Practicing the 13th task of the Unified State Exam in Russian.

For this task you can get 1 point on the Unified State Exam in 2019

Task 13 of the Unified State Exam in the Russian language is used to check the spelling of some words together, separately or with a hyphen. In addition, this exam card teaches you to distinguish homonyms - words that sound and spell the same, but have completely different meanings.

To successfully complete task No. 13 of the Unified State Exam in the Russian language, you will have to distinguish between the spelling of conjunctions and pronouns with prepositions or particles (“I’m sick, so I won’t come” - “I’ll call this phone number”, “It’s cold today, but it’s sunny” - “I Thank you for helping me." The pair of words “also - in the same way” causes particular difficulties for students. The rule here is simple - if the word “also” can be replaced with a synonym “too”, it is written together: “We will also go, we will also go,” and if there is a comparison in the sentence, it must be written separately: “Today is as cold as yesterday". There is also another complex similar pair of words found in exam papers - “so that / that would”. Does the phrase express a purpose? The word can be replaced with a synonym “so that” or a more detailed “in order to”? In this case, it must be written together: “in order not to get sick, so as not to get sick, in order not to get sick.” Can the “would” particle be removed without the sentence losing its meaning? In this case, the writing is separate: “I’m thinking about what to do, I’m thinking about what to do.”

Someone in task No. 13 of the Unified State Exam in the Russian language will encounter a pair of words “what/wherefore”. A word is written together if it can be replaced with a synonym “besides”, in other cases the spelling is separate. The same applies to similar words “besides/besides”.

In task 13 of the profile level of the Unified State Examination in mathematics, it is necessary to solve an equation, but of an increased level of complexity, since tasks of the former level C begin with task 13, and this task can be called C1. Let's move on to consider examples of typical tasks.

Analysis of typical options for tasks No. 13 of the Unified State Exam in mathematics at the profile level

First version of the task (demo version 2018)

a) Solve the equation cos2x = 1-cos(n/2-x)

b) Find all the roots of this equation belonging to the interval [-5n/2;-n].

Solution algorithm:

t
We make the reverse substitution and solve the simplest trigonometric equations.

We build a number axis.
We apply roots to it.
Mark the ends of the segment.
We select those values that lie inside the interval.
We write down the answer.

Solution:

1. Transform the right side of the equality using the reduction formula cos( π/ 2−x)=sin x. We have:

сos2x = 1 – sin x.

Let's transform the left side of the equation using the double argument cosine formula using sine:

cos(2x)=1−2sin 2 x

We get the following equation: 1−sin 2 x=1− sin x

Now there is only one trigonometric function sin in the equation x.

2. Enter the replacement: t= sin x. We solve the resulting quadratic equation:

1−2t 2 =1−t,

−2t 2 +t=0,

t(−2t+1)=0,

t = 0 or -2t + 1 = 0,

t 1 = 0 t 2 = 1/2.

3. Make a reverse replacement:

sin x= 0 or sin x = ½

Let's solve these equations:

sin x =0↔x=πn, nЄZ

sin( x)=1/2↔x= (-1) n ∙( π/6)+πn, nЄZ.

Consequently, we obtain two families of solutions.

1. In the previous paragraph, two families were obtained, each of which has infinitely many solutions. It is necessary to find out which of them are in a given interval. To do this, we build a number line.

2. We apply the roots of both families to it, marking them with green (the first) and blue (the second).

3. Mark the ends of the gap in red.

4. In the indicated interval there are three roots that are three roots: −2 π ;−11π/ 6 and −7 π/ 6.

A) πn, nЄZ;(-1) n ∙( π/6)+πn, nЄZ

b) −2 π ;−11π 6;−7π 6

Second version of the task (from Yashchenko, No. 1)

a) Solve the equation.

Solution algorithm:

We replace this function with a variable t and solve the resulting quadratic equation.
We make the reverse substitution and solve the simplest exponential, then trigonometric equations.

We construct a coordinate plane and a circle of unit radius on it.
We mark the points that are the ends of the segment.
We select those values that lie inside the segment.
We write down the answer.

Solution:

1. We introduce the replacement t = 4 cos x. then the equation will take the form:

We solve the quadratic equation using discriminant and root formulas:

D=b 2 – c = 81 – 4∙4∙2 =49,

t 1 = (9 – 7)/8= ¼, t 2 = (9+7)/8=2.

