answer string | choices dict | id string | kind string | lang string | question string | steps list | subject string | text string | topic string |
|---|---|---|---|---|---|---|---|---|---|
It composes linear maps via row-column dot products, and order matters since maps differ. | null | mat_0051 | concept | en | What is matrix multiplication? | [
"It composes linear maps via row-column dot products, and order matters since maps differ.",
"So the answer stands: It composes linear maps via row-column dot products, and order matters since maps differ."
] | math | Answer: It composes linear maps via row-column dot products, and order matters since maps differ.
Steps:
1. It composes linear maps via row-column dot products, and order matters since maps differ.
2. So the answer stands: It composes linear maps via row-column dot products, and order matters since maps differ. | linear_algebra |
Independent events multiply probabilities, since one's outcome never affects the other's. | null | mat_0052 | concept | en | What is independence? | [
"Independent events multiply probabilities, since one's outcome never affects the other's.",
"So the answer stands: Independent events multiply probabilities, since one's outcome never affects the other's."
] | math | Answer: Independent events multiply probabilities, since one's outcome never affects the other's.
Steps:
1. Independent events multiply probabilities, since one's outcome never affects the other's.
2. So the answer stands: Independent events multiply probabilities, since one's outcome never affects the other's. | probability |
Zbytek je 1. | {
"A": "2",
"B": "1",
"C": "3",
"D": "4"
} | mat_0053 | multiple_choice | cs | Vypočítej (100) mod 9. | [
"100 ÷ 9.",
"Zbytek 1.",
"Vyber možnost B."
] | math | Answer: 1
Steps:
1. 100 ÷ 9.
2. Zbytek 1.
3. Vyber možnost B.
Answer: (B) | number_theory |
C(5,3) is 10. | {
"A": "5",
"B": "15",
"C": "10",
"D": "20"
} | mat_0054 | multiple_choice | en | Compute C(5,3). | [
"5×4×3/6.",
"= 10.",
"Select option C."
] | math | Answer: 10
Steps:
1. 5×4×3/6.
2. = 10.
3. Select option C.
Answer: (C) | combinatorics |
Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers. | null | mat_0055 | concept | en | What is induction? | [
"Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.",
"So the answer stands: Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers."
] | math | Answer: Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.
Steps:
1. Induction proves a base case and shows each case implies the next, so the claim holds for all natural numbers.
2. So the answer stands: Induction proves a base case and shows each case implie... | proofs |
Limit: 8. | null | mat_0056 | worked_example | en | Evaluate the limit of (x² − 4²)/(x − 4) as x → 4. | [
"Factor: (x−4)(x+4).",
"Cancel (x−4).",
"Plug 4: 4+4 = 8."
] | math | Answer: Limit: 8.
Steps:
1. Factor: (x−4)(x+4).
2. Cancel (x−4).
3. Plug 4: 4+4 = 8. | calculus |
The dot product is 11. | {
"A": "7",
"B": "15",
"C": "11",
"D": "19"
} | mat_0057 | multiple_choice | en | Find the dot product (1,2) · (3,4). | [
"1×3 + 2×4.",
"Product = 11.",
"Select option C."
] | math | Answer: 11
Steps:
1. 1×3 + 2×4.
2. Product = 11.
3. Select option C.
Answer: (C) | linear_algebra |
P = 1/7. | null | mat_0058 | worked_example | en | A fair 7-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 7.",
"P = 1/7."
] | math | Answer: P = 1/7.
Steps:
1. Favourable: 1.
2. Possible: 7.
3. P = 1/7. | probability |
gcd = 4. | null | mat_0059 | worked_example | en | Compute gcd(8, 12) step by step. | [
"Factor: 8=4×2, 12=6×2.",
"Common: 2×gcd(4,6).",
"= 2×2 = 4."
] | math | Answer: gcd = 4.
Steps:
1. Factor: 8=4×2, 12=6×2.
2. Common: 2×gcd(4,6).
3. = 2×2 = 4. | number_theory |
P(5,3) = 60. | null | mat_0060 | worked_example | en | Show the computation of 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 |
Half: 6, so 12 is even. | null | mat_0061 | worked_example | en | Show N² + N is even for N = 3 by computing its half. | [
"3² + 3 = 12.",
"12 / 2 = 6.",
"Integer, so even."
] | math | Answer: Half: 6, so 12 is even.
Steps:
1. 3² + 3 = 12.
2. 12 / 2 = 6.
3. Integer, so even. | proofs |
The derivative is 4. | {
"A": "4",
"B": "2",
"C": "6",
"D": "8"
} | mat_0062 | multiple_choice | en | For f(x) = 1x² + 2x, find f'(1). | [
"f'(x) = 2x + 2.",
"At 1: 4.",
"Select option A."
] | math | Answer: 4
Steps:
1. f'(x) = 2x + 2.
2. At 1: 4.
3. Select option A.
Answer: (A) | calculus |
The answer is Nonzero determinant. | {
"A": "Big entries",
"B": "Round shape",
"C": "Long name",
"D": "Nonzero determinant"
} | mat_0063 | multiple_choice | en | Choose: what detects invertibility? | [
"Zero means collapse.",
"So: Nonzero determinant.",
"Select option D."
