answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
The value is 56. | {
"A": "54",
"B": "58",
"C": "56",
"D": "60"
} | mat_0451 | multiple_choice | en | Compute N² + N for N = 7. | [
"7² + 7.",
"Value = 56.",
"Select option C."
] | math | Answer: 56
Steps:
1. 7² + 7.
2. Value = 56.
3. Select option C.
Answer: (C) | proofs |
f'(3) = 20. | null | mat_0452 | worked_example | cs | Pro f(x) = 3x² + 2x spočítej f'(3) krok za krokem. | [
"f'(x) = 6x + 2.",
"Dosaď x = 3.",
"6×3 + 2 = 20."
] | math | Answer: f'(3) = 20.
Steps:
1. f'(x) = 6x + 2.
2. Dosaď x = 3.
3. 6×3 + 2 = 20. | calculus |
A projection drops a vector onto a subspace along perpendiculars, finding the closest point. | null | mat_0453 | concept | en | What is a projection? | [
"A projection drops a vector onto a subspace along perpendiculars, finding the closest point.",
"So the answer stands: A projection drops a vector onto a subspace along perpendiculars, finding the closest point."
] | math | Answer: A projection drops a vector onto a subspace along perpendiculars, finding the closest point.
Steps:
1. A projection drops a vector onto a subspace along perpendiculars, finding the closest point.
2. So the answer stands: A projection drops a vector onto a subspace along perpendiculars, finding the closest point... | linear_algebra |
P = 1/7. | null | mat_0454 | worked_example | cs | Hází se férovou 7-stěnnou kostkou. Ukaž P(jedno konkrétní číslo). | [
"Příznivé: 1.",
"Možné: 7.",
"P = 1/7."
] | math | Answer: P = 1/7.
Steps:
1. Příznivé: 1.
2. Možné: 7.
3. P = 1/7. | probability |
NSD je 9. | {
"A": "6",
"B": "9",
"C": "12",
"D": "15"
} | mat_0455 | multiple_choice | cs | Najdi největšího společného dělitele NSD(18, 27). | [
"Vytkni 3 dvakrát.",
"NSD = 9.",
"Vyber možnost B."
] | math | Answer: 9
Steps:
1. Vytkni 3 dvakrát.
2. NSD = 9.
3. Vyber možnost B.
Answer: (B) | number_theory |
Počítá podmnožiny velikosti r z n objektů. | null | mat_0456 | concept | cs | Co je kombinační číslo? | [
"Počítá podmnožiny velikosti r z n objektů.",
"Odpověď tedy platí: Počítá podmnožiny velikosti r z n objektů."
] | math | Answer: Počítá podmnožiny velikosti r z n objektů.
Steps:
1. Počítá podmnožiny velikosti r z n objektů.
2. Odpověď tedy platí: Počítá podmnožiny velikosti r z n objektů. | combinatorics |
An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately. | null | mat_0457 | concept | en | What is a biconditional proof? | [
"An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.",
"So the answer stands: An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately."
] | math | Answer: An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.
Steps:
1. An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately.
2. So the answer stands: An if-and-only-if proof shows both directions, proving P gives Q and Q gives P separately. | proofs |
The integral is 125. | {
"A": "125",
"B": "120",
"C": "130",
"D": "135"
} | mat_0458 | multiple_choice | en | Compute the integral of 3x² from 0 to 5. | [
"Antiderivative x³.",
"5³ = 125.",
"Select option A."
] | math | Answer: 125
Steps:
1. Antiderivative x³.
2. 5³ = 125.
3. Select option A.
Answer: (A) | calculus |
The dot product is 32. | {
"A": "28",
"B": "32",
"C": "36",
"D": "40"
} | mat_0459 | multiple_choice | en | Find the dot product (4,5) · (3,4). | [
"4×3 + 5×4.",
"Product = 32.",
"Select option B."
] | math | Answer: 32
Steps:
1. 4×3 + 5×4.
2. Product = 32.
3. Select option B.
Answer: (B) | linear_algebra |
P(red) = 10/14. | null | mat_0460 | worked_example | en | A bag holds 10 red and 4 blue marbles. Show P(red). | [
"Red: 10.",
"Total: 14.",
"P = 10/14."
] | math | Answer: P(red) = 10/14.
Steps:
1. Red: 10.
2. Total: 14.
3. P = 10/14. | probability |
gcd = 18. | null | mat_0461 | worked_example | en | Compute gcd(36, 54) step by step. | [
"Factor: 36=4×9, 54=6×9.",
"Common: 9×gcd(4,6).",
"= 9×2 = 18."
] | math | Answer: gcd = 18.
Steps:
1. Factor: 36=4×9, 54=6×9.
2. Common: 9×gcd(4,6).
3. = 9×2 = 18. | number_theory |
C(5,2) je 10. | {
"A": "5",
"B": "10",
"C": "15",
"D": "20"
} | mat_0462 | multiple_choice | cs | Vypočítej kombinační číslo C(5,2). | [
"5×4/2.",
"= 10.",
"Vyber možnost B."
] | math | Answer: 10
Steps:
1. 5×4/2.
2. = 10.
3. Vyber možnost B.
Answer: (B) | combinatorics |
Sum: 105. | null | mat_0463 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 14. | [
"Pair 1+14, 2+13, ...",
"Use 14×15/2.",
"= 105."
] | math | Answer: Sum: 105.
