Spelling verb endings
1. Depending on the personal endings, verbs are divided into two large groups: for verbs of I and II conjugations.
The II conjugation includes:
· verbs in -it (except verbs shave, lay, rest, which belong to the I conjugation),
· 7 verbs for -there are (twirl, see, depend, hate, offend, watch, endure),
· 4 verbs per -at (drive, breathe, hold, hear).
All other verbs belong to the I conjugation.
Personal endings of verbs in the present or future past tense:
2. There are several mixed verbs , not related to either of the two conjugations: want, run, eat, create, give .
units 1 person I read, I take 2nd person read, take 3rd person reads, takes plural 1 person read, take 2nd person read, take 3rd person read, take | Want Want wants we want want to want | I'm running you're running runs let's run run are running | we eat eat are eating | I'll create you will create will create let's create let's create will create | give it will give we'll give give it they will give |
3. If the verb has a prefix un- (obes-) is transitive, then it is conjugated according to the II conjugation, and if intransitive, then according to the I conjugation (for example, compare the conjugation of verbs weaken (someone)) And become weak (oneself)).
4. In verbs of the first conjugation, the ending is written in the form of the future tense -yeah , and in the form of the imperative mood - the ending -ite (You will send this letter tomorrow. - Send this document urgently.)
b (soft sign) in verb forms.
1. b (soft sign) is written:
· V infinitive (write, bake, want, wash ),
· V 2nd person endings singular choose, wash, do, wash ),
· V imperative mood (fix it, hide ), BUT lie down, lie down,
· V return particle , which comes after the vowel ( bent, turned around, I'll be back );
2. b (soft sign) is not written:
· V 3rd person singular form present or simple future tense ( washed, done ).
Spelling verb suffixes
1. B indefinite form verbs most often have the following suffixes: -A- (hear, do); -I- (sow, bark); -e- (to see, to offend); -And- (nursing, nagging).
Remember: 1) a suffix is never written after the vowel letters of the root in the infinitive -e- . In this position, only the suffix can be stressed -I- or -And- (stand, water, milk). IN unstressed position suffix -I- written in verbs winnow, start, repent, bark, cherish, toil, hope, soar, sow, melt, tea, smell, and the suffix -And- - in verbs build, cost, rest, double, triple, glue;
2) after soft consonants (except h ) in unstressed position suffix -I- written in verbs bow, cough, and the suffix -e- - in verbs see, offend, depend, hate, freeze;
3) in suffixes of past tense verb forms the same vowel is retained as in the infinitive. For example: get welle th - get welle l, laI blah blah blahI l, glueAnd t-kleAnd l.
2. Suffixes -yva- (-iva-), -ova- (-eva-).
In verbs with the suffix -yva- (-iva-) (always unstressed), relevant imperfect form (sometimes also multiple actions), written before the letters s or i , For example: roll up, smear, see; overwhelm, talk over, jump up, hear, twist, beg, insist .
Verbs with suffix -yva- (- willow - ) should be distinguished in writing from verbs with the suffix - ova - (- Eve- ). Verbs of these two types form the present tense differently: verbs in -ovate (-eat ) have the form of the 1st person on -yy (-yuyu ) (Without -ov-, -ev - ), For example: I talk - to talk, I manage - to manage, I envy - to envy, I confess - to confess, I preach - to preach, I scourge - to scourge, I grieve - to grieve; verbs have the same - to come (-ive ) the 1st person form ends in -I'm (-Ivayu) (with saving -ыв -, -ive- ), For example: inspecting - inspecting, deploying - deploying, reconnaissance - reconnaissance, visiting - visiting .
3. Vowels before -va - . In imperfective verbs with a suffix -va- , having the form of the 1st person on -va-yu , unstressed vowels before V are checked according to the general rule, for example: overcome(overcome), outdated(outdated), wash down(write down), singing(sing), sow (seed), idea (get started), harden (freeze), outpost (catch).
