sample_index int64 17 251k | generation_id int64 0 9 | difficulty_bucket int64 0 4 | llama8b_solve_rate float64 0 1 | chop_fraction float64 0.2 0.8 | problem stringlengths 14 1.29k | ground_truth stringlengths 1 269 | partial_trace stringlengths 62 31k | continuation stringlengths 28 45.2k | parsed_answer stringlengths 1 11.9k ⌀ | is_correct bool 2
classes | solve_rate float64 0 1 | n_correct int64 0 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
17 | 0 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a cycle.
3^1 = 3, so 3 mod 5 is 3.
3^2 = 9, 9 mod 5 is 4.
3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2).
3^4 = 81, 81 mod 5 is 1 (because 5*16=80, 81-80=1).
Oh, so the cycle length here is 4. Because 3^4 mod 5 is 1, so the powers of 3 modulo 5 repeat every 4.
So 3^4 ≡ 1 m... | 1 | true | 1 | 10 |
17 | 1 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^1, 3^2, 3^3, etc., modulo 5 and see if there's a pattern.
3^1 = 3. 3 mod 5 is 3.
3^2 = 9. 9 mod 5 is 4.
3^3 = 27. 27 mod 5 is 2 (since 5*5=25, 27-25=2).
3^4 = 81. 81 mod 5: 5*16=80, so 81-80=1. Oh, that's 1. So 3^4 ≡ 1 mod 5.
That's a cycle of 4. So the powers of 3 modulo 5 repeat every 4. So 3^4 = 1, 3^5 = 3, 3... | 1 | true | 1 | 10 |
17 | 2 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a cycle. Let's start with n=1: 3^1 = 3, so 3 mod 5 is 3. n=2: 3^2 = 9, 9 mod 5 is 4. n=3: 3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2). n=4: 3^4 = 81, 81 mod 5 is 1 (because 5*16=80, 81-80=1). Oh, that's a cycle! 3, 4, 2, 1, and then it repeats because 3^4 is 1 mod 5. So the c... | 1 | true | 1 | 10 |
17 | 3 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a cycle. Let's start with n=1: 3^1 = 3, so 3 mod 5 is 3. n=2: 3^2 = 9, 9 mod 5 is 4. n=3: 3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2). n=4: 3^4 = 81, 81 mod 5 is 1 (5*16=80, 81-80=1). Oh, so the cycle here is 4. Because 3^4 ≡ 1 mod 5. That makes sense because 3 and 5 are copr... | 1 | true | 1 | 10 |
17 | 4 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a pattern. Let's start with n=1: 3^1 = 3, so 3 mod 5 is 3. n=2: 3^2 = 9, 9 mod 5 is 4. n=3: 3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2). n=4: 3^4 = 81, 81 mod 5 is 1 (because 5*16=80, 81-80=1). Oh, so the cycle here is 4. Let's check n=5: 3^5 = 3^4 * 3 = 1*3 = 3 mod 5, which ... | 1 | true | 1 | 10 |
17 | 5 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a cycle. Let's start with n=1: 3^1 = 3, so 3 mod 5 is 3. n=2: 3^2 = 9, 9 mod 5 is 4. n=3: 3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2). n=4: 3^4 = 81, 81 mod 5 is 1 (because 5*16=80, 81-80=1). Oh, so 3^4 is 1 mod 5. That's a cycle of length 4.
So the cycle is 3, 4, 2, 1, and ... | 1 | true | 1 | 10 |
17 | 6 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a cycle. Let's start with n=1: 3^1 = 3, so 3 mod 5 is 3. n=2: 3^2 = 9, 9 mod 5 is 4. n=3: 3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2). n=4: 3^4 = 81, 81 mod 5 is 1 (5*16=80, 81-80=1). Oh, so the cycle length here is 4. Because 3^4 ≡ 1 mod 5. That's a common cycle for 3 mod 5.... | 1 | true | 1 | 10 |
17 | 7 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a cycle.