1. Construct a coordinate plane and a circle of unit radius on it.

2. Mark the points that are the ends of the segment.

3. Select those values that lie inside the segment..

These are the roots. There are two of them.

Third version of the task (from Yashchenko, No. 6)

a) Solve the equation .

b) Find all the roots of this equation that belong to the segment.

Solution algorithm:

Using trigonometric formulas, we reduce the equation to a form containing only one trigonometric function.
We replace this function with a variable t and solve the resulting quadratic equation.
We make the reverse substitution and solve the simplest exponential and then trigonometric equations.

We solve inequalities for each case.
We write down the answer.

Solution:

1. Using reduction formulas .

2. Then this equation will take the form:

3. We introduce a replacement . We get:

We solve an ordinary quadratic equation using discriminant and root formulas:

Both roots are positive.

3. Return to variable x:

Determine the sentence in which both highlighted words are written CONTINUOUSLY. Open the brackets and write down these two words.

Chopin IMMEDIATELY captivated the Parisian salons with his unique and unusual performance, as well as with his brilliant humor and ingenious improvisations.

In order to more fully experience the flow of life, in the fall of 1877 Tchaikovsky left (ABROAD): he lived for a long time in Italy, Switzerland, and France.

Picasso’s “Self-Portrait of the Artist with a Palette” and “Les Demoiselles d’Avignon” have a lot in common: the SAME facial expressions, the SAME eyes, similar color tones.

(AND) SO, speech etiquette is a universal phenomenon, but at the SAME time, each nation developed its own specific system of rules of speech behavior.

The gift was prepared (IN) SECRET from those around them, (IN) CONVERSING IN A LOW VOICE in the evenings.

Explanation (see also Rule below).

Let's give the correct spelling.

Chopin IMMEDIATELY captivated the Parisian salons with his unique and unusual performance, as well as his brilliant humor and ingenious improvisations.

IN ORDER to more fully experience the flow of life, in the fall of 1877 Tchaikovsky went ABROAD: he lived for a long time in Italy, Switzerland, and France.

Picasso’s “Self-Portrait of the Artist with a Palette” and “Les Demoiselles d’Avignon” have a lot in common: the SAME facial expressions, the SAME eyes, similar color tones.

SO, speech etiquette is a universal phenomenon, but at the SAME time, each nation has developed its own specific system of rules of speech behavior.

The gift was prepared SECRETLY from those around them, talking in a SOTT VOICE in the evenings.

Abroad– a noun, means: foreign countries, for example: trade with abroad, love abroad, live abroad; abroad will help us.

Abroad, abroad– a combination of a preposition with a noun means: in foreign countries, to foreign countries; outside, outside the homeland, for example: live abroad, travel abroad.

Right

Live abroad and live abroad

Rule: Continuous, separate and hyphenated spelling of words. Task 14.

Continuous, separate and hyphenated writing of various parts of speech.

According to the “Specification”, this task tests knowledge of the most voluminous, most diverse and therefore most complex material. In this section “References” the rules of school textbooks will be systematized, as well as supplemented with the information that is necessary for the successful completion of the Unified State Examination task and mastering practical literacy. The set of rules that will be analyzed is not accidental: the creation of the list was preceded by work on studying assignments from past years, FIPI Bank, as well as printed publications, the authors of which are the creators of KIMs (Tsybulko I.P., Egoraeva, Vasiliev I.P. and others ).

Table 1 contains a collection of words; a distinctive feature of many is the presence of homonyms, that is, words that sound the same but have different spellings. To indicate parts of speech and explanations Abbreviations used:

