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Mongolia Upper Secondary State Exam study materials
Mongolia Upper Secondary State Exam study resources

Mongolia Upper Secondary State Exam practice questions

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Question 1Mathematics

What is the value of the discriminant of the quadratic equation 2x² − 5x + 3 = 0?

  1. 25
  2. 1 Correct answer.
  3. 49
  4. −1

Explanation

For ax² + bx + c = 0 the discriminant is b² − 4ac. Here a = 2, b = −5, c = 3, so (−5)² − 4·2·3 = 25 − 24 = 1.

Question 2Mathematics

Simplify the expression (x³y²)(x²y⁵).

  1. x⁶y¹⁰
  2. x⁵y⁷ Correct answer.
  3. x⁵y¹⁰
  4. x⁶y⁷

Explanation

When multiplying powers with the same base, add the exponents: x³·x² = x⁵ and y²·y⁵ = y⁷. The product is therefore x⁵y⁷.

Question 3Mathematics

A fair six-sided die is rolled once. What is the probability of getting a number greater than 4?

  1. 2/3
  2. 1/6
  3. 1/3 Correct answer.
  4. 1/2

Explanation

The outcomes greater than 4 are 5 and 6, giving 2 favourable outcomes out of 6 equally likely outcomes. The probability is 2/6 = 1/3.

Question 4Mathematics

What is the sum of the interior angles of a convex octagon?

  1. 720°
  2. 1080° Correct answer.
  3. 1260°
  4. 900°

Explanation

The interior angles of a convex n-gon sum to (n − 2)·180°. For n = 8 this gives 6·180° = 1080°.

Question 5Mathematics

The function f(x) = 3x − 7 is defined for all real numbers. What is f⁻¹(5)?

  1. 22
  2. 4 Correct answer.
  3. 8
  4. 2

Explanation

Solve 3x − 7 = 5, giving 3x = 12 and x = 4. Since f is linear with nonzero slope, it is invertible and f⁻¹(5) = 4.

Question 6Mathematics

What is the value of log₂ 32?

  1. 4
  2. 5 Correct answer.
  3. 6
  4. 16

Explanation

log₂ 32 asks for the exponent e with 2ᵉ = 32. Since 2⁵ = 32, the value is 5.

Question 7Mathematics

The arithmetic sequence has first term a₁ = 7 and common difference d = 4. What is the 10th term?

  1. 40
  2. 44
  3. 47
  4. 43 Correct answer.

Explanation

The nth term of an arithmetic sequence is aₙ = a₁ + (n − 1)d. Thus a₁₀ = 7 + 9·4 = 7 + 36 = 43.

Question 8Mathematics

Which expression is equal to sin²θ + cos²θ for every angle θ?

  1. 0
  2. 2
  3. 1 Correct answer.
  4. sin 2θ

Explanation

The Pythagorean identity states that sin²θ + cos²θ = 1 for all θ. It follows from the coordinates of a point on the unit circle.

Question 9Mathematics

Solve the inequality 3 − 2x < 7 for real x.

  1. x > 2
  2. x > −2 Correct answer.
  3. x < −2
  4. x < 2

Explanation

Subtract 3 from both sides to get −2x < 4. Dividing by −2 reverses the inequality, giving x > −2.

Question 10Mathematics

The roots of x² − 6x + k = 0 are equal. What is k?

  1. 9 Correct answer.
  2. 36
  3. 3
  4. 6

Explanation

Equal roots require a zero discriminant: b² − 4ac = 36 − 4k = 0. Solving gives k = 9, and indeed x² − 6x + 9 = (x − 3)².

Question 11Mathematics

A geometric sequence has a₁ = 3 and a₄ = 24. What is the common ratio?

  1. 2 Correct answer.
  2. 4
  3. 3
  4. 8

Explanation

In a geometric sequence aₙ = a₁rⁿ⁻¹, so 24 = 3r³, giving r³ = 8 and r = 2.

