’re faced with five choices on a question, you’ve got to be ruthless. Stop internally justifying why one answer could be right, and instead make the shift to asking yourself why that answer could be wrong – play Devil’s Advocate, as cliched as that may sound. I can’t tell you enough how much this shift in thinking has helped me when I have been stuck between two seemingly correct choices. Despite appearances, all choices ARE different and one is certainly the best, or else CollegeBoard would be losing thousands of dollars to successful lawsuits. Keep this in mind. You have got to find the right answer and I will show you how.
Step 3. It is my intention to focus mostly on the long reading passages in this How-To, since that is where the majority of the CR questions lie and since these questions give many test-takers a higher level of grief. For sentence completions, my biggest advice is to stop wasting your time on tricks, to buckle down, and to start attacking vocabulary lists. Direct Hits is vouched for by many and proven to be most effective, though I personally used Princeton Review’s Word Smart I and II cover to cover (perhaps not as efficient as the previously mentioned title). One problem is retention, so what I did was that I made flashcards for every word I didn’t know in the book (it came out to about 1000 words). It takes a long time, but it pays off for the SAT, your reading, your writing, and your life. Only if you really know the words will you be able to confidently answer sentence completions (and consummately schmooze at cocktail parties). For the short passages, it’s all about absorbing the small paragraphs as efficiently as you can before going on to answer the questions. They’re considerably easier if you keep your mind, and obsessing about the short passages (going back to double or triple check) are a huge time drain. Most of the time, it’s a quick fact check paired with a tone question. If you practice a lot on long passages, short passages will be an easy relief for you.
Step 4. Now, onto the long passages. I had loads of trouble with these before I found this method. I am going to give you my step-by-step method of attacking them, which I have found extremely effective, albeit somewhat more time-consuming. Before anything, you MUST read the short blurb before the passage. It gives you a sense (though always limited) not only of what the passage is going to be about, but also of the position and possible tone of the author. You will then be able to perhaps place yourself into the author’s shoes. This is a good point right now to tell you that you MUST love the passage you are reading. Force yourself to love it – throw yourself into the passage with gusto. It works. Though it’s quite ludicrous to be super-enthusiastic about a boy and his alfafa patch, with your enthusiasm comes retention, heightened focus, and an oddly vicarious interest in the passage. My general mental approach was a huge contributing factor in my getting an 800 in CR and a 2400 on the SAT.
Step 5. After you have read that thrilling blurb, don’t start reading the passage yet. Quickly jump to the questions, and as fast as you can, skim every question for line number references (don’t read the choices or the full question yet). On some passages almost every single question has a line reference – on most others it’s about over half. Very rarely will you see a passage with question without any line references (perhaps only rarely on a six question passage). Anyway, once you see a line reference (In lines 23-25 of the passage, the author is saying that…), you should bracket not the lines, but the sentence contained within the lines. This mark-up will allow you to focus in on that sentence once you begin to read the passage. Based on the question, you want to make a small annotation. For this question: (In lines 23-25 of the passage, the author is saying that…), you might make the annotation MEANING next to your marked-up sentence. Other annotations might include: SAYS THIS BECAUSE, REFERS TO, HOW SIMILAR TO PASSAGE 1, BACKS UP WHAT BEFORE (think crude caveman notations – they’re more efficient). Go through all of the questions. Perhaps some of the references will not have any line numbers. If you see (In the last paragraph…), just put brackets around the last paragraph along with an annotation. If you see a general question referring to the passage as a whole, on the question circle the number of the question with a large circle. This means it’s a general question and must be answered AFTER all the specific questions. I find this is always a very comfortable way of attacking the questions based on how CollegeBoard writes these questions.
Step 6. Once you have marked up all the line references as fast as humanly possible, then the real art begins. You must read the passage. There is no way around reading every single word. But HOW you read it is the true art. Read the unmarked sections quickly yet efficiently, absorbing it briefly but not truly pausing to analyze. ONCE you hit a marked section, slow down and absorb it. If you feel that it would not disrupt your flow to answer the corresponding question, do so. If not, keep going a little more. A vast majority of the line reference questions (even complex ones such as inferences) can be answered after reading from the beginning to the point of reference. In a few instances, it may help to read past the point of reference, but NEVER read the whole passage through without pausing to answer questions. Your retention will be terrible and it’s much better to handle the passage in small, manageable chunks. Also, when you answer a question, just circle in the answer in the test booklet. DO NOT BUBBLE IN THE ANSWERS UNTIL YOU FINISH THE ENTIRE PAGE, SOMETIMES EVEN THE PASSAGE. This is a huge time saver and it prevents you from making bubbling mistakes. The time saved is not necessarily the time difference in bubbling, but the time saved because it prevented you from breaking your focus. This is very important in CR. Don’t break focus. If you’re very low on time however, you can bubble as you go.
Step 7. Once you have tackled all the line and paragraph references ruthlessly, you should have already finished reading the entire passage and because you had focused in on the passage in numerous instances, you should also be well-equipped to answer your circled general questions. I always find it’s easier to answer these general question at this point, seeing as how you hit up the passage numerous times already along the way. Remember to never choose an answer unless you can truly back it up with evidence from the passage. Even “inferences” do not stray far from the text. If they did, then the “best answer” would be up in the air. Do not be misled by the word “inference” – it’s a misnomer. A large number of these can actually be pulled straight from the passage. It’s all about the passage – not what you think or have learned thus far in school. Being one with a text and not extracting too much from it is a valuable skill to learn. Don’t put words into the author’s mouth. Another very helpful thing to remember when viewing the choices is that extreme choices (including the words ALWAYS, NEVER, or BEST) are rarely ever correct because they fall under the hard-to-prove category of generalization within inductive reasoning. Though you’ve heard this tip many times and it sounds obvious, it is so helpful (yet easy to forget) and you often find yourself internally justifying these kinds of generalizing answers. Just say no (in a ruthless yet eternally positive way).
Step 8. My method of tackling long passages is somewhat time-consuming, but time is something that can be reduced through assiduous practice. This method is so effective in getting the right answer, and I fully vouch for it from personal expereince. What I also did during practice was that I gave myself twenty minutes instead of twenty-five in the standard CR sections, and I rapidly tried to utilize my developed method. It was extremely difficult to meet the twenty-minute deadline at first but I got better and better at it through practice. While time can be addressed easily through practice, a fundamentally bad approach to the passages cannot.
当一个数列经过一个栈的时候，经由它独特的入栈和出栈操作，这个数列会发生些什么？抛开具体的数值，对于一数列1, 2, 3, … , n, 将它以不同的方式，经过完整的入栈和出栈操作，可以得到哪些数列？
我们来考察一个较为简单的序列1, 2, 3, 4, 5
1, 2, 4, 5, 3
1, 2, 5, 3, 4
很明显，第一个序列是可得到的。我们依次将1入栈，1出栈，2入栈，2出栈, 3, 4入栈，4出栈， 5入栈， 5出栈就可得到该数列。
这样，我们就可以很快知道，下面几个数列都不能由1, 2, 3, …, 9经栈操作得到
1, 3, 2, 5, 4, 8, 6, 7, 9 –> 13|25|48|679
3, 2, 1, 5, 7, 4, 6, 8, 9 –> 3|2|157|4689
证明：设这些数依次为a1, a2, a3, …, an，且记Si为前i项和，规定S0=0。很明显Si单调递增，且对任意i有0<=Si<=32。
我们从这个图中选择一些顶点，将它们的编号排序作为一个可能的序列，比如选择0, 1, 10, 2, 13, 16，
排序后得0, 1, 2, 10, 13, 16，作差得ai依次为