3^1 = 3 mod 5 = 3
3^2 = 9 mod 5 = 4 (since 9 - 5 = 4)
3^3 = 3*4 = 12 mod 5 = 2 (12 - 10 = 2)
3^4 = 3*2 = 6 mod 5 = 1 (6 - 5 = 1)
Oh, 3^4 is 1 mod 5. That's a cycle of length 4. So the powers of 3 modulo 5 repeat every 4. So 3^(4k) ≡ 1 mod 5.
We need 3^20. 20 is 4*5... | 1 | true | 1 | 10 |
17 | 8 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^1 mod 5, 3^2 mod 5, 3^3 mod 5, etc., to see if there's a cycle.
3^1 = 3, so 3 mod 5 is 3.
3^2 = 9, 9 mod 5 is 4.
3^3 = 27, 27 mod 5 is 2 (since 5*5=25, 27-25=2).
3^4 = 81, 81 mod 5 is 1 (because 5*16=80, 81-80=1).
Oh, so the cycle length here is 4. Because 3^4 ≡ 1 mod 5. That's useful.
So the powers of 3 modulo... | 1 | true | 1 | 10 |
17 | 9 | 4 | 1 | 0.2 | The remainder of $3^{20}$ divided by 5 is __________. | 1 | Okay, let's see. I need to find the remainder when 3^20 is divided by 5. Hmm, modular arithmetic, right? So 3^20 mod 5. I remember that there's something called Fermat's little theorem, but maybe I can do it with patterns of remainders.
First, let's list | 3^n mod 5 for small n to see if there's a pattern.
3^1 mod 5 is 3.
3^2 is 9, 9 mod 5 is 4.
3^3 is 3*4 = 12, 12 mod 5 is 2.
3^4 is 3*2 = 6, 6 mod 5 is 1.
Oh, that's a cycle! So the cycle length here is 4. Because 3^4 mod 5 is 1, so then 3^(4k) mod 5 is 1 for any k. Let's check that. 3^4 is 81, 81 divided by 5 is 16... | 1 | true | 1 | 10 |
54 | 0 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. That way, the equation can be rewritten as (y²)/B - (x²)/(-A) = 1, which is the standard form of a hyperbola opening up and down.
Let's write down the denominators here. The denominators are (k - 3) and (k + 3). F... | (-3, 3) | true | 1 | 10 |
54 | 1 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then the x-term is negative, so the hyperbola is (y²)/B - (x²)/|A| = 1, which is a vertical hyperbola (foci on y-axis).
In the given equation, the denominators are (k-3) and (k+3). So, fo... | (-3, 3) | true | 1 | 10 |
54 | 2 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then the x² term is negative, so it's like (y²)/B - (x²)/|A| = 1, which is a hyperbola opening up and down. That makes sense.
In the given equation, A is (k - 3) and B is (k + 3). So for i... | (-3, 3) | true | 1 | 10 |
54 | 3 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then the x² term is negative, so it's like (y²)/B - (x²)/|A| = 1, which is the standard hyperbola opening up and down.
Let me write that down. The given equation is (x²)/(k - 3) + (y²)/(k... | (-3, 3) | true | 1 | 10 |
54 | 4 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then the x² term is negative, so it's like (y²)/B - (x²)/(-A) = 1, which is the standard form of a hyperbola opening up and down (since the y-term is positive).
In the given equation, the... | (-3, 3) | true | 1 | 10 |
54 | 5 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then (x²)/A is like -x²/|A|, so the equation becomes (y²)/B - (x²)/|A| = 1, which is the standard form of a hyperbola opening up and down.
In this problem, A is (k - 3) and B is (k + 3). ... | (-3, 3) | true | 1 | 10 |
54 | 6 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then the x² term is subtracted, so it's (y²)/B - (x²)/|A| = 1, which is a hyperbola opening up and down.
In this problem, the denominators are (k - 3) and (k + 3). So, the equation is (x²... | (-3, 3) | true | 1 | 10 |
54 | 7 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because in standard hyperbola form, it's (y²)/B - (x²)/A = 1 where B and A are positive. So that would mean (x²)/(k-3) is negative, so (k-3) is negative, and (y²)/(k+3) is positive, so (k+3) is positive.