] | math | Answer: Nonzero determinant
Steps:
1. Zero means collapse.
2. So: Nonzero determinant.
3. Select option D.
Answer: (D) | linear_algebra |
Expectation averages outcomes by probability, predicting long-run results of gambles. | null | mat_0064 | concept | en | Why expected value? | [
"Expectation averages outcomes by probability, predicting long-run results of gambles.",
"So the answer stands: Expectation averages outcomes by probability, predicting long-run results of gambles."
] | math | Answer: Expectation averages outcomes by probability, predicting long-run results of gambles.
Steps:
1. Expectation averages outcomes by probability, predicting long-run results of gambles.
2. So the answer stands: Expectation averages outcomes by probability, predicting long-run results of gambles. | probability |
Euclid multiplies known primes and adds one, forcing a new prime factor each time. | null | mat_0065 | concept | en | Why infinitely many primes? | [
"Euclid multiplies known primes and adds one, forcing a new prime factor each time.",
"So the answer stands: Euclid multiplies known primes and adds one, forcing a new prime factor each time."
] | math | Answer: Euclid multiplies known primes and adds one, forcing a new prime factor each time.
Steps:
1. Euclid multiplies known primes and adds one, forcing a new prime factor each time.
2. So the answer stands: Euclid multiplies known primes and adds one, forcing a new prime factor each time. | number_theory |
C(7,3) = 35. | null | mat_0066 | worked_example | en | Show the computation of C(7,3). | [
"n(n−1)(n−2)/6.",
"7×6×5/6.",
"= 35."
] | math | Answer: C(7,3) = 35.
Steps:
1. n(n−1)(n−2)/6.
2. 7×6×5/6.
3. = 35. | combinatorics |
The base case anchors induction, because without a true start the chain of implications proves nothing. | null | mat_0067 | concept | en | Why prove the base case? | [
"The base case anchors induction, because without a true start the chain of implications proves nothing.",
"So the answer stands: The base case anchors induction, because without a true start the chain of implications proves nothing."
] | math | Answer: The base case anchors induction, because without a true start the chain of implications proves nothing.
Steps:
1. The base case anchors induction, because without a true start the chain of implications proves nothing.
2. So the answer stands: The base case anchors induction, because without a true start the cha... | proofs |
The integral is 8. | {
"A": "6",
"B": "8",
"C": "10",
"D": "12"
} | mat_0068 | multiple_choice | en | Compute the integral of 3x² from 0 to 2. | [
"Antiderivative x³.",
"2³ = 8.",
"Select option B."
] | math | Answer: 8
Steps:
1. Antiderivative x³.
2. 2³ = 8.
3. Select option B.
Answer: (B) | calculus |
The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension. | null | mat_0069 | concept | en | Why care about determinant? | [
"The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.",
"So the answer stands: The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension."
] | math | Answer: The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.
Steps:
1. The determinant scales volumes and detects invertibility, with zero meaning collapse to lower dimension.
2. So the answer stands: The determinant scales volumes and detects invertibility, with zero... | linear_algebra |
E = 3.5. | null | mat_0070 | worked_example | en | Compute E of a uniform draw from {1..6}. | [
"Mean of 1..6.",
"(6+1)/2.",
"E = 3.5."
] | math | Answer: E = 3.5.
Steps:
1. Mean of 1..6.
2. (6+1)/2.
3. E = 3.5. | probability |
Remainder: 1. | null | mat_0071 | worked_example | en | Compute (50) mod 7. | [
"Divide 50 by 7.",
"7×7 = 49.",
"50 − 49 = 1."
] | math | Answer: Remainder: 1.
Steps:
1. Divide 50 by 7.
2. 7×7 = 49.
3. 50 − 49 = 1. | number_theory |
C(7,2) = 21. | null | mat_0072 | worked_example | en | Show the computation of C(7,2). | [
"Formula n(n−1)/2.",
"7×6/2.",
"= 21."
] | math | Answer: C(7,2) = 21.
Steps:
1. Formula n(n−1)/2.
2. 7×6/2.
3. = 21. | combinatorics |
Tabulka má 4 řádků. | {
"A": "4",
"B": "2",
"C": "6",
"D": "8"
} | mat_0073 | multiple_choice | cs | Kolik řádků má pravdivostní tabulka s 2 proměnnými? | [
"2^2 řádků.",
"Celkem: 4.",
"Vyber možnost A."
] | math | Answer: 4
Steps:
1. 2^2 řádků.
2. Celkem: 4.
3. Vyber možnost A.
Answer: (A) | proofs |
A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point. | null | mat_0074 | concept | en | What is a derivative? | [
"A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.",
"So the answer stands: A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point."
] | math | Answer: A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.
Steps:
1. A derivative is the instantaneous rate of change, giving the slope of the tangent line at each point.
2. So the answer stands: A derivative is the instantaneous rate of change, giving the slope of the... | calculus |
Dot product: 11. | null | mat_0075 | worked_example | en | Compute (2,1) · (3,5) step by step. | [
"Multiply components.",
"2×3 + 1×5.",
"= 11."
] | math | Answer: Dot product: 11.
Steps:
1. Multiply components.
2. 2×3 + 1×5.