Steps:
1. Pair 1+14, 2+13, ...
2. Use 14×15/2.
3. = 105. | proofs |
f'(2) = 18. | null | mat_0464 | worked_example | en | For f(x) = 2x² + 10x, compute f'(2) step by step. | [
"f'(x) = 4x + 10.",
"Plug x = 2.",
"4×2 + 10 = 18."
] | math | Answer: f'(2) = 18.
Steps:
1. f'(x) = 4x + 10.
2. Plug x = 2.
3. 4×2 + 10 = 18. | calculus |
Skalární součin je 11. | {
"A": "7",
"B": "15",
"C": "11",
"D": "19"
} | mat_0465 | multiple_choice | cs | Najdi skalární součin (1,2) · (3,4). | [
"1×3 + 2×4.",
"Součin = 11.",
"Vyber možnost C."
] | math | Answer: 11
Steps:
1. 1×3 + 2×4.
2. Součin = 11.
3. Vyber možnost C.
Answer: (C) | linear_algebra |
P = 1/14. | null | mat_0466 | worked_example | en | A fair 14-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 14.",
"P = 1/14."
] | math | Answer: P = 1/14.
Steps:
1. Favourable: 1.
2. Possible: 14.
3. P = 1/14. | probability |
NSN = 56. | null | mat_0467 | worked_example | cs | Spočítej NSN(7, 8) krok za krokem. | [
"NSD(7,8) = 1.",
"7×8/1.",
"= 56."
] | math | Answer: NSN = 56.
Steps:
1. NSD(7,8) = 1.
2. 7×8/1.
3. = 56. | number_theory |
It approximates n! via roots and powers, taming huge factorials asymptotically. | null | mat_0468 | concept | en | What is Stirling's formula? | [
"It approximates n! via roots and powers, taming huge factorials asymptotically.",
"So the answer stands: It approximates n! via roots and powers, taming huge factorials asymptotically."
] | math | Answer: It approximates n! via roots and powers, taming huge factorials asymptotically.
Steps:
1. It approximates n! via roots and powers, taming huge factorials asymptotically.
2. So the answer stands: It approximates n! via roots and powers, taming huge factorials asymptotically. | combinatorics |
Half: 55, so 110 is even. | null | mat_0469 | worked_example | en | Show N² + N is even for N = 10 by computing its half. | [
"10² + 10 = 110.",
"110 / 2 = 55.",
"Integer, so even."
] | math | Answer: Half: 55, so 110 is even.
Steps:
1. 10² + 10 = 110.
2. 110 / 2 = 55.
3. Integer, so even. | proofs |
Integral: 1728. | null | mat_0470 | worked_example | en | Evaluate the integral of 3x² from 0 to 12. | [
"Antiderivative: x³.",
"Evaluate 0 to 12.",
"12³ − 0 = 1728."
] | math | Answer: Integral: 1728.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 12.
3. 12³ − 0 = 1728. | calculus |
Least squares fits lines by minimizing squared error, solving unsolvable systems approximately. | null | mat_0471 | concept | en | Why least squares? | [
"Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.",
"So the answer stands: Least squares fits lines by minimizing squared error, solving unsolvable systems approximately."
] | math | Answer: Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.
Steps:
1. Least squares fits lines by minimizing squared error, solving unsolvable systems approximately.
2. So the answer stands: Least squares fits lines by minimizing squared error, solving unsolvable systems appr... | linear_algebra |
The answer is p-value. | {
"A": "Coin age",
"B": "Shoe size",
"C": "Star sign",
"D": "p-value"
} | mat_0472 | multiple_choice | en | Choose: what scores extreme data? | [
"Small casts doubt.",
"So: p-value.",
"Select option D."
] | math | Answer: p-value
Steps:
1. Small casts doubt.
2. So: p-value.
3. Select option D.
Answer: (D) | probability |
Remainder: 1. | null | mat_0473 | worked_example | en | Compute (127) mod 7. | [
"Divide 127 by 7.",
"7×18 = 126.",
"127 − 126 = 1."
] | math | Answer: Remainder: 1.
Steps:
1. Divide 127 by 7.
2. 7×18 = 126.
3. 127 − 126 = 1. | number_theory |
C(14,3) = 364. | null | mat_0474 | worked_example | en | Show the computation of C(14,3). | [
"n(n−1)(n−2)/6.",
"14×13×12/6.",
"= 364."
] | math | Answer: C(14,3) = 364.
Steps:
1. n(n−1)(n−2)/6.
2. 14×13×12/6.
3. = 364. | combinatorics |
The answer is And/or quantifiers. | {
"A": "Numbers only",
"B": "Names only",
"C": "Nothing ever",
"D": "And/or quantifiers"
} | mat_0475 | multiple_choice | en | Choose: what flips under negation? | [
"De Morgan rules it.",
"So: And/or quantifiers.",
"Select option D."
] | math | Answer: And/or quantifiers
Steps:
1. De Morgan rules it.
2. So: And/or quantifiers.
3. Select option D.
Answer: (D) | proofs |
Integrál: 64. | null | mat_0476 | worked_example | cs | Vyčísli integrál 3x² od 0 do 4. | [
"Primitivní: x³.",
"Vyčísli 0 až 4.",
"4³ − 0 = 64."
] | math | Answer: Integrál: 64.
Steps:
1. Primitivní: x³.
2. Vyčísli 0 až 4.
3. 4³ − 0 = 64. | calculus |
Determinant: 130. | null | mat_0477 | worked_example | en | Compute det [[11,1],[2,12]] step by step. | [
"ad − bc.",
"11×12 − 1×2.",
"= 130."
] | math | Answer: Determinant: 130.
Steps:
1. ad − bc.