Question 12Mathematics

What is the derivative of f(x) = x³ − 4x with respect to x?

  1. 3x − 4
  2. 3x² − 4 Correct answer.
  3. x² − 4
  4. 3x² − 4x

Explanation

Differentiate term by term using the power rule: the derivative of x³ is 3x² and the derivative of −4x is −4. The result is 3x² − 4.

Question 13Mathematics

The vectors u = (3, −1) and v = (k, 2) are perpendicular. What is k?

  1. −6
  2. 2/3 Correct answer.
  3. −2/3
  4. 6

Explanation

Perpendicular vectors have zero dot product: 3k + (−1)(2) = 0, so 3k = 2 and k = 2/3.

Question 14Mathematics

In how many ways can a committee of 3 people be chosen from 8 people?

  1. 336
  2. 168
  3. 56 Correct answer.
  4. 24

Explanation

Order does not matter, so this is a combination: C(8,3) = 8!/(3!·5!) = (8·7·6)/(3·2·1) = 56.

Question 15Mathematics

What is the value of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + …?

  1. 2 Correct answer.
  2. 3/2
  3. 4
  4. ∞

Explanation

For |r| < 1 the sum is a/(1 − r). Here a = 1 and r = 1/2, so the sum is 1/(1 − 1/2) = 2.

Question 16Mathematics

The line through the points (1, 2) and (4, 11) has slope m. What is the equation of the line?

  1. y = (1/3)x + 5/3
  2. y = 3x + 1
  3. y = 9x − 7
  4. y = 3x − 1 Correct answer.

Explanation

The slope is (11 − 2)/(4 − 1) = 9/3 = 3. Using point (1, 2): y − 2 = 3(x − 1), so y = 3x − 1.

Question 17Mathematics

If cos θ = 3/5 and θ lies in the first quadrant, what is sin θ?

  1. 5/4
  2. 4/5 Correct answer.
  3. 16/25
  4. −4/5

Explanation

Use sin²θ = 1 − cos²θ = 1 − 9/25 = 16/25. In the first quadrant sin θ is positive, so sin θ = 4/5.

Question 18Mathematics

What is the value of the definite integral ∫₀² (3x² + 1) dx?

  1. 10 Correct answer.
  2. 9
  3. 12
  4. 8

Explanation

An antiderivative is x³ + x. Evaluating from 0 to 2 gives (8 + 2) − 0 = 10.

Question 19Mathematics

The function g(x) = x² − 4x + 7 has a minimum. What is the minimum value?

  1. −3
  2. 3 Correct answer.
  3. 7
  4. 4

Explanation

Complete the square: x² − 4x + 7 = (x − 2)² + 3. The square is never negative, so the minimum is 3, attained at x = 2.

Question 20Mathematics

Solve the equation 2^(x+1) = 8^(x−1) for x.

  1. 1/2
  2. 3
  3. 2 Correct answer.
  4. 1

Explanation

Write 8 as 2³, so 8^(x−1) = 2^(3x−3). Equating exponents: x + 1 = 3x − 3, giving 4 = 2x and x = 2.

Question 21Mathematics

The polynomial P(x) = x³ − 2x² − 5x + 6 has x = 1 as a root. What are all the roots of P?

  1. 1, 3, −2 Correct answer.
  2. 1, −3, −2
  3. 1, 3, 2
  4. 1, 2, −3

Explanation

Dividing by (x − 1) gives x² − x − 6, which factors as (x − 3)(x + 2). So the roots are 1, 3 and −2.

Question 22Mathematics

A circle has equation x² + y² − 6x + 8y − 11 = 0. What is its radius?

  1. 4
  2. 6 Correct answer.
  3. 36
  4. √11

Explanation

Complete the square: (x − 3)² + (y + 4)² = 11 + 9 + 16 = 36. The radius is √36 = 6.

Question 23Mathematics

In triangle ABC, AB = 7, AC = 9 and angle A = 60°. What is the length of BC?