Let me wr... | (-3, 3) | true | 1 | 10 |
54 | 8 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A to be negative and B to be positive. Because if A is negative, then the x² term is negative, so it's like (y²)/B - (x²)/|A| = 1, which is a hyperbola opening up and down. That makes sense.
Let's write down the denominators: the denominator for x² is (... | (-3, 3) | true | 1 | 10 |
54 | 9 | 1 | 0.203125 | 0.2 | If the equation $\frac{x^{2}}{k-3} + \frac{y^{2}}{k+3} = 1$ represents a hyperbola with foci on the $y$-axis, then the range of the real number $k$ is $\_\_\_\_\_\_\_.$ | (-3, 3) | Okay, let's try to solve this problem. The question is about finding the range of k such that the equation (x²)/(k-3) + (y²)/(k+3) = 1 represents a hyperbola with foci on the y-axis. Hmm, first, I need to remember the standard forms of hyperbolas.
Wait, hyperbolas have two forms: one where the x-term is positive and ... | A + (y²)/B = 1 to be a hyperbola with foci on the y-axis, we need A < 0 and B > 0. Because if A is negative, then the x² term is negative, so it's like (y²)/B - (x²)/|A| = 1, which is a hyperbola opening up and down. That makes sense.
In this problem, the denominators are (k-3) and (k+3). So, for the hyperbola to have... | (-3, 3) | true | 1 | 10 |
123 | 0 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 < x < 3 is part of A, but B is up to 4, so B includes 2 to 3, which is part of A.
Let me list the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So the union should be all the numbers in either A or B. Let's see. The leftmost part: B starts at -2, so from -2 to -1 is in B (since B is [-2,4)), and A is (-∞, -1), so th... | (-∞, 4) | false | 0 | 0 |
123 | 1 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2. Wait, no: A is x < -1 OR 2 ≤ x < 3. So A is two separate intervals. B is a single interval from -2 to 4.
So let's list the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So the union would be all numbers that are in either A or B. Let's see. B starts at -2, so from -2 up to 4. A has numbers less than -1 (so from -∞ ... | (-∞, 4) | false | 0 | 0 |
123 | 2 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 ≤ x < 3 is in A, but B goes up to 4, so B includes 2 to 3 as well.
Let me list out the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So the union should be the combination of these. Let's see. Starting from the left: B starts at -2, so the union will start at -2. Then, B goes up to 4, but A has a part from 2 to 3. W... | (-∞, 4) | false | 0 | 0 |
123 | 3 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 ≤ x < 3 is the other part.
Wait, let's list the intervals again.
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So the union should be all the numbers that are in either A or B. Let's see. The B interval goes from -2 to 4. The A interval has two parts: everything left of -1, and [2, 3). So the part of A that's not in B is (-∞, ... | (-∞, 4) | false | 0 | 0 |
123 | 4 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 is in A? Wait, no. Let's be precise.
Let's list the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So the union would be the combination of these. Let's see. The B set starts at -2, so the leftmost part of the union should be -2, since B goes down to -2. Then A has up to -1, so from -2 to -1 is covered by B (since B ... | (-∞, 4) | false | 0 | 0 |
123 | 5 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 ≤ x < 3 is part of A, and x < -1 is part of A. B is from -2 to 4, so B includes from -2 to 3 (since A's second interval is 2 to 3) and then 3 to 4.
Let me list out the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So to find the union, we need to take all the parts that are in A or B. Let's see:
The left part: B st... | (-∞, 4) | false | 0 | 0 |
123 | 6 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2's interval is part of A. Wait, no: A is two separate intervals. So A is (-∞, -1) ∪ [2, 3). B is [-2, 4). So when we take the union, we need to see how these intervals connect.
Let's list the intervals:
A: (-∞, -1) and [2, 3)
B: [-2, 4)
So the union should be all numbers that are in either A or B. Let's see. The B... | (-∞, 4) | false | 0 | 0 |
123 | 7 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 ≤ x < 3 is another part.
Wait, maybe I should write both intervals with their endpoints.
A is two separate intervals: (-∞, -1) and [2, 3).
B is a single interval: [-2, 4).
