answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
The value is 20. | {
"A": "18",
"B": "22",
"C": "24",
"D": "20"
} | mat_0151 | multiple_choice | en | Compute N² + N for N = 4. | [
"4² + 4.",
"Value = 20.",
"Select option D."
] | math | Answer: 20
Steps:
1. 4² + 4.
2. Value = 20.
3. Select option D.
Answer: (D) | proofs |
The answer is Taylor series. | {
"A": "Taylor series",
"B": "Random dots",
"C": "Wild guesses",
"D": "Blank lines"
} | mat_0152 | multiple_choice | en | Choose: what approximates functions? | [
"Polynomials with error bounds.",
"So: Taylor series.",
"Select option A."
] | math | Answer: Taylor series
Steps:
1. Polynomials with error bounds.
2. So: Taylor series.
3. Select option A.
Answer: (A) | calculus |
An eigenvector keeps its direction under the map, scaled only by its eigenvalue. | null | mat_0153 | concept | en | What is an eigenvector? | [
"An eigenvector keeps its direction under the map, scaled only by its eigenvalue.",
"So the answer stands: An eigenvector keeps its direction under the map, scaled only by its eigenvalue."
] | math | Answer: An eigenvector keeps its direction under the map, scaled only by its eigenvalue.
Steps:
1. An eigenvector keeps its direction under the map, scaled only by its eigenvalue.
2. So the answer stands: An eigenvector keeps its direction under the map, scaled only by its eigenvalue. | linear_algebra |
The answer is Variance. | {
"A": "Variance",
"B": "Mean",
"C": "Mode",
"D": "Hope"
} | mat_0154 | multiple_choice | en | Choose: what measures spread? | [
"Squared deviations averaged.",
"So: Variance.",
"Select option A."
] | math | Answer: Variance
Steps:
1. Squared deviations averaged.
2. So: Variance.
3. Select option A.
Answer: (A) | probability |
gcd = 6. | null | mat_0155 | worked_example | en | Compute gcd(12, 18) step by step. | [
"Factor: 12=4×3, 18=6×3.",
"Common: 3×gcd(4,6).",
"= 3×2 = 6."
] | math | Answer: gcd = 6.
Steps:
1. Factor: 12=4×3, 18=6×3.
2. Common: 3×gcd(4,6).
3. = 3×2 = 6. | number_theory |
P(4,2) is 12. | {
"A": "8",
"B": "16",
"C": "12",
"D": "20"
} | mat_0156 | multiple_choice | en | Compute P(4,2). | [
"4×3.",
"= 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. 4×3.
2. = 12.
3. Select option C.
Answer: (C) | combinatorics |
Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí. | null | mat_0157 | concept | cs | Proč důkaz sporem? | [
"Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.",
"Odpověď tedy platí: Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí."
] | math | Answer: Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.
Steps:
1. Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí.
2. Odpověď tedy platí: Spor předpokládá nepravdivost a odvodí nesmysl, tvrzení tedy platí. | proofs |
f'(2) = 12. | null | mat_0158 | worked_example | en | For f(x) = 2x² + 4x, compute f'(2) step by step. | [
"f'(x) = 4x + 4.",
"Plug x = 2.",
"4×2 + 4 = 12."
] | math | Answer: f'(2) = 12.
Steps:
1. f'(x) = 4x + 4.
2. Plug x = 2.
3. 4×2 + 4 = 12. | calculus |
Diagonal form powers matrices easily, since powers act separately on each eigen-direction. | null | mat_0159 | concept | en | Why diagonalize? | [
"Diagonal form powers matrices easily, since powers act separately on each eigen-direction.",
"So the answer stands: Diagonal form powers matrices easily, since powers act separately on each eigen-direction."
] | math | Answer: Diagonal form powers matrices easily, since powers act separately on each eigen-direction.
Steps:
1. Diagonal form powers matrices easily, since powers act separately on each eigen-direction.
2. So the answer stands: Diagonal form powers matrices easily, since powers act separately on each eigen-direction. | linear_algebra |
Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu. | null | mat_0160 | concept | cs | Proč prostor výsledků? | [
"Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.",
"Odpověď tedy platí: Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu."
] | math | Answer: Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.
Steps:
1. Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu.
2. Odpověď tedy platí: Prostor vypisuje všechny výsledky, pravděpodobnosti se vážou na úplnou množinu. | probability |
Remainder: 5. | null | mat_0161 | worked_example | en | Compute (61) mod 7. | [
"Divide 61 by 7.",
"7×8 = 56.",
"61 − 56 = 5."
] | math | Answer: Remainder: 5.
Steps:
1. Divide 61 by 7.
2. 7×8 = 56.
3. 61 − 56 = 5. | number_theory |
It fixes double-counting by alternating union sums, exact for overlapping sets. | null | mat_0162 | concept | en | What is inclusion-exclusion? | [
"It fixes double-counting by alternating union sums, exact for overlapping sets.",
"So the answer stands: It fixes double-counting by alternating union sums, exact for overlapping sets."
] | math | Answer: It fixes double-counting by alternating union sums, exact for overlapping sets.