3. = 11. | linear_algebra |
P(red) = 4/8. | null | mat_0076 | worked_example | en | A bag holds 4 red and 4 blue marbles. Show P(red). | [
"Red: 4.",
"Total: 8.",
"P = 4/8."
] | math | Answer: P(red) = 4/8.
Steps:
1. Red: 4.
2. Total: 8.
3. P = 4/8. | probability |
The lcm is 12. | {
"A": "10",
"B": "14",
"C": "12",
"D": "16"
} | mat_0077 | multiple_choice | en | Find lcm(4, 6). | [
"4×6/gcd.",
"lcm = 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. 4×6/gcd.
2. lcm = 12.
3. Select option C.
Answer: (C) | number_theory |
P(6,3) = 120. | null | mat_0078 | worked_example | en | Show the computation of P(6,3). | [
"n(n−1)(n−2).",
"6×5×4.",
"= 120."
] | math | Answer: P(6,3) = 120.
Steps:
1. n(n−1)(n−2).
2. 6×5×4.
3. = 120. | combinatorics |
Součet: 21. | null | mat_0079 | worked_example | cs | Dokaž vzorcem: najdi součet 1 + 2 + ... + 6. | [
"Páruj 1+6, 2+5, ...",
"Použij 6×7/2.",
"= 21."
] | math | Answer: Součet: 21.
Steps:
1. Páruj 1+6, 2+5, ...
2. Použij 6×7/2.
3. = 21. | proofs |
Integrál je 8. | {
"A": "6",
"B": "8",
"C": "10",
"D": "12"
} | mat_0080 | multiple_choice | cs | Vypočítej integrál 3x² od 0 do 2. | [
"Primitivní x³.",
"2³ = 8.",
"Vyber možnost B."
] | math | Answer: 8
Steps:
1. Primitivní x³.
2. 2³ = 8.
3. Vyber možnost B.
Answer: (B) | calculus |
Determinant: 10. | null | mat_0081 | worked_example | en | Compute det [[3,1],[2,4]] step by step. | [
"ad − bc.",
"3×4 − 1×2.",
"= 10."
] | math | Answer: Determinant: 10.
Steps:
1. ad − bc.
2. 3×4 − 1×2.
3. = 10. | linear_algebra |
Variance averages squared deviations, measuring how spread out a random variable is. | null | mat_0082 | concept | en | What is variance? | [
"Variance averages squared deviations, measuring how spread out a random variable is.",
"So the answer stands: Variance averages squared deviations, measuring how spread out a random variable is."
] | math | Answer: Variance averages squared deviations, measuring how spread out a random variable is.
Steps:
1. Variance averages squared deviations, measuring how spread out a random variable is.
2. So the answer stands: Variance averages squared deviations, measuring how spread out a random variable is. | probability |
The greatest common divisor is the largest integer dividing both numbers at once. | null | mat_0083 | concept | en | What is the gcd? | [
"The greatest common divisor is the largest integer dividing both numbers at once.",
"So the answer stands: The greatest common divisor is the largest integer dividing both numbers at once."
] | math | Answer: The greatest common divisor is the largest integer dividing both numbers at once.
Steps:
1. The greatest common divisor is the largest integer dividing both numbers at once.
2. So the answer stands: The greatest common divisor is the largest integer dividing both numbers at once. | number_theory |
C(8,3) = 56. | null | mat_0084 | worked_example | en | Show the computation of C(8,3). | [
"n(n−1)(n−2)/6.",
"8×7×6/6.",
"= 56."
] | math | Answer: C(8,3) = 56.
Steps:
1. n(n−1)(n−2)/6.
2. 8×7×6/6.
3. = 56. | combinatorics |
The value is 12. | {
"A": "10",
"B": "14",
"C": "12",
"D": "16"
} | mat_0085 | multiple_choice | en | Compute N² + N for N = 3. | [
"3² + 3.",
"Value = 12.",
"Select option C."
] | math | Answer: 12
Steps:
1. 3² + 3.
2. Value = 12.
3. Select option C.
Answer: (C) | proofs |
The limit is 4. | {
"A": "2",
"B": "6",
"C": "4",
"D": "8"
} | mat_0086 | multiple_choice | en | Find the limit of (x² − 2²)/(x − 2) as x → 2. | [
"Factor (x−2)(x+2).",
"Cancel, get 2×2 = 4.",
"Select option C."
] | math | Answer: 4
Steps:
1. Factor (x−2)(x+2).
2. Cancel, get 2×2 = 4.
3. Select option C.
Answer: (C) | calculus |
Trace: 9. | null | mat_0087 | worked_example | en | Compute the trace of [[4,0],[0,5]]. | [
"Sum diagonal.",
"4 + 5.",
"= 9."
] | math | Answer: Trace: 9.
Steps:
1. Sum diagonal.
2. 4 + 5.
3. = 9. | linear_algebra |
The expected value is 2.5. | {
"A": "3.5",
"B": "2.5",
"C": "1.5",
"D": "4.5"
} | mat_0088 | multiple_choice | en | Find the expected value of a uniform draw from {1..4}. | [
"(4+1)/2.",
"E = 2.5.",
"Select option B."
] | math | Answer: 2.5
Steps:
1. (4+1)/2.
