sample_index int64 0 2 | problem stringclasses 3
values | ground_truth stringclasses 3
values | partial_solution stringclasses 9
values | step_number int64 1 37 | total_steps int64 6 37 | pct_complete float64 0.03 1 | model_confidence float64 0 1 | confidence_raw_output stringclasses 3
values | empirical_success_rate float64 0.2 1 | num_continuations int64 10 10 | num_correct int64 2 10 | full_initial_generation stringclasses 3
values | continuations listlengths 10 10 | parsed_answers listlengths 10 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. | [0, \frac{1}{2}] | Step 1: Solve the inequality for \(p\). | 1 | 37 | 0.027027 | 0 | <confidence>0.0</confidence> | 0.4 | 10 | 4 | Step 1: Solve the inequality for \(p\).
\[ |4x-3| \leq 1 \]
This can be broken into a compound inequality:
\[ -1 \leq 4x - 3 \leq 1 \]
Add 3 to all parts of the inequality:
\[ 2 \leq 4x \leq 4 \]
Divide all parts by 4:
\[ \frac{1}{2} \leq x \leq 1 \]
So, the solution set for \(p\) is:
\[ x \in \left[\frac{1}{2... | [
"Step 1: Solve the inequality for \\(p\\).\n\nGiven \\(p\\): \\(|4x-3| \\leqslant 1\\).\n\nTo solve this, we'll break it into two cases based on the definition of absolute value.\n\n\\[ |4x-3| \\leqslant 1 \\]\n\nCase 1: \\(4x - 3 \\leqslant 1\\)\n\\[ 4x \\leqslant 4 \\]\n\\[ x \\leqslant 1 \\]\n\nCase 2: \\(4x - 3... | [
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0 | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. | [0, \frac{1}{2}] | Step 1: Solve the inequality for \(p\).
\[ |4x-3| \leq 1 \]
This can be broken into a compound inequality:
\[ -1 \leq 4x - 3 \leq 1 \]
Add 3 to all parts of the inequality:
\[ 2 \leq 4x \leq 4 \]
Divide all parts by 4:
\[ \frac{1}{2} \leq x \leq 1 \]
So, the solution set for \(p\) is:
\[ x \in \left[\frac{1}{2... | 19 | 37 | 0.513514 | 1 | <confidence>1.0</confidence> | 0.2 | 10 | 2 | Step 1: Solve the inequality for \(p\).
\[ |4x-3| \leq 1 \]
This can be broken into a compound inequality:
\[ -1 \leq 4x - 3 \leq 1 \]
Add 3 to all parts of the inequality:
\[ 2 \leq 4x \leq 4 \]
Divide all parts by 4:
\[ \frac{1}{2} \leq x \leq 1 \]
So, the solution set for \(p\) is:
\[ x \in \left[\frac{1}{2... | [
"Step 3: Solve for the roots:\n\n\\[ x = \\frac{2a + 1 + 1}{2} = \\frac{2a + 2}{2} = a + 1 \\]\n\\[ x = \\frac{2a + 1 - 1}{2} = \\frac{2a}{2} = a \\]\n\nSo, the roots of the quadratic equation are:\n\n\\[ x = a \\quad \\text{and} \\quad x = a + 1 \\]\n\nThe quadratic \\(x^2 - (2a + 1)x + a^2 + a \\leq 0\\) can be f... | [
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0 | Given $p$: $|4x-3|\leqslant 1$ and $q$: $x^{2}-(2a+1)x+a^{2}+a\leqslant 0$, find the range of values for $a$ if $p$ is a necessary but not sufficient condition for $q$. | [0, \frac{1}{2}] | Step 1: Solve the inequality for \(p\).
\[ |4x-3| \leq 1 \]
This can be broken into a compound inequality:
\[ -1 \leq 4x - 3 \leq 1 \]
Add 3 to all parts of the inequality:
\[ 2 \leq 4x \leq 4 \]
Divide all parts by 4:
\[ \frac{1}{2} \leq x \leq 1 \]
So, the solution set for \(p\) is:
\[ x \in \left[\frac{1}{2... | 37 | 37 | 1 | 1 | <confidence>1.0</confidence> | 1 | 10 | 10 | Step 1: Solve the inequality for \(p\).