noun - noun

number - numeral

adv. – adverb

places – pronoun

deepr. – participle

p/p - derived preposition

n/a – non-derivative preposition

v/s – introductory word

fe – phraseological unit

Letter	CONSOLIDATED/SEPARATELY/HYPHEN	EXPLANATION
B	would, b	together only in union so that, so that. I went to bed earlier to don't miss the train. ( union = in order to, would* cannot be moved or removed) To To avoid being left without a certificate, you will have to learn how to spell the particle “would”. So that* Don't forget, I'll write it down in my diary.
		separately in all other cases: What would Should I read? ( local + frequent, would* can be moved or removed.)* Said would earlier; How would don't be late; What would what did I do without you? I'll be back that would neither happened.
IN	after all	according to the rule for writing particles. Always hyphenated.
	at the end unlike (difference) in the dark aside in general (not possible in general) openly all the time doesn't matter through thick and thin	always separately
	at the bottom double (triple...) go home after	always together
	in view of	missed it in view of illnesses (p/p, =because of) have in mind (FE) decoration as butterflies
	up, above	look (where? adv.) up; located ( Where? adverb) up aim (what?) to the top(what?) tree, target ( )
	deep down	leave deep down (Where? nar.) leave deep into (what?) forests ( noun+n/a, there is an explanatory word)
	In the end	become weak In the end (How? adv., =finally) rearrange the phrase In the end (what?) offers ( noun+n/a, there is an explanatory word)
	instead, together to a place, in a place	spoke instead of me ( p/p, =for), together(adverb) with me. You can't: instead of me hit ( into what?) instead of falls, found ( Where?) at location ( noun+n/a, there is an explanatory word)
	far away, far away into the distance, into the distance	look ( where?, adverb) into the distance; appears in the distance (Where? adverb.) into the distance ( what?,n/a+ noun. in Rod. case) seas; appears in the distance (what?) seas ( noun+n/a, there is an explanatory word)
	at first	difficult at first(When? adv.) at first ( what?) books ( noun+n/a, there is an explanatory word
	during	do during (When? nar.) hurts during (what?) sleep ( noun+n/a, there is an explanatory word)
	down, below	fell ( Where? nar.)down; located ( Where? adverb)at the bottom aim ( into what?) down (what?) tree, target ( noun+n/a, there is an explanatory word), at the (very) bottom of the mountain
	right up to	learn up to morning ( p/p, =to) clothe ( into what?) into the flesh and blood ( noun+n/a)
	close in tight	come up close (How? adv. = very close) wrap in tight paper ( which one? adj.+n/a)
	right	to turn right(where, adverb) into what? to the right possession of inheritance ( noun+n/a, there is an explanatory word)
	has the right	has the right to know ( c/s, =has the right) share V(what?) law ownership of an apartment ( noun+n/a, there is an explanatory word), in criminal ( which one?)right
	in continuation in continuation, in continuation	a preposition specifying a period of time. Combines with the words day, day, hour, week, etc.: throughout the day, throughout the lesson, throughout the year (similar to “throughout”) Noun continuation in different cases with the preposition in: in continuation ( into what?) of the novel new characters will be introduced. In the continuation of the novel ( what?) we learn about their fates.
	first	see first (When? nar.) miss first days ( which? number+n/a)
	due to as a consequence, as a result	flight delayed due to (p/p, =because of) bad weather Noun consequence in different cases: to intervene ( into what?) as a consequence(noun+n/a) ; errors in (preliminary) consequence(noun+n/a)
	following	look ( where, what?) following(adv.) the departing train to go after him, behind a friend ( p/p, =for) follow the trail ( noun+n/a)
	during during, during	a preposition specifying a period of time. Combines with the words day, day, hour, week, etc.: during the day, during the lesson, throughout the year (similar to “throughout”) Noun flow in the accusative or prepositional case with the preposition in: during ( into what?) rivers; ( about what) about the flow of the river.
E	if	always the same
AND	Same	Always separated according to the particle rule. The particle contributes an intensifying meaning. Necessary or, Tell or, How or is it possible, like or, This or not true, same thing or, That or at the same time or time, just like that or. Not to be confused with the conjunctions TOO and ALSO (see table)
Z	work and live abroad for lack of time or funds at the expense (treat at the expense of the establishment)	always the same
	before dark	always the same
	then	Adverbs and conjunctions: then (When? adv., =later) we will leave; For what asked? ( adv. = for what purpose?); small, but (union, =but) smart. Pronouns with prepositions: I came for those (which one exactly?) a kitten that I liked; I got in line for those (which one exactly?) by the man who went to the window. Similar: For how(exactly) queue? ( place + n/a, = for what product?) What (exactly) did you come back for? For the keys. Likewise: I thank you for That(for what exactly?) that you helped me; I'm not offended for That (for what exactly?) that you didn’t come, but for that ( for what exactly?) that he deceived me.
	often frequently	often no result ( adv., =often) for frequent change of mood ( for which? adj.+n/a)
AND	from afar	always together
	So	So, let's start the lesson! ( introductory word) twisted So and this way ( How? people + union); So (how so?) many times, So all the time
TO	-KA	Always hyphenated according to the particle rule. Bring it -ka, Tell -ka, look -ka
TO	in a row	Repeatedly in a row (like, adverb = in a row) sneezed To the row(why, noun + n\n.) numbers, to a row acquaintances
	as if as soon as as it were	always the same
L	Lee (particle)	Always the same and separate.
N	against through not far away	Always the same
	for	always the same
	towards to the meeting	go towards(Where? nar.); go towards(to whom?) friend (p/p) go on(long awaited) meeting with a friend ( noun+n/a)
	finally	finally he fell asleep ( adv. = after everything) postpone finally months ( noun+n/a, there is an explanatory word)
