What Happened If You Didn't Take The College Entrance Examination? I Recommend

The first question is an algebra question, an is the sum of a polynomial, prove: when the positive integer n≥2, a(n+1)

When Lu Shixian first saw this question, he was still a little out of ideas, so he stopped there all of a sudden.

After all, pure algebra questions are a test of people's ability to connect logic and thinking.

Can't even solve the first proof question? This is already the easiest.

Lu Shixian suddenly became nervous. If he couldn't even solve the first question, it would definitely be a huge blow to his answers to the following questions.

He let out a breath, slowly forcing himself to calm down.

The more nervous you are, the less anxious you are.

Lu Shixian reviewed the question again, and suddenly found that he had fallen into a misunderstanding. To prove this kind of comparison question, why bother to substitute them separately and then compare them?

He just needs to switch his way of thinking.

The ratio of a to b can also be converted into the difference between a and b or the ratio of a to b.

If the final result of a-b is greater than zero, or the result of a/b is greater than 1, it means that a is greater than b

Thinking of this, Lu Shixian's eyes became brighter and brighter.

He quickly checked the calculation on the draft paper. For formula an, n+1 can be separated separately by using the distributive law of multiplication.

Then get the final simplified formula for any positive integer n≥2, an-a(n+1).

Finally, it is proved that the simplified formula is greater than zero.

Therefore a(n+1)

This question is proved.

After solving this problem, Lu Shixian heaved a sigh of relief and started to look at the next problem.

The second question is a plane analytic geometry.

The main idea of ​​the topic is the two intersection points obtained by the checkmark function and a straight line, and then find out what is the intersection trajectory of the two tangent lines on the checkmark function?

I have to say that if your logical thinking ability is not enough, just looking at the topic is enough to make you dizzy.

But speaking of it, this kind of question is still Lu Shixian's strength. What he is best at in mathematics is converting graphics into algebra.

It is nothing more than finding the coordinates of the intersection point.

According to the given conditional simultaneous equations, it is known from the meaning of the question that the equation has two different real roots x1 and x2 on (0, +∞), so k≠1, and Δ(1)=1 +4(k?1)>0, the sum of two real roots (2) and the product (3) are both greater than zero.

From this, the value range of the slope k of the straight line can be obtained, and finally the tick function is derived

Simplify to obtain the equations (4) and (5) of the straight lines l1 and l2

(4) formula - (5) formula to get the function expression of xp (6) formula

Substitute (2) (3) into (6) to get xp=2

(4) formula + (5) formula to get the function expression of yp (7) formula

Substituting the combined formula of (2) and (3) into formula (7) to get 2yp=(3?2k)xp+2, and xp=2, and yp=4?2k

According to the value range of the slope k 2

That is, the trajectory of point p is the line segment between (2, 2), (2, 25) two points (excluding endpoints)

After Lu Shixian finished writing this question, there were only forty minutes left in the exam.

The second big question is really not difficult. The idea is very simple, but the calculation process is a bit complicated and time-consuming. This question alone took him dozens of minutes.

With no time to complain, Lu Shixian quickly looked at the third question,

Let the function f(x) satisfy f(x+2π)=f(x) for all real numbers x.

Prove: There are 4 functions fi(x)(i=1, 2, 3, 4) satisfying:

(1) For i=1, 2, 3, 4, fi(x) is an even function, and for any real number x, fi(x+π)=fi(x);

(2) For any real number x, f(x)=f1(x)+f2(x)sx+f3(x)sx+f4(x)s2x.

The question seems very simple, but Lu Shixian knows that the final answering process is several times the question, and it may be more than that.

Not much time, Lu Shixian decided to solve the first problem first.

Lu Shixian can figure it out with his ass, everything that is close to pi is basically linked to a periodic function.

He directly instigated g(x) and h(x) of the two generals of the enemy f(x), and g(x) is an even function, h(x) is an odd function, for any x∈r, g( x+2π)=g(x), h(x+2π)=h(x).

Then respectively substitute four functions fi(x), i=1, 2, 3, 4. Get the expressions of four functions f1(x), f2(x), f2(x), f4(x).

So fi(x), i=1, 2, 3, 4 is an even function, and for any x∈r, fi(x+π)=fi(x).

This is rather simple. For a very limited number of verifications, you only need to substitute in the verifications separately, no brains involved.

Lu Shixian feels that as long as the number of times is below 10, he can accept it, and it is nothing more than spending some refills.

After all, it is better than looking at the topic for a long time and not knowing how to start.

However, this question seems to give some leeway to the contestants, because Lu Shixian found that the second question is closely related to the first question.

Substitute the algebraic formula obtained in the first question into f(x)=f1(x)+f2(x)sx+f3(x)sx+f4(x)s2x

Next, the case-by-case discussion is over.

Because f1(x), f2(x), f2(x), and f4(x) have 6 situations because of the value range of x.

There are two of them that do not need to be discussed anymore, and they are already recruited from the ground.

There are still four situations where there is still resistance, and Lu Shixian has no choice but to resort to a hypothetical killing stick.

In the end they were finally beaten into a trick, and thus proved that all six cases were fully established.

To sum up, this formula is established!

Lu Shixian let out a long breath, and when he looked around from the corner of his eye, he was the only one left in Nuoda's classroom.

He suddenly panicked, the time is not over yet, isn't it?

The topic that I spent so much effort to prove, others finished it so quickly?

Is it because I am too old to lift the Dragon Saber, or is it that the kids are too powerful now?

As soon as he looked up, he saw the invigilator staring at him.

What's the meaning? Am I letting you down?

Sorry I apologize, I admit that I am really a math scumbag.

He stood up and handed in the paper rather melancholy, then packed his luggage and prepared to leave this sad place.

Unexpectedly, when he left, there was admiration from the invigilator behind him.

"Oh, not bad! The people in this examination room have long given up and left early, only you are still persisting silently."

Lu Shixian: ????

"Right or wrong, if you can finish it, you are worthy of me staring at you for an hour."

Lu Shixian: e? (?＞灬＜)?3

Lu Shixian's originally low mood gradually picked up again.

This seems to mean that I am not bad, the sword is not old!