Steps:
1. It fixes double-counting by alternating union sums, exact for overlapping sets.
2. So the answer stands: It fixes double-counting by alternating union sums, exact for overlapping sets. | combinatorics |
Polovina: 3, číslo 6 je sudé. | null | mat_0163 | worked_example | cs | Ukaž, že N² + N je sudé pro N = 2, výpočtem poloviny. | [
"2² + 2 = 6.",
"6 / 2 = 3.",
"Celé číslo, tedy sudé."
] | math | Answer: Polovina: 3, číslo 6 je sudé.
Steps:
1. 2² + 2 = 6.
2. 6 / 2 = 3.
3. Celé číslo, tedy sudé. | proofs |
Integral: 216. | null | mat_0164 | worked_example | en | Evaluate the integral of 3x² from 0 to 6. | [
"Antiderivative: x³.",
"Evaluate 0 to 6.",
"6³ − 0 = 216."
] | math | Answer: Integral: 216.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 6.
3. 6³ − 0 = 216. | calculus |
Trace: 13. | null | mat_0165 | worked_example | en | Compute the trace of [[6,0],[0,7]]. | [
"Sum diagonal.",
"6 + 7.",
"= 13."
] | math | Answer: Trace: 13.
Steps:
1. Sum diagonal.
2. 6 + 7.
3. = 13. | linear_algebra |
Distributions list value probabilities compactly, summarizing all chance behaviour at once. | null | mat_0166 | concept | en | Why distributions? | [
"Distributions list value probabilities compactly, summarizing all chance behaviour at once.",
"So the answer stands: Distributions list value probabilities compactly, summarizing all chance behaviour at once."
] | math | Answer: Distributions list value probabilities compactly, summarizing all chance behaviour at once.
Steps:
1. Distributions list value probabilities compactly, summarizing all chance behaviour at once.
2. So the answer stands: Distributions list value probabilities compactly, summarizing all chance behaviour at once. | probability |
lcm = 24. | null | mat_0167 | worked_example | en | Compute lcm(6, 8) step by step. | [
"gcd(6,8) = 2.",
"6×8/2.",
"= 24."
] | math | Answer: lcm = 24.
Steps:
1. gcd(6,8) = 2.
2. 6×8/2.
3. = 24. | number_theory |
P(3,2) je 6. | {
"A": "3",
"B": "6",
"C": "9",
"D": "12"
} | mat_0168 | multiple_choice | cs | Vypočítej permutační číslo P(3,2). | [
"3×2.",
"= 6.",
"Vyber možnost B."
] | math | Answer: 6
Steps:
1. 3×2.
2. = 6.
3. Vyber možnost B.
Answer: (B) | combinatorics |
For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy. | null | mat_0169 | concept | en | Why state quantifiers clearly? | [
"For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.",
"So the answer stands: For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy."
] | math | Answer: For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.
Steps:
1. For-all and exists claims need different proofs, so naming the quantifier picks the right proof strategy.
2. So the answer stands: For-all and exists claims need different proofs, so naming the qu... | proofs |
Derivování nachází maxima, minima a rychlosti. | null | mat_0170 | concept | cs | Proč derivovat? | [
"Derivování nachází maxima, minima a rychlosti.",
"Odpověď tedy platí: Derivování nachází maxima, minima a rychlosti."
] | math | Answer: Derivování nachází maxima, minima a rychlosti.
Steps:
1. Derivování nachází maxima, minima a rychlosti.
2. Odpověď tedy platí: Derivování nachází maxima, minima a rychlosti. | calculus |
A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra. | null | mat_0171 | concept | en | What is a vector space? | [
"A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.",
"So the answer stands: A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra."
] | math | Answer: A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.
Steps:
1. A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra.
2. So the answer stands: A vector space closes under addition and scaling, with axioms guaranteeing familiar algebra... | linear_algebra |
The probability is 1/7. | {
"A": "1/6",
"B": "1/7",
"C": "1/8",
"D": "2/7"
} | mat_0172 | multiple_choice | en | A fair 7-sided die is rolled. What is the probability of one specific number? | [
"1 face of 7.",
"P = 1/7.",
"Select option B."
] | math | Answer: 1/7
Steps:
1. 1 face of 7.
2. P = 1/7.
3. Select option B.
Answer: (B) | probability |
It computes with remainders upon division, wrapping numbers around a modulus clock. | null | mat_0173 | concept | en | What is modular arithmetic? | [
"It computes with remainders upon division, wrapping numbers around a modulus clock.",
"So the answer stands: It computes with remainders upon division, wrapping numbers around a modulus clock."
] | math | Answer: It computes with remainders upon division, wrapping numbers around a modulus clock.
Steps:
1. It computes with remainders upon division, wrapping numbers around a modulus clock.
2. So the answer stands: It computes with remainders upon division, wrapping numbers around a modulus clock. | number_theory |
More items than boxes forces sharing, proving existence without construction. | null | mat_0174 | concept | en | Why pigeonhole? | [
"More items than boxes forces sharing, proving existence without construction.",
"So the answer stands: More items than boxes forces sharing, proving existence without construction."
] | math | Answer: More items than boxes forces sharing, proving existence without construction.
Steps:
1. More items than boxes forces sharing, proving existence without construction.
2. So the answer stands: More items than boxes forces sharing, proving existence without construction. | combinatorics |
A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example. | null | mat_0175 | concept | en | What is a constructive proof? | [
"A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.",
"So the answer stands: A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example."