2. E = 2.5.
3. Select option B.
Answer: (B) | probability |
The answer is Primes. | {
"A": "Evens",
"B": "Squares",
"C": "Cubes",
"D": "Primes"
} | mat_0089 | multiple_choice | en | Choose: what are multiplication atoms? | [
"Unique factorization.",
"So: Primes.",
"Select option D."
] | math | Answer: Primes
Steps:
1. Unique factorization.
2. So: Primes.
3. Select option D.
Answer: (D) | number_theory |
The answer is Factorial. | {
"A": "Sum",
"B": "Average",
"C": "Guess",
"D": "Factorial"
} | mat_0090 | multiple_choice | en | Choose: what counts orderings? | [
"Choices shrink per slot.",
"So: Factorial.",
"Select option D."
] | math | Answer: Factorial
Steps:
1. Choices shrink per slot.
2. So: Factorial.
3. Select option D.
Answer: (D) | combinatorics |
A counterexample is one case where a universal claim fails, and a single one disproves the whole claim. | null | mat_0091 | concept | en | What is a counterexample? | [
"A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.",
"So the answer stands: A counterexample is one case where a universal claim fails, and a single one disproves the whole claim."
] | math | Answer: A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.
Steps:
1. A counterexample is one case where a universal claim fails, and a single one disproves the whole claim.
2. So the answer stands: A counterexample is one case where a universal claim fails, and a sin... | proofs |
f'(4) = 35. | null | mat_0092 | worked_example | en | For f(x) = 4x² + 3x, compute f'(4) step by step. | [
"f'(x) = 8x + 3.",
"Plug x = 4.",
"8×4 + 3 = 35."
] | math | Answer: f'(4) = 35.
Steps:
1. f'(x) = 8x + 3.
2. Plug x = 4.
3. 8×4 + 3 = 35. | calculus |
Rank counts independent rows or columns, measuring the true dimension of the map's image. | null | mat_0093 | concept | en | What is rank? | [
"Rank counts independent rows or columns, measuring the true dimension of the map's image.",
"So the answer stands: Rank counts independent rows or columns, measuring the true dimension of the map's image."
] | math | Answer: Rank counts independent rows or columns, measuring the true dimension of the map's image.
Steps:
1. Rank counts independent rows or columns, measuring the true dimension of the map's image.
2. So the answer stands: Rank counts independent rows or columns, measuring the true dimension of the map's image. | linear_algebra |
P(red) is 2/7. | {
"A": "5/7",
"B": "2/8",
"C": "2/7",
"D": "3/7"
} | mat_0094 | multiple_choice | en | A bag holds 2 red and 5 blue marbles. What is P(red)? | [
"2 of 7.",
"P = 2/7.",
"Select option C."
] | math | Answer: 2/7
Steps:
1. 2 of 7.
2. P = 2/7.
3. Select option C.
Answer: (C) | probability |
Repeated remainders shrink the pair fast, computing gcds in logarithmic steps. | null | mat_0095 | concept | en | Why Euclid's algorithm? | [
"Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.",
"So the answer stands: Repeated remainders shrink the pair fast, computing gcds in logarithmic steps."
] | math | Answer: Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.
Steps:
1. Repeated remainders shrink the pair fast, computing gcds in logarithmic steps.
2. So the answer stands: Repeated remainders shrink the pair fast, computing gcds in logarithmic steps. | number_theory |
nCr counts subsets of size r from n objects, ignoring order via division by r!. | null | mat_0096 | concept | en | What is nCr? | [
"nCr counts subsets of size r from n objects, ignoring order via division by r!.",
"So the answer stands: nCr counts subsets of size r from n objects, ignoring order via division by r!."
] | math | Answer: nCr counts subsets of size r from n objects, ignoring order via division by r!.
Steps:
1. nCr counts subsets of size r from n objects, ignoring order via division by r!.
2. So the answer stands: nCr counts subsets of size r from n objects, ignoring order via division by r!. | combinatorics |
The answer is Base case. | {
"A": "Fancy notation",
"B": "Long chains",
"C": "Big numbers",
"D": "Base case"
} | mat_0097 | multiple_choice | en | Choose: what anchors induction? | [
"Without base, chain proves nothing.",
"So: Base case.",
"Select option D."
] | math | Answer: Base case
Steps:
1. Without base, chain proves nothing.
2. So: Base case.
3. Select option D.
Answer: (D) | proofs |
Integral: 125. | null | mat_0098 | worked_example | en | Evaluate the integral of 3x² from 0 to 5. | [
"Antiderivative: x³.",
"Evaluate 0 to 5.",
"5³ − 0 = 125."
] | math | Answer: Integral: 125.
Steps:
1. Antiderivative: x³.
2. Evaluate 0 to 5.
3. 5³ − 0 = 125. | calculus |
Row reduction exposes pivots and solutions, solving linear systems mechanically every time. | null | mat_0099 | concept | en | Why row-reduce? | [
"Row reduction exposes pivots and solutions, solving linear systems mechanically every time.",
"So the answer stands: Row reduction exposes pivots and solutions, solving linear systems mechanically every time."
] | math | Answer: Row reduction exposes pivots and solutions, solving linear systems mechanically every time.
Steps:
1. Row reduction exposes pivots and solutions, solving linear systems mechanically every time.
2. So the answer stands: Row reduction exposes pivots and solutions, solving linear systems mechanically every time. | linear_algebra |
P = 1/8. | null | mat_0100 | worked_example | en | A fair 8-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 8.",
"P = 1/8."
