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Science & Engineering

SuperGPQA

SuperGPQA: graduate-level multiple-choice knowledge & reasoning across 285 disciplines (26,529 questions, avg 9.67 options/question). Per-(model, question) verdicts for 49 LLMs from the upstream zero-shot records dump; response is the binary correct/incorrect outcome.

26,529items
49subjects
100%observed
CC-BY-NC-SA-4.0license
generaldomain
reasoningdomain
sciencedomain
textmodality

Response matrix

Every model, scored item by item.

Each row is an AI model and each column an item, ordered so the strongest models and easiest items gather toward one corner. 49 subjects × 26,529 items, 100% of cells evaluated.

Fit to width. Hover for subject & item; click a cell for details.

SuperGPQA response matrix: AI models (rows) against items (columns)
Correct (1)Incorrect (0)Unobserved

Scale: 1 = correct · 0 = incorrect

Sample items

What the questions look like — and how subjects answer.

A spread of items across the difficulty range, each shown with a few subjects' actual answers.

Item 10% solve rate

Which of the following statements about the necessary conditions for producing wheel peripheral traction is correct? (Choose the option that includes all the correct statements) I. The locomotive driving wheels receive torque transmitted from the power transmission device. II. The output shaft of the diesel engine can only rotate in one direction. III. Friction exists between the driving wheels and the steel rails. IV. The power of the diesel engine is proportional to n, and the diesel engine generally operates near the condition n = n_{\text{max}}. Options:

  • A.

    I, III, IV

  • B.

    I, IV

  • C.

    I, II, III, IV, V

  • D.

    I, II, IV

  • E.

    II, IV

  • F.

    II, III, IV

  • G.

    I, II, III, IV

  • H.

    I,II I. I, III

How subjects answered

  • DeepSeek-R1-Zero incorrect
    <think> To solve for the necessary conditions for producing wheel peripheral traction, let us analyze the given statements one by one.

    I. "The locomotive driving wheels receive torque transmitted from the power transmission device." This statement is true. For a locomotive to produce wheel peripheral traction, the driving wheels need to receive torque from the power transmission device which usually comes from the diesel engine through a transmission system.

    II. "The output shaft of the diesel engine can only rotate in one direction." This statement seems a bit odd. Typically, a diesel engine's output shaft rotates in one direction (usually clockwise when viewed from the front), but it is …

  • DeepSeek-R1 incorrect
    <think> Okay, so I need to figure out which statements about the necessary conditions for producing wheel peripheral traction are correct. The options are I, II, III, IV, and the answer choices are combinations of these. Let me think through each statement one by one.

    The question is about the conditions needed for a locomotive's wheels to have peripheral traction. Peripheral traction here probably refers to the force that allows the wheels to push against the rails without slipping. So, the basic idea is that the wheels need to have enough friction (adhesion) to convert the torque into forward motion.

    Statement I: The locomotive driving wheels receive torque transmitted from the power …

  • DeepSeek-V3 incorrect

    To determine the correct answer, let's analyze each statement:

    I. The locomotive driving wheels receive torque transmitted from the power transmission device.
    This is correct. Torque is transmitted from the power transmission device to the driving wheels, enabling them to rotate and produce traction.

    II. The output shaft of the diesel engine can only rotate in one direction.
    This is incorrect. The output shaft of a diesel engine can rotate in both directions, depending on the design and application. For example, in some locomotives, the engine can reverse direction.

    III. Friction exists between the driving wheels and the steel rails.
    This is correct. Friction is …

  • phi-4 incorrect

    To determine the necessary conditions for producing wheel peripheral traction, let's evaluate each statement:

    I. The locomotive driving wheels receive torque transmitted from the power transmission device.

    • This is necessary for traction, as torque is required to turn the wheels.

    II. The output shaft of the diesel engine can only rotate in one direction.

    • This is not necessary for traction. Many locomotives can reverse direction, which requires the ability to rotate the output shaft in both directions.

    III. Friction exists between the driving wheels and the steel rails.

    • This is essential for traction, as friction is what allows the wheels to grip the rails and move the locomotive …
  • qwen-max-2024-09-19 incorrect

    To determine the correct answer, let's analyze each statement:

    I. The locomotive driving wheels receive torque transmitted from the power transmission device.

    • This is a necessary condition for producing wheel peripheral traction as the torque is what drives the wheels.

    II. The output shaft of the diesel engine can only rotate in one direction.

    • This is not necessarily a requirement for producing wheel peripheral traction. The ability to rotate in one direction is more about the design and operation of the engine rather than a direct condition for traction.

    III. Friction exists between the driving wheels and the steel rails.

    • Friction is essential for traction. Without friction, the …
  • qwen-max-2025-01-25 incorrect

    To determine the correct answer, let's analyze each statement:

    I. The locomotive driving wheels receive torque transmitted from the power transmission device.

    • This statement is correct. For a locomotive to produce peripheral traction, the driving wheels must receive torque from the power transmission system.

    II. The output shaft of the diesel engine can only rotate in one direction.

    • This statement is incorrect. The direction of rotation of the output shaft of a diesel engine does not inherently limit the production of wheel peripheral traction. The ability to rotate in one direction or both depends on the specific design and application, but it is not a necessary condition for …
Item 24% solve rate

What is the feed ratio of hexamethylenediamine to adipic acid required to obtain nylon-66 with a number-average molecular weight of 10,000 when the reaction extent is at 99.5%? Options:

  • A.
    1:1.1091 : 1 . 1 0 9
  • B.
    1:1.0121 : 1 . 0 1 2
  • C.
    1:0.9741 : 0 . 9 7 4
  • D.
    1:1.0561 : 1 . 0 5 6
  • E.
    1:1.0051 : 1 . 0 0 5
  • F.
    1:0.9871 : 0. 9 8 7
  • G.
    1:0.9251 : 0 . 9 2 5
  • H.
    1:0.9531 : 0 . 9 5 3

    I. $$ 1 : 1 . 0 8 0

    J. $$ 1 : 0 . 9 6 8

How subjects answered

  • Yi-Lighting correct

    To determine the feed ratio of hexamethylenediamine (HMDA) to adipic acid (AA) required for obtaining nylon-66 with a specific number-average molecular weight (Mˉn\bar{M}_n) and reaction extent, we need to consider the stoichiometry of the polymerization reaction and the relationship between the molecular weight and the extent of reaction.

    Step-by-Step Solution:

    1. Reaction Stoichiometry:
      • Nylon-66 is formed by the condensation reaction between hexamethylenediamine (HMDA) and adipic acid (AA).
      • The reaction can be represented as: nHMDA+nAANylon-66+nH2On \, \text{HMDA} + n \, \text{AA} \rightarrow \text{Nylon-66} + n \, \text{H}_2\text{O}
      • For a 1:1 molar ratio of …
  • Yi-1.5-6B-Chat correct

    To find the correct feed ratio of hexamethylenediamine to adipic acid for nylon-66 with a number-average molecular weight of 10,000, we need to use the Markownikow rule, which states that the ratio of the number-average molecular weights of the reactants to the number-average molecular weight of the product is equal to the ratio of the stoichiometric coefficients of the reactants.

