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BCD | The filament of a light bulb has surface area $64 \mathrm{~mm}^{2}$. The filament can be considered as a black body at temperature $2500 \mathrm{~K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \mathrm{~m}$. Assume the pupil of the eyes of the... | math | daman1209arora/jeebench/phy | {'subject': 'phy', 'description': 'JEE Adv 2020 Paper 1', 'index': 8, 'type': 'MCQ(multiple)'} | Let's break down the problem into smaller steps.
First, we need to calculate the power radiated by the filament. We are given that the filament acts as a black body. The Stefan-Boltzmann law states that the power radiated by a black body is given by $P = \sigma A T^4$, where $\sigma$ is the Stefan-Boltzmann constant, ... | Solution:
(A) The power radiated by the filament is given by the Stefan-Boltzmann law:
$P = \sigma A T^4$
$P = (5.67 \times 10^{-8} \mathrm{Wm}^{-2} \mathrm{~K}^{-4}) \times (64 \times 10^{-6} \mathrm{~m}^{2}) \times (2500 \mathrm{~K})^4$
$P = 5.67 \times 64 \times 10^{-14} \times 39.0625 \times 10^{12} \mathrm{~W}$
$P... | Okay, so I have this problem about a light bulb filament, and I need to check the validity of four statements (A, B, C, D). Let's try to go through each part step by step.
First, the given data: The filament has a surface area of 64 mm2, which is 64 square millimeters. Since 1 mm2 is 1e-6 m2, 64 mm2 is 64e-6 m2, righ... | Given the problem about a light bulb filament, we need to verify the validity of four statements (A, B, C, D) based on the provided data and constants.
**Given Data:**
- Filament surface area: \(64 \, \text{mm}^2 = 6.4 \times 10^{-5} \, \text{m}^2\)
- Temperature: \(2500 \, \text{K}\)
- Distance from observer: \(100 \... | Yes | I need to compare the student's attempt with the correct answer by evaluating each option.
(A) The student calculated the power radiated by the filament using the Stefan-Boltzmann law:
P = σAT⁴ = 5.67 × 10⁻⁸ × 64 × 10⁻⁶ × (2500)⁴
The student calculated 141.7125 W, which falls outside the range of 642 W to 645 W, so th... | Yes | I need to evaluate the student's attempt against the correct answer by analyzing each part of the solution.
For this problem, the student needed to determine which of the given statements (A, B, C, D) are correct regarding a light bulb filament emitting radiation.
Let's check the student's answers against the correct... | [
{
"from": "human",
"value": "The filament of a light bulb has surface area $64 \\mathrm{~mm}^{2}$. The filament can be considered as a black body at temperature $2500 \\mathrm{~K}$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of $100 \\math... |
923 | Triangle $ABC$ has $AB=21$ , $AC=22$ and $BC=20$ . Points $D$ and $E$ are located on $\overline{AB}$ and $\overline{AC}$ , respectively, such that $\overline{DE}$ is parallel to $\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$ . Then $DE=\frac{m}{n}$ , where $m$ and $n$ are relatively p... | math | qq8933/AIME_1983_2024 | {'ID': '2001-I-7', 'Year': 2001, 'Problem Number': 7, 'Part': 'I'} |
The problem asks for the length of the segment $DE$ which is parallel to $BC$ and passes through the incenter of triangle $ABC$. We are given the side lengths of the triangle $AB=21$, $AC=22$, and $BC=20$.
First, let's find the properties of triangle $ABC$, such as its semi-perimeter, area, and the radius of the insc... | Solution:
Let the triangle be $ABC$ with side lengths $AB=21$, $AC=22$, and $BC=20$.
The semi-perimeter of the triangle is $s = \frac{21+22+20}{2} = \frac{63}{2}$.