	in advance on the front (rarely!)	I know everything in advance(How? adv., = in advance) fell on the front cars ( )
	like like	figure like ball ( p/p, =like) task ( for what?) like triangles ( noun+n/a, there is an explanatory word)
	For example For example	Was, For example, (introductory word) such a case. note on(this) example (noun+n/a)
	half by half	destroyed half (adv., =partially) claim ( for what?) by half Houses ( noun + n/a, there is an explanatory word)
	about	to know about (p/p, =o) tickets put money on(bank) check(noun+n/a), don't take it personally (FE)
	back on the back (rarely!)	turn around back (Where? nar.) look on(broken) ass car ( noun+n/a)
	upstairs, upstairs to the top, to the top*	rise up(Where? nar.), upstairs (Where? nar.) it was cold rise to the top (what?) mountains ( noun+n/a, there is an explanatory word), on ( how?) at the top of the building, at the top of bliss (figurative meaning)
	by force	by force held (how? adv. = with great difficulty) hope on(my) force (for what? noun+n/a)
	how much for how long	How much It's right? ( adv., = to what extent?) How long will they increase their pension? (local+n/a)
	so so much	so I'm so tired I fell asleep ( to what extent? nar.) What number so much is it less? ( =for the same number of seats.+n/a)
	Maybe Maybe	Maybe, it will be raining. ( introductory word = probably) Maybe let's go! ( adj.+n/a, what’s the matter? correct)
	toDeath to death	fight to death(How? adv. = until death) They were sent on(true) death.(noun+n/a)
	on the head on the head	enemies are defeated completely (How? adv., =finally) bandage on the head (for what? noun+n/a)
	vying with each other for interruptions	started talking vying with each other (How? adv. = interrupting each other) complain for (permanent) interruptions water supply ( for what? noun+n/a, there is an explanatory word)
	frankly for cleanliness	talk frankly (How? adv. = frankly) note for cleanliness in room ( for what? noun + n/a, there is an explanatory word)
	on the face	data on the face (adv., = available) apply cream on(yours) face (for what? noun+n/a)
	for a long time	leave for a long time(Nar., for how long?) look for a long time dancing girl ( dancing how? long, adv.+n/a)
	forever forever	leave forever(Nar., for how long?) look on Always neat girl ( neat when? always, adv.+n/a)
	despite despite	ran despite fatigue ( p/p, = in spite of) ran despite under your feet ( ger., = without looking)
	regardless of regardless	Regardless of tired, we went to explore the city ( p/p, = in spite of) I tried to sit regardless of those around you and hiding your eyes ( ger., = without looking)
ABOUT	from there	always the same
	from this	always separately
	that's why	I didn't receive the summons that's why and didn't show up. ( adv. = for this reason) From that, who is not nice, and the gift is hateful. ( from whom? place + n/a, = from person) Pushed off from that shores ( from which one? local + n/a, = from adj..)
	from what	From what You did not sleep? ( adv. = for what reason?) Because he worked. From what did you refuse? ( local+n/a) From work, from a task.
	partly	Partly You are right. ( how much? adv., =partially) He refused from(more) parts income. ( from what? noun+n/a, there is an explanatory word)
P	behind simply because the little by little one by one	always together
	because compared with) with the flow	always separately
	and	The thing is beautiful and inexpensive. ( union, = besides, in addition) What does it have to do with will you stay? ( place + n/a, = with what?) What does it have to do with ( local+n/a) parents here, if it’s your fault?
	besides	The thing is beautiful besides inexpensive. ( union, = besides, in addition) At volume The manual has an appendix. ( at what? local+n/a, =adj.)
	That's why	He stayed That's why, that he wants to know the truth. ( Why? adv. = for that reason) go That's why shore ( on what? local+n/a, =adj.) guessed That's why what I saw ( local + n/a, = according to what he saw)
	Why	Why are you silent? ( adv. = for what reason?) Because I don't want to talk. Why are you working out? ( place + n/a, = for what benefit?) Based on Rosenthal's textbook. Why what do you miss the most? In the summer, in the sun.
	That's why	worked a lot, That's why tired ( Nar., why? for what reason?) study That's why textbook ( place + n/a, according to what? =adj.)
	truly truly	This truly a rare thing. ( adv. = actually) People are bored By(present) truth. (noun+n/a)
	at first at the beginning	miss at first (adv. = first time) guess By(to himself) the beginning novel ( noun+n/a, there is an explanatory word)
WITH	with time	always separately
	right away, right away	always the same
	at first	at first think ( When? nar.) Shouldn't we start a fairy tale? With(himself) started? (why? noun+n/a)
	at all	at all not enough sleep ( How? adv. = absolutely) left at all property ( with which? local+n/a)
	shoulder	swung shoulder(How? adv. =immediately) take off With(his) shoulder parrot ( why? noun+n/a)
T	Yes	Can be written either with a hyphen or separately. Broke -yes, again -yes, directly -yes after verbs, adverbs, hyphenated particles Me after all got hired, I after all late - in all other cases separately
	Immediately	appeared immediately (How? adv. =immediately) has arrived that hour (place + noun, what time is it? That) when I saw you again
		Homonyms: there are conjunctions too, also I Same, like you, I learn foreign languages. ( conjunction, =and, SAME cannot be omitted). I have a cat and so do you Same.. I'm into music and Also I really like to read. ( conjunction, =and, SAME cannot be omitted). The cat loves to sleep, and Also hide in new packages. Homonyms: pronoun with particle or. It enhances the meaning, see SAME Meet me at Same time. ( local + frequent, what time exactly?) - at that time; or Brother advised Same, as you. ( local + frequent, advised what exactly?) - That; or You can omit it, replace it with a union and you can’t! You Also as young as ten years ago. (popular, how young?) - so; or can be omitted, replaced with a conjunction and cannot).
	right	always separately
X	at least, at least	always separately