    For nylon-66 production, the reaction is: [n(\text{hexamethylenediamine}) : n(\text{adipic acid}) = M(\text{adipic acid}) : M(\text{hexamethylenediamine})]

    Given:

    • (M(\text{hexamethylenediamine}) = 116) g/mol
    • (M(\text{adipic acid}) = 146.14) g/mol
    • (M(\text{nylon-66}) = 10,000) g/mol

    The ratio of …

  • DeepSeek-V3 incorrect

    To determine the feed ratio of hexamethylenediamine (HMD) to adipic acid (AA) required to obtain nylon-66 with a number-average molecular weight (( \overline{M_n} )) of 10,000 at a reaction extent of 99.5%, we need to use the Carothers equation for step-growth polymerization:

    [ \overline{M_n} = \frac{M_0}{1 - p} ]

    Where:

    • ( \overline{M_n} ) is the number-average molecular weight.
    • ( M_0 ) is the average molecular weight of the repeating unit.
    • ( p ) is the extent of the reaction.

    For nylon-66, the repeating unit consists of one molecule of HMD and one molecule of AA. The molecular weight of the repeating unit (( M_0 )) is:

    [ M_0 = \text{Molecular weight of HMD} + …

  • phi-4 incorrect

    To determine the feed ratio of hexamethylenediamine to adipic acid required to obtain nylon-66 with a number-average molecular weight (( \overline{M}_n )) of 10,000 when the reaction extent is at 99.5%, we can use the following relationship for step-growth polymerization:

    [ \overline{M}_n = \frac{M_0}{1 - p} ]

    where:

    • ( M_0 ) is the molecular weight of the repeating unit.
    • ( p ) is the extent of reaction.

    For nylon-66, the repeating unit is formed from hexamethylenediamine (molecular weight = 116 g/mol) and adipic acid (molecular weight = 146 g/mol). The molecular weight of the repeating unit ( M_0 ) is:

    [ M_0 = 116 + 146 - 18 = 244 , \text{g/mol} ]

    The extent of …

  • qwen-max-2024-09-19 incorrect

    To solve this problem, we need to understand the relationship between the feed ratio of hexamethylenediamine (HMD) to adipic acid (AA), the extent of reaction, and the resulting molecular weight of nylon-66.

    The number-average molecular weight ((M_n)) of a polymer can be related to the extent of reaction ((p)) and the functionality of the monomers. For nylon-66, which is a condensation polymer, the number-average degree of polymerization ((X_n)) is given by:

    [ X_n = \frac{1}{1 - p} ]

    Given:

    • The number-average molecular weight ((M_n)) is 10,000.
    • The molar mass of the repeat unit of nylon-66 is approximately 226 g/mol.
    • The extent of reaction ((p)) is 99.5%, or …
  • qwen-max-2025-01-25 incorrect

    To solve this problem, we need to calculate the feed ratio of hexamethylenediamine to adipic acid required to achieve a nylon-66 polymer with a number-average molecular weight (MnM_n) of 10,000 at a reaction extent (pp) of 99.5%.

    Key Concepts:

    1. Polymerization Reaction: Nylon-66 is synthesized through a condensation polymerization reaction between hexamethylenediamine and adipic acid. Each reaction step eliminates one water molecule.

    2. Carothers Equation: The number-average degree of polymerization (XnX_n) is related to the reaction extent (pp) by the Carothers equation:

      Xn=1+r1+r2rpX_n = \frac{1 + r}{1 + r - 2rp}

      where rr is the molar ratio of the reactants …

Item 38% solve rate

Early closure of fontanelles in children can lead to which of the following deformities? Options:

  • A.

    Hydrocephaly (fluid in brain)

  • B.

    Brachycephaly (short head)

  • C.

    Scaphocephaly (boat-shaped head)

  • D.

    Anencephaly (absence of a major portion of brain, skull, and scalp)

  • E.

    Acrocephaly (pointed skull)

  • F.

    Exencephaly (brain outside skull)

  • G.

    Microcephaly (small head)

  • H.

    Macrocephaly (large head) I. Dolichocephaly (long head) J. Plagiocephaly (asymmetric head)

How subjects answered

  • Doubao-1.5-pro-32k-250115 correct
    1. First, understand the role of fontanelles:
      • Fontanelles are the soft - spots on a baby's skull. They allow for the growth and expansion of the brain during the early stages of development. When the fontanelles close too early (craniosynostosis), it restricts the normal growth of the skull in the affected area, and the skull may grow in an abnormal shape to accommodate the growing brain.
    2. Analyze each option:
      • A) Hydrocephaly: Hydrocephaly is caused by an abnormal accumulation of cerebrospinal fluid in the brain, not by early closure of fontanelles. So, option A is incorrect.
      • B) Brachycephaly: Brachycephaly can be caused by positional factors (such as a baby …
  • claude-3-5-sonnet-20241022 correct

    Let me help you solve this question about the effects of early fontanelle closure.

    Fontanelles are the soft spots on a baby's skull that allow for rapid brain growth during infancy. When these fontanelles close too early (a condition called craniosynostosis), it restricts normal skull growth and forces the brain to grow in other directions, leading to abnormal head shapes.

    Microcephaly (option G) is a condition where the head is abnormally small, and this is directly related to premature fusion of cranial sutures and early closure of fontanelles. When fontanelles close too early:

    1. The brain cannot grow normally in all directions
    2. This restricts overall skull growth
    3. Results in a …
  • qwen-max-2025-01-25 correct

    Early closure of fontanelles in children, also known as craniosynostosis, can lead to various skull deformities depending on which sutures close prematurely. Among the options provided, Microcephaly (small head) is a condition that could be associated with early closure of the fontanelles. This occurs when the brain doesn't grow properly, leading to a smaller than normal head size.

    Answer: G

  • o3-mini-2025-01-31-medium incorrect

    Early closure of the fontanelles is generally an indication of premature fusion of the cranial sutures (craniosynostosis). When the sutures close too early, the skull cannot expand properly to accommodate brain growth, leading to an abnormal head shape. One of the classic deformities seen in affected children is acrocephaly—a skull with a pointed, peaked appearance often associated with premature suture closure (as seen in some craniosynostosis syndromes such as Apert syndrome).

    Answer: E

  • phi-4 incorrect

    Early closure of fontanelles, also known as craniosynostosis, can lead to various cranial deformities depending on which sutures close prematurely. When the sagittal suture closes early, it can result in scaphocephaly, which is characterized by a long, narrow head shape. If the coronal sutures close early, it can lead to brachycephaly, which results in a short, broad head. Premature closure of the lambdoid suture can cause plagiocephaly, leading to an asymmetric head shape. Acrocephaly, characterized by a pointed skull, can occur due to the early closure of multiple sutures.