2. 11×12 − 1×2.
3. = 130. | linear_algebra |
E = 7. | null | mat_0478 | worked_example | en | Compute E of a uniform draw from {1..13}. | [
"Mean of 1..13.",
"(13+1)/2.",
"E = 7."
] | math | Answer: E = 7.
Steps:
1. Mean of 1..13.
2. (13+1)/2.
3. E = 7. | probability |
Zbytek je 8. | {
"A": "0",
"B": "1",
"C": "8",
"D": "2"
} | mat_0479 | multiple_choice | cs | Vypočítej (107) mod 9. | [
"107 ÷ 9.",
"Zbytek 8.",
"Vyber možnost C."
] | math | Answer: 8
Steps:
1. 107 ÷ 9.
2. Zbytek 8.
3. Vyber možnost C.
Answer: (C) | number_theory |
P(5,3) = 60. | null | mat_0480 | worked_example | cs | Ukaž výpočet P(5,3). | [
"n(n−1)(n−2).",
"5×4×3.",
"= 60."
] | math | Answer: P(5,3) = 60.
Steps:
1. n(n−1)(n−2).
2. 5×4×3.
3. = 60. | combinatorics |
Polovina: 10, číslo 20 je sudé. | null | mat_0481 | worked_example | cs | Ukaž, že N² + N je sudé pro N = 4, výpočtem poloviny. | [
"4² + 4 = 20.",
"20 / 2 = 10.",
"Celé číslo, tedy sudé."
] | math | Answer: Polovina: 10, číslo 20 je sudé.
Steps:
1. 4² + 4 = 20.
2. 20 / 2 = 10.
3. Celé číslo, tedy sudé. | proofs |
Limita: 8. | null | mat_0482 | worked_example | cs | Vyčísli limitu (x² − 4²)/(x − 4) pro x → 4. | [
"Rozlož: (x−4)(x+4).",
"Pokrať (x−4).",
"Dosaď 4: 4+4 = 8."
] | math | Answer: Limita: 8.
Steps:
1. Rozlož: (x−4)(x+4).
2. Pokrať (x−4).
3. Dosaď 4: 4+4 = 8. | calculus |
The answer is Eigenvector. | {
"A": "Random vector",
"B": "Zero vector",
"C": "Eigenvector",
"D": "Long vector"
} | mat_0483 | multiple_choice | en | Choose: what keeps direction? | [
"Scaled by eigenvalue.",
"So: Eigenvector.",
"Select option C."
] | math | Answer: Eigenvector
Steps:
1. Scaled by eigenvalue.
2. So: Eigenvector.
3. Select option C.
Answer: (C) | linear_algebra |
Covariance measures joint variation of two variables, positive when they rise together. | null | mat_0484 | concept | en | What is covariance? | [
"Covariance measures joint variation of two variables, positive when they rise together.",
"So the answer stands: Covariance measures joint variation of two variables, positive when they rise together."
] | math | Answer: Covariance measures joint variation of two variables, positive when they rise together.
Steps:
1. Covariance measures joint variation of two variables, positive when they rise together.
2. So the answer stands: Covariance measures joint variation of two variables, positive when they rise together. | probability |
They tell which numbers are squares mod p, decided fast by Euler's criterion. | null | mat_0485 | concept | en | Why quadratic residues? | [
"They tell which numbers are squares mod p, decided fast by Euler's criterion.",
"So the answer stands: They tell which numbers are squares mod p, decided fast by Euler's criterion."
] | math | Answer: They tell which numbers are squares mod p, decided fast by Euler's criterion.
Steps:
1. They tell which numbers are squares mod p, decided fast by Euler's criterion.
2. So the answer stands: They tell which numbers are squares mod p, decided fast by Euler's criterion. | number_theory |
P(4,2) je 12. | {
"A": "8",
"B": "16",
"C": "12",
"D": "20"
} | mat_0486 | multiple_choice | cs | Vypočítej permutační číslo P(4,2). | [
"4×3.",
"= 12.",
"Vyber možnost C."
] | math | Answer: 12
Steps:
1. 4×3.
2. = 12.
3. Vyber možnost C.
Answer: (C) | combinatorics |
Řádků: 32. | null | mat_0487 | worked_example | cs | Důkaz prochází všechny případy 5 booleovských proměnných. Kolik řádků? | [
"2 volby každá.",
"2^5 = 32."
] | math | Answer: Řádků: 32.
Steps:
1. 2 volby každá.
2. 2^5 = 32. | proofs |
Inflection points mark where bending flips, revealing structural changes in the graph. | null | mat_0488 | concept | en | Why find inflection points? | [
"Inflection points mark where bending flips, revealing structural changes in the graph.",
"So the answer stands: Inflection points mark where bending flips, revealing structural changes in the graph."
] | math | Answer: Inflection points mark where bending flips, revealing structural changes in the graph.
Steps:
1. Inflection points mark where bending flips, revealing structural changes in the graph.
2. So the answer stands: Inflection points mark where bending flips, revealing structural changes in the graph. | calculus |
Trace: 25. | null | mat_0489 | worked_example | en | Compute the trace of [[12,0],[0,13]]. | [
"Sum diagonal.",
"12 + 13.",
"= 25."
] | math | Answer: Trace: 25.
Steps:
1. Sum diagonal.
2. 12 + 13.
3. = 25. | linear_algebra |
Correlation scales covariance to [-1,1], giving a universal strength of linear relation. | null | mat_0490 | concept | en | Why correlation? | [
"Correlation scales covariance to [-1,1], giving a universal strength of linear relation.",
"So the answer stands: Correlation scales covariance to [-1,1], giving a universal strength of linear relation."
