answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
Determinant: 88. | null | mat_0351 | worked_example | en | Compute det [[9,1],[2,10]] step by step. | [
"ad − bc.",
"9×10 − 1×2.",
"= 88."
] | math | Answer: Determinant: 88.
Steps:
1. ad − bc.
2. 9×10 − 1×2.
3. = 88. | linear_algebra |
A p-value is the chance of data this extreme under the null, small values casting doubt. | null | mat_0352 | concept | en | What is a p-value? | [
"A p-value is the chance of data this extreme under the null, small values casting doubt.",
"So the answer stands: A p-value is the chance of data this extreme under the null, small values casting doubt."
] | math | Answer: A p-value is the chance of data this extreme under the null, small values casting doubt.
Steps:
1. A p-value is the chance of data this extreme under the null, small values casting doubt.
2. So the answer stands: A p-value is the chance of data this extreme under the null, small values casting doubt. | probability |
Remainder: 3. | null | mat_0353 | worked_example | en | Compute (94) mod 7. | [
"Divide 94 by 7.",
"7×13 = 91.",
"94 − 91 = 3."
] | math | Answer: Remainder: 3.
Steps:
1. Divide 94 by 7.
2. 7×13 = 91.
3. 94 − 91 = 3. | number_theory |
They count trees, brackets, and paths, popping up across recursive structures. | null | mat_0354 | concept | en | Why Catalan numbers? | [
"They count trees, brackets, and paths, popping up across recursive structures.",
"So the answer stands: They count trees, brackets, and paths, popping up across recursive structures."
] | math | Answer: They count trees, brackets, and paths, popping up across recursive structures.
Steps:
1. They count trees, brackets, and paths, popping up across recursive structures.
2. So the answer stands: They count trees, brackets, and paths, popping up across recursive structures. | combinatorics |
Proof by cases splits possibilities into an exhaustive list and proves the claim under each one. | null | mat_0355 | concept | en | What is proof by cases? | [
"Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.",
"So the answer stands: Proof by cases splits possibilities into an exhaustive list and proves the claim under each one."
] | math | Answer: Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.
Steps:
1. Proof by cases splits possibilities into an exhaustive list and proves the claim under each one.
2. So the answer stands: Proof by cases splits possibilities into an exhaustive list and proves the claim un... | proofs |
Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds. | null | mat_0356 | concept | en | Why learn Taylor series? | [
"Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.",
"So the answer stands: Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds."
] | math | Answer: Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.
Steps:
1. Taylor series approximate functions by polynomials, giving accurate local estimates with error bounds.
2. So the answer stands: Taylor series approximate functions by polynomials, giving accurate loc... | calculus |
Determinant škáluje objemy a detekuje regularitu. | null | mat_0357 | concept | cs | Proč determinant? | [
"Determinant škáluje objemy a detekuje regularitu.",
"Odpověď tedy platí: Determinant škáluje objemy a detekuje regularitu."
] | math | Answer: Determinant škáluje objemy a detekuje regularitu.
Steps:
1. Determinant škáluje objemy a detekuje regularitu.
2. Odpověď tedy platí: Determinant škáluje objemy a detekuje regularitu. | linear_algebra |
P(red) = 8/12. | null | mat_0358 | worked_example | en | A bag holds 8 red and 4 blue marbles. Show P(red). | [
"Red: 8.",
"Total: 12.",
"P = 8/12."
] | math | Answer: P(red) = 8/12.
Steps:
1. Red: 8.
2. Total: 12.
3. P = 8/12. | probability |
It merges congruences into one solution modulo the product, splitting hard mods apart. | null | mat_0359 | concept | en | Why Chinese remainder? | [
"It merges congruences into one solution modulo the product, splitting hard mods apart.",
"So the answer stands: It merges congruences into one solution modulo the product, splitting hard mods apart."
] | math | Answer: It merges congruences into one solution modulo the product, splitting hard mods apart.
Steps:
1. It merges congruences into one solution modulo the product, splitting hard mods apart.
2. So the answer stands: It merges congruences into one solution modulo the product, splitting hard mods apart. | number_theory |
P(10,3) = 720. | null | mat_0360 | worked_example | en | Show the computation of P(10,3). | [
"n(n−1)(n−2).",
"10×9×8.",
"= 720."
] | math | Answer: P(10,3) = 720.
Steps:
1. n(n−1)(n−2).
2. 10×9×8.
3. = 720. | combinatorics |
The sum is 36. | {
"A": "36",
"B": "28",
"C": "44",
"D": "52"
} | mat_0361 | multiple_choice | en | What is the sum 1 + 2 + ... + 8? | [
"8×9/2.",
"Sum = 36.",
"Select option A."
] | math | Answer: 36
Steps:
1. 8×9/2.
2. Sum = 36.
3. Select option A.
Answer: (A) | proofs |
It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones. | null | mat_0362 | concept | en | What is L'Hôpital's rule? | [
"It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.",
"So the answer stands: It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones."
] | math | Answer: It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.
Steps:
1. It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy ones.
2. So the answer stands: It resolves 0/0 forms by differentiating top and bottom, turning stuck limits into easy o... | calculus |
Hodnost počítá nezávislé řádky a měří dimenzi obrazu. | null | mat_0363 | concept | cs | Co je hodnost? | [
"Hodnost počítá nezávislé řádky a měří dimenzi obrazu.",
"Odpověď tedy platí: Hodnost počítá nezávislé řádky a měří dimenzi obrazu."