    Given these possibilities, the deformity most directly associated with early closure of fontanelles is …

  • qwen-max-2024-09-19 incorrect

    Early closure of fontanelles, also known as craniosynostosis, can lead to various deformities depending on which suture closes prematurely. Let's evaluate the options:

    • A) Hydrocephaly (fluid in brain): This condition is related to an abnormal accumulation of cerebrospinal fluid and is not directly caused by early fontanelle closure.
    • B) Brachycephaly (short head): This can occur if the coronal sutures close early, leading to a short, wide head.
    • C) Scaphocephaly (boat-shaped head): This results from premature closure of the sagittal suture, leading to a long, narrow head.
    • D) Anencephaly (absence of a major portion of brain, skull, and scalp): This is a neural tube …
Item 414% solve rate

A α\alpha particle beam with a current of 200 microamperes passes through 1 cubic centimeter of nitrogen at 76 cmHg and 0°C. If all the nitrogen nuclei in 1 cubic centimeter are concentrated on a 1 square centimeter surface, the number of nuclear reactions is 6.2×108/\mathbf{6.2 \times 10^{8}}/ second. Calculate the effective cross-section (expressed in barns). Options:

  • A.

    7.3 barns

  • B.

    3.1 barns

  • C.

    9.1 barns

  • D.

    5.2 barns

  • E.

    0.04 bar

  • F.

    8.0 barns

  • G.

    6.5 barns

  • H.

    1.8 barns I. 4.7 barns J. 2.9 barns

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for the effective cross-section of the nuclear reaction, we need to first recall what the question is asking us. The question states that a beam of $\alpha$ particles with a current of $200$ microamperes passes through $1$ cubic centimeter of nitrogen gas at a pressure of $76$ cmHg and a temperature of $0^\circ C$. The question also states that the number of nuclear reactions per second is $6.2 \times 10^8$. The goal is to find the effective cross-section in barns (where $1$ barn = $10^{-24}$ cm²).

    The current II of the α\alpha particle beam is given as 200200 microamperes, which is:

    I=200×106 A=200×106 C/s.I = 200 \times 10^{-6} \text{ A} = 200 \times 10^{-6} \text{ C/s}.

    The charge of an …

  • DeepSeek-R1 correct

    To calculate the effective cross-section (σ) for the nuclear reactions, the following steps were performed:

    1. Number of Nitrogen Nuclei in 1 cm³:
      At STP (76 cmHg and 0°C), 1 mole of gas occupies 22.4 L (22400 cm³). For 1 cm³:
      [ \text{Moles of N}_2 = \frac{1}{22400}, \quad \text{Molecules of N}_2 = \frac{1}{22400} \times 6.022 \times 10^{23} \approx 2.688 \times 10^{19}. ]
      Each N₂ molecule contains 2 nitrogen nuclei, so:
      [ \text{Total nuclei} = 2 \times 2.688 \times 10^{19} = 5.376 \times 10^{19} , \text{nuclei/cm}^3. ]
      When concentrated on 1 cm², the surface density becomes ( n = 5.376 \times 10^{19} , \text{nuclei/cm}^2 ).

    2. **Alpha …

  • o3-mini-2025-01-31-low correct

    We begin by computing the number of α‐particles arriving per second. With a beam current I = 200 μA (that is, 2.0×10⁻⁴ C/s) and each α‐particle carrying a charge of 2e (where e = 1.60×10⁻¹⁹ C), the number of α‐particles per second is

      Nₐ = I/(2e) = (2.0×10⁻⁴ C/s) / (2×1.60×10⁻¹⁹ C)      = (2.0×10⁻⁴)/(3.2×10⁻¹⁹)      ≈ 6.25×10¹⁴ α‐particles/s.

    The problem states that if all the nitrogen nuclei in 1 cm³ are “concentrated” on a 1 cm² target area, then the number of nuclear reactions per second (i.e. the reaction rate) is 6.2×10⁸ s⁻¹. In a thin‐target approximation the probability for an α‐particle to react is given by

      P = (target areal density)×(cross‐section, σ),

    and the reaction rate …

  • phi-4 incorrect

    To solve this problem, we need to calculate the effective cross-section for the nuclear reactions occurring between the (\alpha) particles and nitrogen nuclei.

    First, let's determine the number of (\alpha) particles per second. The current (I) is given as 200 microamperes, which is (200 \times 10^{-6}) amperes. The charge of one (\alpha) particle (which is a helium nucleus) is (2e), where (e = 1.6 \times 10^{-19}) coulombs. Therefore, the number of (\alpha) particles per second (n_{\alpha}) is:

    [ n_{\alpha} = \frac{I}{2e} = \frac{200 \times 10^{-6}}{2 \times 1.6 \times 10^{-19}} = \frac{200 \times 10^{-6}}{3.2 \times 10^{-19}} = 6.25 \times 10^{14} \text{ …

  • qwen-max-2024-09-19 incorrect

    To solve this problem, we need to calculate the effective cross-section ((\sigma)) for the nuclear reactions. The formula that relates the number of reactions per second (R), the current (I), the charge of the particle (q), the number density of target nuclei (N), and the effective cross-section ((\sigma)) is given by:

    [ R = I \cdot \frac{q}{e} \cdot N \cdot \sigma ]

    Where:

    • (R) is the number of nuclear reactions per second,
    • (I) is the beam current,
    • (q) is the charge of the (\alpha) particle,
    • (e) is the elementary charge,
    • (N) is the number density of nitrogen nuclei,
    • (\sigma) is the effective cross-section.

    Given:

    • (R = 6.2 \times 10^8) …
  • qwen-max-2025-01-25 incorrect

    To solve this problem, we need to calculate the effective cross-section of the nitrogen nuclei using the given data. Let's break it down step by step.


    Step 1: Understand the formula for reaction rate

    The number of nuclear reactions per second (RR) is related to the beam current, the density of target nuclei, and the effective cross-section (σ\sigma) as follows:

    R=InσR = I \cdot n \cdot \sigma

    Where:

    • RR is the reaction rate (given as 6.2×108reactions/second6.2 \times 10^8 \, \text{reactions/second}),
    • II is the beam current (in particles per second),
    • nn is the number density of nitrogen nuclei in the target,
    • σ\sigma is the effective cross-section (to be calculated).

    We rearrange this …

Item 520% solve rate

Who accompanied Lilian Ngoyi on an illegal journey to Lausanne, Switzerland, to participate in the World Congress of Mothers held by the Women's International Democratic Federation (WIDF)? Options:

  • A.

    Dora Nyembe

  • B.

    Doris Sisulu

  • C.

    Doris Nyembe

  • D.

    Dorothy Nyembe

  • E.

    Helen Tamana

  • F.

    Dora Tamana

  • G.

    Doris Tamana

  • H.

    Albertina Sisulu I. Doris Joseph J. Helen Joseph

How subjects answered

  • Llama-3.1-70B-Instruct correct

    Dora Tamana accompanied Lilian Ngoyi on an illegal journey to Lausanne, Switzerland, to participate in the World Congress of Mothers held by the Women's International Democratic Federation (WIDF).

    Answer: F

  • Llama-3.3-70B-Instruct correct

    To answer this question, we need to identify the correct individual who accompanied Lilian Ngoyi on her journey. The correct name associated with this event is Dora Tamana, but since the options provided include various similar names, it's essential to select the one that matches historical records. Dora Tamana is known to have accompanied Lilian Ngoyi to the World Congress of Mothers in Lausanne, Switzerland.