] | math | Answer: Correlation scales covariance to [-1,1], giving a universal strength of linear relation.
Steps:
1. Correlation scales covariance to [-1,1], giving a universal strength of linear relation.
2. So the answer stands: Correlation scales covariance to [-1,1], giving a universal strength of linear relation. | probability |
The gcd is 18. | {
"A": "18",
"B": "15",
"C": "21",
"D": "24"
} | mat_0491 | multiple_choice | en | Find gcd(36, 54). | [
"Factor 3 out twice.",
"gcd = 18.",
"Select option A."
] | math | Answer: 18
Steps:
1. Factor 3 out twice.
2. gcd = 18.
3. Select option A.
Answer: (A) | number_theory |
C(14,2) = 91. | null | mat_0492 | worked_example | en | Show the computation of C(14,2). | [
"Formula n(n−1)/2.",
"14×13/2.",
"= 91."
] | math | Answer: C(14,2) = 91.
Steps:
1. Formula n(n−1)/2.
2. 14×13/2.
3. = 91. | combinatorics |
Rows: 2048. | null | mat_0493 | worked_example | en | A proof checks all cases of 11 boolean variables. How many rows? | [
"2 choices each.",
"2^11 = 2048."
] | math | Answer: Rows: 2048.
Steps:
1. 2 choices each.
2. 2^11 = 2048. | proofs |
Limit: 24. | null | mat_0494 | worked_example | en | Evaluate the limit of (x² − 12²)/(x − 12) as x → 12. | [
"Factor: (x−12)(x+12).",
"Cancel (x−12).",
"Plug 12: 12+12 = 24."
] | math | Answer: Limit: 24.
Steps:
1. Factor: (x−12)(x+12).
2. Cancel (x−12).
3. Plug 12: 12+12 = 24. | calculus |
The null space holds all vectors mapped to zero, measuring how much the map collapses. | null | mat_0495 | concept | en | What is the null space? | [
"The null space holds all vectors mapped to zero, measuring how much the map collapses.",
"So the answer stands: The null space holds all vectors mapped to zero, measuring how much the map collapses."
] | math | Answer: The null space holds all vectors mapped to zero, measuring how much the map collapses.
Steps:
1. The null space holds all vectors mapped to zero, measuring how much the map collapses.
2. So the answer stands: The null space holds all vectors mapped to zero, measuring how much the map collapses. | linear_algebra |
Odpověď je prostor výsledků. | {
"A": "seznam přání",
"B": "pytel hádání",
"C": "truhla nadějí",
"D": "prostor výsledků"
} | mat_0496 | multiple_choice | cs | Vyber: co vypisuje všechny výsledky? | [
"Jevy jsou podmnožiny.",
"Tedy: prostor výsledků.",
"Vyber možnost D."
] | math | Answer: prostor výsledků
Steps:
1. Jevy jsou podmnožiny.
2. Tedy: prostor výsledků.
3. Vyber možnost D.
Answer: (D) | probability |
lcm = 24. | null | mat_0497 | worked_example | en | Compute lcm(12, 8) step by step. | [
"gcd(12,8) = 4.",
"12×8/4.",
"= 24."
] | math | Answer: lcm = 24.
Steps:
1. gcd(12,8) = 4.
2. 12×8/4.
3. = 24. | number_theory |
The answer is Pigeonhole. | {
"A": "Big boxes",
"B": "Few items",
"C": "Pigeonhole",
"D": "Kind hosts"
} | mat_0498 | multiple_choice | en | Choose: what forces sharing? | [
"Items exceed boxes.",
"So: Pigeonhole.",
"Select option C."
] | math | Answer: Pigeonhole
Steps:
1. Items exceed boxes.
2. So: Pigeonhole.
3. Select option C.
Answer: (C) | combinatorics |
Sum: 120. | null | mat_0499 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 15. | [
"Pair 1+15, 2+14, ...",
"Use 15×16/2.",
"= 120."
] | math | Answer: Sum: 120.
Steps:
1. Pair 1+15, 2+14, ...
2. Use 15×16/2.
3. = 120. | proofs |
The limit is 10. | {
"A": "8",
"B": "10",
"C": "12",
"D": "14"
} | mat_0500 | multiple_choice | en | Find the limit of (x² − 5²)/(x − 5) as x → 5. | [
"Factor (x−5)(x+5).",
"Cancel, get 2×5 = 10.",
"Select option B."
] | math | Answer: 10
Steps:
1. Factor (x−5)(x+5).
2. Cancel, get 2×5 = 10.
3. Select option B.
Answer: (B) | calculus |
The answer is Trace. | {
"A": "Rank",
"B": "Norm",
"C": "Size",
"D": "Trace"
} | mat_0501 | multiple_choice | en | Choose: what sums the diagonal? | [
"Eigenvalue sum equal.",
"So: Trace.",
"Select option D."
] | math | Answer: Trace
Steps:
1. Eigenvalue sum equal.
2. So: Trace.
3. Select option D.
Answer: (D) | linear_algebra |
It jumps between states with fixed probabilities depending only on the current state. | null | mat_0502 | concept | en | What is a Markov chain? | [
"It jumps between states with fixed probabilities depending only on the current state.",
"So the answer stands: It jumps between states with fixed probabilities depending only on the current state."
] | math | Answer: It jumps between states with fixed probabilities depending only on the current state.