\[ |4x-3| \leq 1 \]
This can be broken into a compound inequality:
\[ -1 \leq 4x - 3 \leq 1 \]
Add 3 to all parts of the inequality:
\[ 2 \leq 4x \leq 4 \]
Divide all parts by 4:
\[ \frac{1}{2} \leq x \leq 1 \]
So, the solution set for \(p\) is:
\[ x \in \left[\frac{1}{2... | [
"Step 1: Solve the inequality for \\(p\\).\n\n\\[ |4x-3| \\leq 1 \\]\n\nThis can be broken into a compound inequality:\n\n\\[ -1 \\leq 4x - 3 \\leq 1 \\]\n\nAdd 3 to all parts of the inequality:\n\n\\[ 2 \\leq 4x \\leq 4 \\]\n\nDivide all parts by 4:\n\n\\[ \\frac{1}{2} \\leq x \\leq 1 \\]\n\nSo, the solution set f... | [
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)",
"\\([0, \\frac{1}{2}] \\)"
] |
1 | In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$. | \sqrt{21} | Step 1: Given the sides of the triangle, $a = 4$, $b = 5$, and the area of the triangle is $5 \sqrt{3}$. We need to find the length of side $c$, denoted as $|AB|$, which we will refer to as $c$. | 1 | 18 | 0.055556 | 0.9 | <confidence>0.9</confidence> | 1 | 10 | 10 | Step 1: Given the sides of the triangle, $a = 4$, $b = 5$, and the area of the triangle is $5 \sqrt{3}$. We need to find the length of side $c$, denoted as $|AB|$, which we will refer to as $c$.
Step 2: Using the formula for the area of a triangle, which is $\frac{1}{2}ab\sin C$, we can find the sine of angle $C$.
\[... | [
"Step 2: We can use the formula for the area of a triangle given two sides and the included angle, which is $\\frac{1}{2}ab\\sin C$. Here, the area is given as $5\\sqrt{3}$, $a=4$, and $b=5$. So we can set up the equation:\n\\[ \\frac{1}{2} \\times 4 \\times 5 \\times \\sin C = 5\\sqrt{3} \\]\n\\[ 10\\sin C = 10\\s... | [
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] |
1 | In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$. | \sqrt{21} | Step 1: Given the sides of the triangle, $a = 4$, $b = 5$, and the area of the triangle is $5 \sqrt{3}$. We need to find the length of side $c$, denoted as $|AB|$, which we will refer to as $c$.
Step 2: Using the formula for the area of a triangle, which is $\frac{1}{2}ab\sin C$, we can find the sine of angle $C$.
\[... | 10 | 18 | 0.555556 | 1 | <confidence>1.0</confidence> | 1 | 10 | 10 | Step 1: Given the sides of the triangle, $a = 4$, $b = 5$, and the area of the triangle is $5 \sqrt{3}$. We need to find the length of side $c$, denoted as $|AB|$, which we will refer to as $c$.
Step 2: Using the formula for the area of a triangle, which is $\frac{1}{2}ab\sin C$, we can find the sine of angle $C$.
\[... | [
"Step 5: The Law of Cosines states:\n\n\\[\nc^2 = a^2 + b^2 - 2ab \\cos C\n\\]\n\nSince $\\angle C = \\frac{\\pi}{3}$, we know $\\cos C = \\cos \\frac{\\pi}{3} = \\frac{1}{2}$. Plugging in the known values:\n\n\\[\nc^2 = 4^2 + 5^2 - 2 \\times 4 \\times 5 \\times \\frac{1}{2}\n\\]\n\n\\[\nc^2 = 16 + 25 - 20\n\\]\n\n... | [
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] |
1 | In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $a=4$, $b=5$, and the area of $\triangle ABC$ is $5 \sqrt {3}$, find $|AB|$. | \sqrt{21} | Step 1: Given the sides of the triangle, $a = 4$, $b = 5$, and the area of the triangle is $5 \sqrt{3}$. We need to find the length of side $c$, denoted as $|AB|$, which we will refer to as $c$.