] | math | Answer: Hodnost počítá nezávislé řádky a měří dimenzi obrazu.
Steps:
1. Hodnost počítá nezávislé řádky a měří dimenzi obrazu.
2. Odpověď tedy platí: Hodnost počítá nezávislé řádky a měří dimenzi obrazu. | linear_algebra |
The answer is Large numbers law. | {
"A": "Small samples",
"B": "Wild hopes",
"C": "Large numbers law",
"D": "Quick peeks"
} | mat_0364 | multiple_choice | en | Choose: what settles averages? | [
"Trials many, mean neared.",
"So: Large numbers law.",
"Select option C."
] | math | Answer: Large numbers law
Steps:
1. Trials many, mean neared.
2. So: Large numbers law.
3. Select option C.
Answer: (C) | probability |
Odpověď je čínská zbytková věta. | {
"A": "čínská zbytková věta",
"B": "slepé štěstí",
"C": "divoké hádání",
"D": "dlouhý spánek"
} | mat_0365 | multiple_choice | cs | Vyber: co slučuje kongruence? | [
"Jedno řešení modulo součin.",
"Tedy: čínská zbytková věta.",
"Vyber možnost A."
] | math | Answer: čínská zbytková věta
Steps:
1. Jedno řešení modulo součin.
2. Tedy: čínská zbytková věta.
3. Vyber možnost A.
Answer: (A) | number_theory |
C(12,3) = 220. | null | mat_0366 | worked_example | en | Show the computation of C(12,3). | [
"n(n−1)(n−2)/6.",
"12×11×10/6.",
"= 220."
] | math | Answer: C(12,3) = 220.
Steps:
1. n(n−1)(n−2)/6.
2. 12×11×10/6.
3. = 220. | combinatorics |
The value is 42. | {
"A": "40",
"B": "42",
"C": "44",
"D": "46"
} | mat_0367 | multiple_choice | en | Compute N² + N for N = 6. | [
"6² + 6.",
"Value = 42.",
"Select option B."
] | math | Answer: 42
Steps:
1. 6² + 6.
2. Value = 42.
3. Select option B.
Answer: (B) | proofs |
The derivative is 22. | {
"A": "20",
"B": "24",
"C": "22",
"D": "26"
} | mat_0368 | multiple_choice | en | For f(x) = 3x² + 4x, find f'(3). | [
"f'(x) = 6x + 4.",
"At 3: 22.",
"Select option C."
] | math | Answer: 22
Steps:
1. f'(x) = 6x + 4.
2. At 3: 22.
3. Select option C.
Answer: (C) | calculus |
Determinant: 10. | null | mat_0369 | worked_example | cs | Spočítej det [[3,1],[2,4]] krok za krokem. | [
"ad − bc.",
"3×4 − 1×2.",
"= 10."
] | math | Answer: Determinant: 10.
Steps:
1. ad − bc.
2. 3×4 − 1×2.
3. = 10. | linear_algebra |
Past independent outcomes never change future odds, so streaks carry no memory at all. | null | mat_0370 | concept | en | Why avoid gambler's fallacy? | [
"Past independent outcomes never change future odds, so streaks carry no memory at all.",
"So the answer stands: Past independent outcomes never change future odds, so streaks carry no memory at all."
] | math | Answer: Past independent outcomes never change future odds, so streaks carry no memory at all.
Steps:
1. Past independent outcomes never change future odds, so streaks carry no memory at all.
2. So the answer stands: Past independent outcomes never change future odds, so streaks carry no memory at all. | probability |
It seeks integer solutions only, turning algebra into number puzzles with gcd conditions. | null | mat_0371 | concept | en | What is a Diophantine equation? | [
"It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.",
"So the answer stands: It seeks integer solutions only, turning algebra into number puzzles with gcd conditions."
] | math | Answer: It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.
Steps:
1. It seeks integer solutions only, turning algebra into number puzzles with gcd conditions.
2. So the answer stands: It seeks integer solutions only, turning algebra into number puzzles with gcd conditions. | number_theory |
It counts graphs, paths, and colorings, blending combinatorics with networks. | null | mat_0372 | concept | en | What is graph counting? | [
"It counts graphs, paths, and colorings, blending combinatorics with networks.",
"So the answer stands: It counts graphs, paths, and colorings, blending combinatorics with networks."
] | math | Answer: It counts graphs, paths, and colorings, blending combinatorics with networks.
Steps:
1. It counts graphs, paths, and colorings, blending combinatorics with networks.
2. So the answer stands: It counts graphs, paths, and colorings, blending combinatorics with networks. | combinatorics |
Missing one case breaks the proof, so the split must cover every possibility with no gaps. | null | mat_0373 | concept | en | Why check all cases? | [
"Missing one case breaks the proof, so the split must cover every possibility with no gaps.",
"So the answer stands: Missing one case breaks the proof, so the split must cover every possibility with no gaps."
] | math | Answer: Missing one case breaks the proof, so the split must cover every possibility with no gaps.