] | math | Answer: A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.
Steps:
1. A constructive proof exhibits the claimed object explicitly, giving both truth and a concrete example.
2. So the answer stands: A constructive proof exhibits the claimed object explicitly, giving bot... | proofs |
Limit: 12. | null | mat_0176 | worked_example | en | Evaluate the limit of (x² − 6²)/(x − 6) as x → 6. | [
"Factor: (x−6)(x+6).",
"Cancel (x−6).",
"Plug 6: 6+6 = 12."
] | math | Answer: Limit: 12.
Steps:
1. Factor: (x−6)(x+6).
2. Cancel (x−6).
3. Plug 6: 6+6 = 12. | calculus |
The answer is Dot product. | {
"A": "Dot product",
"B": "Cross guess",
"C": "Wild sum",
"D": "Blind hope"
} | mat_0177 | multiple_choice | en | Choose: what measures angles? | [
"Zero means orthogonal.",
"So: Dot product.",
"Select option A."
] | math | Answer: Dot product
Steps:
1. Zero means orthogonal.
2. So: Dot product.
3. Select option A.
Answer: (A) | linear_algebra |
It counts successes in fixed trials with constant p, modelling repeated yes-no experiments. | null | mat_0178 | concept | en | What is the binomial distribution? | [
"It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.",
"So the answer stands: It counts successes in fixed trials with constant p, modelling repeated yes-no experiments."
] | math | Answer: It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.
Steps:
1. It counts successes in fixed trials with constant p, modelling repeated yes-no experiments.
2. So the answer stands: It counts successes in fixed trials with constant p, modelling repeated yes-no experiments. | probability |
Congruences preserve sums and products, letting big-number questions shrink to small cases. | null | mat_0179 | concept | en | Why congruences? | [
"Congruences preserve sums and products, letting big-number questions shrink to small cases.",
"So the answer stands: Congruences preserve sums and products, letting big-number questions shrink to small cases."
] | math | Answer: Congruences preserve sums and products, letting big-number questions shrink to small cases.
Steps:
1. Congruences preserve sums and products, letting big-number questions shrink to small cases.
2. So the answer stands: Congruences preserve sums and products, letting big-number questions shrink to small cases. | number_theory |
C(9,3) = 84. | null | mat_0180 | worked_example | en | Show the computation of C(9,3). | [
"n(n−1)(n−2)/6.",
"9×8×7/6.",
"= 84."
] | math | Answer: C(9,3) = 84.
Steps:
1. n(n−1)(n−2)/6.
2. 9×8×7/6.
3. = 84. | combinatorics |
Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier. | null | mat_0181 | concept | en | Why use contrapositive? | [
"Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.",
"So the answer stands: Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier."
] | math | Answer: Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.
Steps:
1. Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is often easier.
2. So the answer stands: Proving not-Q implies not-P is equivalent to P implies Q, and the flipped form is oft... | proofs |
A definite integral sums infinite thin slices, measuring area under a curve exactly. | null | mat_0182 | concept | en | What is an integral? | [
"A definite integral sums infinite thin slices, measuring area under a curve exactly.",
"So the answer stands: A definite integral sums infinite thin slices, measuring area under a curve exactly."
] | math | Answer: A definite integral sums infinite thin slices, measuring area under a curve exactly.
Steps:
1. A definite integral sums infinite thin slices, measuring area under a curve exactly.
2. So the answer stands: A definite integral sums infinite thin slices, measuring area under a curve exactly. | calculus |
The determinant is 4. | {
"A": "1",
"B": "4",
"C": "7",
"D": "10"
} | mat_0183 | multiple_choice | en | Find the determinant of [[2,2],[3,5]]. | [
"2×5 − 2×3.",
"Det = 4.",
"Select option B."
] | math | Answer: 4
Steps:
1. 2×5 − 2×3.
2. Det = 4.
3. Select option B.
Answer: (B) | linear_algebra |
Averages of many variables turn normal, so it models errors and natural variation. | null | mat_0184 | concept | en | Why the normal distribution? | [
"Averages of many variables turn normal, so it models errors and natural variation.",
"So the answer stands: Averages of many variables turn normal, so it models errors and natural variation."
] | math | Answer: Averages of many variables turn normal, so it models errors and natural variation.
Steps:
1. Averages of many variables turn normal, so it models errors and natural variation.
2. So the answer stands: Averages of many variables turn normal, so it models errors and natural variation. | probability |
The answer is Euclid's argument. | {
"A": "Euclid's argument",
"B": "Big computers",
"C": "Long lists",
"D": "Loud claims"
} | mat_0185 | multiple_choice | en | Choose: what forces new primes? | [
"Product plus one.",
"So: Euclid's argument.",
"Select option A."
] | math | Answer: Euclid's argument
Steps:
1. Product plus one.
2. So: Euclid's argument.
3. Select option A.
Answer: (A) | number_theory |
Kombinatorika počítá uspořádání a výběry. | null | mat_0186 | concept | cs | Co je kombinatorika? | [
"Kombinatorika počítá uspořádání a výběry.",
"Odpověď tedy platí: Kombinatorika počítá uspořádání a výběry."
] | math | Answer: Kombinatorika počítá uspořádání a výběry.