Steps:
1. It jumps between states with fixed probabilities depending only on the current state.
2. So the answer stands: It jumps between states with fixed probabilities depending only on the current state. | probability |
It encodes residue status as 1, -1, or 0, compressing square questions to algebra. | null | mat_0503 | concept | en | What is Legendre's symbol? | [
"It encodes residue status as 1, -1, or 0, compressing square questions to algebra.",
"So the answer stands: It encodes residue status as 1, -1, or 0, compressing square questions to algebra."
] | math | Answer: It encodes residue status as 1, -1, or 0, compressing square questions to algebra.
Steps:
1. It encodes residue status as 1, -1, or 0, compressing square questions to algebra.
2. So the answer stands: It encodes residue status as 1, -1, or 0, compressing square questions to algebra. | number_theory |
The answer is Gray code. | {
"A": "Wild order",
"B": "Blind list",
"C": "Vague chain",
"D": "Gray code"
} | mat_0504 | multiple_choice | en | Choose: what orders flipping one bit? | [
"Neighbours differ once.",
"So: Gray code.",
"Select option D."
] | math | Answer: Gray code
Steps:
1. Neighbours differ once.
2. So: Gray code.
3. Select option D.
Answer: (D) | combinatorics |
The answer is Assuming conclusion. | {
"A": "Assuming conclusion",
"B": "Round diagrams",
"C": "Long proofs",
"D": "Many cases"
} | mat_0505 | multiple_choice | en | Choose: what is circular reasoning? | [
"Proves nothing at all.",
"So: Assuming conclusion.",
"Select option A."
] | math | Answer: Assuming conclusion
Steps:
1. Proves nothing at all.
2. So: Assuming conclusion.
3. Select option A.
Answer: (A) | proofs |
f'(4) = 35. | null | mat_0506 | worked_example | cs | Pro f(x) = 4x² + 3x spočítej f'(4) krok za krokem. | [
"f'(x) = 8x + 3.",
"Dosaď x = 4.",
"8×4 + 3 = 35."
] | math | Answer: f'(4) = 35.
Steps:
1. f'(x) = 8x + 3.
2. Dosaď x = 4.
3. 8×4 + 3 = 35. | calculus |
Odpověď je nenulový determinant. | {
"A": "velké prvky",
"B": "kulatý tvar",
"C": "dlouhé jméno",
"D": "nenulový determinant"
} | mat_0507 | multiple_choice | cs | Vyber: co detekuje regularitu? | [
"Nula znamená zhroucení.",
"Tedy: nenulový determinant.",
"Vyber možnost D."
] | math | Answer: nenulový determinant
Steps:
1. Nula znamená zhroucení.
2. Tedy: nenulový determinant.
3. Vyber možnost D.
Answer: (D) | linear_algebra |
They give long-run state fractions, predicting where chains settle over time. | null | mat_0508 | concept | en | Why stationary distributions? | [
"They give long-run state fractions, predicting where chains settle over time.",
"So the answer stands: They give long-run state fractions, predicting where chains settle over time."
] | math | Answer: They give long-run state fractions, predicting where chains settle over time.
Steps:
1. They give long-run state fractions, predicting where chains settle over time.
2. So the answer stands: They give long-run state fractions, predicting where chains settle over time. | probability |
Největší společný dělitel je největší číslo dělící obě čísla. | null | mat_0509 | concept | cs | Co je NSD? | [
"Největší společný dělitel je největší číslo dělící obě čísla.",
"Odpověď tedy platí: Největší společný dělitel je největší číslo dělící obě čísla."
] | math | Answer: Největší společný dělitel je největší číslo dělící obě čísla.
Steps:
1. Největší společný dělitel je největší číslo dělící obě čísla.
2. Odpověď tedy platí: Největší společný dělitel je největší číslo dělící obě čísla. | number_theory |
Counting one set two ways proves identities, a favourite elegant technique. | null | mat_0510 | concept | en | Why double counting? | [
"Counting one set two ways proves identities, a favourite elegant technique.",
"So the answer stands: Counting one set two ways proves identities, a favourite elegant technique."
] | math | Answer: Counting one set two ways proves identities, a favourite elegant technique.
Steps:
1. Counting one set two ways proves identities, a favourite elegant technique.
2. So the answer stands: Counting one set two ways proves identities, a favourite elegant technique. | combinatorics |
Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements. | null | mat_0511 | concept | en | Why use Venn diagrams? | [
"Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.",
"So the answer stands: Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements."
] | math | Answer: Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.
Steps:
1. Venn diagrams picture set relations, guiding proofs about unions, intersections, and complements.
2. So the answer stands: Venn diagrams picture set relations, guiding proofs about unions, intersections, ... | proofs |
The answer is Fundamental theorem. | {
"A": "Guessing rule",
"B": "Lucky axiom",
"C": "Fundamental theorem",
"D": "Long division"
} | mat_0512 | multiple_choice | en | Choose: what links both operations? | [
"Integrals via antiderivatives.",
"So: Fundamental theorem.",
"Select option C."
] | math | Answer: Fundamental theorem
Steps:
1. Integrals via antiderivatives.
2. So: Fundamental theorem.
3. Select option C.
Answer: (C) | calculus |
Rank plus nullity equals columns, linking image size to collapse in one equation. | null | mat_0513 | concept | en | Why rank-nullity? | [
"Rank plus nullity equals columns, linking image size to collapse in one equation.",
"So the answer stands: Rank plus nullity equals columns, linking image size to collapse in one equation."
] | math | Answer: Rank plus nullity equals columns, linking image size to collapse in one equation.