Steps:
1. Missing one case breaks the proof, so the split must cover every possibility with no gaps.
2. So the answer stands: Missing one case breaks the proof, so the split must cover every possibility with no gaps. | proofs |
Integral: 1000. | null | mat_0374 | worked_example | en | Evaluate the integral of 3x² from 0 to 10. | [
"Antiderivative: x³.",
"Evaluate 0 to 10.",
"10³ − 0 = 1000."
] | math | Answer: Integral: 1000.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 10.
3. 10³ − 0 = 1000. | calculus |
Dot products measure angles and lengths, turning geometry into algebra on coordinates. | null | mat_0375 | concept | en | Why use dot products? | [
"Dot products measure angles and lengths, turning geometry into algebra on coordinates.",
"So the answer stands: Dot products measure angles and lengths, turning geometry into algebra on coordinates."
] | math | Answer: Dot products measure angles and lengths, turning geometry into algebra on coordinates.
Steps:
1. Dot products measure angles and lengths, turning geometry into algebra on coordinates.
2. So the answer stands: Dot products measure angles and lengths, turning geometry into algebra on coordinates. | linear_algebra |
The probability is 1/9. | {
"A": "1/8",
"B": "1/10",
"C": "2/9",
"D": "1/9"
} | mat_0376 | multiple_choice | en | A fair 9-sided die is rolled. What is the probability of one specific number? | [
"1 face of 9.",
"P = 1/9.",
"Select option D."
] | math | Answer: 1/9
Steps:
1. 1 face of 9.
2. P = 1/9.
3. Select option D.
Answer: (D) | probability |
They model cycles and schedules, solved by inverses when gcd permits. | null | mat_0377 | concept | en | Why solve linear congruences? | [
"They model cycles and schedules, solved by inverses when gcd permits.",
"So the answer stands: They model cycles and schedules, solved by inverses when gcd permits."
] | math | Answer: They model cycles and schedules, solved by inverses when gcd permits.
Steps:
1. They model cycles and schedules, solved by inverses when gcd permits.
2. So the answer stands: They model cycles and schedules, solved by inverses when gcd permits. | number_theory |
C(12,2) = 66. | null | mat_0378 | worked_example | en | Show the computation of C(12,2). | [
"Formula n(n−1)/2.",
"12×11/2.",
"= 66."
] | math | Answer: C(12,2) = 66.
Steps:
1. Formula n(n−1)/2.
2. 12×11/2.
3. = 66. | combinatorics |
Rows: 512. | null | mat_0379 | worked_example | en | A proof checks all cases of 9 boolean variables. How many rows? | [
"2 choices each.",
"2^9 = 512."
] | math | Answer: Rows: 512.
Steps:
1. 2 choices each.
2. 2^9 = 512. | proofs |
The integral is 64. | {
"A": "60",
"B": "68",
"C": "72",
"D": "64"
} | mat_0380 | multiple_choice | en | Compute the integral of 3x² from 0 to 4. | [
"Antiderivative x³.",
"4³ = 64.",
"Select option D."
] | math | Answer: 64
Steps:
1. Antiderivative x³.
2. 4³ = 64.
3. Select option D.
Answer: (D) | calculus |
Trace: 21. | null | mat_0381 | worked_example | en | Compute the trace of [[10,0],[0,11]]. | [
"Sum diagonal.",
"10 + 11.",
"= 21."
] | math | Answer: Trace: 21.
Steps:
1. Sum diagonal.
2. 10 + 11.
3. = 21. | linear_algebra |
The expected value is 4. | {
"A": "4",
"B": "5",
"C": "3",
"D": "6"
} | mat_0382 | multiple_choice | en | Find the expected value of a uniform draw from {1..7}. | [
"(7+1)/2.",
"E = 4.",
"Select option A."
] | math | Answer: 4
Steps:
1. (7+1)/2.
2. E = 4.
3. Select option A.
Answer: (A) | probability |
It generates all nonzero residues by powers, giving the multiplicative group one engine. | null | mat_0383 | concept | en | What is a primitive root? | [
"It generates all nonzero residues by powers, giving the multiplicative group one engine.",
"So the answer stands: It generates all nonzero residues by powers, giving the multiplicative group one engine."
] | math | Answer: It generates all nonzero residues by powers, giving the multiplicative group one engine.
Steps:
1. It generates all nonzero residues by powers, giving the multiplicative group one engine.
2. So the answer stands: It generates all nonzero residues by powers, giving the multiplicative group one engine. | number_theory |
It guarantees order inside chaos, finding monochromatic cliques in all big graphs. | null | mat_0384 | concept | en | Why Ramsey theory? | [
"It guarantees order inside chaos, finding monochromatic cliques in all big graphs.",
"So the answer stands: It guarantees order inside chaos, finding monochromatic cliques in all big graphs."
] | math | Answer: It guarantees order inside chaos, finding monochromatic cliques in all big graphs.
Steps:
1. It guarantees order inside chaos, finding monochromatic cliques in all big graphs.
2. So the answer stands: It guarantees order inside chaos, finding monochromatic cliques in all big graphs. | combinatorics |
Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém. | null | mat_0385 | concept | cs | Co je důkaz rozborem případů? | [
"Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.",
"Odpověď tedy platí: Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém."
] | math | Answer: Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.