However, in the following verbs - va-t b(in 1st person -va-yu) is written with a special suffix -Eve- with a letter e in place of an untested unstressed vowel: overshadow, prolong, corrupt(eclipse, prolong, corrupt), get stuck, get stuck(get stuck, get stuck); overwhelm, exhort, intend, doubt .
4. -e(t), -i(t). The verbs differ -eat And -it b. Verbs to -eat (in 1st person -by her ) – intransitive I conjugations – have the meaning ‘to become something, acquire a characteristic’, for example: become weak, become weak‘become powerless, lose strength’, to numb, to harden‘become harsh’. Verbs to - it (in 1st person And absent) – transitive II conjugations – have the meaning ‘to make something, to endow with a characteristic’, for example: weaken, weaken‘to make someone powerless, to deprive someone of strength’, anesthetize, neutralize, numb, weaken. Compare the same stressed suffixes in verbs like ( By)white And ( By)whiteness, (O)weakness , (once)have fun.
5. -ene(t), – eni(t). The acquisition of some characteristic, the transition to another state is also expressed in intransitive verbs I conjugation on -nope , in which there is an unstressed vowel before n transmitted by letter e : freeze, ossify, stiffen, become numb, traverse, turf, become bloody, glazed over, frenzied, dumbfounded, numb. Correlative transitive verbs II conjugation on -eni-t, denoting endowment with a characteristic, are also written with the letter e : freeze, numb, ossify, bloody, frenzied, numb and so on.
Writing verbs in - go crazy And -enit does not agree with the spelling of the corresponding relative adjectives, in which (if any) is written in the suffix before n letter I : ice, bone, blood, grass, wood and so on.
Exception: in verbs scarlet And purple(option: crimson thread) the letter is written I as in adjective scarlet.
The verb is one of the most difficult parts of speech in the Russian language to learn and write. The most difficult spelling in this topic is the spelling of verb endings in unstressed position. Their correct writing depends on such a concept as conjugation.
What is conjugation?
Conjugation is linguistic term, denoting a change in person and number of a verb. In other words, this is the set of personal endings that a word acquires when changing person and number. Fortunately for students, there are only two types of conjugation in the Russian language. Whatever verb out of several million existing ones is taken as an example, when changed it will acquire inflections:
- U/YU, EAT, ET in the singular and EAT, ETE, UT/YUT in the plural.
- U/YU, ISH, IT, in the singular and IM, IT, AT/YAT in the plural.
Let's conjugate the verb sleep: I sleep, we sleep, you sleep, you sleep, he sleeps, they sleep. Its endings, when changed, show that it belongs to the second conjugation. Another word - live. I live, we live, you live, you live, he lives, they live. Therefore, it is a first conjugation verb.
Prefix you- and verb conjugation
Verbs that have a prefix belong to the same conjugation as the non-prefixed verbs with the same root. This information is very important for words with the prefix You-, which has the property of “pulling off” the emphasis. For example, to determine the conjugation of a word will fly out, you need to remove the prefix, and you get a verb flies with a stressed personal ending, which refers to the second conjugation.
Thus, the endings of verbs in personal forms, under stress. But what to do if the accent is not the ending, but the stem of the word? Indeed, in this case the desired letter is not clearly heard, it is in weak position, and you can make a mistake. A rule comes to the rescue.
How to determine the conjugation if the personal ending is unstressed?
To determine which of the two types of conjugation available in the Russian language a verb belongs to, if the stress falls not on the ending, but on the stem, you need to form the initial form of the word. This form is called an infinitive and you can ask questions to it what to do? (perfect view) And what to do?(imperfect species). Next, you should look at what the resulting word ends with and apply the rule. In other words, unstressed endings of verbs can be written correctly only if their conjugation is determined.
The 2nd conjugation includes verbs that have unstressed personal endings, which in the infinitive form end in -it.
The 1st conjugation includes verbs with personal endings in unstressed position, which in the infinitive end in any letters except -it. These could be combinations -et, -at, -ot, -t, -ch and many others.