    Answer: F

  • qwen-max-2024-09-19 correct

    The correct person who accompanied Lilian Ngoyi on the illegal journey to Lausanne, Switzerland, to participate in the World Congress of Mothers held by the Women's International Democratic Federation (WIDF) was Dora Tamana.

    Answer: F

  • o1-mini-2024-09-12 incorrect

    Lilian Ngoyi, a prominent South African anti-apartheid activist, was known for her collaboration with other key figures in the struggle against apartheid. Albertina Sisulu, another leading activist and a close associate of Ngoyi, accompanied her on the illegal journey to Lausanne, Switzerland. Their participation in the World Congress of Mothers organized by the Women's International Democratic Federation was a significant event in the internationalization of the anti-apartheid movement.

    Answer: H

  • phi-4 incorrect

    Lilian Ngoyi, a prominent South African anti-apartheid activist, was known for her significant contributions to the struggle against apartheid. In 1953, she made an illegal journey to Lausanne, Switzerland, to attend the World Congress of Mothers organized by the Women's International Democratic Federation (WIDF). She was accompanied by Doris Sisulu, who was the wife of Walter Sisulu, another key figure in the anti-apartheid movement. This journey was significant as it highlighted the international dimension of the struggle against apartheid and the role of women in this movement.

    Answer: B

  • qwen-max-2025-01-25 incorrect

    Lilian Ngoyi was accompanied by Helen Joseph on an illegal journey to Lausanne, Switzerland, to participate in the World Congress of Mothers held by the Women's International Democratic Federation (WIDF).

    Answer: J

Item 627% solve rate

The residue of the function f(z)=tan(z1)/(z1)f(z) = \tan(z-1) / (z-1) at its finite singularity is (). Options:

  • A.

    33,3(1+k)π\frac{-3} {\left( \, 1+k \, \right) \pi}.

  • B.

    1-1,2(k)π\frac{-2} {\left( \, k \, \right) \pi}.

  • C.

    11,3(3k+1)π\frac{3} {\left( \, 3 k+1 \, \right) \pi}.

  • D.

    00,1(2k+1)π\frac{1} {\left( \, 2 k+1 \, \right) \pi}.

  • E.

    00,1(2k1)π\frac{1} {\left( \, 2 k-1 \, \right) \pi}.

  • F.

    00,1(2k+2)π\frac{-1} {\left( \, 2 k+2 \, \right) \pi}.

  • G.

    00,2(2k+1)π\frac{-2} {\left( \, 2 k+1 \, \right) \pi}.

  • H.

    00,2(4k+2)π\frac{2} {\left( \, 4 k+2 \, \right) \pi}. I. 00,2(2k)π\frac{-2} {\left( \, 2 k \, \right) \pi}. J. 22,1(k+1)π\frac{-1} {\left( \, k+1 \, \right) \pi}.

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for the residue of the function $f(z) = \frac{\tan(z-1)}{z-1}$ at its finite singularities, we need to identify the singularities of $f(z)$.

    The function f(z)=tan(z1)z1f(z) = \frac{\tan(z-1)}{z-1} can be written as:

    f(z)=tan(z1)z1=sin(z1)(z1)cos(z1).f(z) = \frac{\tan(z-1)}{z-1} = \frac{\sin(z-1)}{(z-1)\cos(z-1)}.

    The singularities of f(z)f(z) occur where the denominator is zero. Thus, we need to solve for zz such that:

    (z1)cos(z1)=0.(z-1)\cos(z-1) = 0.

    This leads to two possible conditions:

    z1=0orcos(z1)=0.z-1 = 0 \quad \text{or} \quad \cos(z-1) = 0.

    The first condition z1=0z-1=0 gives us z=1z=1. However, we need to check what happens at z=1z=1 explicitly:

    limz1f(z)=limz1tan(z1)z1.\lim_{z \to 1} f(z) = \lim_{z \to 1} \frac{\tan(z-1)}{z-1}.

    Let $u = …

  • DeepSeek-R1 correct

    To determine the residue of the function ( f(z) = \frac{\tan(z-1)}{z-1} ) at its finite singularities, we first identify the singular points. The singularities occur where the denominator ((z-1)\cos(z-1)) is zero. This happens at:

    1. ( z = 1 ): Here, both the numerator (\sin(z-1)) and denominator ((z-1)\cos(z-1)) vanish. However, near ( z = 1 ), the function simplifies to ( \frac{\sin(z-1)}{(z-1)\cos(z-1)} ), which behaves like ( \frac{(z-1)}{(z-1)} = 1 ) as ( z \to 1 ). Thus, ( z = 1 ) is a removable singularity, and the residue here is ( 0 ).

    2. ( z = 1 + \frac{(2k+1)\pi}{2} ) (for integer ( k )): At these points, (\cos(z-1) = 0), leading to simple poles. …

  • DeepSeek-V3 correct

    To find the residue of the function ( f(z) = \frac{\tan(z-1)}{z-1} ) at its finite singularity, we first identify the singularities of the function.

    The function ( \tan(z-1) ) has singularities where its denominator is zero, i.e., at ( z-1 = \frac{(2k+1)\pi}{2} ) for any integer ( k ). However, the function ( f(z) ) has a singularity at ( z = 1 ) due to the denominator ( z-1 ).

    To find the residue at ( z = 1 ), we consider the Laurent series expansion of ( f(z) ) around ( z = 1 ). Since ( \tan(z-1) ) is analytic at ( z = 1 ) and has a simple zero there, we can write: [ \tan(z-1) = (z-1) + \frac{(z-1)^3}{3} + \cdots ] Thus, the function ( f(z) ) can be expressed …

  • gemma-2-9b-it incorrect

    Here's how to find the residue:

    • Identify the singularity: The function has a finite singularity at z = 1 because the denominator (z-1) becomes zero.

    • Type of singularity: Since the numerator doesn't also become zero at z=1, this is a simple pole.

    • Residue formula: For a simple pole at z=a, the residue is given by: Res(f, a) = lim (z->a) [ (z-a) * f(z) ]

    • Apply the formula: Res(f, 1) = lim (z->1) [ (z-1) * tan(z-1) / (z-1) ] Res(f, 1) = lim (z->1) tan(z-1) Res(f, 1) = tan(0) = 0

    Answer: E

  • phi-4 incorrect

    To find the residue of the function ( f(z) = \frac{\tan(z-1)}{z-1} ) at its finite singularity, we first identify the singularity. The function (\tan(z-1)) has singularities where (z-1 = \frac{\pi}{2} + k\pi) for (k \in \mathbb{Z}). Thus, the singularities of (f(z)) are at (z = 1 + \frac{\pi}{2} + k\pi).

    The singularity at (z = 1) is removable because the factor (z-1) in the denominator cancels with the zero of (\tan(z-1)) at (z = 1). Therefore, we need to find the residue at the other singularities (z = 1 + \frac{\pi}{2} + k\pi).