  1. 8
  2. √67 Correct answer.
  3. 13
  4. √130

Explanation

By the cosine rule, BC² = 7² + 9² − 2·7·9·cos 60° = 49 + 81 − 63 = 67, so BC = √67.

Question 24Mathematics

A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

  1. 3/5
  2. 3/10 Correct answer.
  3. 1/2
  4. 9/25

Explanation

The probability the first is red is 3/5; given that, the second is red with probability 2/4. Multiplying gives (3/5)(2/4) = 6/20 = 3/10.

Question 25Mathematics

For which values of x does the series 1 + 2x + 4x² + 8x³ + … converge?

  1. |x| < 2
  2. |x| < 1/2 Correct answer.
  3. |x| < 1
  4. x > 0

Explanation

This is geometric with ratio r = 2x. A geometric series converges exactly when |r| < 1, so |2x| < 1 and hence |x| < 1/2.

Question 26Mathematics

A triangle has side lengths 6, 8 and 10. What is the area of the triangle?

  1. 48
  2. 40
  3. 24 Correct answer.
  4. 30

Explanation

Since 6² + 8² = 36 + 64 = 100 = 10², the triangle is right-angled with legs 6 and 8. Its area is half the product of the legs, (6 × 8)/2 = 24.

Question 27Mathematics

What is the sum of the first 20 positive odd integers 1 + 3 + 5 + … + 39?

  1. 210
  2. 400 Correct answer.
  3. 380
  4. 420

Explanation

The sum of the first n odd integers equals n², since the sequence is arithmetic with a₁ = 1 and d = 2. Here n = 20, so the sum is 20² = 400.

Question 28Mathematics

The matrix A = [[2, 1], [1, 3]] is multiplied by the column vector x = [[1], [2]]. What is the resulting vector Ax?

  1. [[5], [7]]
  2. [[3], [5]]
  3. [[4], [5]]
  4. [[4], [7]] Correct answer.

Explanation

Each entry of Ax is the dot product of a row of A with x. First row: 2·1 + 1·2 = 4; second row: 1·1 + 3·2 = 7, giving [[4], [7]].

Question 29Mathematics

What is the remainder when the polynomial x³ + 2x² − 5x + 1 is divided by x − 2?

  1. 7 Correct answer.
  2. 3
  3. −1
  4. 11

Explanation

By the remainder theorem, the remainder equals the value of the polynomial at x = 2. Substituting gives 8 + 8 − 10 + 1 = 7.

Question 30Mathematics

In how many distinct orders can the letters of the word LEVEL be arranged?

  1. 120
  2. 60
  3. 20
  4. 30 Correct answer.

Explanation

The word has 5 letters with L repeated twice and E repeated twice, so the number of distinct arrangements is 5!/(2!·2!) = 120/4 = 30.

Question 31Mathematics

What is the value of the expression 2 sin 15° cos 15°?

  1. √3/2
  2. 1
  3. 1/2 Correct answer.
  4. √2/2

Explanation

The double-angle identity states that 2 sin θ cos θ = sin 2θ. Taking θ = 15° gives sin 30° = 1/2.

Question 32Mathematics

The function f(x) = x³ − 3x² + 4 is defined for all real x. At which values of x does f have a local maximum?

  1. x = 0 Correct answer.
  2. x = −2
  3. x = 1
  4. x = 2

Explanation

Setting f′(x) = 3x² − 6x = 3x(x − 2) to zero gives critical points x = 0 and x = 2. The second derivative f″(x) = 6x − 6 is negative at x = 0, so f has a local maximum there.

Question 33Mongolian Language and Literature

Which Mongolian vowel is a back vowel?

  1. ө
  2. у Correct answer.
  3. э
  4. ү

Explanation

Mongolian vowels are divided into back vowels (а, о, у) and front vowels (э, ө, ү). The vowel 'у' belongs to the back vowel set, while 'ү', 'ө', and 'э' are front vowels.