For example, the word dreaming. The emphasis falls not on the ending, but on the stem of the word. To determine the conjugation, we put the verb in the infinitive form: what to do? - dream. Ends with - at. This means that this verb belongs to 1 conjugation, and, changing in persons and numbers, it will take the corresponding endings.
You're sawing. Let's form the initial form - to nag. At the end of the word - it, therefore, it is a second conjugation verb, and the set of personal endings will be appropriate.
So, in order to correctly write vowels at the endings of verbs in an unstressed position, you need to put the word in the infinitive form and, depending on what it ends in, determine the conjugation. It seems simple. But if in the Russian language there weren’t a dozen exceptions for every rule, it would not be considered one of the most difficult languages in the world to learn. And in simple rule There are also insidious exception words about conjugation.
Exception verbs
Exceptions are those words that do not follow the general rule. There are 14 exception verbs in the conjugation rule that must be remembered in order to avoid spelling errors.
There are 7 words in -et, which, changing in person and number, take on the endings of verbs of 2 conjugations: these are words offend, depend,endure, hate, watch,twirlsee. For example, the word tolerate accepts the following endings: I tolerate, we tolerate, you tolerate, you tolerate, he tolerates, they tolerate.
4 verbs in - at, which are also not conjugated in accordance with the rule and have personal endings of 2 conjugations: drive, hear, hold, breathe. I hear, we hear, you hear, you hear, he hears, they hear.
And finally, 3 verbs starting with - it - lay,shave And rest on- have personal endings for verbs of 1st conjugation: I am laying, we are laying, you are laying, you are laying, he is laying, they are laying.
The endings of these 14 verbs need to be known by heart, because they do not obey the general rule.
Variably conjugated verbs
It is noteworthy that in the Russian language there are 2 interesting verbs, which are called heteroconjugated, because, when the person and number change, they acquire the endings of the first conjugation in some forms, and the second in others, without obeying any rules. Fortunately, their personal endings are stressed, so there are no errors in their spelling. But they still need to be remembered. These are verbs want And run. When they are conjugated, the following picture is obtained.
Want: I want, we want, you want, you want, he wants, they want(in the singular the verb changes according to type 1 conjugation, in the plural - according to type 2). Run: I'm running, we're running, you're running, you're running, he's running, they're running(in 3rd person form plural the verb has the ending of the 2nd conjugation, in all other forms - the endings characteristic of the 1st conjugation).
Algorithm for choosing verb endings
So, the spelling of verb endings follows a fairly simple and logical rule; it is important to simply understand it. To correctly write the endings of verbs, you need to be able to determine their conjugation. To do this, you should use a clear algorithm.
1. See where the emphasis falls: on the ending or on the stem of the word (don’t forget about the insidious prefix You: if it exists, it is necessary to determine the conjugation by an unprefixed synonym).
2. If the emphasis falls on the personal ending, then the conjugation is determined from it, in accordance with the rule.
3. If the base is stressed, then it is necessary to put the verb in the infinitive form. Three will be important last letters educated form.
- The verb ends in -it (except for three exceptions), which means it is of the second conjugation. Here we also include 7 verbs ending in - there, and 4 verbs -at.
- A verb in its initial form ends in any other letters (except for the 11 verbs listed above), which means it is conjugated according to the type of first conjugation. Let's add here 3 exception verbs in -it.
4. Verbs want And run- differently conjugated, they cannot be attributed to either the first or the second conjugation, and this feature just needs to be remembered.
Let's sum it up
Spelling verb endings requires knowledge of the rules and the application of a clear algorithm. The main thing is to correctly determine which of the two conjugations the word belongs to, and depending on this write the desired letter in an unstressed personal ending. Verbs are insidious, among them there are many exceptions that do not want to obey general rules, but paying attention to the word and applying a simple rule will help you avoid many mistakes!
Instructions
Transitive verbs are capable of forming verb phrases in conjunction with a noun, numeral or pronoun in the accusative case without a preposition:
- “buy (“who?”, “what?”) book”;
- “take (“who?”, “what?”) her with you”;
- “get (“who?”, “what?”) five.”