    Near (z = 1 + \frac{\pi}{2} + k\pi), we have: [ \tan(z-1) \approx \frac{1}{z - \left(1 + \frac{\pi}{2} + k\pi\right)} ] Thus, near …

  • qwen-max-2024-09-19 incorrect

    To find the residue of the function ( f(z) = \frac{\tan(z-1)}{z-1} ) at its finite singularities, we need to identify where these singularities occur and then compute the residue at those points.

    The function (\tan(z-1)) has singularities (poles) when (z-1 = \frac{(2k+1)\pi}{2}), for any integer (k). This means that the singularities of (f(z)) are at: [ z = 1 + \frac{(2k+1)\pi}{2} ]

    These are simple poles because (\tan(z-1)) has simple poles at these points. To find the residue at a simple pole, we can use the formula for the residue of a function (g(z)/h(z)) at a point (z_0) where (h(z_0) = 0) and (h'(z_0) \neq 0): [ \text{Res}(f, z_0) = \frac{g(z_0)}{h'(z_0)} …

Item 733% solve rate

In which region is it advisable to use high-grade and low viscosity asphalt for the construction of asphalt pavement? Options:

  • A.

    Channelized Traffic Sections

  • B.

    Western dry regions

  • C.

    Hot climate regions

  • D.

    Northern cold regions

  • E.

    Eastern rainy regions

  • F.

    Low altitude zones

  • G.

    Heavy Traffic Section

  • H.

    Southern humid regions I. Coastal flood areas J. Urban residential areas

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for which region it is advisable to use high-grade and low viscosity asphalt for the construction of asphalt pavement, we need to understand what "high-grade" and "low viscosity" asphalt means.
    1. High-grade asphalt typically refers to asphalt that has a higher performance grade (PG), which means it can withstand higher temperatures without becoming too soft and lower temperatures without becoming too brittle.

    2. Low viscosity asphalt refers to asphalt that flows more easily at a given temperature. This makes it easier to work with and can be particularly useful in colder climates where the asphalt needs to remain workable at lower temperatures.

    Based on this, …

  • DeepSeek-R1 correct

    To determine the appropriate region for using high-grade, low viscosity asphalt in pavement construction, it is essential to consider the relationship between asphalt properties and environmental conditions. Asphalt viscosity refers to its resistance to flow, with low viscosity indicating a softer, more flexible binder. High-grade asphalt typically implies superior performance characteristics, such as enhanced durability or temperature resistance.

    In hot climate regions (Option C), asphalt is prone to rutting due to high temperatures. Here, a stiffer (higher viscosity) binder is preferred to resist deformation under heat. Conversely, in cold regions (Option D), asphalt faces the …

  • Doubao-1.5-pro-32k-250115 correct
    1. First, understand the properties of high - grade and low - viscosity asphalt:
      • High - grade asphalt generally has better performance characteristics such as higher softening point and better resistance to deformation. Low - viscosity asphalt is more fluid at lower temperatures, which helps in better workability during construction and also provides good flexibility at low temperatures.
      • In cold regions, the asphalt pavement needs to be flexible enough to withstand the expansion and contraction caused by temperature changes. Low - viscosity asphalt can remain flexible in cold conditions, reducing the risk of cracking.
    2. Analyze each option:
      • **Option A: Channelized Traffic …
  • o3-mini-2025-01-31-high incorrect

    High-grade, low viscosity asphalt is typically recommended for use in regions experiencing very high ambient temperatures. In hot climates, the asphalt binder is required to withstand high service temperatures without softening excessively, while still being fluid enough at production and laying temperatures to ensure proper mixing and compaction. This combination of high-quality performance (high-grade) and low viscosity helps in achieving a durable pavement that resists rutting and other high-temperature distresses.

    Answer: C

  • phi-4 incorrect

    High-grade and low viscosity asphalt is typically used in regions where the pavement is exposed to high temperatures. This type of asphalt is designed to resist rutting and deformation under heavy loads and high temperatures, making it suitable for hot climate regions. In such areas, the asphalt needs to maintain its structural integrity despite the heat, which can cause softer asphalt to deform.

    Let's evaluate the options:

    A) Channelized Traffic Sections - This refers to specific traffic patterns rather than climate conditions.

    B) Western dry regions - While these regions can be hot, the term "dry" does not specifically indicate high temperatures.

    C) Hot climate regions - This directly …

  • o3-mini-2025-01-31-medium incorrect

    High-grade, low-viscosity asphalt is typically recommended for regions with high ambient temperatures. In hot climate regions, asphalt can become overly soft and may lead to rutting or deformation under heavy traffic. Using a high-grade binder ensures the material has the necessary performance characteristics to withstand high temperatures and traffic loads, while low viscosity improves workability during laying and compaction.

    Answer: C

Item 841% solve rate

Which of the following Chinese generals has never served as a senior leadership official in the Chinese Academy of Military Science? Options:

  • A.

    Yu Su

  • B.

    Wenhan Zheng

  • C.

    He Zheng

  • D.

    Tian Qin

  • E.

    Jianying Ye

  • F.

    Yunrui Wang

  • G.

    Shilun Song

  • H.

    Dehuai Peng I. Fuke Li

How subjects answered

  • DeepSeek-R1 correct

    To determine which Chinese general never served as a senior leadership official in the Chinese Academy of Military Science (AMS), we analyze the historical roles of the listed individuals:

    • H) Dehuai Peng (Peng Dehuai): Peng Dehuai was a prominent marshal and Minister of National Defense until his purge in 1959. The AMS was established in 1958, and its first president was Ye Jianying. Peng Dehuai, despite his high-ranking military roles, was not associated with AMS leadership. His dismissal in 1959 further supports the conclusion that he did not serve in AMS leadership.

    • G) Shilun Song (Song Shilun): Song Shilun served as the president of AMS from 1972 to 1985, making him a …

  • Llama-3.1-8B-Instruct correct

    I couldn't find any information on the specific individuals mentioned in the options A) Yu Su, B) Wenhan Zheng, C) He Zheng, D) Tian Qin, E) Jianying Ye, F) Yunrui Wang, G) Shilun Song, H) Dehuai Peng, I) Fuke Li. However, I can tell you that Peng Dehuai was a senior leadership official in the Chinese Academy of Military Science.

    Since I couldn't verify the information about the other individuals, I will choose one of the options that I know is incorrect.

    Answer: H

  • DeepSeek-V3 correct

    To determine which Chinese general has never served as a senior leadership official in the Chinese Academy of Military Science, we need to analyze the historical roles of each individual listed. Here's a brief overview of each:

    • Yu Su: Not a well-documented figure in the context of the Chinese Academy of Military Science.
    • Wenhan Zheng: No significant records of serving in a senior leadership role at the Academy.
    • He Zheng: Known for his military contributions but not specifically linked to the Academy.
    • Tian Qin: No notable association with the Academy.
    • Jianying Ye: A prominent military figure, but not specifically tied to the Academy.
    • Yunrui Wang: No …
  • o3-mini-2025-01-31-low incorrect

    Based on available information it appears that among the listed names, Tian Qin is not known to have served as a senior leadership official at the Chinese Academy of Military Science.

