MikaStars39/nano-eval huggingface dataset:
Consider a super-Gaussian pulse with the following power distribution:
$$
P ( t )=P_{0} \operatorname{e x p} [-( t / T_{0} )^{2 m} ]
$$
where parameter $m$ controls the pulse shape. Calculate the ratio $T_{r} / T_{\mathrm{F W H M}}$, where $T_{\mathrm{F W H M}}$ is the full width at half maximum, and indicate what this ratio equals for a Gaussian pulse $( \, m=1 \, )$?

Options:
A. $0.923$
B. $0.577$
C. $0.693$
D. $0.716$
E. $0.943$
F. $0.628$
G. $0.842$
H. $0.785$
I. $0.651$
J. $0.814$
Answer: D
If there are currently three birds here, and at least one of them is male, what is the probability that among these four birds (after adding one more male bird), at least two birds are female?

Options:
A. 6/11
B. 1/2
C. 5/11
D. 3/5
Answer: A
If more than half of the vertices of an n-dimensional hypercube are colored blue, what is the minimum number of other blue vertices that at least one blue vertex must be connected to?

Options:
A. 1
B. √n
C. n+1
D. n/2
E. 0
F. log₂(n)
G. n
H. n!
I. 2ⁿ
J. n-1
Answer: B
Consider the following statements about complete graphs. How many of these statements are TRUE?
1. Kₙ is the graph with n vertices that has the fewest edges.
2. Kₙ is a simple graph.
3. If n is prime, then Kₙ is a bipartite graph.
4. Kₙ is n-regular.
5. Kₙ has n(n-1)/2 edges.
6. Kₙ is the graph with n vertices having the maximum number of edges among disconnected graphs.
7. The number of edges in Kₙ is given by the combination formula C(n, 2).
8. Kₙ is the graph with n vertices that has the maximum number of edges among all graphs (including non-simple graphs).
9. The complete graph K₁₀₁ contains a 1-factor.
10. The complete graph K₁₀₂ contains a 2-factor.

Options:
A. 4
B. 8
C. 6
D. 7
E. 9
F. 10
G. 5
H. 0
I. 2
J. 1
Answer: G
The media divides the newspaper market into the 'reader market, advertising market, news resource market, and investor market'. Who proposed this?

Options:
A. Shram
B. Syndicate
C. Benjamin Day
D. McManus
Answer: D
According to Clark's three-part classification of economics, economics should be divided into which of the following?

Options:
A. Production economics, distribution economics, exchange economics
B. Production economics, distribution economics, consumption economics
C. Pure economics, static economics, dynamic economics
D. General economics, static economics, dynamic economics
Answer: D
Which of the following statements about groups is the most difficult to verify directly without resorting to advanced theorems?

Options:
A. If a set has at least one element, then it is a group.
B. The intersection of sets has both identity and inverse elements.
C. If G is a group and H and K are subgroups of G such that H is a subgroup of K, and K is a subgroup of H, then H and K are the same subgroup.
D. There exists an empty group.
E. The identity element is unique in a group.
F. A group with 13 elements has subgroups.
G. If x is an element of a group G, and e is the identity element of G, then x * e = x.
H. The universal set R on which a group is defined is closed under the group operation.
I. The set of all subsets of a set with 21 elements forms a group under the operation of set intersection.
J. Equivalence relations define a group under the operation of composition.
Answer: H
During a debate, crucial information was lost. Among the participants-Zhang, Wang, Li, and Zhao-only one person is telling the truth. They made the following statements:
Zhang: "My statement is not true."
Wang: "My statement is true."
Li: "My statement is true."
Zhao: "My statement has no truth value."
After some discussion, it is determined that exactly one person is telling the truth, while the others are lying. Based on these conditions, which of the following lines of reasoning correctly identifies the truth-teller?