The category of transitivity/intransitivity of a verb, although it refers to features, is closely related to its lexical meaning in a specific statement. The same verb in can be both intransitive and transitive depending on contextual meaning. The list of such verbs tends to expand. Compare: “walking down the street - walking the dog.”
The ending is an important morpheme in the Russian language, unlike most foreign languages. It is she who connects words with each other, turning them into phrases and sentences. The ability to put the ending correctly is important for writing and correct speech.
Instructions
In the Russian language, all parts of speech have an ending, except adverbs and gerunds (which is why they do not change). With the help of a correctly posed question, you can easily determine how to write unstressed ending at specific word.
If you want to determine whether an adjective has, then the auxiliary word-question “which?” will help you. According to him stressed ending it is clear which letter should be written in the word being checked. For example: " kind person it’s hard not to notice” - ask the question: “a (what kind of) kind person.” The correct ending is "oh".
The same applies to participles and ordinal numbers, answering the questions “which?/which?” For example: “From the leaving platform” - “platform (which one?) leaving.”
The endings are soft and hard. Therefore, sometimes the question and the word being tested may differ slightly in . In such cases, look at the meaning. Often interchangeable, for example, “oh-y-yy”, “oh-oh-him”, “oh-oh-him”, “ym-im”, “om-em”.
IN possessive adjectives(except i.p. and v.p.) and those from names calendar months, there is a soft sign before the ending. For example, “fox”, “October”.
Also correct asked question helps determine the ending of nouns and verbs. Answering the questions “who?, what?” (for nouns) and “?, what to do?” (for verbs), you get the initial form of the word, which you need to use to determine how the ending in the word is.
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Be careful, the question is asked from the main word (usually a noun) to the dependent one. If you do not preserve this logic, you may get the wrong result.
The term "verb" came into our speech from Ancient Rus'. In those distant times, the Slavs called their alphabet “Glagolitic”. IN modern language This part of speech occupies an important place. Verb words are often found in sentences, together with the subject they form grammatical basis. The verb has a number grammatical features, may be the main and minor member offers.
Instructions
The action and state of an object are conveyed with the help of , having unchangeable signs of perfect or imperfect form, transitivity - intransitivity, reflexivity - irrevocability and conjugation.
The imperfect is more common in our speech. Usually morphemes help to form the perfect from it: “look - look”, “shout - shout”. But it also happens the other way around: “-to stitch”, “to decide-to decide.” These verb variants represent species pairs.
If verbs can control the nouns that accompany them in the form accusative case, and the connection between them is expressed without the help of a preposition, which means they will be considered: “show”, “cook”, “”. Intransitives are not characterized by such subordinating connection: “to be absent”, “to take a closer look”, “to sit”.
The suffix -sya (-s) indicates that the verb is reflexive. Non-returnables do not have such a suffix. It should be remembered that reflexivity indicates intransitivity.
Conjugation is indicated by a set of endings when changing according to persons and numbers. It’s easy to recognize this sign if the personal ending of the verb is stressed. If the conjugation is not established, you need to pay attention to the infinitive. All, excluding “shave” and “lay”, verbs ending in -it, and several excluded from this list (on -et, -at) - constitute II conjugation. The rest represent the I conjugation. Among the verbs, several different conjugations stand out: “to want”, “to run”, “to honor”.
Existing category The mood of the verb helps to establish how the actions performed relate to reality. Verb words in each mood have a certain set signs. Verbs indicative mood actions that actually occur are conveyed. The concept of the category of time is applied to them. The present and future tenses tend to change according to persons and numbers, and the past, instead of a person, according to gender. The imperative contains a call to action. Similar form the verb can form a unity with the words “yes”, “give (those)”, “let”. The possibility and certain conditions of action are indicated conditional mood, in which the verb is necessarily in the past tense and has the particle “would (b)”.