Using Heron's formula, the area of triangle $ABC$ is
Area $= \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{\frac{63}{2} \cdot \frac{23}{2} \cdot \frac{19}{2} \cdot \frac{... | Okay, let's tackle this problem step by step. It's about triangle ABC with sides AB = 21, AC = 22, and BC = 20. We need to find the length of DE, which is parallel to BC and passes through the incenter of the triangle. The answer should be in the form m/n where m and n are coprime, and then find m + n. Alright, let's b... | Triangle \(ABC\) has \(AB = 21\), \(AC = 22\), and \(BC = 20\). We need to find the length of \(DE\) where \(D\) and \(E\) are on \(\overline{AB}\) and \(\overline{AC}\) respectively, such that \(\overline{DE}\) is parallel to \(\overline{BC}\) and contains the incenter of triangle \(ABC\).
First, we calculate the coo... | Yes | I'll evaluate the student's attempt to solve this problem.
The student approached the problem by:
1. Finding the area of triangle ABC using Heron's formula
2. Calculating the inradius of the triangle
3. Finding the altitude from vertex A to side BC
4. Using similarity principles since DE is parallel to BC
5. Determini... | Yes | I'm going to evaluate the student's attempt to find the value of $m + n$ where $DE = \frac{m}{n}$.
The student's approach is to:
1. Set up a coordinate system with $A$ at the origin
2. Calculate the coordinates of points $B$, $C$, and the incenter $I$
3. Find the equation of line $DE$ passing through $I$ and parallel ... | [
{
"from": "human",
"value": "Triangle $ABC$ has $AB=21$ , $AC=22$ and $BC=20$ . Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$ , respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$ . Then $DE=\\fr... |
"1. We start with the given function:\n \\[\n f(k) = \\int_{-\\pi}^{\\pi} (\\sin kx - a \\sin mx(...TRUNCATED) | "For integers $k\\ (0\\leq k\\leq 5)$ , positive numbers $m,\\ n$ and real numbers $a,\\ b$ , le(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': 'For integers $k\\\\ (0\\\\leq k\\\\leq 5)$ , po(...TRUNCATED) | "\nThe problem asks for the values of $m, n, a, b$ that minimize the expectation $E$ of the function(...TRUNCATED) | "Solution:\nLet $f(k)=\\int_{-\\pi}^{\\pi}(\\sin kx-a\\sin mx-b\\sin nx)^{2}\\ dx$.\nExpanding the s(...TRUNCATED) | "Okay, so I need to find the values of m, n, a, b that minimize the expected value E, which is a sum(...TRUNCATED) | "To minimize the expected value \\( E = \\sum_{k=0}^{5} p(k) f(k) \\), where \\( p(k) \\) is a binom(...TRUNCATED) | Yes | "I'll compare the student's attempt with the correct answer to determine if the attempt is correct.\(...TRUNCATED) | Yes | "I need to analyze the student's attempt and compare it with the correct solution.\n\nThe student co(...TRUNCATED) | [{"from":"human","value":"For integers $k\\ (0\\leq k\\leq 5)$ , positive numbers $m,\\ n$ and re(...TRUNCATED) |
"1. We start with the given series:\n \\[\n \\sum_{k=1, k \\neq m}^{\\infty} \\frac{1}{(k+m)(k-m(...TRUNCATED) | "For a given positive integer $m$ , the series $$ \\sum_{k=1,k\\neq m}^{\\infty}\\frac{1}{(k+m)(k(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': 'For a given positive integer $m$ , the series (...TRUNCATED) | "The problem asks for the evaluation of the series $\\sum_{k=1,k\\neq m}^{\\infty}\\frac{1}{(k+m)(k-(...TRUNCATED) | "We are asked to evaluate the series $S = \\sum_{k=1,k\\neq m}^{\\infty}\\frac{1}{(k+m)(k-m)}$.\nUsi(...TRUNCATED) | "Alright, let's tackle this problem. So, we have a series sum from k=1 to infinity, but k is not equ(...TRUNCATED) | "To evaluate the series \\( \\sum_{k=1,k\\neq m}^{\\infty}\\frac{1}{(k+m)(k-m)} \\) for a given posi(...TRUNCATED) | Yes | "I'll evaluate the student's attempt by comparing it with the correct answer.\n\nThe student's appro(...TRUNCATED) | No | "I'll assess whether the student's attempt is correct by comparing it with the correct answer.\n\nTh(...TRUNCATED) | [{"from":"human","value":"For a given positive integer $m$ , the series $$ \\sum_{k=1,k\\neq m}^{(...TRUNCATED) |