Steps:
1. Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém.
2. Odpověď tedy platí: Rozbor rozdělí možnosti na úplný seznam a tvrzení dokáže v každém. | proofs |
Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them. | null | mat_0386 | concept | en | Why check endpoints? | [
"Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.",
"So the answer stands: Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them."
] | math | Answer: Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.
Steps:
1. Maxima on closed intervals can sit at endpoints, so interior critical points alone miss them.
2. So the answer stands: Maxima on closed intervals can sit at endpoints, so interior critical points alone miss t... | calculus |
Eliminace odhaluje pivoty a řešení soustav mechanicky. | null | mat_0387 | concept | cs | Proč Gaussova eliminace? | [
"Eliminace odhaluje pivoty a řešení soustav mechanicky.",
"Odpověď tedy platí: Eliminace odhaluje pivoty a řešení soustav mechanicky."
] | math | Answer: Eliminace odhaluje pivoty a řešení soustav mechanicky.
Steps:
1. Eliminace odhaluje pivoty a řešení soustav mechanicky.
2. Odpověď tedy platí: Eliminace odhaluje pivoty a řešení soustav mechanicky. | linear_algebra |
Switching doors wins twice as often, because the host's reveal concentrates probability. | null | mat_0388 | concept | en | What is Monty Hall? | [
"Switching doors wins twice as often, because the host's reveal concentrates probability.",
"So the answer stands: Switching doors wins twice as often, because the host's reveal concentrates probability."
] | math | Answer: Switching doors wins twice as often, because the host's reveal concentrates probability.
Steps:
1. Switching doors wins twice as often, because the host's reveal concentrates probability.
2. So the answer stands: Switching doors wins twice as often, because the host's reveal concentrates probability. | probability |
The lcm is 42. | {
"A": "40",
"B": "42",
"C": "44",
"D": "46"
} | mat_0389 | multiple_choice | en | Find lcm(7, 6). | [
"7×6/gcd.",
"lcm = 42.",
"Select option B."
] | math | Answer: 42
Steps:
1. 7×6/gcd.
2. lcm = 42.
3. Select option B.
Answer: (B) | number_theory |
A partition splits integers into summands, counted by deep generating functions. | null | mat_0390 | concept | en | What is a partition? | [
"A partition splits integers into summands, counted by deep generating functions.",
"So the answer stands: A partition splits integers into summands, counted by deep generating functions."
] | math | Answer: A partition splits integers into summands, counted by deep generating functions.
Steps:
1. A partition splits integers into summands, counted by deep generating functions.
2. So the answer stands: A partition splits integers into summands, counted by deep generating functions. | combinatorics |
The answer is Not-Q to not-P. | {
"A": "Q to P",
"B": "P to P",
"C": "Not-Q to not-P",
"D": "Not-P to Q"
} | mat_0391 | multiple_choice | en | Choose: what is the contrapositive? | [
"Equivalent to original.",
"So: Not-Q to not-P.",
"Select option C."
] | math | Answer: Not-Q to not-P
Steps:
1. Equivalent to original.
2. So: Not-Q to not-P.
3. Select option C.
Answer: (C) | proofs |
Limit: 20. | null | mat_0392 | worked_example | en | Evaluate the limit of (x² − 10²)/(x − 10) as x → 10. | [
"Factor: (x−10)(x+10).",
"Cancel (x−10).",
"Plug 10: 10+10 = 20."
] | math | Answer: Limit: 20.
Steps:
1. Factor: (x−10)(x+10).
2. Cancel (x−10).
3. Plug 10: 10+10 = 20. | calculus |
Dot product: 67. | null | mat_0393 | worked_example | en | Compute (9,8) · (3,5) step by step. | [
"Multiply components.",
"9×3 + 8×5.",
"= 67."
] | math | Answer: Dot product: 67.
Steps:
1. Multiply components.
2. 9×3 + 8×5.
3. = 67. | linear_algebra |
Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny. | null | mat_0394 | concept | cs | Co je rozptyl? | [
"Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.",
"Odpověď tedy platí: Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny."
] | math | Answer: Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.
Steps:
1. Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny.
2. Odpověď tedy platí: Rozptyl průměruje čtvercové odchylky a měří rozptýlenost veličiny. | probability |
lcm = 72. | null | mat_0395 | worked_example | en | Compute lcm(9, 8) step by step. | [
"gcd(9,8) = 1.",
"9×8/1.",
"= 72."
] | math | Answer: lcm = 72.
Steps:
1. gcd(9,8) = 1.
2. 9×8/1.
3. = 72. | number_theory |
The answer is Double counting. | {
"A": "Double talking",
"B": "Double hoping",
"C": "Double counting",
"D": "Double tasting"
} | mat_0396 | multiple_choice | en | Choose: what proves by two counts? | [
"One set, two views.",
"So: Double counting.",
"Select option C."
] | math | Answer: Double counting
Steps:
1. One set, two views.
2. So: Double counting.
3. Select option C.
Answer: (C) | combinatorics |
Sum: 91. | null | mat_0397 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 13. | [
"Pair 1+13, 2+12, ...",
"Use 13×14/2.",
"= 91."
] | math | Answer: Sum: 91.