Steps:
1. Rank plus nullity equals columns, linking image size to collapse in one equation.
2. So the answer stands: Rank plus nullity equals columns, linking image size to collapse in one equation. | linear_algebra |
Odpověď je rozptyl. | {
"A": "rozptyl",
"B": "průměr",
"C": "modus",
"D": "naděje"
} | mat_0514 | multiple_choice | cs | Vyber: co měří rozptyl? | [
"Čtvercové odchylky průměrovány.",
"Tedy: rozptyl.",
"Vyber možnost A."
] | math | Answer: rozptyl
Steps:
1. Čtvercové odchylky průměrovány.
2. Tedy: rozptyl.
3. Vyber možnost A.
Answer: (A) | probability |
It swaps primes in Legendre symbols, making huge residue questions computable. | null | mat_0515 | concept | en | Why quadratic reciprocity? | [
"It swaps primes in Legendre symbols, making huge residue questions computable.",
"So the answer stands: It swaps primes in Legendre symbols, making huge residue questions computable."
] | math | Answer: It swaps primes in Legendre symbols, making huge residue questions computable.
Steps:
1. It swaps primes in Legendre symbols, making huge residue questions computable.
2. So the answer stands: It swaps primes in Legendre symbols, making huge residue questions computable. | number_theory |
A bijection pairs sets one-to-one, proving equal size without any counting. | null | mat_0516 | concept | en | What is a bijection? | [
"A bijection pairs sets one-to-one, proving equal size without any counting.",
"So the answer stands: A bijection pairs sets one-to-one, proving equal size without any counting."
] | math | Answer: A bijection pairs sets one-to-one, proving equal size without any counting.
Steps:
1. A bijection pairs sets one-to-one, proving equal size without any counting.
2. So the answer stands: A bijection pairs sets one-to-one, proving equal size without any counting. | combinatorics |
Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky. | null | mat_0517 | concept | cs | Co je lemma? | [
"Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.",
"Odpověď tedy platí: Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky."
] | math | Answer: Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.
Steps:
1. Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky.
2. Odpověď tedy platí: Lemma je malé pomocné tvrzení, jež dělí důkaz na kontrolovatelné kroky. | proofs |
f'(3) = 29. | null | mat_0518 | worked_example | en | For f(x) = 3x² + 11x, compute f'(3) step by step. | [
"f'(x) = 6x + 11.",
"Plug x = 3.",
"6×3 + 11 = 29."
] | math | Answer: f'(3) = 29.
Steps:
1. f'(x) = 6x + 11.
2. Plug x = 3.
3. 6×3 + 11 = 29. | calculus |
A linear system is equations linear in unknowns, solved together by elimination. | null | mat_0519 | concept | en | What is a linear system? | [
"A linear system is equations linear in unknowns, solved together by elimination.",
"So the answer stands: A linear system is equations linear in unknowns, solved together by elimination."
] | math | Answer: A linear system is equations linear in unknowns, solved together by elimination.
Steps:
1. A linear system is equations linear in unknowns, solved together by elimination.
2. So the answer stands: A linear system is equations linear in unknowns, solved together by elimination. | linear_algebra |
Pravděpodobnost je 1/7. | {
"A": "1/6",
"B": "1/7",
"C": "1/8",
"D": "2/7"
} | mat_0520 | multiple_choice | cs | Hází se férovou 7-stěnnou kostkou. Jaká je pravděpodobnost jednoho konkrétního čísla? | [
"1 stěna z 7.",
"P = 1/7.",
"Vyber možnost B."
] | math | Answer: 1/7
Steps:
1. 1 stěna z 7.
2. P = 1/7.
3. Vyber možnost B.
Answer: (B) | probability |
gcd = 20. | null | mat_0521 | worked_example | en | Compute gcd(40, 60) step by step. | [
"Factor: 40=4×10, 60=6×10.",
"Common: 10×gcd(4,6).",
"= 10×2 = 20."
] | math | Answer: gcd = 20.
Steps:
1. Factor: 40=4×10, 60=6×10.
2. Common: 10×gcd(4,6).
3. = 10×2 = 20. | number_theory |
C(8,2) is 28. | {
"A": "28",
"B": "20",
"C": "36",
"D": "44"
} | mat_0522 | multiple_choice | en | Compute C(8,2). | [
"8×7/2.",
"= 28.",
"Select option A."
] | math | Answer: 28
Steps:
1. 8×7/2.
2. = 28.
3. Select option A.
Answer: (A) | combinatorics |
Součet: 45. | null | mat_0523 | worked_example | cs | Dokaž vzorcem: najdi součet 1 + 2 + ... + 9. | [
"Páruj 1+9, 2+8, ...",
"Použij 9×10/2.",
"= 45."
] | math | Answer: Součet: 45.
Steps:
1. Páruj 1+9, 2+8, ...
2. Použij 9×10/2.
3. = 45. | proofs |
A partial derivative varies one variable while freezing others, extending calculus to surfaces. | null | mat_0524 | concept | en | What is a partial derivative? | [
"A partial derivative varies one variable while freezing others, extending calculus to surfaces.",
"So the answer stands: A partial derivative varies one variable while freezing others, extending calculus to surfaces."
] | math | Answer: A partial derivative varies one variable while freezing others, extending calculus to surfaces.
Steps:
1. A partial derivative varies one variable while freezing others, extending calculus to surfaces.
2. So the answer stands: A partial derivative varies one variable while freezing others, extending calculus to... | calculus |
Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu. | null | mat_0525 | concept | cs | Co je inverzní matice? | [
"Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.",
"Odpověď tedy platí: Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu."