Options:
A. Assume Wang is telling the truth. Then Li is also telling the truth, which contradicts the condition that only one person is telling the truth. Therefore, Wang is lying.
B. Since only one person is truthful, at least three must be lying. Therefore, consider only those who claim to be truthful. Both Wang and Li make such claims. As only one can be truthful, both Wang and Li must be lying, leaving Zhao as the only possible truth-teller.
C. Assume Li is lying. Then Wang must also be lying. If Wang is lying, then Zhang could be telling the truth. No contradiction arises, so Li is lying.
D. If Zhang is telling the truth, his statement creates a paradox. If Wang is truthful, then Li is also truthful. Hence, neither Zhang nor Wang is telling the truth, meaning Li or Zhao might be truthful. Since Zhao's statement is paradoxical, Li must be truthful.
E. If we assume Wang is truthful, then Li's statement is also true. However, this is impossible; thus, Wang and Li are lying, meaning Zhao is truthful.
F. Assume Zhang is telling the truth. Then his statement "My statement is not true" is true, which is a contradiction. Therefore, Zhang is lying.
G. If Zhang is truthful, his statement is paradoxical. If Wang is truthful, Li is truthful, creating a contradiction. If Li is truthful, Wang is also truthful, creating a contradiction. Therefore, by elimination, Zhao must be truthful.
H. Assume Li is telling the truth. Since only one person can be truthful, both Wang and Zhao must be lying. This doesn't lead to any contradiction, so Li is telling the truth.
I. Assume Zhao is telling the truth. Then his statement "My statement has no truth value" is true, which is a contradiction. Therefore, Zhao is lying.
J. Since only one person can be telling the truth, assume it's Zhang. This directly contradicts his statement, so Zhang must be lying.
Answer: A
Cotton and linen both readily catch fire. A batch of towels is composed of both cotton and linen, and is known to have caught fire. If it is known that the towels were ignited by a lit cigarette, which of the following arguments utilizes the most appropriate form of reasoning?

Options:
A. Eliminating all other unlikely or improbable causes, such as spontaneous combustion or electrical faults, it is highly probable, given the presence of flammable materials (cotton and linen towels) and an ignition source (lit cigarette), that the cigarette caused the fire.
B. There was a lit cigarette near flammable towels, and there was a fire involving those same towels. The most likely explanation is that the cigarette caused the towels to ignite, and consequently, the fire. Therefore, the cigarette was the cause of the fire.
C. Most fires involving fabric materials like towels start from an open flame, like a cigarette. Since the towels are made of cotton and linen, and a lit cigarette was found, the fire most likely originated within the cotton and linen mixture due to the presence of the open flame.
D. Over 40% of household fires are started by cigarettes, and a lit cigarette was confirmed to be present. Given that the towels are composed of 40% cotton and 40% linen, and it's highly likely a cigarette started the fire, it can be concluded that the fire started in either the cotton or linen portions of the towels.
E. Towels are made to absorb water, yet cotton and linen are flammable materials. It is therefore inherently risky to have a lit cigarette near cotton/linen towels, so it is almost certain that the lit cigarette was the cause of the fire.
F. While not ruling out other less probable sources, the simultaneous presence of a lit cigarette and a pile of towels made of cotton and linen suggests a likely connection between the two. Since most fires in the home are due to easily preventable accidents, it is probable that the cigarette ignited the towels.
G. Fires require three elements: heat, fuel, and oxygen. The towels, being made of flammable cotton and linen, provided the fuel. The air supplied the oxygen. The lit cigarette supplied the heat. Therefore, the cigarette caused the fire.
H. Statistically, cigarettes are the leading cause of house fires. Therefore, combined with the known flammable materials present in the towels, the probability that the cigarette was the cause of the fire is very high.
I. It is common knowledge that cigarettes can start fires. Therefore, since a lit cigarette was near the towels, and those towels are known to be flammable, the fire must have started there due to the presence of the lit cigarette.
J. Because a lit cigarette was the only ignition source present near the towels made of combustible cotton and linen, it can be concluded that the cigarette was the cause of the fire.
Answer: J