"1. **Define the function \\( g(t) \\):**\n \\[\n g(t) = -\\log\\left( \\int_K \\rho(\\mathbf{x}(...TRUNCATED) | "Let $\\rho:\\mathbb{R}^n\\to \\mathbb{R}$ , $\\rho(\\mathbf{x})=e^{-||\\mathbf{x}||^2}$ , and let(...TRUNCATED) | math | AI-MO/NuminaMath-CoT/aops_forum | "{'source': 'aops_forum', 'messages': [{'content': 'Let $\\\\rho:\\\\mathbb{R}^n\\\\to \\\\mathbb{R(...TRUNCATED) | "Let $K$ be a convex body in $\\mathbb{R}^n$.\nLet $\\rho(\\mathbf{x}) = e^{-\\|\\mathbf{x}\\|^2}$.\(...TRUNCATED) | "Let $\\mathbf{s}_{K+\\mathbf{a}}$ and $\\mathbf{s}_{K+\\mathbf{b}}$ be the barycenters of the trans(...TRUNCATED) | "Okay, so I need to prove that if we take any two different translates of a convex body K in R^n, th(...TRUNCATED) | "To prove that the translates of a convex body \\( K \\subset \\mathbb{R}^n \\) have pairwise distin(...TRUNCATED) | No | "I need to evaluate whether the student's attempt correctly proves that translates of a convex body (...TRUNCATED) | Yes | "I need to evaluate whether the student's attempt correctly proves that the translates of a convex b(...TRUNCATED) | [{"from":"human","value":"Let $\\rho:\\mathbb{R}^n\\to \\mathbb{R}$ , $\\rho(\\mathbf{x})=e^{-||\\(...TRUNCATED) |
"A. In this reaction, (E)-penta-1,3-diene acts as the diene, and acrylonitrile acts as the dienophil(...TRUNCATED) | "In a cycloaddition reaction, two π systems combine to form a single-ring structure. These reaction(...TRUNCATED) | science | Idavidrein/gpqa | "{'Pre-Revision Question': 'In a cycloaddition reaction, two π systems combine to form a single-rin(...TRUNCATED) | "Let's break down each reaction and predict the product.\n\n**Reaction 1: (E)-penta-1,3-diene + acry(...TRUNCATED) | "Let's analyze each reaction to determine the cycloaddition products.\n\n**Reaction 1: (E)-penta-1,3(...TRUNCATED) | "Okay, let's see. So the problem is about cycloaddition reactions. Cycloadditions... I remember they(...TRUNCATED) | "The reactions involve cycloadditions under thermal conditions. For the first reaction, **(E)-penta-(...TRUNCATED) | Yes | "I need to compare the student's attempt with the correct answer for this cycloaddition reaction pro(...TRUNCATED) | No | "I need to evaluate whether the student's attempt matches the correct answer for this cycloaddition (...TRUNCATED) | [{"from":"human","value":"In a cycloaddition reaction, two π systems combine to form a single-ring (...TRUNCATED) |
C | "Consider the ellipse\n\n\\[\n\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\n\\]\n\nLet $H(\\alpha, 0), 0<\\al(...TRUNCATED) | math | daman1209arora/jeebench/math | {'subject': 'math', 'description': 'JEE Adv 2022 Paper 1', 'index': 18, 'type': 'MCQ'} | "\nThe problem involves an ellipse, its auxiliary circle, and properties of tangents and angles. We (...TRUNCATED) | "Solution:\nThe equation of the ellipse is $\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1$. The auxiliary circ(...TRUNCATED) | "Okay, let me see how to approach this problem. We have an ellipse given by the equation \\(\\frac{x(...TRUNCATED) | "Given the ellipse \\(\\frac{x^2}{4} + \\frac{y^2}{3} = 1\\) and a point \\(H(\\alpha, 0)\\) where \(...TRUNCATED) | Yes | "I'll evaluate the student's approach and compare it with the correct answer.\n\nThe student correct(...TRUNCATED) | Yes | "I need to evaluate whether the student's attempt correctly addresses the problem of finding the are(...TRUNCATED) | [{"from":"human","value":"Consider the ellipse\n\n\\[\n\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\n\\]\n\nL(...TRUNCATED) |