So the union would be all the numbers in either A or B. Let's see. B starts at -2, so the leftmost part of the union should be from -2. Becaus... | (-\infty, 4) | false | 0 | 0 |
123 | 8 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | x in [-2, -1) are in A? Wait, no. Wait, A is x < -1, so that's all numbers less than -1, which includes [-2, -1). Because -2 is less than -1, so [-2, -1) is part of A. But B is [-2, 4), so the union of A and B would be all the numbers that are in either A or B.
Let me list the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2... | (-\infty, 4) | false | 0 | 0 |
123 | 9 | 2 | 0.421875 | 0.2 | Given sets $A = \{x\,|\,x < -1 \text{ or } 2 \leq x < 3\}$ and $B = \{x\,|\,-2 \leq x < 4\}$, find the union $A \cup B$. | \\{x | x < 4\\} | Okay, let's try to solve this problem. So, we have two sets A and B, and we need to find their union. First, I need to remember what the union of two sets means. The union A ∪ B is all the elements that are in A, in B, or in both. So, I should probably write out what each set looks like on a number line and then see wh... | 2 ≤ x < 3 is part of A, and x < -1 is the other part. B is -2 ≤ x < 4, so B includes from -2 to 4. So when we take the union, we need to combine all the parts.
Let's list the intervals:
A: (-∞, -1) ∪ [2, 3)
B: [-2, 4)
So the union would be all numbers that are in either A or B. Let's see. The leftmost part of B is ... | (-∞, 4) | false | 0 | 0 |
136 | 0 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. So, y/4 = 1 - x/3, which means y = 4(1 - x/3) = 4 - (4x)/3. Since x and y are positive, x has to be between 0 and 3, right? Because if x is 3, then y would be 0, and if x is 0, y is 4. So x ∈ (0, 3), y ∈ (0, 4).
Now, we need to maximize xy. So s... | 3 | true | 1 | 10 |
136 | 1 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since x and y are positive, x has to be less than 3 (because 12 - 4x > 0 => x < 3) and y has to be positive, so (12 - 4x)/3 > 0 => 12... | 3 | true | 1 | 10 |
136 | 2 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since x and y are positive, x has to be less than 3 (because 12 - 4x > 0 implies x < 3) and y has to be positive, so 12 - 4x > 0, whi... | 3 | true | 1 | 10 |
136 | 3 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since x and y are positive, 12 - 4x has to be positive, so 4x < 12, so x < 3. Also, x is positive, so 0 < x < 3.
Now, the product xy... | 3 | true | 1 | 10 |
136 | 4 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3 = 4 - (4x)/3. Since x and y are positive, x has to be less than 3 (because y has to be positive, so 4 - (4x)/3 > 0 => 4x/3 < 4 => x < ... | 3 | true | 1 | 10 |
136 | 5 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since x and y are positive, x has to be less than 3 (because 12 - 4x > 0 => x < 3) and y has to be positive, so (12 - 4x)/3 > 0 => 12... | 3 | true | 1 | 10 |
136 | 6 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. That simplifies to y = 4 - (4x)/3. Since x and y are positive, x has to be less than 3 (because if x were 3, y would be 0, but they h... | 3 | true | 1 | 10 |
136 | 7 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since y has to be positive, (12 - 4x)/3 > 0, so 12 - 4x > 0, which implies x < 3. Also, x is positive, so x is in (0, 3).
Now, xy = ... | 3 | true | 1 | 10 |
136 | 8 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since x and y are positive, 12 - 4x has to be positive, so 12 - 4x > 0 => x < 3. Also, x > 0, so x is in (0, 3).