Steps:
1. Pair 1+13, 2+12, ...
2. Use 13×14/2.
3. = 91. | proofs |
f'(4) = 41. | null | mat_0398 | worked_example | en | For f(x) = 4x² + 9x, compute f'(4) step by step. | [
"f'(x) = 8x + 9.",
"Plug x = 4.",
"8×4 + 9 = 41."
] | math | Answer: f'(4) = 41.
Steps:
1. f'(x) = 8x + 9.
2. Plug x = 4.
3. 8×4 + 9 = 41. | calculus |
Determinant: 108. | null | mat_0399 | worked_example | en | Compute det [[10,1],[2,11]] step by step. | [
"ad − bc.",
"10×11 − 1×2.",
"= 108."
] | math | Answer: Determinant: 108.
Steps:
1. ad − bc.
2. 10×11 − 1×2.
3. = 108. | linear_algebra |
P = 1/12. | null | mat_0400 | worked_example | en | A fair 12-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 12.",
"P = 1/12."
] | math | Answer: P = 1/12.
Steps:
1. Favourable: 1.
2. Possible: 12.
3. P = 1/12. | probability |
gcd = 14. | null | mat_0401 | worked_example | en | Compute gcd(28, 42) step by step. | [
"Factor: 28=4×7, 42=6×7.",
"Common: 7×gcd(4,6).",
"= 7×2 = 14."
] | math | Answer: gcd = 14.
Steps:
1. Factor: 28=4×7, 42=6×7.
2. Common: 7×gcd(4,6).
3. = 7×2 = 14. | number_theory |
C(7,2) is 21. | {
"A": "14",
"B": "28",
"C": "35",
"D": "21"
} | mat_0402 | multiple_choice | en | Compute C(7,2). | [
"7×6/2.",
"= 21.",
"Select option D."
] | math | Answer: 21
Steps:
1. 7×6/2.
2. = 21.
3. Select option D.
Answer: (D) | combinatorics |
The answer is QED mark. | {
"A": "New question",
"B": "Blank page",
"C": "Loud sigh",
"D": "QED mark"
} | mat_0403 | multiple_choice | en | Choose: what ends a proof? | [
"Claim now secured.",
"So: QED mark.",
"Select option D."
] | math | Answer: QED mark
Steps:
1. Claim now secured.
2. So: QED mark.
3. Select option D.
Answer: (D) | proofs |
The limit is 8. | {
"A": "8",
"B": "6",
"C": "10",
"D": "12"
} | mat_0404 | multiple_choice | en | Find the limit of (x² − 4²)/(x − 4) as x → 4. | [
"Factor (x−4)(x+4).",
"Cancel, get 2×4 = 8.",
"Select option A."
] | math | Answer: 8
Steps:
1. Factor (x−4)(x+4).
2. Cancel, get 2×4 = 8.
3. Select option A.
Answer: (A) | calculus |
The determinant is 22. | {
"A": "19",
"B": "25",
"C": "28",
"D": "22"
} | mat_0405 | multiple_choice | en | Find the determinant of [[4,2],[3,7]]. | [
"4×7 − 2×3.",
"Det = 22.",
"Select option D."
] | math | Answer: 22
Steps:
1. 4×7 − 2×3.
2. Det = 22.
3. Select option D.
Answer: (D) | linear_algebra |
P(červená) je 2/7. | {
"A": "5/7",
"B": "2/8",
"C": "2/7",
"D": "3/7"
} | mat_0406 | multiple_choice | cs | Sáček má 2 červených a 5 modrých kuliček. Jaká je P(červená)? | [
"2 z 7.",
"P = 2/7.",
"Vyber možnost C."
] | math | Answer: 2/7
Steps:
1. 2 z 7.
2. P = 2/7.
3. Vyber možnost C.
Answer: (C) | probability |
Remainder: 0. | null | mat_0407 | worked_example | en | Compute (105) mod 7. | [
"Divide 105 by 7.",
"7×15 = 105.",
"105 − 105 = 0."
] | math | Answer: Remainder: 0.
Steps:
1. Divide 105 by 7.
2. 7×15 = 105.
3. 105 − 105 = 0. | number_theory |
P(11,3) = 990. | null | mat_0408 | worked_example | en | Show the computation of P(11,3). | [
"n(n−1)(n−2).",
"11×10×9.",
"= 990."
] | math | Answer: P(11,3) = 990.
Steps:
1. n(n−1)(n−2).
2. 11×10×9.
3. = 990. | combinatorics |
Řádků: 16. | null | mat_0409 | worked_example | cs | Důkaz prochází všechny případy 4 booleovských proměnných. Kolik řádků? | [
"2 volby každá.",
"2^4 = 16."
] | math | Answer: Řádků: 16.
Steps:
1. 2 volby každá.
2. 2^4 = 16. | proofs |
Odpověď je Taylorova řada. | {
"A": "Taylorova řada",
"B": "náhodné tečky",
"C": "divoké hádání",
"D": "prázdné řádky"
} | mat_0410 | multiple_choice | cs | Vyber: co aproximuje funkce? | [
"Polynomy s odhadem chyby.",
"Tedy: Taylorova řada.",
"Vyber možnost A."