Steps:
1. Kombinatorika počítá uspořádání a výběry.
2. Odpověď tedy platí: Kombinatorika počítá uspořádání a výběry. | combinatorics |
The answer is One counterexample. | {
"A": "One counterexample",
"B": "Two examples",
"C": "Loud denial",
"D": "Long proof"
} | mat_0187 | multiple_choice | en | Choose: what disproves a for-all claim? | [
"Single failure kills it.",
"So: One counterexample.",
"Select option A."
] | math | Answer: One counterexample
Steps:
1. Single failure kills it.
2. So: One counterexample.
3. Select option A.
Answer: (A) | proofs |
f'(3) = 23. | null | mat_0188 | worked_example | en | For f(x) = 3x² + 5x, compute f'(3) step by step. | [
"f'(x) = 6x + 5.",
"Plug x = 3.",
"6×3 + 5 = 23."
] | math | Answer: f'(3) = 23.
Steps:
1. f'(x) = 6x + 5.
2. Plug x = 3.
3. 6×3 + 5 = 23. | calculus |
The trace is 9. | {
"A": "7",
"B": "11",
"C": "9",
"D": "13"
} | mat_0189 | multiple_choice | en | Find the trace of [[3,1],[2,6]]. | [
"3 + 6.",
"Trace = 9.",
"Select option C."
] | math | Answer: 9
Steps:
1. 3 + 6.
2. Trace = 9.
3. Select option C.
Answer: (C) | linear_algebra |
Střední hodnota je 2,5. | {
"A": "3,5",
"B": "2,5",
"C": "1,5",
"D": "4,5"
} | mat_0190 | multiple_choice | cs | Najdi střední hodnotu rovnoměrného tahu z {1..4}. | [
"(4+1)/2.",
"E = 2,5.",
"Vyber možnost B."
] | math | Answer: 2,5
Steps:
1. (4+1)/2.
2. E = 2,5.
3. Vyber možnost B.
Answer: (B) | probability |
The answer is Diophantine equation. | {
"A": "Real guess",
"B": "Diophantine equation",
"C": "Complex hope",
"D": "Vague wish"
} | mat_0191 | multiple_choice | en | Choose: what seeks integer solutions? | [
"Gcd conditions apply.",
"So: Diophantine equation.",
"Select option B."
] | math | Answer: Diophantine equation
Steps:
1. Gcd conditions apply.
2. So: Diophantine equation.
3. Select option B.
Answer: (B) | number_theory |
C(9,2) = 36. | null | mat_0192 | worked_example | en | Show the computation of C(9,2). | [
"Formula n(n−1)/2.",
"9×8/2.",
"= 36."
] | math | Answer: C(9,2) = 36.
Steps:
1. Formula n(n−1)/2.
2. 9×8/2.
3. = 36. | combinatorics |
A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable. | null | mat_0193 | concept | en | What is a lemma? | [
"A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.",
"So the answer stands: A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable."
] | math | Answer: A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.
Steps:
1. A lemma is a small proven helper claim, and breaking proofs into lemmas keeps each step checkable.
2. So the answer stands: A lemma is a small proven helper claim, and breaking proofs into lemmas keeps e... | proofs |
Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace. | null | mat_0194 | concept | cs | Co je řetízkové pravidlo? | [
"Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.",
"Odpověď tedy platí: Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace."
] | math | Answer: Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.
Steps:
1. Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace.
2. Odpověď tedy platí: Řetízkové pravidlo derivuje složeniny násobením vnější a vnitřní derivace. | calculus |
Independence means no vector is redundant, so each adds a new dimension to the span. | null | mat_0195 | concept | en | Why check linear independence? | [
"Independence means no vector is redundant, so each adds a new dimension to the span.",
"So the answer stands: Independence means no vector is redundant, so each adds a new dimension to the span."
] | math | Answer: Independence means no vector is redundant, so each adds a new dimension to the span.
Steps:
1. Independence means no vector is redundant, so each adds a new dimension to the span.
2. So the answer stands: Independence means no vector is redundant, so each adds a new dimension to the span. | linear_algebra |
Aktualizuje šanci při nové evidenci přes P(A|B). | null | mat_0196 | concept | cs | Co je podmíněná pravděpodobnost? | [
"Aktualizuje šanci při nové evidenci přes P(A|B).",
"Odpověď tedy platí: Aktualizuje šanci při nové evidenci přes P(A|B)."
] | math | Answer: Aktualizuje šanci při nové evidenci přes P(A|B).
Steps:
1. Aktualizuje šanci při nové evidenci přes P(A|B).
2. Odpověď tedy platí: Aktualizuje šanci při nové evidenci přes P(A|B). | probability |
gcd = 8. | null | mat_0197 | worked_example | en | Compute gcd(16, 24) step by step. | [
"Factor: 16=4×4, 24=6×4.",
"Common: 4×gcd(4,6).",
"= 4×2 = 8."
] | math | Answer: gcd = 8.
Steps:
1. Factor: 16=4×4, 24=6×4.
2. Common: 4×gcd(4,6).
3. = 4×2 = 8. | number_theory |
P(4,3) = 24. | null | mat_0198 | worked_example | cs | Ukaž výpočet P(4,3). | [
"n(n−1)(n−2).",
"4×3×2.",
"= 24."