Table 1 is based on materials from Russian language teacher M.A. Kryukova. (website “Up to your ears in the OGE and the Unified State Exam!”)

Adverbs formed with a hyphen are written:

1) from full adjectives or pronouns using a prefix By- and ending with -mu, -him, -tski, -ski, -i	act in a friendly way sing in Kazakh live in a new way do in your own way in a hare's way cowardly
2) from ordinal numbers to -y, -them using the attachment in- (in-)	secondly thirdly fifthly
3) by repeating the same word or root, complicated by prefixes and suffixes	barely, just like every day, more or less willy-nilly
4) by connecting synonyms	unexpectedly, I'll say hello to you
5) using an attachment some and suffixes -this, -either, -something	somewhere, sometime, somewhere, somewhere

Semi- is always written together: crescent, half-year.

Unified State Exam. Russian language.

Task 13. How easy is it to complete?

Task No. 13- one of the most difficult. This is due to the fact that you need to know a lot of rules for fused, separate, and hyphenated spelling of words. In addition, there are a lot of words that you just need to remember. So there are difficulties.

I suggest the easiest way to complete this task.

Algorithm for completing task No. 13

Integrated, separate, hyphenated spelling of words

Read the assignment carefully. You need to find a sentence out of five proposed, in which the highlighted words are written seamlessly or apart. Even if the books you study with mostly ask you to find fused writing words, an exam is an exam, you need to be prepared for anything. So it is with a careful reading of the task that its implementation begins.

In each sentence, eliminate the words that are written with hyphen. Most often this is:

Words with suffixes EITHER, EITHER and prefix CFU

Words after all, exactly the same.

Adverbs with prefix BY and suffixes OMU, HIM, SKI, YI:

in our opinion, in the fox's way.

Adjectives meaning shades of colors, taste(bright red, sweet and sour)

Cardinal directions: southwest.

Words with roots floor: start with L(half a lemon) with a vowel(half an apple) capitalized(half of Europe).