Now, the expression... | 3 | true | 1 | 10 |
136 | 9 | 1 | 0.234375 | 0.2 | Given $x, y \in R^{+}$, and they satisfy the equation $\frac{x}{3} + \frac{y}{4} = 1$, find the maximum value of $xy$. | 3 | Okay, let's see. I need to find the maximum value of xy given that x and y are positive real numbers and they satisfy x/3 + y/4 = 1. Hmm, so this is a constrained optimization problem. I remember that for these kinds of problems, you can use substitution or maybe the AM-GM inequality. Let me try substitution first beca... | 1, I can solve for one variable in terms of the other. Let's solve for y. Multiply both sides by 12 to eliminate denominators: 4x + 3y = 12. So 3y = 12 - 4x, which means y = (12 - 4x)/3. Since x and y are positive, x has to be less than 3 (because 12 - 4x > 0 implies x < 3) and y has to be positive, so (12 - 4x)/3 > 0,... | 3 | true | 1 | 10 |
174 | 0 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | usually rationalize it by multiplying the numerator and denominator by the conjugate. The conjugate of (2 - √3) is (2 + √3), right? So I'll multiply both top and bottom by (2 + √3) to get rid of the square root in the denominator.
So, 1/(2 - √3) * (2 + √3)/(2 + √3) = (2 + √3)/[(2 - √3)(2 + √3)]. Let's compute the den... | 1 | true | 1 | 10 |
174 | 1 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | rationalize the denominator. That means multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of (2 - √3) is (2 + √3), right? So I'll multiply both top and bottom by (2 + √3).
So, 1/(2 - √3) times (2 + √3)/(2 + √3) equals (2 + √3)/[(2 - √3)(2 + √3)]. Let's compute the denominato... | 1 | true | 1 | 10 |
174 | 2 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | usually rationalize it. So to rationalize 1/(2 - √3), I should multiply the numerator and denominator by the conjugate of the denominator, which is (2 + √3). Let's do that.
So, (1*(2 + √3)) / [(2 - √3)(2 + √3)]. The denominator is a difference of squares: (2)^2 - (√3)^2 = 4 - 3 = 1. So the denominator becomes 1, whic... | 1 | true | 1 | 10 |
174 | 3 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | rationalize the denominator. So, to rationalize 1/(2 - √3), I should multiply the numerator and denominator by the conjugate of the denominator, which is (2 + √3). Let's do that.
So, (1)(2 + √3) divided by (2 - √3)(2 + √3). The denominator becomes (2)^2 - (√3)^2, which is 4 - 3 = 1. So the denominator is 1, and the n... | 1 | true | 1 | 10 |
174 | 4 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | usually rationalize the denominator. So to rationalize 1/(2 - √3), I can multiply the numerator and denominator by the conjugate of the denominator, which is (2 + √3). Let me do that.
So, (1 * (2 + √3)) / [(2 - √3)(2 + √3)] = (2 + √3) / (4 - (√3)^2). Since (√3)^2 is 3, the denominator becomes 4 - 3 = 1. So that simpl... | 1 | true | 1 | 10 |
174 | 5 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | rationalize it by multiplying the numerator and denominator by the conjugate. The conjugate of (2 - √3) is (2 + √3), right? So if I multiply numerator and denominator by (2 + √3), that should eliminate the square root in the denominator.
Let's do that: [1*(2 + √3)] / [(2 - √3)(2 + √3)]. The denominator becomes (2)^2 ... | 1 | true | 1 | 10 |
174 | 6 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | usually rationalize the denominator. So to rationalize 1/(2 - √3), I should multiply the numerator and denominator by the conjugate of the denominator, which is (2 + √3). Let me do that.
So, 1/(2 - √3) * (2 + √3)/(2 + √3) = (2 + √3)/[(2 - √3)(2 + √3)]. The denominator is a difference of squares: (2)^2 - (√3)^2 = 4 - ... | 1 | true | 1 | 10 |
174 | 7 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | rationalize the denominator. That means multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of (2 - √3) is (2 + √3), right? So I'll multiply both top and bottom by (2 + √3).
So that would be [1*(2 + √3)] / [(2 - √3)(2 + √3)]. Let's compute the denominator first. (2 - √3)(2 + √... | 1 | true | 1 | 10 |
174 | 8 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | rationalize the denominator. That means multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of (2 - √3) is (2 + √3). So, let's do that.