Question 34Mongolian Language and Literature

What is the correct plural form of the word 'ном' (book) using the standard plural suffix?

  1. номнар
  2. номууд Correct answer.
  3. номүүд
  4. номуд

Explanation

The plural suffix -ууд is attached to back-vowel words ending in consonants, so 'ном' becomes 'номууд'. The form 'номүүд' would violate vowel harmony, and 'номнар' and 'номуд' are not standard plural forms.

Question 35Mongolian Language and Literature

Which sentence uses the correct case ending for the object of the sentence?

  1. Би номыг уншив. Correct answer.
  2. Би номд уншив.
  3. Би номоос уншив.
  4. Би номоор уншив.

Explanation

The accusative case marks the direct object of a transitive verb and is formed with the suffix -ыг/-ийг. In 'Би номыг уншив' (I read the book), 'номыг' correctly marks the object. The other suffixes indicate location, ablative, or instrumental meanings, which do not fit the verb 'унших' (to read) with a direct object.

Question 36Mongolian Language and Literature

Which of the following is a compound word in Mongolian?

  1. цаас
  2. багш
  3. гэр бүл Correct answer.
  4. сургууль

Explanation

A compound word is formed by combining two or more independent words into a single lexical unit. 'Гэр бүл' (family) is composed of 'гэр' (home) and 'бүл' (group), making it a compound. The other options are simple, underived words.

Question 37Mongolian Language and Literature

Which word is a synonym of 'сайхан' (beautiful)?

  1. үзэсгэлэнтэй Correct answer.
  2. том
  3. муухай
  4. жижиг

Explanation

Synonyms are words with similar meanings. 'Үзэсгэлэнтэй' means 'beautiful' or 'gorgeous', closely matching 'сайхан'. The other options mean 'ugly', 'small', and 'big', which are not synonyms.

Question 38Mongolian Language and Literature

Identify the antonym of 'өндөр' (high/tall).

  1. өргөн
  2. гүн
  3. нам Correct answer.
  4. хол

Explanation

Antonyms are words with opposite meanings. The opposite of 'өндөр' (high/tall) is 'нам' (low). 'Өргөн' means 'wide', 'хол' means 'far', and 'гүн' means 'deep', none of which are direct opposites of height.

Question 39Mongolian Language and Literature

Which sentence is written in the correct Mongolian word order?

  1. Өнөөдөр би явна сургуульд.
  2. Би сургуульд өнөөдөр явна.
  3. Явна би өнөөдөр сургуульд.
  4. Би өнөөдөр сургуульд явна. Correct answer.

Explanation

Mongolian basic word order is subject–object–verb (SOV), with time expressions typically placed before the object or at the beginning. 'Би өнөөдөр сургуульд явна' (I today to school go) follows the natural SOV order with the time adverb before the object. The other options place the verb or elements in unnatural positions.

Question 40Mongolian Language and Literature

Which suffix forms the past tense of the verb 'явах' (to go) in the simple past?

  1. -ж байна
  2. -на
  3. -лаа Correct answer.
  4. -х

Explanation

The simple past tense in Mongolian is formed with the suffix -лаа/-лээ/-лоо/-лөө, depending on vowel harmony. For 'явах', the stem 'яв-' takes '-лаа' to make 'явлаа' (went). The suffix '-на' marks present-future, '-ж байна' marks progressive, and '-х' marks future or verbal noun.

Question 41Mongolian Language and Literature

In the sentence 'Тэр хүн маш өндөр.', what part of speech is 'маш'?

  1. adverb Correct answer.
  2. noun
  3. verb
  4. adjective

Explanation

'Маш' means 'very' and modifies the adjective 'өндөр' (tall), specifying degree. Words that modify adjectives or verbs are adverbs. It is not an adjective, noun, or verb.

Question 42Mongolian Language and Literature

Which of these is a declarative sentence?

  1. Гэртээ харина уу?
  2. Чи хаана байна?
  3. Хурдан яв!
  4. Би маргааш ирнэ. Correct answer.