Adjectives formed from homogeneous members, a conjunction can be placed between them AND(magazine-newspaper - that is, magazine and newspaper)

The first step has been taken. There will definitely be a word in some sentence that is written with a hyphen. Therefore, the number of offers is decreasing.

As if

In view of

Keep in mind

During

In continuation

Due to

Subsequently

Because

Whereas

That is

In order to

Despite

Regardless of

Immediately

As if

The third step is the most crucial. You need to clearly distinguish the words written seamlessly or apart.

So that - what would

Same too

Also - same

But for that

Why - from what

Because - because

What does it have to do with it

About (= about) – to an account (in a bank)

Remember: if a word has logical stress, you highlight it with intonation, it is pronounced firmly, with some slowing down of intonation, and most importantly, you can specifically imagine something, then this word is written APART.

If none of the above is present, then this is an ordinary conjunction, it is written FULL.

Compare.

TO should I give it to you for your birthday? (The emphasis falls on the word; we present the gift we want to buy).

We met, TO discuss current affairs. (The word is pronounced quickly, as if casually; we cannot imagine anything when we say the word SO THAT)

FOR THAT I got five for the task.

He prepared for a long time BUT passed the exam well.

Remember: if after SO SAME There is HOW AND, then it is always written separately. (The work was done with the SAME quality AS ALWAYS.)

Word SO written together, if this is an ordinary introductory word, it sums up something.( SO, the work was completed before the holiday)

If we have an adverb and a conjunction before us, then we write them separately, we can ask a question How?(So he spent all his free time (HOW did he spend it? – SO).

Remember that negative adverbs are always written seamlessly: nowhere, no way, not at all, nowhere, nowhere etc.

These are the main cases that need to be remembered first.

All rules are on this site. Pay special attention to the tables with the spelling of adverbs, memorize the words.

EXAMPLE

Identify the sentence in which both highlighted words are written FULL. Open the brackets and write down these two words.

Everything was (STILL) THE SAME, (THAT) HAS NOT changed at all.

(SO) TO arrive (TO) THE MEETING on time, we left early in the morning.

(SOME) WHERE (IN) DISTANCE lights of huts could be seen.

He disappeared (AS) as suddenly as he appeared.

(AND) SO let's start with the fact that I (FINALLY) met you.

EXPLANATION

We find sentences in which words are written with a hyphen. This is the first and third - SOMEWHERE, STILL. Let's exclude them. 3 offers left.

We find words that you have no doubt about spelling separately. This THAT IS(the first sentence, however, it has already been deleted)

There are 3 sentences left in which the words can be written correctly by thinking about their meaning.

Sentence 2: where did we go? – TO THE MEETING(for example, to a long-awaited meeting). That is, we clearly imagine the meeting to which our heroes are going. We write apart. Word TO here it is written together, since there is no lexical meaning in the word "What" No).

Sentence 4 is easy, it contains AS WELL AS, which means I write the word separately.

That leaves number 5 - this is the correct answer: SO- introductory word, FINALLY- adverb when?

Complete more tasks, and you will definitely succeed

Good luck!

Material prepared by: Melnikova Vera Aleksandrovna

Unified State Examination in mathematics profile level

The work consists of 19 tasks.
Part 1:
8 short answer tasks of basic difficulty level.
Part 2:
4 short answer tasks
7 tasks with detailed answers of a high level of difficulty.

Running time - 3 hours 55 minutes.

Examples of Unified State Examination tasks

Solving the Unified State Examination task in mathematics.

Problem with solution:

In a regular triangular pyramid ABCS with base ABC, the following edges are known: AB = 5 roots of 3, SC = 13.
Find the angle formed by the base plane and the straight line passing through the middle of the edges AS and BC.

Solution:

1. Since SABC is a regular pyramid, ABC is an equilateral triangle, and the remaining faces are equal isosceles triangles.
That is, all sides of the base are equal to 5 sqrt(3), and all side edges are equal to 13.

2. Let D be the midpoint of BC, E the midpoint of AS, SH the height descended from point S to the base of the pyramid, EP the height descended from point E to the base of the pyramid.

3. Find AD from the right triangle CAD using the Pythagorean theorem. It turns out 15/2 = 7.5.

4. Since the pyramid is regular, point H is the point of intersection of the altitudes/medians/bisectors of triangle ABC, and therefore divides AD in the ratio 2:1 (AH = 2 AD).