There may be no person or object performing the action. The purpose of such verb words is to convey various states nature or man. They have a corresponding name - “impersonal”. Examples of the use of such verbs in impersonal sentences: “It was getting dark outside”, “I’m shivering.”
The usual purpose of a verb in sentences is to act as a predicate. Syntactic functions expand when used in: here it can be the subject, perform the function of a sentence. Consider different variants: “Whistle (tale) everyone up!”, “Tourists began to carefully move (part of the predicate) forward”, “Learning (subject) will always come in handy”, “The guests asked to turn on (additional) music louder”, “The boy expressed a desire take (def.) volleyball seriously”, “I came to see (obv.) you.”
note
Linguistic scientists have two points of view regarding participles and gerunds formed from verbs: they are distinguished as independent parts of speech or verb forms.
Sources:
- general characteristics verb as part of speech
The verb is one of significant parts speeches, which denotes a procedural feature of an object, that is, an action, state or attitude. The verb is characterized grammatical categories type, voice, mood, tense and person.
Spelling endings
All verbs are usually divided into two groups: verbs that belong to the first conjugation, and verbs that belong to the second conjugation. The second conjugation includes all verbs in –it (exceptions are “shave”, “lay”, “”), as well as exception verbs in –et and –at (“drive”, “breathe”, “look”, “ see”, “hear”, “turn”, “offend”, “endure”, “depend”, “hate”, “hold”). All other verbs are usually classified as first conjugation.
Note: there are several differently conjugated verbs that cannot be classified as either the first or the second conjugation: “give”, “create”, “eat”, “run”, “want”.
If the verb has the prefix obez-, ob-transitive, it is in the second conjugation, in otherwise- according to the first conjugation.
If the verb of the first conjugation is in the future tense, then the ending is –ete. If you put such a verb in , the ending changes to -ite. For example: “You will send a letter this week,” but “Send the documents urgently.”
Soft sign(s) in verbs
Mild in several cases. First - initial form verb. The second is when the verb is placed in imperative mood. Third - second person singular verbs in the present and simple future tenses. The fourth is in reflexive verbs.
For example: “write”, “correct”, “”, “bent”.
Soft sign It is not written in the third person singular present or simple future tense.
For example: “washes his face.”
Spelling suffixes
Verbs with the suffixes –ыва- and –iva-, which have the meaning of the imperfect form, are written with the vowels –ы- and –и-.
For example: “smear”, “beg”, “insist”, “roll up”, “fill up”, “throw away”.
Imperfective verbs with suffixes –va-, which are in the first person form, can be checked by writing the vowel before the letter “v”.
For example: “zast-a-va-t - zast-a-t.”
Note: in some exception verbs, the suffix –eva- is written in –vat in place of an unchecked unstressed vowel: “eclipse-eva-t - to overshadow”; “extend - extend”; “vstr-eva-t - butt in”; “bur-eva-t - clean up”, etc.
There are also verbs ending in -et and -it. Verbs ending in –et are intransitive verbs of the 1st conjugation. They mean “to acquire someone’s attribute, to become something.”
For example: “to become more severe”, i.e. “become harsh”; “to weaken”, i.e. "to become powerless"; “de-soul”, i.e. “become soulless”, etc.
Verbs that end in –it are transitive verbs. They mean “to endow with some attribute, to make an object something.”
For example: “numb the pain”, i.e. “stop the pain”; “to weaken”, i.e. “to deprive of strength”; “to neutralize, i.e. stop the harm”, etc.
Verbs that end in -enet and -enit do not agree in their spelling with the corresponding relative adjectives, in which the letter “I” is written, for example: “blood”, “herbal”, “wooden”, etc. The exception is the verbs “crimson” and “crimson”, in which the letter “ya” is written in the same way as in “crimson”.