] | math | Answer: Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.
Steps:
1. Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu.
2. Odpověď tedy platí: Inverzní matice ruší původní zobrazení a existuje při nenulovém determinantu. | linear_algebra |
It counts rare events in fixed intervals, modelling arrivals and decays. | null | mat_0526 | concept | en | What is the Poisson distribution? | [
"It counts rare events in fixed intervals, modelling arrivals and decays.",
"So the answer stands: It counts rare events in fixed intervals, modelling arrivals and decays."
] | math | Answer: It counts rare events in fixed intervals, modelling arrivals and decays.
Steps:
1. It counts rare events in fixed intervals, modelling arrivals and decays.
2. So the answer stands: It counts rare events in fixed intervals, modelling arrivals and decays. | probability |
NSD = 10. | null | mat_0527 | worked_example | cs | Spočítej NSD(20, 30) krok za krokem. | [
"Rozlož: 20=4×5, 30=6×5.",
"Společné: 5×NSD(4,6).",
"= 5×2 = 10."
] | math | Answer: NSD = 10.
Steps:
1. Rozlož: 20=4×5, 30=6×5.
2. Společné: 5×NSD(4,6).
3. = 5×2 = 10. | number_theory |
P(13,3) = 1716. | null | mat_0528 | worked_example | en | Show the computation of P(13,3). | [
"n(n−1)(n−2).",
"13×12×11.",
"= 1716."
] | math | Answer: P(13,3) = 1716.
Steps:
1. n(n−1)(n−2).
2. 13×12×11.
3. = 1716. | combinatorics |
The table has 128 rows. | {
"A": "126",
"B": "128",
"C": "130",
"D": "132"
} | mat_0529 | multiple_choice | en | How many rows does a truth table with 7 variables have? | [
"2^7 rows.",
"Total: 128.",
"Select option B."
] | math | Answer: 128
Steps:
1. 2^7 rows.
2. Total: 128.
3. Select option B.
Answer: (B) | proofs |
Základní věta spojuje derivace a integrály přes primitivní funkce. | null | mat_0530 | concept | cs | Co je základní věta kalkulu? | [
"Základní věta spojuje derivace a integrály přes primitivní funkce.",
"Odpověď tedy platí: Základní věta spojuje derivace a integrály přes primitivní funkce."
] | math | Answer: Základní věta spojuje derivace a integrály přes primitivní funkce.
Steps:
1. Základní věta spojuje derivace a integrály přes primitivní funkce.
2. Odpověď tedy platí: Základní věta spojuje derivace a integrály přes primitivní funkce. | calculus |
The determinant is 34. | {
"A": "34",
"B": "31",
"C": "37",
"D": "40"
} | mat_0531 | multiple_choice | en | Find the determinant of [[5,2],[3,8]]. | [
"5×8 − 2×3.",
"Det = 34.",
"Select option A."
] | math | Answer: 34
Steps:
1. 5×8 − 2×3.
2. Det = 34.
3. Select option A.
Answer: (A) | linear_algebra |
Střední hodnota je 3. | {
"A": "4",
"B": "2",
"C": "3",
"D": "5"
} | mat_0532 | multiple_choice | cs | Najdi střední hodnotu rovnoměrného tahu z {1..5}. | [
"(5+1)/2.",
"E = 3.",
"Vyber možnost C."
] | math | Answer: 3
Steps:
1. (5+1)/2.
2. E = 3.
3. Vyber možnost C.
Answer: (C) | probability |
NSN je 30. | {
"A": "28",
"B": "32",
"C": "34",
"D": "30"
} | mat_0533 | multiple_choice | cs | Najdi nejmenší společný násobek NSN(5, 6). | [
"5×6/NSD.",
"NSN = 30.",
"Vyber možnost D."
] | math | Answer: 30
Steps:
1. 5×6/NSD.
2. NSN = 30.
3. Vyber možnost D.
Answer: (D) | number_theory |
P(7,2) is 42. | {
"A": "35",
"B": "42",
"C": "49",
"D": "56"
} | mat_0534 | multiple_choice | en | Compute P(7,2). | [
"7×6.",
"= 42.",
"Select option B."
] | math | Answer: 42
Steps:
1. 7×6.
2. = 42.
3. Select option B.
Answer: (B) | combinatorics |
Half: 66, so 132 is even. | null | mat_0535 | worked_example | en | Show N² + N is even for N = 11 by computing its half. | [
"11² + 11 = 132.",
"132 / 2 = 66.",
"Integer, so even."
] | math | Answer: Half: 66, so 132 is even.
Steps:
1. 11² + 11 = 132.
2. 132 / 2 = 66.
3. Integer, so even. | proofs |
Taylorovy řady aproximují funkce polynomy s odhadem chyby. | null | mat_0536 | concept | cs | Proč Taylorovy řady? | [
"Taylorovy řady aproximují funkce polynomy s odhadem chyby.",
"Odpověď tedy platí: Taylorovy řady aproximují funkce polynomy s odhadem chyby."
] | math | Answer: Taylorovy řady aproximují funkce polynomy s odhadem chyby.
Steps:
1. Taylorovy řady aproximují funkce polynomy s odhadem chyby.
2. Odpověď tedy platí: Taylorovy řady aproximují funkce polynomy s odhadem chyby. | calculus |
Dot product: 83. | null | mat_0537 | worked_example | en | Compute (11,10) · (3,5) step by step. | [
"Multiply components.",
"11×3 + 10×5.",
"= 83."
] | math | Answer: Dot product: 83.