] | math | Answer: P = 1/8.
Steps:
1. Favourable: 1.
2. Possible: 8.
3. P = 1/8. | probability |
Zbytek: 1. | null | mat_0101 | worked_example | cs | Spočítej (50) mod 7. | [
"Děl 50 sedmi.",
"7×7 = 49.",
"50 − 49 = 1."
] | math | Answer: Zbytek: 1.
Steps:
1. Děl 50 sedmi.
2. 7×7 = 49.
3. 50 − 49 = 1. | number_theory |
C(4,2) je 6. | {
"A": "6",
"B": "2",
"C": "10",
"D": "14"
} | mat_0102 | multiple_choice | cs | Vypočítej kombinační číslo C(4,2). | [
"4×3/2.",
"= 6.",
"Vyber možnost A."
] | math | Answer: 6
Steps:
1. 4×3/2.
2. = 6.
3. Vyber možnost A.
Answer: (A) | combinatorics |
Rows: 16. | null | mat_0103 | worked_example | en | A proof checks all cases of 4 boolean variables. How many rows? | [
"2 choices each.",
"2^4 = 16."
] | math | Answer: Rows: 16.
Steps:
1. 2 choices each.
2. 2^4 = 16. | proofs |
Limita je hodnota, k níž se funkce blíží, i když v bodě není definována. | null | mat_0104 | concept | cs | Co je limita? | [
"Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.",
"Odpověď tedy platí: Limita je hodnota, k níž se funkce blíží, i když v bodě není definována."
] | math | Answer: Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.
Steps:
1. Limita je hodnota, k níž se funkce blíží, i když v bodě není definována.
2. Odpověď tedy platí: Limita je hodnota, k níž se funkce blíží, i když v bodě není definována. | calculus |
Dot product: 19. | null | mat_0105 | worked_example | en | Compute (3,2) · (3,5) step by step. | [
"Multiply components.",
"3×3 + 2×5.",
"= 19."
] | math | Answer: Dot product: 19.
Steps:
1. Multiply components.
2. 3×3 + 2×5.
3. = 19. | linear_algebra |
E = 4. | null | mat_0106 | worked_example | en | Compute E of a uniform draw from {1..7}. | [
"Mean of 1..7.",
"(7+1)/2.",
"E = 4."
] | math | Answer: E = 4.
Steps:
1. Mean of 1..7.
2. (7+1)/2.
3. E = 4. | probability |
The answer is Chinese remainder. | {
"A": "Chinese remainder",
"B": "Blind luck",
"C": "Wild guess",
"D": "Long nap"
} | mat_0107 | multiple_choice | en | Choose: what merges congruences? | [
"One solution modulo product.",
"So: Chinese remainder.",
"Select option A."
] | math | Answer: Chinese remainder
Steps:
1. One solution modulo product.
2. So: Chinese remainder.
3. Select option A.
Answer: (A) | number_theory |
It builds nCr rows by addition, revealing symmetry and binomial patterns visually. | null | mat_0108 | concept | en | Why Pascal's triangle? | [
"It builds nCr rows by addition, revealing symmetry and binomial patterns visually.",
"So the answer stands: It builds nCr rows by addition, revealing symmetry and binomial patterns visually."
] | math | Answer: It builds nCr rows by addition, revealing symmetry and binomial patterns visually.
Steps:
1. It builds nCr rows by addition, revealing symmetry and binomial patterns visually.
2. So the answer stands: It builds nCr rows by addition, revealing symmetry and binomial patterns visually. | combinatorics |
Sum: 36. | null | mat_0109 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 8. | [
"Pair 1+8, 2+7, ...",
"Use 8×9/2.",
"= 36."
] | math | Answer: Sum: 36.
Steps:
1. Pair 1+8, 2+7, ...
2. Use 8×9/2.
3. = 36. | proofs |
Limit: 10. | null | mat_0110 | worked_example | en | Evaluate the limit of (x² − 5²)/(x − 5) as x → 5. | [
"Factor: (x−5)(x+5).",
"Cancel (x−5).",
"Plug 5: 5+5 = 10."
] | math | Answer: Limit: 10.
Steps:
1. Factor: (x−5)(x+5).
2. Cancel (x−5).
3. Plug 5: 5+5 = 10. | calculus |
Determinant: 18. | null | mat_0111 | worked_example | en | Compute det [[4,1],[2,5]] step by step. | [
"ad − bc.",
"4×5 − 1×2.",
"= 18."
] | math | Answer: Determinant: 18.
Steps:
1. ad − bc.
2. 4×5 − 1×2.
3. = 18. | linear_algebra |
Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování. | null | mat_0112 | concept | cs | Co je pravděpodobnost? | [
"Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.",
"Odpověď tedy platí: Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování."
] | math | Answer: Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.
Steps:
1. Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování.
2. Odpověď tedy platí: Pravděpodobnost měří šanci od 0 do 1 a axiomy zaručují soudržné uvažování. | probability |
The gcd is 9. | {
"A": "6",
"B": "9",
"C": "12",
"D": "15"
} | mat_0113 | multiple_choice | en | Find gcd(18, 27). | [
"Factor 3 out twice.",
"gcd = 9.",
"Select option B."