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increasing on the interval \(X\) if for any \(x_1, x_2\in X\) such that \(x_1 The function is called non-decreasing \(\blacktriangleright\) The function \(f(x)\) is called decreasing on the interval \(X\) if for any \(x_1, x_2\in X\) such that \(x_1 The function is called non-increasing on the interval \(X\) if for any \(x_1, x_2\in X\) such that \(x_1 \(\blacktriangleright\) Increasing and decreasing functions are called strictly monotonous, and non-increasing and non-decreasing are simply monotonous. \(\blacktriangleright\) Basic properties: I. If the function \(f(x)\) is strictly monotone on \(X\) , then from the equality \(x_1=x_2\) (\(x_1,x_2\in X\) ) it follows \(f(x_1)= f(x_2)\) , and vice versa. Example: the function \(f(x)=\sqrt x\) is strictly increasing for all \(x\in \) , therefore the equation \(x^2=9\) has at most one solution on this interval, or rather one: \(x=-3\) . the function \(f(x)=-\dfrac 1(x+1)\) is strictly increasing for all \(x\in (-1;+\infty)\), so the equation \(-\dfrac 1(x +1)=0\) has no more than one solution on this interval, or rather none, because the numerator of the left-hand side can never be equal to zero. III. If the function \(f(x)\) is non-decreasing (non-increasing) and continuous on the segment \(\), and at the ends of the segment it takes the values \(f(a)=A, f(b)=B\) , then for \(C\in \) (\(C\in \) ) the equation \(f(x)=C\) always has at least one solution. Example: the function \(f(x)=x^3\) is strictly increasing (that is, strictly monotone) and continuous for all \(x\in\mathbb(R)\) , therefore for any \(C\in ( -\infty;+\infty)\) the equation \(x^3=C\) has exactly one solution: \(x=\sqrt(C)\) . Task 1 #3153 Task level: Easier than the Unified State Exam has exactly two roots. Let's rewrite the equation as: \[(3x^2)^3+3x^2=(x-a)^3+(x-a)\] Consider the function \(f(t)=t^3+t\) . Then the equation will be rewritten in the form: \ Let's study the function \(f(t)\) . \ Consequently, the function \(f(t)\) increases for all \(t\) . This means that each value of the function \(f(t)\) corresponds to exactly one value of the argument \(t\) . Therefore, in order for the equation to have roots, it is necessary: \
For the resulting equation to have two roots, its discriminant must be positive: \
Answer: \(\left(-\infty;\dfrac1(12)\right)\) Task 2 #2653 Task level: Equal to the Unified State Exam Find all values of the parameter \(a\) for which the equation \
has two roots. (Task from subscribers.) Let's make a replacement: \(ax^2-2x=t\) , \(x^2-1=u\) . Then the equation will take the form: \
Consider the function \(f(w)=7^w+\sqrtw\) . Then our equation will take the form: \ Let's find the derivative \
Note that for all \(w\ne 0\) the derivative is \(f"(w)>0\) , since \(7^w>0\) , \(w^6>0\) . Note also that the function \(f(w)\) itself is defined for all \(w\). Since, in addition, \(f(w)\) is continuous, we can conclude that \(f (w)\) increases on the whole \(\mathbb(R)\) . \
In order for this equation to have two roots, it must be square and its discriminant must be positive: \[\begin(cases) a-1\ne 0\\ 4-4(a-1)>0\end(cases) \quad\Leftrightarrow\quad \begin(cases)a\ne1\\a<2\end{cases}\]