So, 1/(2 - √3) multiplied by (2 + √3)/(2 + √3) equals (2 + √3)/[(2 - √3)(2 + √3)]. The denominator is a difference of squares, right? (a - b)... | 1 | true | 1 | 10 |
174 | 9 | 1 | 0.21875 | 0.2 | Evaluate: $$\frac {1}{2- \sqrt {3}}$$\-π<sup>0</sup>\-2cos30°= \_\_\_\_\_\_ ． | 1 | Okay, let's try to solve this problem. Hmm, the problem is to evaluate 1 divided by (2 minus the square root of 3) minus pi to the power of 0 minus 2 times cosine of 30 degrees. Alright, let's break this down step by step.
First, I need to handle each part separately. Let's start with the first term: 1/(2 - √3). I rem... | usually rationalize it. So to rationalize 1/(2 - √3), I can multiply the numerator and the denominator by the conjugate of the denominator, which is (2 + √3). Let's do that.
So, (1*(2 + √3)) / [(2 - √3)(2 + √3)]. The denominator becomes (2)^2 - (√3)^2, which is 4 - 3 = 1. So that simplifies to (2 + √3)/1 = 2 + √3. Ok... | 1 | true | 1 | 10 |
345 | 0 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's recall that the absolute value |a| is a if a ≥ 0, and -a if a < 0. So here, the expression inside the absolute value is (x - 2). When x = 1, that's 1 - 2, which is -1. So |x - 2| when x = 1 is |-1|, which is 1.
Then the function f(x) is x squared plus that absolut... | 2 | true | 1 | 10 |
345 | 1 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's recall that the absolute value |a| is a if a ≥ 0, and -a if a < 0. So here, the expression inside the absolute value is (x - 2). When x is 1, that's 1 - 2 = -1. So |x - 2| when x = 1 is |-1|, which is 1. Right?
So f(1) should be (1)² + |1 - 2|. Let's compute that.... | 2 | true | 1 | 10 |
345 | 2 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's recall that |x - 2| is the absolute value of (x minus 2). So when x is less than 2, (x - 2) is negative, right? Because 1 is less than 2, so (1 - 2) is -1. The absolute value of -1 is 1. So |1 - 2| = 1.
Then, x squared when x is 1 is 1² = 1. So f(1) should be 1 + ... | 2 | true | 1 | 10 |
345 | 3 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's recall that |a| is a if a is non-negative, and -a if a is negative. So for x = 1, the expression inside the absolute value is 1 - 2, which is -1. So |1 - 2| is |-1|, which is 1. That makes sense because the absolute value of a negative number is its positive counte... | 2 | true | 1 | 10 |
345 | 4 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's recall that the absolute value |a| is a if a ≥ 0, and -a if a < 0. So here, the expression inside the absolute value is (x - 2). When x = 1, that's 1 - 2 = -1. Since -1 is negative, the absolute value of that would be the negative of -1, which is 1. So |1 - 2| = |-... | 2 | true | 1 | 10 |
345 | 5 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe I can do that directly.
First, let's recall that |a| is a if a is non-negative, and -a if a is negative. So for x = 1, the expression inside the absolute value is 1 - 2, which is -1. So |1 - 2| is |-1|, which is 1. That makes sense because the absolute value of a negative number is its positive counte... | 2 | true | 1 | 10 |
345 | 6 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe it's straightforward.
First, let's recall that the absolute value |a| is equal to a if a is non-negative, and -a if a is negative. So here, the expression inside the absolute value is (x - 2). When x is 1, x - 2 is 1 - 2, which is -1. So |x - 2| when x = 1 is |-1|, which is 1. Right?
So f(1) should b... | 2 | true | 1 | 10 |
345 | 7 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe I don't have to worry about the general case. Let me think.
First, the function is x squared plus the absolute value of (x minus 2). So when x is 1, I can substitute 1 into both parts. Let's do that step by step.
First, compute x squared: 1 squared is 1. Then, compute the absolute value part: |1 - 2|... | 2 | true | 1 | 10 |
345 | 8 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's recall that the absolute value |a| is a if a ≥ 0, and -a if a < 0. So for |x - 2|, when x = 1, x - 2 is 1 - 2 = -1. Since -1 is negative, the absolute value of that is 1. So |1 - 2| = |-1| = 1.