    Answer: D

  • qwen-max-2024-09-19 incorrect

    To determine which of the listed Chinese generals has never served as a senior leadership official in the Chinese Academy of Military Science, we need to consider their known roles and positions.

    • Yu Su: Known for his contributions to military science and education.
    • Wenhan Zheng: Not widely recognized in the context of the Chinese Academy of Military Science.
    • He Zheng: Has held significant positions in the military, but not specifically noted for a role in the Academy.
    • Tian Qin: Information is limited, but no strong association with the Academy.
    • Jianying Ye: Known for other military roles, but not prominently associated with the Academy.
    • Yunrui Wang: Served in various military …
  • qwen-max-2025-01-25 incorrect

    To answer this question, we need to evaluate the careers of the listed Chinese generals and determine which one has never served in a senior leadership role at the Chinese Academy of Military Science (AMS). The AMS is one of China's most prestigious military institutions, responsible for strategic research, doctrine development, and high-level military education. Many prominent Chinese military leaders have held positions there.

    Analysis of Each General:

    1. Yu Su (A): No significant records indicate that Yu Su has been associated with the AMS in a senior capacity.
    2. Wenhan Zheng (B): Wenhan Zheng has had a distinguished military career but does not appear to have served in a …
Item 947% solve rate

Archaeological evidence for the collapse of civilizations suggests which of the following is the most important variable? Options:

  • A.

    the existence of a written language

  • B.

    the civilization's trade networks with other societies

  • C.

    the civilization's religious beliefs and practices

  • D.

    the ability of bureaucracies to store food and feed its people

  • E.

    how societies respond to challenges

  • F.

    changing environmental conditions

  • G.

    the development of new technologies

  • H.

    the size of the civilization's population I. whether warfare can be ended J. the number of years the civilization has existed

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for the most important variable suggested by archaeological evidence for the collapse of civilizations, we need to analyze what the question is asking. The question is asking us to identify which variable is most important based on archaeological evidence for the collapse of civilizations.
    1. A) The existence of a written language:

      • While a written language can help in record-keeping and administration, it is not typically cited as a primary factor in the collapse of civilizations.
    2. B) The civilization's trade networks with other societies:

      • Trade networks can be important for economic stability and cultural exchange, but they are usually not the most …
  • DeepSeek-R1 correct
    <think> Okay, so I need to figure out which of these options is the most important variable in the collapse of civilizations according to archaeological evidence. Let's think through each option.

    The question is about what's the main factor leading to collapse. Let's start by recalling some examples of collapsed civilizations. Like the Mayans, the Roman Empire, maybe the Anasazi. What do I know about why they collapsed?

    Option A: Existence of a written language. Hmm. Well, having a written language might help in record-keeping and administration, but does that prevent collapse? Or maybe civilizations without writing might be less complex and thus less prone to collapse? Not sure. But I …

  • DeepSeek-V3 correct

    To determine the most important variable for the collapse of civilizations based on archaeological evidence, we need to consider factors that have historically played a significant role in societal decline. While many of the options listed (e.g., trade networks, religious beliefs, population size) can influence a civilization's stability, the most critical factor often cited by archaeologists and historians is how societies respond to challenges, particularly environmental and social stressors. This includes their ability to adapt to changing conditions, manage resources, and address internal and external pressures.

    Key points:

    • Environmental changes (e.g., droughts, floods, …
  • granite-3.1-2b-instruct incorrect

    Answer: F

  • o3-mini-2025-01-31-high incorrect

    Archaeological studies have repeatedly shown that environmental stressors such as droughts, climate shifts, and resource depletion can critically undermine a civilization's sustainability. While factors like food management and social organization are important, evidence often points to changing environmental conditions as the trigger affecting a range of social responses. Thus, among the options provided, changing environmental conditions (option F) are considered the most important variable in the collapse of civilizations.

    Answer: F

  • o1-mini-2024-09-12 incorrect

    Archaeological evidence frequently highlights that changing environmental conditions, such as climate change, resource depletion, and natural disasters, play a critical role in the collapse of civilizations. While how societies respond to challenges is also important, it is often the environmental stresses that act as the primary catalyst for collapse.

    Answer: F

Item 1053% solve rate

A compound (C5H8)({ C }_{ 5 }{ H }_{ 8 }) reacts with ammonical CuCl2{ Cu }{ Cl }_{ 2 } to give a red precipitate and reacts with excess of KMnO4KMnO_{ 4 } solution to give (CH3)2CHCOOH.(CH_{ 3 })_{ 2 }CH- COOH. The compound is Options:

  • A.

    (CH3)(CH2)2C=CH(CH_{ 3 })(CH_{ 2 })_{ 2 }C=CH

  • B.

    CH2=CHCH2CH2CH3{ CH }_{ 2 }=CH-CH_{ 2 }-CH_{ 2 }-CH_{ 3 }

  • C.

    (CH3)2CHC=CH(CH_{ 3 })_{ 2 }CH-C=CH

  • D.

    (CH3)2C=C=CH2{ (CH }_{ 3 })_{ 2 }C=C=CH_{ 2 }

  • E.

    CH2=CHCH=CHCH2CH3{ CH }_{ 2 }=CH-CH=CH-CH_{ 2 }-CH_{ 3 }

  • F.

    CH_{ 2 }=CH-CH=CH-CH_{ 2 }-CH_{ 3 }

  • G.

    (CH3)2CH=CHCH3(CH_{ 3 })_{ 2 }CH=CH-CH_{ 3 }

  • H.

    CH2=CHCH2CH=CH2{ CH }_{ 2 }=CH-CH_{ 2 }-CH=CH_{ 2 } I. CH_{ 2 }=CH-CH_{ 2 }-CH=CH_{ 2 }

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for the compound $C_5H_8$ that reacts with ammonical $Cu_2Cl_2$ to give a red precipitate and reacts with excess $KMnO_4$ solution to give $(CH_3)_2CH-COOH$, we need to analyze the given reactions and the provided options.
    1. Reaction with ammonical Cu2Cl2Cu_2Cl_2 (which is actually a test for terminal alkynes, i.e., alkynes where the triple bond is at the terminal position, RCCHR-C\equiv CH):

      The reaction of a terminal alkyne with ammonical Cu2Cl2Cu_2Cl_2 (also known as ammoniacal cuprous chloride) results in the formation of a red precipitate of copper acetylide. This reaction is a characteristic test for terminal alkynes where the alkyne has a CCHC\equiv CH group.

  • DeepSeek-R1 correct
    <think> Okay, so I've got this organic chemistry problem here. Let me read through it again. The compound is C5H8. It reacts with ammoniacal CuCl2 to give a red precipitate. And when it reacts with excess KMnO4 solution, it gives (CH3)2CH-COOH. The question is asking which of the options A to I is the correct structure.