] | math | Answer: P(4,3) = 24.
Steps:
1. n(n−1)(n−2).
2. 4×3×2.
3. = 24. | combinatorics |
Řádků: 8. | null | mat_0199 | worked_example | cs | Důkaz prochází všechny případy 3 booleovských proměnných. Kolik řádků? | [
"2 volby každá.",
"2^3 = 8."
] | math | Answer: Řádků: 8.
Steps:
1. 2 volby každá.
2. 2^3 = 8. | proofs |
Integral: 343. | null | mat_0200 | worked_example | en | Evaluate the integral of 3x² from 0 to 7. | [
"Antiderivative: x³.",
"Evaluate 0 to 7.",
"7³ − 0 = 343."
] | math | Answer: Integral: 343.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 7.
3. 7³ − 0 = 343. | calculus |
Matice kódují lineární zobrazení a soustavy kompaktně. | null | mat_0201 | concept | cs | Proč matice? | [
"Matice kódují lineární zobrazení a soustavy kompaktně.",
"Odpověď tedy platí: Matice kódují lineární zobrazení a soustavy kompaktně."
] | math | Answer: Matice kódují lineární zobrazení a soustavy kompaktně.
Steps:
1. Matice kódují lineární zobrazení a soustavy kompaktně.
2. Odpověď tedy platí: Matice kódují lineární zobrazení a soustavy kompaktně. | linear_algebra |
E = 4.5. | null | mat_0202 | worked_example | en | Compute E of a uniform draw from {1..8}. | [
"Mean of 1..8.",
"(8+1)/2.",
"E = 4.5."
] | math | Answer: E = 4.5.
Steps:
1. Mean of 1..8.
2. (8+1)/2.
3. E = 4.5. | probability |
Remainder: 2. | null | mat_0203 | worked_example | en | Compute (72) mod 7. | [
"Divide 72 by 7.",
"7×10 = 70.",
"72 − 70 = 2."
] | math | Answer: Remainder: 2.
Steps:
1. Divide 72 by 7.
2. 7×10 = 70.
3. 72 − 70 = 2. | number_theory |
Faktoriály počítají pořadí různých objektů. | null | mat_0204 | concept | cs | Proč faktoriály? | [
"Faktoriály počítají pořadí různých objektů.",
"Odpověď tedy platí: Faktoriály počítají pořadí různých objektů."
] | math | Answer: Faktoriály počítají pořadí různých objektů.
Steps:
1. Faktoriály počítají pořadí různých objektů.
2. Odpověď tedy platí: Faktoriály počítají pořadí různých objektů. | combinatorics |
Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla. | null | mat_0205 | concept | cs | Co je indukce? | [
"Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.",
"Odpověď tedy platí: Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla."
] | math | Answer: Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.
Steps:
1. Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla.
2. Odpověď tedy platí: Indukce dokáže základ a krok, tvrzení pak platí pro všechna přirozená čísla. | proofs |
Limit: 14. | null | mat_0206 | worked_example | en | Evaluate the limit of (x² − 7²)/(x − 7) as x → 7. | [
"Factor: (x−7)(x+7).",
"Cancel (x−7).",
"Plug 7: 7+7 = 14."
] | math | Answer: Limit: 14.
Steps:
1. Factor: (x−7)(x+7).
2. Cancel (x−7).
3. Plug 7: 7+7 = 14. | calculus |
The dot product is 18. | {
"A": "14",
"B": "22",
"C": "26",
"D": "18"
} | mat_0207 | multiple_choice | en | Find the dot product (2,3) · (3,4). | [
"2×3 + 3×4.",
"Product = 18.",
"Select option D."
] | math | Answer: 18
Steps:
1. 2×3 + 3×4.
2. Product = 18.
3. Select option D.
Answer: (D) | linear_algebra |
P(red) = 6/10. | null | mat_0208 | worked_example | en | A bag holds 6 red and 4 blue marbles. Show P(red). | [
"Red: 6.",
"Total: 10.",
"P = 6/10."
] | math | Answer: P(red) = 6/10.
Steps:
1. Red: 6.
2. Total: 10.
3. P = 6/10. | probability |
The gcd is 12. | {
"A": "9",
"B": "15",
"C": "12",
"D": "18"
} | mat_0209 | multiple_choice | en | Find gcd(24, 36). | [
"Factor 3 out twice.",
"gcd = 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. Factor 3 out twice.
2. gcd = 12.
3. Select option C.
Answer: (C) | number_theory |
C(5,3) je 10. | {
"A": "5",
"B": "15",
"C": "10",
"D": "20"
} | mat_0210 | multiple_choice | cs | Vypočítej kombinační číslo C(5,3). | [
"5×4×3/6.",
"= 10.",
"Vyber možnost C."
] | math | Answer: 10
Steps:
1. 5×4×3/6.
2. = 10.
3. Vyber možnost C.
Answer: (C) | combinatorics |
Half: 15, so 30 is even. | null | mat_0211 | worked_example | en | Show N² + N is even for N = 5 by computing its half. | [
"5² + 5 = 30.",
"30 / 2 = 15.",
"Integer, so even."
] | math | Answer: Half: 15, so 30 is even.
Steps:
1. 5² + 5 = 30.
2. 30 / 2 = 15.