Answer: \((-\infty;1)\cup(1;2)\) Task 3 #3921 Task level: Equal to the Unified State Exam Find all positive values of the parameter \(a\) for which the equation has at least \(2\) solutions. Let's move all terms containing \(ax\) to the left, and those containing \(x^2\) to the right, and consider the function Then the original equation will take the form: Let's find the derivative: Because \((t-2)^2 \geqslant 0, \e^t>0, \1+\cos(2t) \geqslant 0\), then \(f"(t)\geqslant 0\) for any \(t\in \mathbb(R)\) . Moreover, \(f"(t)=0\) if \((t-2)^2=0\) and \(1+\cos(2t)=0\) at the same time, which is not true for any \ (t\). Therefore, \(f"(t)> 0\) for any \(t\in \mathbb(R)\) . Thus, the function \(f(t)\) is strictly increasing for all \(t\in \mathbb(R)\) . This means that the equation \(f(ax)=f(x^2)\) is equivalent to the equation \(ax=x^2\) . The equation \(x^2-ax=0\) for \(a=0\) has one root \(x=0\), and for \(a\ne 0\) it has two various roots\(x_1=0\) and \(x_2=a\) . Answer: \((0;+\infty)\) . Task 4 #1232 Task level: Equal to the Unified State Exam Find all values of the parameter \(a\) , for each of which the equation \
has a unique solution. Let's multiply the right and left sides of the equation by \(2^(\sqrt(x+1))\) (since \(2^(\sqrt(x+1))>0\) ) and rewrite the equation in the form : \
Consider the function \(y=2^t\cdot \log_(\frac(1)(9))((t+2))\) for \(t\geqslant 0\) (since \(\sqrt(x+1)\geqslant 0\) ). Derivative \(y"=\left(-2^t\cdot \log_9((t+2))\right)"=-\dfrac(2^t)(\ln9)\cdot \left(\ln 2\cdot \ln((t+2))+\dfrac(1)(t+2)\right)\). Because \(2^t>0, \ \dfrac(1)(t+2)>0, \ \ln((t+2))>0\) for all \(t\geqslant 0\) , then \(y"<0\)
при всех \(t\geqslant 0\)
. Consequently, as \(t\geqslant 0\) the function \(y\) decreases monotonically. The equation can be considered in the form \(y(t)=y(z)\) , where \(z=ax, t=\sqrt(x+1)\) . From the monotonicity of the function it follows that equality is possible only if \(t=z\) . This means that the equation is equivalent to the equation: \(ax=\sqrt(x+1)\), which in turn is equivalent to the system: \[\begin(cases) a^2x^2-x-1=0\\ ax \geqslant 0 \end(cases)\] When \(a=0\) the system has one solution \(x=-1\) that satisfies the condition \(ax\geqslant 0\) . Consider the case \(a\ne 0\) . Discriminant of the first equation of the system \(D=1+4a^2>0\) for all \(a\) . Consequently, the equation always has two roots \(x_1\) and \(x_2\), and they are of different signs (since according to Vieta’s theorem \(x_1\cdot x_2=-\dfrac(1)(a^2)<0\)
). This means that for \(a<0\)
условию \(ax\geqslant 0\)
подходит отрицательный корень, при \(a>0\) the condition is satisfied by a positive root. Therefore, the system always has a unique solution. So, \(a\in \mathbb(R)\) . Answer: \(a\in \mathbb(R)\) . Task 5 #1234 Task level: Equal to the Unified State Exam Find all values of the parameter \(a\) , for each of which the equation \
has at least one root from the segment \([-1;0]\) . Consider the function \(f(x)=2x^3-3x(ax+x-a^2-1)-3a-a^3\) for some fixed \(a\) . Let's find its derivative: \(f"(x)=6x^2-6ax-6x+3a^2+3=3(x^2-2ax+a^2+x^2-2x+1)=3((x-a)^2 +(x-1)^2)\). Note that \(f"(x)\geqslant 0\) for all values of \(x\) and \(a\) , and is equal to \(0\) only for \(x=a=1\). But for \(a=1\) : This means that for all \(a\ne 1\) the function \(f(x)\) is strictly increasing, therefore, the equation \(f(x)=0\) can have no more than one root. Taking into account the properties of the cubic function, the graph of \(f(x)\) for some fixed \(a\) will look like this: This means that in order for the equation to have a root from the segment \([-1;0]\), it is necessary: \[\begin(cases) f(0)\geqslant 0\\ f(-1)\leqslant 0 \end(cases) \Rightarrow \begin(cases) a(a^2+3)\leqslant 0\\ ( a+2)(a^2+a+4)\geqslant 0 \end(cases) \Rightarrow \begin(cases) a\leqslant 0\\ a\geqslant -2 \end(cases) \Rightarrow -2\leqslant a\leqslant 0\] Thus, \(a\in [-2;0]\) . Answer: \(a\in [-2;0]\) . Task 6 #2949 Task level: Equal to the Unified State Exam Find all values of the parameter \(a\) , for each of which the equation \[(\sin^2x-5\sin x-2a(\sin x-3)+6)\cdot (\sqrt2a+8x\sqrt(2x-2x^2))=0\] has roots. (Task from subscribers) ODZ equations: \(2x-2x^2\geqslant 0 \quad\Leftrightarrow\quad x\in \). Therefore, in order for an equation to have roots, it is necessary that at least one of the equations \[\sin^2x-5\sin x-2a(\sin x-3)+6=0 \quad (\small(\text(or)))\quad \sqrt2a+8x\sqrt(2x-2x^ 2)=0\] had decisions on ODZ. 1) Consider the first equation \[\sin^2x-5\sin x-2a(\sin x-3)+6=0 \quad\Leftrightarrow\quad \left[\begin(gathered)\begin(aligned) &\sin x=2a+ 2\\ &\sin x=3\\ \end(aligned) \end(gathered)\right. \quad\Leftrightarrow\quad \sin x=2a+2\] This equation must have roots in \(\) . Consider a circle: Thus, we see that for any \(2a+2\in [\sin 0;\sin 1]\) the equation will have one solution, and for all others it will have no solutions. Therefore, when \(a\in \left[-1;-1+\sin 1\right]\) the equation has solutions. 2) Consider the second equation \[\sqrt2a+8x\sqrt(2x-2x^2)=0 \quad\Leftrightarrow\quad 8x\sqrt(x-x^2)=-a\] Consider the function \(f(x)=8x\sqrt(x-x^2)\) . Let's find its derivative: \
On the ODZ, the derivative has one zero: \(x=\frac34\) , which is also the maximum point of the function \(f(x)\) . Therefore, in order for the equation to have solutions, it is necessary that the graph \(f(x)\) intersect with the straight line \(y=-a\) (the figure shows one of the suitable options). That is, it is necessary that \
. For these \(x\) : The function \(y_1=\sqrt(x-1)\) is strictly increasing. The graph of the function \(y_2=5x^2-9x\) is a parabola, the vertex of which is at the point \(x=\dfrac(9)(10)\) . Consequently, for all \(x\geqslant 1\), the function \(y_2\) is also strictly increasing (the right branch of the parabola). Because the sum of strictly increasing functions is strictly increasing, then \(f_a(x)\) is strictly increasing (the constant \(3a+8\) does not affect the monotonicity of the function). The function \(g_a(x)=\dfrac(a^2)(x)\) for all \(x\geqslant 1\) represents part of the right branch of the hyperbola and is strictly decreasing. Solving the equation \(f_a(x)=g_a(x)\) means finding the intersection points of the functions \(f\) and \(g\) . From their opposite monotonicity it follows that the equation can have at most one root. When \(x\geqslant 1\) \(f_a(x)\geqslant 3a+4, \ \ \ 0 \\cup Answer: \(a\in (-\infty;-1]\cup)
This means that the equality \(f(t)=f(u)\) is possible if and only if \(t=u\) . Let's return to the original variables and solve the resulting equation:
\
\
\
We need to find the values of \(a\) at which the equation will have at least two roots, also taking into account the fact that \(a>0\) .
Therefore, the answer is: \(a\in (0;+\infty)\) .
\(f"(x)=6(x-1)^2 \Rightarrow f(x)=2(x-1)^3 \Rightarrow\) the equation \(2(x-1)^3=0\) has a single root \(x=1\) that does not satisfy the condition. Therefore, \(a\) cannot be equal to \(1\) .
Note that \(f(0)=f(1)=0\) . So, schematically the graph \(f(x)\) looks like this: 