    Hmm. Let me think. First, the formula C5H8. Let's calculate the degree of unsaturation. For a hydrocarbon, the formula is CnH2n+2 for alkanes. So here, 5 carbons would be C5H12 for an alkane. The actual is C5H8. So 12-8=4. Divide by 2 gives 2. So two degrees of unsaturation. That could be two double bonds, one double bond and a ring, or two rings, or a triple bond. But …

  • DeepSeek-V3 correct

    To solve this problem, we need to analyze the given reactions and deduce the structure of the compound ( C_5H_8 ).

    1. Reaction with ammonical ( CuCl_2 ):

      • The formation of a red precipitate indicates the presence of a terminal alkyne (( -C \equiv CH )). Terminal alkynes react with ammonical ( CuCl_2 ) to form copper acetylides, which are red in color.
    2. Reaction with excess ( KMnO_4 ):

      • The product ( (CH_3)_2CH-COOH ) suggests that the compound has a structure that, upon oxidation, yields isobutyric acid. This implies the presence of a ( (CH_3)_2CH- ) group attached to a carbon-carbon triple bond.

    Now, let's analyze the options:

    • Option A: ( …
  • gpt-4o-2024-11-20 incorrect

    To solve the question, let us carefully analyze the information provided in the problem and the possible structures.

    Clues from the question:

    1. The compound has the molecular formula C(_5)H(_8). This indicates it is unsaturated, with two degrees of unsaturation (rings or multiple bonds).

    2. Reacts with ammoniacal CuCl(_2) to give a red precipitate:

      • This reaction is characteristic of terminal alkynes (≡C-H group) because terminal alkynes form a precipitate with Cu(^+) ions.
    3. Reacts with excess KMnO(_4) to give (CH(_3))(_2)CH-COOH (isobutyric acid):

      • This suggests that upon oxidation with KMnO(_4), the compound breaks down into the …
  • gemma-2-2b-it incorrect

    The compound is A) (CH3)(CH2)2C=CH. Here's why:

    • Reaction with ammonical CuCl2: This reaction indicates the presence of an alkene (a carbon-carbon double bond). The reaction with ammonical CuCl2 leads to the formation of a red precipitate, which is a characteristic of the formation of a conjugated system.
    • Reaction with KMnO4: The reaction with excess KMnO4 indicates the presence of a carbonyl group (C=O). The product formed is a carboxylic acid, which is consistent with the formation of a conjugated system.

    Let me know if you'd like a more detailed explanation of the reaction mechanisms!

    Answer: A

  • qwen-max-2024-09-19 incorrect

    To solve this problem, we need to identify the compound ( C_5H_8 ) that reacts with ammonical ( CuCl_2 ) to give a red precipitate and with excess ( KMnO_4 ) to form ( (CH_3)_2CH-COOH ).

    1. Ammonical ( CuCl_2 ) Test:

      • A red precipitate with ammonical ( CuCl_2 ) indicates the presence of an acetylenic group (triple bond, -C≡C-).
    2. Excess ( KMnO_4 ) Reaction:

      • The product ( (CH_3)_2CH-COOH ) suggests that the original compound has a methyl group and a double bond that can be oxidized to a carboxylic acid.

    Given these observations, let's analyze each option:

    • A) ( (CH_3)(CH_2)_2C=CH ):
      • This is 1-pentyne, which has a triple bond but does not match …
Item 1161% solve rate

Let the matrix A=(1002102a1)A=\left( \begin{matrix} {{{-1}}} & {{{0}}} & {{{0}}} \\ {{{-2}}} & {{{1}}} & {{{0}}} \\ {{{-2}}} & {{{a}}} & {{{1}}} \\ \end{matrix} \right), so what is the value of aa when the matrix AA can be diagonalized similarly? Options:

  • A.
    a=3a=-3
  • B.
    a=4a=-4
  • C.
    a=0a=0
  • D.
    a=2a=2
  • E.
    a=1a=1
  • F.
    a=2a=-2
  • G.
    a=1a=-1
  • H.
    a=3a=3

    I. $$ a=5

    J. $$ a=4

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for when the matrix $A$ can be diagonalized, we need to find the eigenvalues of $A$ and check if the algebraic multiplicity of each eigenvalue equals its geometric multiplicity.

    The given matrix is:

    A=(1002102a1).A = \begin{pmatrix} -1 & 0 & 0 \\ -2 & 1 & 0 \\ -2 & a & 1 \end{pmatrix}.

    The characteristic polynomial of AA is given by:

    det(λIA)=det(λ+1002λ102aλ1).\det(\lambda I - A) = \det \begin{pmatrix} \lambda + 1 & 0 & 0 \\ 2 & \lambda - 1 & 0 \\ 2 & -a & \lambda - 1 \end{pmatrix}.

    This is a lower triangular block matrix where the top left block is (λ+1)(\lambda + 1) and the bottom right block is:

    (λ10aλ1).\begin{pmatrix} \lambda - 1 & 0 \\ -a & \lambda - 1 \end{pmatrix}.

    The determinant of a triangular …

  • DeepSeek-R1 correct

    To determine the value of ( a ) that makes the matrix ( A ) diagonalizable, we analyze its eigenvalues and eigenvectors. The matrix ( A ) is lower triangular, so its eigenvalues are the diagonal entries: ( -1, 1, 1 ). The eigenvalue ( 1 ) has algebraic multiplicity 2, so the matrix is diagonalizable if the geometric multiplicity of ( 1 ) is also 2.

    The geometric multiplicity of ( \lambda = 1 ) is the dimension of the null space of ( A - I ), where ( I ) is the identity matrix. Subtracting ( I ) from ( A ), we get:

    [ A - I = \begin{pmatrix} -2 & 0 & 0 \ -2 & 0 & 0 \ -2 & a & 0 \end{pmatrix}. ]

    To find the null space, solve ( (A - I)\mathbf{x} = \mathbf{0} …

  • DeepSeek-V3 correct

    To determine the value of ( a ) for which the matrix ( A ) can be diagonalized, we need to analyze its eigenvalues and eigenvectors. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors.

    The matrix ( A ) is given by: [ A = \begin{pmatrix} -1 & 0 & 0 \ -2 & 1 & 0 \ -2 & a & 1 \end{pmatrix} ]

    Step 1: Find the eigenvalues of ( A )

    The eigenvalues are found by solving the characteristic equation ( \det(A - \lambda I) = 0 ), where ( I ) is the identity matrix and ( \lambda ) is an eigenvalue.

    The characteristic polynomial is: [ \det(A - \lambda I) = \det \begin{pmatrix} -1 - \lambda & 0 & 0 \ -2 & 1 - \lambda & 0 \ -2 & a & 1 - …

  • gemma-2-27b-it incorrect

    A matrix can be diagonalized if and only if it has three linearly independent eigenvectors. This means the determinant of the matrix (AλI)(A-\lambda I) must be zero for three distinct values of λ\lambda.