3. Integer, so even. | proofs |
f'(4) = 38. | null | mat_0212 | worked_example | en | For f(x) = 4x² + 6x, compute f'(4) step by step. | [
"f'(x) = 8x + 6.",
"Plug x = 4.",
"8×4 + 6 = 38."
] | math | Answer: f'(4) = 38.
Steps:
1. f'(x) = 8x + 6.
2. Plug x = 4.
3. 8×4 + 6 = 38. | calculus |
Dot product: 35. | null | mat_0213 | worked_example | en | Compute (5,4) · (3,5) step by step. | [
"Multiply components.",
"5×3 + 4×5.",
"= 35."
] | math | Answer: Dot product: 35.
Steps:
1. Multiply components.
2. 5×3 + 4×5.
3. = 35. | linear_algebra |
The expected value is 3. | {
"A": "4",
"B": "2",
"C": "3",
"D": "5"
} | mat_0214 | multiple_choice | en | Find the expected value of a uniform draw from {1..5}. | [
"(5+1)/2.",
"E = 3.",
"Select option C."
] | math | Answer: 3
Steps:
1. (5+1)/2.
2. E = 3.
3. Select option C.
Answer: (C) | probability |
lcm = 56. | null | mat_0215 | worked_example | en | Compute lcm(7, 8) step by step. | [
"gcd(7,8) = 1.",
"7×8/1.",
"= 56."
] | math | Answer: lcm = 56.
Steps:
1. gcd(7,8) = 1.
2. 7×8/1.
3. = 56. | number_theory |
C(6,3) = 20. | null | mat_0216 | worked_example | cs | Ukaž výpočet C(6,3). | [
"n(n−1)(n−2)/6.",
"6×5×4/6.",
"= 20."
] | math | Answer: C(6,3) = 20.
Steps:
1. n(n−1)(n−2)/6.
2. 6×5×4/6.
3. = 20. | combinatorics |
Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity. | null | mat_0217 | concept | en | Why define terms first? | [
"Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.",
"So the answer stands: Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity."
] | math | Answer: Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.
Steps:
1. Proofs rest on definitions, so stating exact meanings first prevents hidden assumptions and ambiguity.
2. So the answer stands: Proofs rest on definitions, so stating exact meanings first prevents hi... | proofs |
Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou. | null | mat_0218 | concept | cs | Co je integrál? | [
"Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.",
"Odpověď tedy platí: Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou."
] | math | Answer: Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.
Steps:
1. Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou.
2. Odpověď tedy platí: Určitý integrál sčítá nekonečně tenké plátky a měří plochu pod křivkou. | calculus |
The answer is Rank. | {
"A": "Rank",
"B": "Size",
"C": "Color",
"D": "Age"
} | mat_0219 | multiple_choice | en | Choose: what counts image dimension? | [
"Independent rows tallied.",
"So: Rank.",
"Select option A."
] | math | Answer: Rank
Steps:
1. Independent rows tallied.
2. So: Rank.
3. Select option A.
Answer: (A) | linear_algebra |
P(red) is 3/10. | {
"A": "7/10",
"B": "3/11",
"C": "4/10",
"D": "3/10"
} | mat_0220 | multiple_choice | en | A bag holds 3 red and 7 blue marbles. What is P(red)? | [
"3 of 10.",
"P = 3/10.",
"Select option D."
] | math | Answer: 3/10
Steps:
1. 3 of 10.
2. P = 3/10.
3. Select option D.
Answer: (D) | probability |
NSD = 6. | null | mat_0221 | worked_example | cs | Spočítej NSD(12, 18) krok za krokem. | [
"Rozlož: 12=4×3, 18=6×3.",
"Společné: 3×NSD(4,6).",
"= 3×2 = 6."
] | math | Answer: NSD = 6.
Steps:
1. Rozlož: 12=4×3, 18=6×3.
2. Společné: 3×NSD(4,6).
3. = 3×2 = 6. | number_theory |
P(8,3) = 336. | null | mat_0222 | worked_example | en | Show the computation of P(8,3). | [
"n(n−1)(n−2).",
"8×7×6.",
"= 336."
] | math | Answer: P(8,3) = 336.
Steps:
1. n(n−1)(n−2).
2. 8×7×6.
3. = 336. | combinatorics |
Rows: 64. | null | mat_0223 | worked_example | en | A proof checks all cases of 6 boolean variables. How many rows? | [
"2 choices each.",
"2^6 = 64."
] | math | Answer: Rows: 64.
Steps:
1. 2 choices each.
2. 2^6 = 64. | proofs |
The derivative is 11. | {
"A": "9",
"B": "11",
"C": "13",
"D": "15"
} | mat_0224 | multiple_choice | en | For f(x) = 2x² + 3x, find f'(2). | [
"f'(x) = 4x + 3.",
"At 2: 11.",
"Select option B."
] | math | Answer: 11
Steps:
1. f'(x) = 4x + 3.
2. At 2: 11.
3. Select option B.
Answer: (B) | calculus |
Skalární součin: 11. | null | mat_0225 | worked_example | cs | Spočítej (2,1) · (3,5) krok za krokem. | [
"Násob složky.",
"2×3 + 1×5.",
"= 11."
] | math | Answer: Skalární součin: 11.