] | math | Answer: 9
Steps:
1. Factor 3 out twice.
2. gcd = 9.
3. Select option B.
Answer: (B) | number_theory |
C(8,2) = 28. | null | mat_0114 | worked_example | en | Show the computation of C(8,2). | [
"Formula n(n−1)/2.",
"8×7/2.",
"= 28."
] | math | Answer: C(8,2) = 28.
Steps:
1. Formula n(n−1)/2.
2. 8×7/2.
3. = 28. | combinatorics |
The answer is Witness or counting. | {
"A": "Witness or counting",
"B": "Long names",
"C": "Big fonts",
"D": "Hope alone"
} | mat_0115 | multiple_choice | en | Choose: what proves existence? | [
"Build or count it.",
"So: Witness or counting.",
"Select option A."
] | math | Answer: Witness or counting
Steps:
1. Build or count it.
2. So: Witness or counting.
3. Select option A.
Answer: (A) | proofs |
Differentiation finds maxima, minima, and speeds, turning change questions into algebra. | null | mat_0116 | concept | en | Why differentiate? | [
"Differentiation finds maxima, minima, and speeds, turning change questions into algebra.",
"So the answer stands: Differentiation finds maxima, minima, and speeds, turning change questions into algebra."
] | math | Answer: Differentiation finds maxima, minima, and speeds, turning change questions into algebra.
Steps:
1. Differentiation finds maxima, minima, and speeds, turning change questions into algebra.
2. So the answer stands: Differentiation finds maxima, minima, and speeds, turning change questions into algebra. | calculus |
Trace: 11. | null | mat_0117 | worked_example | en | Compute the trace of [[5,0],[0,6]]. | [
"Sum diagonal.",
"5 + 6.",
"= 11."
] | math | Answer: Trace: 11.
Steps:
1. Sum diagonal.
2. 5 + 6.
3. = 11. | linear_algebra |
Pravděpodobnost je 1/6. | {
"A": "1/6",
"B": "1/5",
"C": "1/7",
"D": "2/6"
} | mat_0118 | multiple_choice | cs | Hází se férovou 6-stěnnou kostkou. Jaká je pravděpodobnost jednoho konkrétního čísla? | [
"1 stěna z 6.",
"P = 1/6.",
"Vyber možnost A."
] | math | Answer: 1/6
Steps:
1. 1 stěna z 6.
2. P = 1/6.
3. Vyber možnost A.
Answer: (A) | probability |
NSN = 40. | null | mat_0119 | worked_example | cs | Spočítej NSN(5, 8) krok za krokem. | [
"NSD(5,8) = 1.",
"5×8/1.",
"= 40."
] | math | Answer: NSN = 40.
Steps:
1. NSD(5,8) = 1.
2. 5×8/1.
3. = 40. | number_theory |
The answer is Recurrence. | {
"A": "Recurrence",
"B": "Prophecy",
"C": "Rumour",
"D": "Dream"
} | mat_0120 | multiple_choice | en | Choose: what defines from the past? | [
"Base plus step.",
"So: Recurrence.",
"Select option A."
] | math | Answer: Recurrence
Steps:
1. Base plus step.
2. So: Recurrence.
3. Select option A.
Answer: (A) | combinatorics |
Half: 10, so 20 is even. | null | mat_0121 | worked_example | en | Show N² + N is even for N = 4 by computing its half. | [
"4² + 4 = 20.",
"20 / 2 = 10.",
"Integer, so even."
] | math | Answer: Half: 10, so 20 is even.
Steps:
1. 4² + 4 = 20.
2. 20 / 2 = 10.
3. Integer, so even. | proofs |
The chain rule differentiates compositions by multiplying outer and inner derivatives together. | null | mat_0122 | concept | en | What is the chain rule? | [
"The chain rule differentiates compositions by multiplying outer and inner derivatives together.",
"So the answer stands: The chain rule differentiates compositions by multiplying outer and inner derivatives together."
] | math | Answer: The chain rule differentiates compositions by multiplying outer and inner derivatives together.
Steps:
1. The chain rule differentiates compositions by multiplying outer and inner derivatives together.
2. So the answer stands: The chain rule differentiates compositions by multiplying outer and inner derivatives... | calculus |
An inverse matrix undoes the original map, existing exactly when the determinant is nonzero. | null | mat_0123 | concept | en | What is an inverse? | [
"An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.",
"So the answer stands: An inverse matrix undoes the original map, existing exactly when the determinant is nonzero."
] | math | Answer: An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.
Steps:
1. An inverse matrix undoes the original map, existing exactly when the determinant is nonzero.
2. So the answer stands: An inverse matrix undoes the original map, existing exactly when the determinant is nonzero... | linear_algebra |
It returns spread to original units, making variability interpretable in context. | null | mat_0124 | concept | en | Why standard deviation? | [
"It returns spread to original units, making variability interpretable in context.",
"So the answer stands: It returns spread to original units, making variability interpretable in context."
] | math | Answer: It returns spread to original units, making variability interpretable in context.