5. Find SH from right triangle ASH. AH = AD 2/3 = 5, AS = 13, according to the Pythagorean theorem SH = sqrt(13 2 -5 2) = 12.

6. Triangles AEP and ASH are both right angles and have a common angle A, hence similar. By condition, AE = AS/2, which means AP = AH/2 and EP = SH/2.

7. It remains to consider the right triangle EDP (we are just interested in the angle EDP).
EP = SH/2 = 6;
DP = AD 2/3 = 5;

Angle tangent EDP = EP/DP = 6/5,
Angle EDP = arctan(6/5)

Answer:

Do you know what?

Among all the figures with the same perimeter, the circle will have the largest area. Conversely, among all shapes with the same area, the circle will have the smallest perimeter.

Leonardo da Vinci derived a rule according to which the square of the diameter of a tree trunk is equal to the sum of the squares of the diameters of the branches taken at a common fixed height. Later studies confirmed it with only one difference - the degree in the formula is not necessarily equal to 2, but lies in the range from 1.8 to 2.3. Traditionally, it was believed that this pattern is explained by the fact that a tree with such a structure has an optimal mechanism for supplying its branches with nutrients. However, in 2010, American physicist Christophe Alloy found a simpler mechanical explanation for the phenomenon: if we consider a tree as a fractal, then Leonardo’s law minimizes the likelihood of branches breaking under the influence of wind.

Laboratory studies have shown that bees are able to choose the optimal route. After localizing the flowers placed in different places, the bee makes a flight and returns back in such a way that the final path turns out to be the shortest. Thus, these insects effectively cope with the classic “traveling salesman problem” from computer science, which modern computers, depending on the number of points, can spend more than one day solving.

If you multiply your age by 7, then multiply by 1443, the result will be your age written three times in a row.

We think of negative numbers as something natural, but this was not always the case. Negative numbers were first legalized in China in the 3rd century, but were used only for exceptional cases, as they were considered, in general, meaningless. A little later, negative numbers began to be used in India to denote debts, but in the west they did not take root - the famous Diophantus of Alexandria argued that the equation 4x+20=0 was absurd.

American mathematician George Dantzig, while a graduate student at the university, was late for class one day and mistook the equations written on the blackboard for homework. It seemed more difficult to him than usual, but after a few days he was able to complete it. It turned out that he solved two “unsolvable” problems in statistics that many scientists had struggled with.

In Russian mathematical literature, zero is not a natural number, but in Western literature, on the contrary, it belongs to the set of natural numbers.

The decimal number system we use arose because humans have 10 fingers. The ability for abstract counting did not appear in people right away, and it turned out to be most convenient to use fingers for counting. The Mayan civilization and, independently of them, the Chukchi historically used the twenty-digit number system, using fingers not only on the hands, but also on the toes. The duodecimal and sexagesimal systems common in ancient Sumer and Babylon were also based on the use of hands: the phalanges of the other fingers of the palm, the number of which is 12, were counted with the thumb.

One lady friend asked Einstein to call her, but warned that her phone number was very difficult to remember: - 24-361. Do you remember? Repeat! Surprised, Einstein replied: “Of course I remember!” Two dozen and 19 squared.

Stephen Hawking is one of the leading theoretical physicists and popularizer of science. In his story about himself, Hawking mentioned that he became a mathematics professor without receiving any mathematics education since high school. When Hawking began teaching mathematics at Oxford, he read the textbook two weeks ahead of his own students.

The maximum number that can be written in Roman numerals without violating Shvartsman's rules (rules for writing Roman numerals) is 3999 (MMMCMXCIX) - you cannot write more than three digits in a row.

There are many parables about how one person invites another to pay him for some service in the following way: on the first square of the chessboard he will put one grain of rice, on the second - two, and so on: on each subsequent square twice as much as on the previous one. As a result, the one who pays in this way will certainly go bankrupt. This is not surprising: it is estimated that the total weight of rice will be more than 460 billion tons.

In many sources, often with the purpose of encouraging poorly performing students, there is a statement that Einstein failed mathematics at school or, moreover, generally studied very poorly in all subjects. In fact, everything was not like that: Albert began to show talent in mathematics at an early age and knew it far beyond the school curriculum.