Steps:
1. Násob složky.
2. 2×3 + 1×5.
3. = 11. | linear_algebra |
The answer is Conditional probability. | {
"A": "Conditional probability",
"B": "Blind guess",
"C": "Stubborn prior",
"D": "Wild hope"
} | mat_0226 | multiple_choice | en | Choose: what updates with evidence? | [
"Space shrinks, ratio taken.",
"So: Conditional probability.",
"Select option A."
] | math | Answer: Conditional probability
Steps:
1. Space shrinks, ratio taken.
2. So: Conditional probability.
3. Select option A.
Answer: (A) | probability |
The remainder is 6. | {
"A": "7",
"B": "8",
"C": "0",
"D": "6"
} | mat_0227 | multiple_choice | en | Find (114) mod 9. | [
"114 ÷ 9.",
"Remainder 6.",
"Select option D."
] | math | Answer: 6
Steps:
1. 114 ÷ 9.
2. Remainder 6.
3. Select option D.
Answer: (D) | number_theory |
C(6,3) is 20. | {
"A": "14",
"B": "26",
"C": "32",
"D": "20"
} | mat_0228 | multiple_choice | en | Compute C(6,3). | [
"6×5×4/6.",
"= 20.",
"Select option D."
] | math | Answer: 20
Steps:
1. 6×5×4/6.
2. = 20.
3. Select option D.
Answer: (D) | combinatorics |
Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí. | null | mat_0229 | concept | cs | Co je protipříklad? | [
"Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.",
"Odpověď tedy platí: Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí."
] | math | Answer: Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.
Steps:
1. Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí.
2. Odpověď tedy platí: Protipříklad je případ, kde obecné tvrzení selže, a sám je vyvrátí. | proofs |
The integral is 27. | {
"A": "24",
"B": "30",
"C": "27",
"D": "33"
} | mat_0230 | multiple_choice | en | Compute the integral of 3x² from 0 to 3. | [
"Antiderivative x³.",
"3³ = 27.",
"Select option C."
] | math | Answer: 27
Steps:
1. Antiderivative x³.
2. 3³ = 27.
3. Select option C.
Answer: (C) | calculus |
The answer is Projection. | {
"A": "Guessing",
"B": "Projection",
"C": "Hoping",
"D": "Waiting"
} | mat_0231 | multiple_choice | en | Choose: what finds closest point? | [
"Error orthogonal.",
"So: Projection.",
"Select option B."
] | math | Answer: Projection
Steps:
1. Error orthogonal.
2. So: Projection.
3. Select option B.
Answer: (B) | linear_algebra |
P = 1/10. | null | mat_0232 | worked_example | en | A fair 10-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 10.",
"P = 1/10."
] | math | Answer: P = 1/10.
Steps:
1. Favourable: 1.
2. Possible: 10.
3. P = 1/10. | probability |
Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá. | null | mat_0233 | concept | cs | Proč prvočísla? | [
"Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.",
"Odpověď tedy platí: Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá."
] | math | Answer: Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.
Steps:
1. Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá.
2. Odpověď tedy platí: Prvočísla jsou atomy násobení, každé číslo se jednoznačně rozkládá. | number_theory |
The answer is nCr. | {
"A": "nCr",
"B": "n+r",
"C": "n*r",
"D": "n+r!"
} | mat_0234 | multiple_choice | en | Choose: what counts subsets? | [
"Order divided out.",
"So: nCr.",
"Select option A."
] | math | Answer: nCr
Steps:
1. Order divided out.
2. So: nCr.
3. Select option A.
Answer: (A) | combinatorics |
The answer is Every possibility. | {
"A": "Easy ones",
"B": "Every possibility",
"C": "First two",
"D": "Odd cases"
} | mat_0235 | multiple_choice | en | Choose: what must cases cover? | [
"Gaps break proofs.",
"So: Every possibility.",
"Select option B."
] | math | Answer: Every possibility
Steps:
1. Gaps break proofs.
2. So: Every possibility.
3. Select option B.
Answer: (B) | proofs |
The limit is 6. | {
"A": "4",
"B": "8",
"C": "10",
"D": "6"
} | mat_0236 | multiple_choice | en | Find the limit of (x² − 3²)/(x − 3) as x → 3. | [
"Factor (x−3)(x+3).",
"Cancel, get 2×3 = 6.",
"Select option D."
] | math | Answer: 6
Steps:
1. Factor (x−3)(x+3).
2. Cancel, get 2×3 = 6.
3. Select option D.
Answer: (D) | calculus |
A basis is an independent spanning set, giving every vector unique coordinates. | null | mat_0237 | concept | en | What is a basis? | [
"A basis is an independent spanning set, giving every vector unique coordinates.",
"So the answer stands: A basis is an independent spanning set, giving every vector unique coordinates."
] | math | Answer: A basis is an independent spanning set, giving every vector unique coordinates.
Steps:
1. A basis is an independent spanning set, giving every vector unique coordinates.
2. So the answer stands: A basis is an independent spanning set, giving every vector unique coordinates. | linear_algebra |
E = 5. | null | mat_0238 | worked_example | en | Compute E of a uniform draw from {1..9}. | [
"Mean of 1..9.",
"(9+1)/2.",
"E = 5."
] | math | Answer: E = 5.
Steps:
1. Mean of 1..9.
2. (9+1)/2.