Steps:
1. It returns spread to original units, making variability interpretable in context.
2. So the answer stands: It returns spread to original units, making variability interpretable in context. | probability |
The least common multiple is the smallest positive integer both numbers divide. | null | mat_0125 | concept | en | What is the lcm? | [
"The least common multiple is the smallest positive integer both numbers divide.",
"So the answer stands: The least common multiple is the smallest positive integer both numbers divide."
] | math | Answer: The least common multiple is the smallest positive integer both numbers divide.
Steps:
1. The least common multiple is the smallest positive integer both numbers divide.
2. So the answer stands: The least common multiple is the smallest positive integer both numbers divide. | number_theory |
P(7,3) = 210. | null | mat_0126 | worked_example | en | Show the computation of P(7,3). | [
"n(n−1)(n−2).",
"7×6×5.",
"= 210."
] | math | Answer: P(7,3) = 210.
Steps:
1. n(n−1)(n−2).
2. 7×6×5.
3. = 210. | combinatorics |
Rows: 32. | null | mat_0127 | worked_example | en | A proof checks all cases of 5 boolean variables. How many rows? | [
"2 choices each.",
"2^5 = 32."
] | math | Answer: Rows: 32.
Steps:
1. 2 choices each.
2. 2^5 = 32. | proofs |
The product rule splits (fg)' into f'g + fg', handling products no single rule covers. | null | mat_0128 | concept | en | Why learn the product rule? | [
"The product rule splits (fg)' into f'g + fg', handling products no single rule covers.",
"So the answer stands: The product rule splits (fg)' into f'g + fg', handling products no single rule covers."
] | math | Answer: The product rule splits (fg)' into f'g + fg', handling products no single rule covers.
Steps:
1. The product rule splits (fg)' into f'g + fg', handling products no single rule covers.
2. So the answer stands: The product rule splits (fg)' into f'g + fg', handling products no single rule covers. | calculus |
Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour. | null | mat_0129 | concept | en | Why study eigenvalues? | [
"Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.",
"So the answer stands: Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour."
] | math | Answer: Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.
Steps:
1. Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour.
2. So the answer stands: Eigenvalues reveal stretch directions of a map, unlocking stability and long-term behaviour... | linear_algebra |
P(red) = 5/9. | null | mat_0130 | worked_example | en | A bag holds 5 red and 4 blue marbles. Show P(red). | [
"Red: 5.",
"Total: 9.",
"P = 5/9."
] | math | Answer: P(red) = 5/9.
Steps:
1. Red: 5.
2. Total: 9.
3. P = 5/9. | probability |
The remainder is 8. | {
"A": "0",
"B": "1",
"C": "8",
"D": "2"
} | mat_0131 | multiple_choice | en | Find (107) mod 9. | [
"107 ÷ 9.",
"Remainder 8.",
"Select option C."
] | math | Answer: 8
Steps:
1. 107 ÷ 9.
2. Remainder 8.
3. Select option C.
Answer: (C) | number_theory |
It expands (a+b)^n into nCr-weighted terms, linking algebra to counting. | null | mat_0132 | concept | en | What is the binomial theorem? | [
"It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.",
"So the answer stands: It expands (a+b)^n into nCr-weighted terms, linking algebra to counting."
] | math | Answer: It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.
Steps:
1. It expands (a+b)^n into nCr-weighted terms, linking algebra to counting.
2. So the answer stands: It expands (a+b)^n into nCr-weighted terms, linking algebra to counting. | combinatorics |
The table has 8 rows. | {
"A": "6",
"B": "8",
"C": "10",
"D": "12"
} | mat_0133 | multiple_choice | en | How many rows does a truth table with 3 variables have? | [
"2^3 rows.",
"Total: 8.",
"Select option B."
] | math | Answer: 8
Steps:
1. 2^3 rows.
2. Total: 8.
3. Select option B.
Answer: (B) | proofs |
Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu. | null | mat_0134 | concept | cs | Proč spojitost? | [
"Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.",
"Odpověď tedy platí: Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu."
] | math | Answer: Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.
Steps:
1. Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu.
2. Odpověď tedy platí: Spojité funkce nemají skoky, malé změny vstupu dávají malé změny výstupu. | calculus |
Dot product: 27. | null | mat_0135 | worked_example | en | Compute (4,3) · (3,5) step by step. | [
"Multiply components.",
"4×3 + 3×5.",
"= 27."
] | math | Answer: Dot product: 27.
Steps:
1. Multiply components.
2. 4×3 + 3×5.
3. = 27. | linear_algebra |
P = 1/9. | null | mat_0136 | worked_example | en | A fair 9-sided die is rolled. Show P(one specific number). | [
"Favourable: 1.",
"Possible: 9.",
"P = 1/9."
] | math | Answer: P = 1/9.
Steps:
1. Favourable: 1.
2. Possible: 9.
3. P = 1/9. | probability |
The lcm is 30. | {
"A": "28",
"B": "32",
"C": "34",
"D": "30"
} | mat_0137 | multiple_choice | en | Find lcm(5, 6). | [
"5×6/gcd.",
"lcm = 30.",
"Select option D."
] | math | Answer: 30
Steps:
1. 5×6/gcd.
2. lcm = 30.
3. Select option D.
Answer: (D) | number_theory |
C(5,2) = 10. | null | mat_0138 | worked_example | cs | Ukaž výpočet C(5,2). | [
"Vzorec n(n−1)/2.",
"5×4/2.",
"= 10."