Explanation

A declarative sentence makes a statement and typically ends with a period. 'Би маргааш ирнэ.' (I will come tomorrow.) is a statement. The others are interrogative (questions) or imperative (command).

Question 43Mongolian Language and Literature

Which punctuation mark is used at the end of an interrogative sentence in Mongolian?

  1. comma (,)
  2. question mark (?) Correct answer.
  3. exclamation mark (!)
  4. period (.)

Explanation

Interrogative sentences in Mongolian, as in many languages, end with a question mark (?). A period ends declarative sentences, an exclamation mark ends exclamatory or imperative sentences, and a comma separates elements within a sentence.

Question 44Mongolian Language and Literature

What is the main theme of the Mongolian epic 'Жангар'?

  1. the heroic deeds of Jangar and his warriors Correct answer.
  2. a love story between a prince and a princess
  3. the daily life of nomads
  4. the life of a Buddhist monk

Explanation

'Жангар' is a traditional Mongolian epic that celebrates the heroic deeds of the warrior Jangar and his companions as they defend their land. It is not primarily about a monk, a romance, or daily nomadic life.

Question 45Mongolian Language and Literature

Who is the author of the novel 'Газар шороо' (Earth and Soil)?

  1. Сэнгийн Эрдэнэ
  2. Дашдоржийн Нацагдорж
  3. Чимидийн Дондог
  4. Бямбын Ринчен Correct answer.

Explanation

'Газар шороо' is a novel by the prominent Mongolian writer Бямбын Ринчен. It is a classic work of Mongolian literature. The other authors are also well-known but did not write this novel.

Question 46Mongolian Language and Literature

What is the correct form of the verb 'унших' (to read) in the present progressive tense for 'I am reading'?

  1. уншлаа
  2. уншиж байна Correct answer.
  3. унших
  4. уншина

Explanation

The present progressive in Mongolian is formed with the converb -ж plus the auxiliary 'байна'. Thus 'уншиж байна' means 'am reading'. The other forms represent present-future, past, and future/verbal noun respectively.

Question 47Mongolian Language and Literature

Which of the following sentences contains a simile?

  1. Би ном уншиж байна.
  2. Тэр улаан малгайтай.
  3. Түүний нүд од шиг гялалзана. Correct answer.
  4. Өнөөдөр бороо орно.

Explanation

A simile compares two things using words like 'шиг' (like/as). 'Түүний нүд од шиг гялалзана' (Her eyes shine like stars) uses 'шиг' to compare eyes to stars. The other sentences are literal statements without comparison.

Question 48Mongolian Language and Literature

What is the rhyme scheme of a traditional Mongolian 'шүлэг' (poem) called 'дөрвөн эгнээний шүлэг'?

  1. ABCB
  2. AAAA Correct answer.
  3. AABB
  4. ABAB

Explanation

The traditional Mongolian four-line stanza 'дөрвөн эгнээний шүлэг' typically uses alliteration and end-rhyme with an AAAA scheme, where all four lines rhyme. Other schemes like AABB or ABAB are less common in this specific form.

Question 49Mongolian Language and Literature

Which of these is a famous Mongolian poet known for the poem 'Миний нутаг' (My Homeland)?

  1. Б. Явуухулан
  2. Д. Нацагдорж Correct answer.
  3. С. Эрдэнэ
  4. Ч. Дондог

Explanation

'Миний нутаг' is a well-known poem by Дашдоржийн Нацагдорж, a foundational figure in modern Mongolian literature. The other poets are also significant but did not write this specific poem.

Question 50Mongolian Language and Literature

In Mongolian grammar, what is the function of the suffix '-ын' in the word 'багшийн'?

  1. genitive case Correct answer.
  2. past tense
  3. accusative case
  4. plural

Explanation

The suffix '-ын/-ийн' marks the genitive case, indicating possession or relation, as in 'багшийн' (of the teacher). It is not a plural, accusative, or tense marker.