Step 2: Using the formula for the area of a triangle, which is $\frac{1}{2}ab\sin C$, we can find the sine of angle $C$.
\[... | 18 | 18 | 1 | 1 | <confidence>1.0</confidence> | 1 | 10 | 10 | Step 1: Given the sides of the triangle, $a = 4$, $b = 5$, and the area of the triangle is $5 \sqrt{3}$. We need to find the length of side $c$, denoted as $|AB|$, which we will refer to as $c$.
Step 2: Using the formula for the area of a triangle, which is $\frac{1}{2}ab\sin C$, we can find the sine of angle $C$.
\[... | [
"Step 1: Given the sides of the triangle, \\(a = 4\\), \\(b = 5\\), and the area of the triangle is \\(5 \\sqrt{3}\\). We need to find the length of side \\(c\\), denoted as \\(|AB|\\), which we will refer to as \\(c\\).\n\nStep 2: Using the formula for the area of a triangle, which is \\(\\frac{1}{2}ab\\sin C\\), ... | [
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"\\sqrt{21}",
"\\sqrt{21}",
"\\sqrt{21}",
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"\\sqrt{21}"
] |
2 | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. | y = 2x | Step 1: Understand the original linear function.
The original linear function is given by \( y = 2x - 3 \). | 1 | 6 | 0.166667 | 1 | <confidence>1.0</confidence> | 0.9 | 10 | 9 | Step 1: Understand the original linear function.
The original linear function is given by \( y = 2x - 3 \).
Step 2: Identify the shift applied to the function.
It is stated that the function is shifted 3 units upwards. Shifting a function upwards by a certain number of units means adding that number to the output of t... | [
"Step 2: Determine the effect of shifting the function upwards.\nShifting the function \\( y = 2x - 3 \\) upwards by 3 units means adding 3 to the current \\( y \\)-value of the function. Mathematically, this can be expressed as:\n\\[ y = 2x - 3 + 3 \\]\nStep 3: Simplify the expression.\nSimplify the expression obt... | [
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] |
2 | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. | y = 2x | Step 1: Understand the original linear function.
The original linear function is given by \( y = 2x - 3 \).
Step 2: Identify the shift applied to the function.
It is stated that the function is shifted 3 units upwards. Shifting a function upwards by a certain number of units means adding that number to the output of t... | 4 | 6 | 0.666667 | 1 | <confidence>1.0</confidence> | 1 | 10 | 10 | Step 1: Understand the original linear function.
The original linear function is given by \( y = 2x - 3 \).
Step 2: Identify the shift applied to the function.
It is stated that the function is shifted 3 units upwards. Shifting a function upwards by a certain number of units means adding that number to the output of t... | [
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2 | If the linear function is $y=2x-3$, and it is shifted $3$ units upwards, the new equation of the graph is ______. | y = 2x | Step 1: Understand the original linear function.
The original linear function is given by \( y = 2x - 3 \).
Step 2: Identify the shift applied to the function.
It is stated that the function is shifted 3 units upwards. Shifting a function upwards by a certain number of units means adding that number to the output of t... | 6 | 6 | 1 | 1 | <confidence>1.0</confidence> | 1 | 10 | 10 | Step 1: Understand the original linear function.
The original linear function is given by \( y = 2x - 3 \).
Step 2: Identify the shift applied to the function.
It is stated that the function is shifted 3 units upwards. Shifting a function upwards by a certain number of units means adding that number to the output of t... | [
"Step 1: Understand the original linear function.\nThe original linear function is given by \\( y = 2x - 3 \\).\n\nStep 2: Identify the shift applied to the function.\nIt is stated that the function is shifted 3 units upwards. Shifting a function upwards by a certain number of units means adding that number to the ... | [
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