"Using the numbers from the problem, the average human produces\n\n$$\nu=\\frac{100}{66400 \\times 1(...TRUNCATED) | "An often-repeated fun fact is that humans produce more power per unit volume than stars. If the sun(...TRUNCATED) | science | GAIR/OlympicArena/Astronomy | "{'id': 'Astronomy_855', 'prompt': 'You are participating in an international Astronomy competition (...TRUNCATED) | "\nThe problem asks for the surface temperature of the sun if it had the same size but produced powe(...TRUNCATED) | "Solution:\nThe power produced by an average human is $P_{human} = 100 \\mathrm{~W}$, and the volume(...TRUNCATED) | "Okay, so there's this fun fact saying that humans produce more power per unit volume than stars. Th(...TRUNCATED) | "To determine the surface temperature of the Sun if it produced the same amount of power per unit vo(...TRUNCATED) | Yes | "I need to evaluate whether the student's attempt correctly solves the problem.\n\nThe student appro(...TRUNCATED) | Yes | "I need to assess whether the student's attempt correctly solves this problem.\n\nLooking at their w(...TRUNCATED) | [{"from":"human","value":"An often-repeated fun fact is that humans produce more power per unit volu(...TRUNCATED) |
"Note that $c = \\tfrac{1000}{a}$ and $d = \\tfrac{1000}{b}$. Substituting $c$ and $d$ results in $\(...TRUNCATED) | "Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\\begin{array}{c@{\\hspace{3pt}}c@{\\(...TRUNCATED) | math | qfq/openaimath/Intermediate Algebra | "{'answer': '126', 'subject': 'Intermediate Algebra', 'level': 5, 'unique_id': 'train/intermediate_a(...TRUNCATED) | "\nThe problem asks for the value of $\\lfloor S\\rfloor$, where $S=a+b+c+d$, and $a, b, c, d$ are p(...TRUNCATED) | "Solution:\nLet the given equations be\n\\begin{align*} a^2+b^2 &= 2008 \\quad &(1) \\\\ c^2+d^2 &= (...TRUNCATED) | "Okay, let's see. I have this problem where there are four positive real numbers a, b, c, and d. The(...TRUNCATED) | "Given positive real numbers \\(a, b, c,\\) and \\(d\\) satisfying the equations:\n\n\\[\na^2 + b^2 (...TRUNCATED) | Yes | "I'll evaluate the student's attempt by comparing it with the correct answer.\n\nThe student's appro(...TRUNCATED) | Yes | "I'll analyze the student's attempt by comparing it with the correct answer.\n\nThe student's approa(...TRUNCATED) | [{"from":"human","value":"Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\\begin{arra(...TRUNCATED) |
"We want to prove an inequality of the form\n\\[\\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2} \\le k,\(...TRUNCATED) | "Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers. Find the maximum value of\n\\[\\frac{wx + xy(...TRUNCATED) | math | qfq/openaimath/Intermediate Algebra | "{'answer': '\\\\frac{1 + \\\\sqrt{5}}{4}', 'subject': 'Intermediate Algebra', 'level': 5, 'unique_i(...TRUNCATED) | "\nThe problem asks for the maximum value of the expression $\\frac{wx + xy + yz}{w^2 + x^2 + y^2 + (...TRUNCATED) | "Solution:\nLet the given expression be $E = \\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.$\nWe want (...TRUNCATED) | "Alright, so I need to find the maximum value of the expression (wx + xy + yz) divided by (w2 + x2 +(...TRUNCATED) | "To find the maximum value of the expression \\(\\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}\\) where(...TRUNCATED) | Yes | "I'll compare the student's attempt with the correct answer to determine if it's correct.\n\nThe stu(...TRUNCATED) | Yes | "I need to evaluate whether the student's attempt is correct by comparing it with the provided corre(...TRUNCATED) | [{"from":"human","value":"Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers. Find the maximum va(...TRUNCATED) |
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