] | math | Answer: Taylorova řada
Steps:
1. Polynomy s odhadem chyby.
2. Tedy: Taylorova řada.
3. Vyber možnost A.
Answer: (A) | calculus |
Trace: 23. | null | mat_0411 | worked_example | en | Compute the trace of [[11,0],[0,12]]. | [
"Sum diagonal.",
"11 + 12.",
"= 23."
] | math | Answer: Trace: 23.
Steps:
1. Sum diagonal.
2. 11 + 12.
3. = 23. | linear_algebra |
E = 6. | null | mat_0412 | worked_example | en | Compute E of a uniform draw from {1..11}. | [
"Mean of 1..11.",
"(11+1)/2.",
"E = 6."
] | math | Answer: E = 6.
Steps:
1. Mean of 1..11.
2. (11+1)/2.
3. E = 6. | probability |
The answer is Modulus clock. | {
"A": "Big hammer",
"B": "Long rope",
"C": "Modulus clock",
"D": "Wide net"
} | mat_0413 | multiple_choice | en | Choose: what wraps numbers? | [
"Remainders kept.",
"So: Modulus clock.",
"Select option C."
] | math | Answer: Modulus clock
Steps:
1. Remainders kept.
2. So: Modulus clock.
3. Select option C.
Answer: (C) | number_theory |
Odpověď je rekurence. | {
"A": "rekurence",
"B": "proroctví",
"C": "fáma",
"D": "sen"
} | mat_0414 | multiple_choice | cs | Vyber: co definuje z minulosti? | [
"Základ plus krok.",
"Tedy: rekurence.",
"Vyber možnost A."
] | math | Answer: rekurence
Steps:
1. Základ plus krok.
2. Tedy: rekurence.
3. Vyber možnost A.
Answer: (A) | combinatorics |
Half: 45, so 90 is even. | null | mat_0415 | worked_example | en | Show N² + N is even for N = 9 by computing its half. | [
"9² + 9 = 90.",
"90 / 2 = 45.",
"Integer, so even."
] | math | Answer: Half: 45, so 90 is even.
Steps:
1. 9² + 9 = 90.
2. 90 / 2 = 45.
3. Integer, so even. | proofs |
Integral: 1331. | null | mat_0416 | worked_example | en | Evaluate the integral of 3x² from 0 to 11. | [
"Antiderivative: x³.",
"Evaluate 0 to 11.",
"11³ − 0 = 1331."
] | math | Answer: Integral: 1331.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 11.
3. 11³ − 0 = 1331. | calculus |
Stopa je 7. | {
"A": "5",
"B": "7",
"C": "9",
"D": "11"
} | mat_0417 | multiple_choice | cs | Vypočítej stopu matice [[2,1],[2,5]]. | [
"2 + 5.",
"Stopa = 7.",
"Vyber možnost B."
] | math | Answer: 7
Steps:
1. 2 + 5.
2. Stopa = 7.
3. Vyber možnost B.
Answer: (B) | linear_algebra |
P(red) = 9/13. | null | mat_0418 | worked_example | en | A bag holds 9 red and 4 blue marbles. Show P(red). | [
"Red: 9.",
"Total: 13.",
"P = 9/13."
] | math | Answer: P(red) = 9/13.
Steps:
1. Red: 9.
2. Total: 13.
3. P = 9/13. | probability |
lcm = 40. | null | mat_0419 | worked_example | en | Compute lcm(10, 8) step by step. | [
"gcd(10,8) = 2.",
"10×8/2.",
"= 40."
] | math | Answer: lcm = 40.
Steps:
1. gcd(10,8) = 2.
2. 10×8/2.
3. = 40. | number_theory |
P(6,2) is 30. | {
"A": "30",
"B": "24",
"C": "36",
"D": "42"
} | mat_0420 | multiple_choice | en | Compute P(6,2). | [
"6×5.",
"= 30.",
"Select option A."
] | math | Answer: 30
Steps:
1. 6×5.
2. = 30.
3. Select option A.
Answer: (A) | combinatorics |
Součet: 36. | null | mat_0421 | worked_example | cs | Dokaž vzorcem: najdi součet 1 + 2 + ... + 8. | [
"Páruj 1+8, 2+7, ...",
"Použij 8×9/2.",
"= 36."
] | math | Answer: Součet: 36.
Steps:
1. Páruj 1+8, 2+7, ...
2. Použij 8×9/2.
3. = 36. | proofs |
The answer is Definite integral. | {
"A": "Derivative",
"B": "Definite integral",
"C": "Limit point",
"D": "Equation set"
} | mat_0422 | multiple_choice | en | Choose: what sums slices? | [
"Area under curve.",
"So: Definite integral.",
"Select option B."
] | math | Answer: Definite integral
Steps:
1. Area under curve.
2. So: Definite integral.
3. Select option B.
Answer: (B) | calculus |
The trace is 13. | {
"A": "13",
"B": "11",
"C": "15",
"D": "17"
} | mat_0423 | multiple_choice | en | Find the trace of [[5,1],[2,8]]. | [
"5 + 8.",
"Trace = 13.",
"Select option A."
] | math | Answer: 13
Steps:
1. 5 + 8.
2. Trace = 13.
3. Select option A.
Answer: (A) | linear_algebra |
Simulation estimates hard probabilities by repeated random trials on computers. | null | mat_0424 | concept | en | Why simulate? | [
"Simulation estimates hard probabilities by repeated random trials on computers.",
"So the answer stands: Simulation estimates hard probabilities by repeated random trials on computers."