Steps:
1. Multiply components.
2. 11×3 + 10×5.
3. = 83. | linear_algebra |
It models waiting times with no memory, fitting lifetimes and queues. | null | mat_0538 | concept | en | Why exponential distribution? | [
"It models waiting times with no memory, fitting lifetimes and queues.",
"So the answer stands: It models waiting times with no memory, fitting lifetimes and queues."
] | math | Answer: It models waiting times with no memory, fitting lifetimes and queues.
Steps:
1. It models waiting times with no memory, fitting lifetimes and queues.
2. So the answer stands: It models waiting times with no memory, fitting lifetimes and queues. | probability |
Remainder: 5. | null | mat_0539 | worked_example | en | Compute (138) mod 7. | [
"Divide 138 by 7.",
"7×19 = 133.",
"138 − 133 = 5."
] | math | Answer: Remainder: 5.
Steps:
1. Divide 138 by 7.
2. 7×19 = 133.
3. 138 − 133 = 5. | number_theory |
C(15,3) = 455. | null | mat_0540 | worked_example | en | Show the computation of C(15,3). | [
"n(n−1)(n−2)/6.",
"15×14×13/6.",
"= 455."
] | math | Answer: C(15,3) = 455.
Steps:
1. n(n−1)(n−2)/6.
2. 15×14×13/6.
3. = 455. | combinatorics |
Rows: 4096. | null | mat_0541 | worked_example | en | A proof checks all cases of 12 boolean variables. How many rows? | [
"2 choices each.",
"2^12 = 4096."
] | math | Answer: Rows: 4096.
Steps:
1. 2 choices each.
2. 2^12 = 4096. | proofs |
Integral: 2197. | null | mat_0542 | worked_example | en | Evaluate the integral of 3x² from 0 to 13. | [
"Antiderivative: x³.",
"Evaluate 0 to 13.",
"13³ − 0 = 2197."
] | math | Answer: Integral: 2197.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 13.
3. 13³ − 0 = 2197. | calculus |
Augmenting appends answers to coefficients, keeping elimination tidy in one array. | null | mat_0543 | concept | en | Why augmented matrices? | [
"Augmenting appends answers to coefficients, keeping elimination tidy in one array.",
"So the answer stands: Augmenting appends answers to coefficients, keeping elimination tidy in one array."
] | math | Answer: Augmenting appends answers to coefficients, keeping elimination tidy in one array.
Steps:
1. Augmenting appends answers to coefficients, keeping elimination tidy in one array.
2. So the answer stands: Augmenting appends answers to coefficients, keeping elimination tidy in one array. | linear_algebra |
P(red) = 11/15. | null | mat_0544 | worked_example | en | A bag holds 11 red and 4 blue marbles. Show P(red). | [
"Red: 11.",
"Total: 15.",
"P = 11/15."
] | math | Answer: P(red) = 11/15.
Steps:
1. Red: 11.
2. Total: 15.
3. P = 11/15. | probability |
The remainder is 2. | {
"A": "3",
"B": "2",
"C": "4",
"D": "5"
} | mat_0545 | multiple_choice | en | Find (128) mod 9. | [
"128 ÷ 9.",
"Remainder 2.",
"Select option B."
] | math | Answer: 2
Steps:
1. 128 ÷ 9.
2. Remainder 2.
3. Select option B.
Answer: (B) | number_theory |
C(9,3) is 84. | {
"A": "75",
"B": "93",
"C": "84",
"D": "102"
} | mat_0546 | multiple_choice | en | Compute C(9,3). | [
"9×8×7/6.",
"= 84.",
"Select option C."
] | math | Answer: 84
Steps:
1. 9×8×7/6.
2. = 84.
3. Select option C.
Answer: (C) | combinatorics |
De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B). | null | mat_0547 | concept | en | What is De Morgan's law? | [
"De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).",
"So the answer stands: De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B)."
] | math | Answer: De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).
Steps:
1. De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B).
2. So the answer stands: De Morgan's laws flip and/or under negation, turning not-(A and B) into (not A) or (not B). | proofs |
The answer is Partial derivative. | {
"A": "Total guess",
"B": "Full stop",
"C": "Side note",
"D": "Partial derivative"
} | mat_0548 | multiple_choice | en | Choose: what varies one variable? | [
"Others frozen.",
"So: Partial derivative.",
"Select option D."
] | math | Answer: Partial derivative
Steps:
1. Others frozen.
2. So: Partial derivative.
3. Select option D.
Answer: (D) | calculus |
The trace is 15. | {
"A": "13",
"B": "15",
"C": "17",
"D": "19"
} | mat_0549 | multiple_choice | en | Find the trace of [[6,1],[2,9]]. | [
"6 + 9.",
"Trace = 15.",
"Select option B."
] | math | Answer: 15
Steps:
1. 6 + 9.
2. Trace = 15.
3. Select option B.
Answer: (B) | linear_algebra |
Průměry mnoha veličin jsou normální, modeluje chyby a variace. | null | mat_0550 | concept | cs | Proč normální rozdělení? | [
"Průměry mnoha veličin jsou normální, modeluje chyby a variace.",
"Odpověď tedy platí: Průměry mnoha veličin jsou normální, modeluje chyby a variace."
] | math | Answer: Průměry mnoha veličin jsou normální, modeluje chyby a variace.
Steps:
1. Průměry mnoha veličin jsou normální, modeluje chyby a variace.
2. Odpověď tedy platí: Průměry mnoha veličin jsou normální, modeluje chyby a variace. | probability |
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