Then the function f(x) is x squared plus that absolute value. So f(1) ... | 2 | true | 1 | 10 |
345 | 9 | 4 | 0.984375 | 0.2 | Given the function $f(x) = x^2 + |x - 2|$, find the value of $f(1)$. | 2 | Okay, let's see. I need to find f(1) for the function f(x) = x² + |x - 2|. Hmm, absolute value functions can sometimes be tricky because they depend on whether the inside is positive or negative. But since I just need to plu | in x = 1, maybe that's straightforward.
First, let's remember that |x - 2| is the absolute value of (x minus 2). So when x is 1, x - 2 is 1 - 2, which is -1. The absolute value of -1 is 1, right? Because absolute value makes it positive. So |1 - 2| = |-1| = 1.
Then, x² when x is 1 is 1² = 1. So f(1) should be 1 + 1,... | 2 | true | 1 | 10 |
352 | 0 | 3 | 0.796875 | 0.2 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | subtraction: 459 - 357. Let's calculate that. 459 minus 300 is 159, then minus 57 more is 102. So 459 - 357 = 102. Now the two numbers are 357 and 102.
Next, I need to subtract the smaller number (102) from the larger one (357). Let's do that: 357 - 102. 357 minus 100 is 257, minus 2 more is 255. So 357 - 102 = 255. ... | 51 | true | 1 | 10 |
352 | 1 | 3 | 0.796875 | 0.2 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | calculation. 459 minus 357... Let's see, 459 - 300 is 159, then subtract 57 more: 159 - 57 = 102. So now we have 357 and 102.
Next, since 357 is larger than 102, I subtract 102 from 357. Let's calculate that. 357 divided by 102 is 3 times (since 102*3=306), so 357 - 306 = 51. Now we have 102 and 51.
Now, 102 is larg... | 51 | true | 1 | 10 |
352 | 2 | 3 | 0.796875 | 0.2 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | calculation: 459 minus 357. Let's see, 459 - 357. 459 minus 300 is 159, then minus 57 is 102. So 459 - 357 = 102. Now the two numbers are 357 and 102.
Next, since 357 is larger than 102, I need to subtract 102 from 357. Let's do that: 357 - 102. 357 minus 100 is 257, minus 2 more is 255. So 357 - 102 = 255. Now the n... | 51 | true | 1 | 10 |
352 | 3 | 3 | 0.796875 | 0.2 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | subtraction: 459 - 357. Let's calculate that. 459 minus 300 is 159, then minus 57 more is 102. So 459 - 357 = 102. Now the numbers are 357 and 102.
Next, since 357 is larger than 102, I subtract 102 from 357. How many times can I do that? Let's see. 102 times 3 is 306 (because 100*3=300, 2*3=6), so 357 - 306 = 51. So... | 51 | true | 1 | 10 |
352 | 4 | 3 | 0.796875 | 0.2 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | calculation: 459 minus 357. Let's see, 459 - 357. 459 minus 300 is 159, then minus 57 more is 102. So 459 - 357 = 102. Now the numbers are 357 and 102.
Next, since 357 is larger than 102, I subtract 102 from 357. Let's calculate that: 357 - 102. 357 minus 100 is 257, minus 2 is 255. So 357 - 102 = 255. Now the number... | 51 | true | 1 | 10 |
352 | 5 | 3 | 0.796875 | 0.2 | Use the method of successive subtraction to find the greatest common divisor of 459 and 357. | 51 | Okay, let's see. I need to find the greatest common divisor (GCD) of 459 and 357 using the method of successive subtraction. Hmm, I remember that the method of successive subtraction involves subtracting the smaller number from the larger one repeatedly until both numbers become equal. That common value is the GCD. Let... | subtraction: 459 - 357. Let's calculate that. 459 minus 300 is 159, then minus 57 more is 102. So 459 - 357 = 102. Now, the two numbers are 357 and 102.
Next, I need to subtract the smaller number (102) from the larger one (357). Let's do that: 357 - 102. 357 minus 100 is 257, then minus 2 is 255. So 357 - 102 = 255.... | 51 | true | 1 | 10 |
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