] | math | Answer: Simulation estimates hard probabilities by repeated random trials on computers.
Steps:
1. Simulation estimates hard probabilities by repeated random trials on computers.
2. So the answer stands: Simulation estimates hard probabilities by repeated random trials on computers. | probability |
gcd = 16. | null | mat_0425 | worked_example | en | Compute gcd(32, 48) step by step. | [
"Factor: 32=4×8, 48=6×8.",
"Common: 8×gcd(4,6).",
"= 8×2 = 16."
] | math | Answer: gcd = 16.
Steps:
1. Factor: 32=4×8, 48=6×8.
2. Common: 8×gcd(4,6).
3. = 8×2 = 16. | number_theory |
C(13,3) = 286. | null | mat_0426 | worked_example | en | Show the computation of C(13,3). | [
"n(n−1)(n−2)/6.",
"13×12×11/6.",
"= 286."
] | math | Answer: C(13,3) = 286.
Steps:
1. n(n−1)(n−2)/6.
2. 13×12×11/6.
3. = 286. | combinatorics |
Rows: 1024. | null | mat_0427 | worked_example | en | A proof checks all cases of 10 boolean variables. How many rows? | [
"2 choices each.",
"2^10 = 1024."
] | math | Answer: Rows: 1024.
Steps:
1. 2 choices each.
2. 2^10 = 1024. | proofs |
The answer is Second derivative. | {
"A": "First guess",
"B": "Last digit",
"C": "Second derivative",
"D": "Middle value"
} | mat_0428 | multiple_choice | en | Choose: what shows bending? | [
"Sign gives shape.",
"So: Second derivative.",
"Select option C."
] | math | Answer: Second derivative
Steps:
1. Sign gives shape.
2. So: Second derivative.
3. Select option C.
Answer: (C) | calculus |
Dot product: 75. | null | mat_0429 | worked_example | en | Compute (10,9) · (3,5) step by step. | [
"Multiply components.",
"10×3 + 9×5.",
"= 75."
] | math | Answer: Dot product: 75.
Steps:
1. Multiply components.
2. 10×3 + 9×5.
3. = 75. | linear_algebra |
P(red) is 5/16. | {
"A": "11/16",
"B": "5/16",
"C": "5/17",
"D": "6/16"
} | mat_0430 | multiple_choice | en | A bag holds 5 red and 11 blue marbles. What is P(red)? | [
"5 of 16.",
"P = 5/16.",
"Select option B."
] | math | Answer: 5/16
Steps:
1. 5 of 16.
2. P = 5/16.
3. Select option B.
Answer: (B) | probability |
Remainder: 4. | null | mat_0431 | worked_example | en | Compute (116) mod 7. | [
"Divide 116 by 7.",
"7×16 = 112.",
"116 − 112 = 4."
] | math | Answer: Remainder: 4.
Steps:
1. Divide 116 by 7.
2. 7×16 = 112.
3. 116 − 112 = 4. | number_theory |
C(13,2) = 78. | null | mat_0432 | worked_example | en | Show the computation of C(13,2). | [
"Formula n(n−1)/2.",
"13×12/2.",
"= 78."
] | math | Answer: C(13,2) = 78.
Steps:
1. Formula n(n−1)/2.
2. 13×12/2.
3. = 78. | combinatorics |
Součet je 15. | {
"A": "10",
"B": "15",
"C": "20",
"D": "25"
} | mat_0433 | multiple_choice | cs | Kolik je součet 1 + 2 + ... + 5? | [
"5×6/2.",
"Součet = 15.",
"Vyber možnost B."
] | math | Answer: 15
Steps:
1. 5×6/2.
2. Součet = 15.
3. Vyber možnost B.
Answer: (B) | proofs |
Concavity from the second derivative shows where graphs bend up or down around points. | null | mat_0434 | concept | en | What is concavity? | [
"Concavity from the second derivative shows where graphs bend up or down around points.",
"So the answer stands: Concavity from the second derivative shows where graphs bend up or down around points."
] | math | Answer: Concavity from the second derivative shows where graphs bend up or down around points.
Steps:
1. Concavity from the second derivative shows where graphs bend up or down around points.
2. So the answer stands: Concavity from the second derivative shows where graphs bend up or down around points. | calculus |
Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions. | null | mat_0435 | concept | en | What is orthogonality? | [
"Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.",
"So the answer stands: Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions."
] | math | Answer: Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.
Steps:
1. Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions.
2. So the answer stands: Orthogonal vectors meet at right angles with zero dot product, giving clean decompositions. | linear_algebra |
The answer is Probabilities. | {
"A": "Outcomes",
"B": "Names",
"C": "Probabilities",
"D": "Colors"
} | mat_0436 | multiple_choice | en | Choose: what multiplies for independent? | [
"Joint equals product.",
"So: Probabilities.",
"Select option C."
] | math | Answer: Probabilities
Steps:
1. Joint equals product.
2. So: Probabilities.
3. Select option C.
Answer: (C) | probability |
The answer is Hard factoring. | {
"A": "Big primes secret",
"B": "Long keys",
"C": "Fast computers",
"D": "Hard factoring"
} | mat_0437 | multiple_choice | en | Choose: what secures RSA? | [
"Multiply easy, split hard.",
"So: Hard factoring.",
"Select option D."