    Let's calculate the determinant of (AλI)(A-\lambda I):

    det(AλI)=det(1λ0021λ02a1λ)=(1λ)((1λ)(1λ)0)=(1λ)(1λ)2\begin{aligned} \text{det}(A-\lambda I) &= \text{det}\left( \begin{matrix} {-1-\lambda} & {0} & {0} \\ {-2} & {1-\lambda} & {0} \\ {-2} & {a} & {1-\lambda} \\ \end{matrix} \right) \\ &= (-1-\lambda)((1-\lambda)(1-\lambda) - 0) \\ &= (-1-\lambda)(1-\lambda)^2 \end{aligned}

    For the matrix to be diagonalizable, this determinant must have three distinct roots. This means the equation (1λ)(1λ)2=0(-1-\lambda)(1-\lambda)^2 = 0 must have three …

  • gemma-2-9b-it incorrect

    A matrix can be diagonalized if and only if it has a full set of linearly independent eigenvectors. The eigenvalues of AA are 1-1, 1, and 1. The eigenvector corresponding to the eigenvalue 1-1 is (122)\begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}. The eigenvectors corresponding to the eigenvalue 1 are of the form (0xx)\begin{pmatrix} 0 \\ x \\ -x \end{pmatrix}. Thus, the matrix AA can be diagonalized if and only if a2a \neq 2.

    Answer: F

  • gpt-4o-2024-05-13 incorrect

    To determine the value of (a) for which the matrix (A) can be diagonalized, we need to consider the conditions under which a matrix is diagonalizable. A matrix (A) is diagonalizable if and only if it has a complete set of linearly independent eigenvectors. This generally corresponds to the geometric multiplicity of each eigenvalue equaling its algebraic multiplicity.

    The characteristic polynomial of matrix (A) is obtained by solving the determinant of (A - \lambda I):

    [ A - \lambda I = \left( \begin{matrix} -1 - \lambda & 0 & 0 \ -2 & 1 - \lambda & 0 \ -2 & a & 1 - \lambda \ \end{matrix} \right) ]

    The characteristic polynomial is found by calculating the determinant of …

Item 1269% solve rate

The P wave originating from the sinoatrial node must be inverted in the _______. Options:

  • A.

    AVR lead

  • B.

    V2 lead

  • C.

    Lead I

  • D.

    V5 lead

  • E.

    Lead II

  • F.

    V1 lead

  • G.

    V6 lead

  • H.

    Lead III I. V3 lead J. V4 lead

How subjects answered

  • DeepSeek-R1-Zero correct
    <think> To solve for which lead the P wave originating from the sinoatrial (SA) node should be inverted, we need to recall the normal electrical conduction pathway in the heart. The SA node is located in the right atrium near the junction of the superior vena cava. The normal depolarization wave spreads from the SA node through the right atrium to the left atrium. The main direction of this depolarization wave is generally from right to left and from top to bottom (superior to inferior).

    The P wave on an electrocardiogram (ECG) represents atrial depolarization. Normally, the P wave is upright in leads where the depolarization wave travels towards the positive electrode of that lead. …

  • DeepSeek-R1 correct

    The P wave on an ECG represents atrial depolarization, which originates from the sinoatrial (SA) node in the right atrium. The electrical impulse spreads downward and to the left through the atria, creating a characteristic direction of the depolarization vector. The orientation of ECG leads determines whether the P wave appears upright or inverted.

    In a normal sinus rhythm, the P wave is upright in leads oriented in the direction of the depolarization wave (e.g., leads I, II, and aVF). Conversely, in leads oriented opposite to this direction, the P wave is inverted. Lead aVR, positioned on the right arm, is oriented away from the main direction of atrial depolarization. As a result, the …

  • DeepSeek-V3 correct

    The P wave represents atrial depolarization, which originates from the sinoatrial (SA) node and spreads through the atria. In the AVR lead, the electrical activity of the heart is viewed from the perspective of the right shoulder, which is opposite to the direction of the normal depolarization wave. As a result, the P wave in the AVR lead is typically inverted.

    The other leads listed (V2, Lead I, V5, Lead II, V1, V6, Lead III, V3, V4) do not typically show an inverted P wave under normal conditions. Therefore, the correct answer is:

    Answer: A

  • granite-3.1-2b-instruct incorrect

    Answer: E

  • gemma-2-2b-it incorrect

    The answer is (C) Lead I.

    Here's why:

    • P waves represent atrial depolarization, which is the electrical signal that causes the atria to contract.
    • Sinoatrial (SA) node is the natural pacemaker of the heart.
    • Inverted P waves in the Lead I are a characteristic finding of a normal sinus rhythm.

    Let me know if you have any other questions! Answer: C

  • phi-4 incorrect

    The P wave represents atrial depolarization, which normally originates from the sinoatrial (SA) node located in the right atrium. The electrical activity of the heart is recorded in different leads, each providing a unique perspective on the heart's electrical activity.

    • Lead I and Lead II typically show a positive P wave because the depolarization wave moves towards these leads.
    • Lead III also usually shows a positive P wave for the same reason.
    • V1 lead is positioned on the right side of the sternum, near the right atrium. Since the depolarization wave moves away from V1, the P wave is often inverted in this lead.
    • V2 to V6 leads are placed on the left side of the …

Subjects

The models, agents, and reward models evaluated.

49 subjects, ranked by mean response (accuracy) across this benchmark's items.

  1. 1DeepSeek-R10.6182
  2. 2o1-2024-12-170.6024
  3. 3DeepSeek-R1-Zero0.6024
  4. 4o3-mini-2025-01-31-high0.5522
  5. 5Doubao-1.5-pro-32k-2501150.5509
  6. 6o3-mini-2025-01-31-medium0.5269
  7. 7Doubao-1.5-pro-32k-2412250.5093
  8. 8qwen-max-2025-01-250.5008
  9. 9claude-3-5-sonnet-202410220.4816
  10. 10o3-mini-2025-01-31-low0.4803
  11. 11gemini-2.0-flash0.4773
  12. 12DeepSeek-V30.474
  13. 13o1-mini-2024-09-120.4522
  14. 14MiniMax-Text-010.4511
  15. 15gpt-4o-2024-11-200.444
  16. 16QwQ0.4359
  17. 17Llama-3.1-405B-Instruct0.4314
  18. 18gpt-4o-2024-08-060.4164
  19. 19Qwen2.5-72B-Instruct0.4075
  20. 20Mistral-Large-Instruct-24110.4065
  21. 21qwen-max-2024-09-190.3996
  22. 22gpt-4o-2024-05-130.3976
  23. 23Qwen2.5-32B-Instruct0.3876
  24. 24Llama-3.3-70B-Instruct0.3769
  25. 25phi-40.3765
  26. 26Qwen2.5-14B-Instruct0.3515
  27. 27Llama-3.1-70B-Instruct0.3486
  28. 28Yi-Lighting0.3342
  29. 29Mixtral-8x22B-Instruct-v0.10.2923
  30. 30Qwen2.5-7B-Instruct0.2878
  31. 31gemma-2-27b-it0.2743
  32. 32Yi-1.5-34B-Chat0.2603
  33. 33Mistral-Small-Instruct-24090.2589
  34. 34gemma-2-9b-it0.2404
  35. 35Qwen2.5-3B-Instruct0.2331
  36. 36Yi-1.5-9B-Chat0.2317