3. E = 5. | probability |
Zbytek: 5. | null | mat_0239 | worked_example | cs | Spočítej (61) mod 7. | [
"Děl 61 sedmi.",
"7×8 = 56.",
"61 − 56 = 5."
] | math | Answer: Zbytek: 5.
Steps:
1. Děl 61 sedmi.
2. 7×8 = 56.
3. 61 − 56 = 5. | number_theory |
The answer is Bijection. | {
"A": "Wild map",
"B": "Bijection",
"C": "Blind guess",
"D": "Vague hope"
} | mat_0240 | multiple_choice | en | Choose: what pairs sets equally? | [
"Inverse proves size.",
"So: Bijection.",
"Select option B."
] | math | Answer: Bijection
Steps:
1. Inverse proves size.
2. So: Bijection.
3. Select option B.
Answer: (B) | combinatorics |
The table has 16 rows. | {
"A": "14",
"B": "18",
"C": "16",
"D": "20"
} | mat_0241 | multiple_choice | en | How many rows does a truth table with 4 variables have? | [
"2^4 rows.",
"Total: 16.",
"Select option C."
] | math | Answer: 16
Steps:
1. 2^4 rows.
2. Total: 16.
3. Select option C.
Answer: (C) | proofs |
Integrál: 27. | null | mat_0242 | worked_example | cs | Vyčísli integrál 3x² od 0 do 3. | [
"Primitivní: x³.",
"Vyčísli 0 až 3.",
"3³ − 0 = 27."
] | math | Answer: Integrál: 27.
Steps:
1. Primitivní: x³.
2. Vyčísli 0 až 3.
3. 3³ − 0 = 27. | calculus |
Dimension is the basis size, fixing how many numbers describe every vector exactly. | null | mat_0243 | concept | en | Why count dimension? | [
"Dimension is the basis size, fixing how many numbers describe every vector exactly.",
"So the answer stands: Dimension is the basis size, fixing how many numbers describe every vector exactly."
] | math | Answer: Dimension is the basis size, fixing how many numbers describe every vector exactly.
Steps:
1. Dimension is the basis size, fixing how many numbers describe every vector exactly.
2. So the answer stands: Dimension is the basis size, fixing how many numbers describe every vector exactly. | linear_algebra |
Sample averages converge to expectation as trials grow, justifying statistics from data. | null | mat_0244 | concept | en | What is the law of large numbers? | [
"Sample averages converge to expectation as trials grow, justifying statistics from data.",
"So the answer stands: Sample averages converge to expectation as trials grow, justifying statistics from data."
] | math | Answer: Sample averages converge to expectation as trials grow, justifying statistics from data.
Steps:
1. Sample averages converge to expectation as trials grow, justifying statistics from data.
2. So the answer stands: Sample averages converge to expectation as trials grow, justifying statistics from data. | probability |
gcd = 10. | null | mat_0245 | worked_example | en | Compute gcd(20, 30) step by step. | [
"Factor: 20=4×5, 30=6×5.",
"Common: 5×gcd(4,6).",
"= 5×2 = 10."
] | math | Answer: gcd = 10.
Steps:
1. Factor: 20=4×5, 30=6×5.
2. Common: 5×gcd(4,6).
3. = 5×2 = 10. | number_theory |
A permutation orders objects, and n distinct objects admit exactly n! orders. | null | mat_0246 | concept | en | What is a permutation? | [
"A permutation orders objects, and n distinct objects admit exactly n! orders.",
"So the answer stands: A permutation orders objects, and n distinct objects admit exactly n! orders."
] | math | Answer: A permutation orders objects, and n distinct objects admit exactly n! orders.
Steps:
1. A permutation orders objects, and n distinct objects admit exactly n! orders.
2. So the answer stands: A permutation orders objects, and n distinct objects admit exactly n! orders. | combinatorics |
Sum: 55. | null | mat_0247 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 10. | [
"Pair 1+10, 2+9, ...",
"Use 10×11/2.",
"= 55."
] | math | Answer: Sum: 55.
Steps:
1. Pair 1+10, 2+9, ...
2. Use 10×11/2.
3. = 55. | proofs |
Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy. | null | mat_0248 | concept | cs | Proč integrovat? | [
"Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.",
"Odpověď tedy platí: Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy."
] | math | Answer: Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.
Steps:
1. Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy.
2. Odpověď tedy platí: Integrace obnovuje úhrny z rychlostí, počítá plochy a objemy. | calculus |
Determinant: 40. | null | mat_0249 | worked_example | en | Compute det [[6,1],[2,7]] step by step. | [
"ad − bc.",
"6×7 − 1×2.",
"= 40."
] | math | Answer: Determinant: 40.
Steps:
1. ad − bc.
2. 6×7 − 1×2.
3. = 40. | linear_algebra |
The answer is Normal distribution. | {
"A": "Wild guess",
"B": "Normal distribution",
"C": "Flat line",
"D": "Single dot"
} | mat_0250 | multiple_choice | en | Choose: what models error bells? | [
"Averages turn normal.",
"So: Normal distribution.",
"Select option B."
] | math | Answer: Normal distribution
Steps:
1. Averages turn normal.
2. So: Normal distribution.
3. Select option B.
Answer: (B) | probability |
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