] | math | Answer: Hard factoring
Steps:
1. Multiply easy, split hard.
2. So: Hard factoring.
3. Select option D.
Answer: (D) | number_theory |
P(12,3) = 1320. | null | mat_0438 | worked_example | en | Show the computation of P(12,3). | [
"n(n−1)(n−2).",
"12×11×10.",
"= 1320."
] | math | Answer: P(12,3) = 1320.
Steps:
1. n(n−1)(n−2).
2. 12×11×10.
3. = 1320. | combinatorics |
The table has 64 rows. | {
"A": "64",
"B": "62",
"C": "66",
"D": "68"
} | mat_0439 | multiple_choice | en | How many rows does a truth table with 6 variables have? | [
"2^6 rows.",
"Total: 64.",
"Select option A."
] | math | Answer: 64
Steps:
1. 2^6 rows.
2. Total: 64.
3. Select option A.
Answer: (A) | proofs |
The derivative is 13. | {
"A": "11",
"B": "15",
"C": "17",
"D": "13"
} | mat_0440 | multiple_choice | en | For f(x) = 1x² + 5x, find f'(4). | [
"f'(x) = 2x + 5.",
"At 4: 13.",
"Select option D."
] | math | Answer: 13
Steps:
1. f'(x) = 2x + 5.
2. At 4: 13.
3. Select option D.
Answer: (D) | calculus |
Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work. | null | mat_0441 | concept | en | Why Gram-Schmidt? | [
"Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.",
"So the answer stands: Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work."
] | math | Answer: Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.
Steps:
1. Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work.
2. So the answer stands: Gram-Schmidt turns any basis orthonormal, simplifying projections and numerical work. | linear_algebra |
P = 1/13. | null | mat_0442 | worked_example | en | A fair 13-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 13.",
"P = 1/13."
] | math | Answer: P = 1/13.
Steps:
1. Favourable: 1.
2. Possible: 13.
3. P = 1/13. | probability |
Zbytek: 2. | null | mat_0443 | worked_example | cs | Spočítej (72) mod 7. | [
"Děl 72 sedmi.",
"7×10 = 70.",
"72 − 70 = 2."
] | math | Answer: Zbytek: 2.
Steps:
1. Děl 72 sedmi.
2. 7×10 = 70.
3. 72 − 70 = 2. | number_theory |
They count set partitions, tallying every way to group objects into blocks. | null | mat_0444 | concept | en | Why Bell numbers? | [
"They count set partitions, tallying every way to group objects into blocks.",
"So the answer stands: They count set partitions, tallying every way to group objects into blocks."
] | math | Answer: They count set partitions, tallying every way to group objects into blocks.
Steps:
1. They count set partitions, tallying every way to group objects into blocks.
2. So the answer stands: They count set partitions, tallying every way to group objects into blocks. | combinatorics |
The sum is 45. | {
"A": "36",
"B": "45",
"C": "54",
"D": "63"
} | mat_0445 | multiple_choice | en | What is the sum 1 + 2 + ... + 9? | [
"9×10/2.",
"Sum = 45.",
"Select option B."
] | math | Answer: 45
Steps:
1. 9×10/2.
2. Sum = 45.
3. Select option B.
Answer: (B) | proofs |
Limit: 22. | null | mat_0446 | worked_example | en | Evaluate the limit of (x² − 11²)/(x − 11) as x → 11. | [
"Factor: (x−11)(x+11).",
"Cancel (x−11).",
"Plug 11: 11+11 = 22."
] | math | Answer: Limit: 22.
Steps:
1. Factor: (x−11)(x+11).
2. Cancel (x−11).
3. Plug 11: 11+11 = 22. | calculus |
Stopa: 9. | null | mat_0447 | worked_example | cs | Spočítej stopu matice [[4,0],[0,5]]. | [
"Sečti diagonálu.",
"4 + 5.",
"= 9."
] | math | Answer: Stopa: 9.
Steps:
1. Sečti diagonálu.
2. 4 + 5.
3. = 9. | linear_algebra |
E = 6.5. | null | mat_0448 | worked_example | en | Compute E of a uniform draw from {1..12}. | [
"Mean of 1..12.",
"(12+1)/2.",
"E = 6.5."
] | math | Answer: E = 6.5.
Steps:
1. Mean of 1..12.
2. (12+1)/2.
3. E = 6.5. | probability |
lcm = 88. | null | mat_0449 | worked_example | en | Compute lcm(11, 8) step by step. | [
"gcd(11,8) = 1.",
"11×8/1.",
"= 88."
] | math | Answer: lcm = 88.
Steps:
1. gcd(11,8) = 1.
2. 11×8/1.
3. = 88. | number_theory |
C(8,3) is 56. | {
"A": "48",
"B": "56",
"C": "64",
"D": "72"
} | mat_0450 | multiple_choice | en | Compute C(8,3). | [
"8×7×6/6.",
"= 56.",
"Select option B."
] | math | Answer: 56
Steps:
1. 8×7×6/6.
2. = 56.
3. Select option B.
Answer: (B) | combinatorics |
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