] | math | Answer: C(5,2) = 10.
Steps:
1. Vzorec n(n−1)/2.
2. 5×4/2.
3. = 10. | combinatorics |
Sum: 45. | null | mat_0139 | worked_example | en | Prove by formula: find the sum 1 + 2 + ... + 9. | [
"Pair 1+9, 2+8, ...",
"Use 9×10/2.",
"= 45."
] | math | Answer: Sum: 45.
Steps:
1. Pair 1+9, 2+8, ...
2. Use 9×10/2.
3. = 45. | proofs |
Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě. | null | mat_0140 | concept | cs | Co je derivace? | [
"Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.",
"Odpověď tedy platí: Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě."
] | math | Answer: Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.
Steps:
1. Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě.
2. Odpověď tedy platí: Derivace je okamžitá rychlost změny, směrnice tečny v každém bodě. | calculus |
Vektor je veličina se směrem a velikostí, sčítá se po složkách. | null | mat_0141 | concept | cs | Co je vektor? | [
"Vektor je veličina se směrem a velikostí, sčítá se po složkách.",
"Odpověď tedy platí: Vektor je veličina se směrem a velikostí, sčítá se po složkách."
] | math | Answer: Vektor je veličina se směrem a velikostí, sčítá se po složkách.
Steps:
1. Vektor je veličina se směrem a velikostí, sčítá se po složkách.
2. Odpověď tedy platí: Vektor je veličina se směrem a velikostí, sčítá se po složkách. | linear_algebra |
A random variable maps outcomes to numbers, letting chance be studied with functions. | null | mat_0142 | concept | en | What is a random variable? | [
"A random variable maps outcomes to numbers, letting chance be studied with functions.",
"So the answer stands: A random variable maps outcomes to numbers, letting chance be studied with functions."
] | math | Answer: A random variable maps outcomes to numbers, letting chance be studied with functions.
Steps:
1. A random variable maps outcomes to numbers, letting chance be studied with functions.
2. So the answer stands: A random variable maps outcomes to numbers, letting chance be studied with functions. | probability |
lcm = 40. | null | mat_0143 | worked_example | en | Compute lcm(5, 8) step by step. | [
"gcd(5,8) = 1.",
"5×8/1.",
"= 40."
] | math | Answer: lcm = 40.
Steps:
1. gcd(5,8) = 1.
2. 5×8/1.
3. = 40. | number_theory |
It counts distributions of identical items into boxes via separator placements. | null | mat_0144 | concept | en | Why stars and bars? | [
"It counts distributions of identical items into boxes via separator placements.",
"So the answer stands: It counts distributions of identical items into boxes via separator placements."
] | math | Answer: It counts distributions of identical items into boxes via separator placements.
Steps:
1. It counts distributions of identical items into boxes via separator placements.
2. So the answer stands: It counts distributions of identical items into boxes via separator placements. | combinatorics |
The sum is 21. | {
"A": "15",
"B": "27",
"C": "21",
"D": "33"
} | mat_0145 | multiple_choice | en | What is the sum 1 + 2 + ... + 6? | [
"6×7/2.",
"Sum = 21.",
"Select option C."
] | math | Answer: 21
Steps:
1. 6×7/2.
2. Sum = 21.
3. Select option C.
Answer: (C) | proofs |
The answer is Instant rate. | {
"A": "Total area",
"B": "Long length",
"C": "Big volume",
"D": "Instant rate"
} | mat_0146 | multiple_choice | en | Choose: what does f' measure? | [
"Tangent slope read.",
"So: Instant rate.",
"Select option D."
] | math | Answer: Instant rate
Steps:
1. Tangent slope read.
2. So: Instant rate.
3. Select option D.
Answer: (D) | calculus |
Determinant: 28. | null | mat_0147 | worked_example | en | Compute det [[5,1],[2,6]] step by step. | [
"ad − bc.",
"5×6 − 1×2.",
"= 28."
] | math | Answer: Determinant: 28.
Steps:
1. ad − bc.
2. 5×6 − 1×2.
3. = 28. | linear_algebra |
The answer is Sample space. | {
"A": "Wish list",
"B": "Guess bag",
"C": "Hope chest",
"D": "Sample space"
} | mat_0148 | multiple_choice | en | Choose: what lists all outcomes? | [
"Events are subsets.",
"So: Sample space.",
"Select option D."
] | math | Answer: Sample space
Steps:
1. Events are subsets.
2. So: Sample space.
3. Select option D.
Answer: (D) | probability |
It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving. | null | mat_0149 | concept | en | Why Bézout's identity? | [
"It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.",
"So the answer stands: It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving."
] | math | Answer: It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.
Steps:
1. It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving.
2. So the answer stands: It writes gcd(a,b) as a combination ax+by, unlocking inverses and Diophantine solving. | number_theory |
C(5,2) is 10. | {
"A": "5",
"B": "10",
"C": "15",
"D": "20"
} | mat_0150 | multiple_choice | en | Compute C(5,2). | [
"5×4/2.",
"= 10.",
"Select option B."
] | math | Answer: 10
Steps:
1. 5×4/2.
2. = 10.
3. Select option B.
Answer: (B) | combinatorics |
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