MikaStars39/nano-eval huggingface dataset:
If the method of completing the square is used to transform the quadratic form `f(x_1,x_2,x_3)=2x_1^2+x_2^2+4x_3^2+2x_1x_2-2x_2x_3` into its canonical form, then the canonical form and the transformation matrix used are

Options:
A. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} & -1/sqrt{2} 0 & sqrt{2} & -sqrt{2} 0 & 0 & 1/sqrt{2} end{bmatrix}
B. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} &1/sqrt{2} &-1/sqrt{2} 0& sqrt{2}& sqrt{2} 0 & 0 & 1/sqrt{2} end{bmatrix}
C. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} &-1/sqrt{2} 0& sqrt{2}& sqrt{2} 0 & 0 & 1/sqrt{2} end{bmatrix}
D. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} & -1/sqrt{2} 0 & -sqrt{2} & -sqrt{2} 0 & 0 & 1/sqrt{2} end{bmatrix}
E. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} &-1/sqrt{2} 0& -sqrt{2}& -sqrt{2} 0 & 0 & -1/sqrt{2} end{bmatrix}
F. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} -1/sqrt{2} & -1/sqrt{2} &-1/sqrt{2} 0& sqrt{2}& sqrt{2} 0 & 0 & -1/sqrt{2} end{bmatrix}
G. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} &-1/sqrt{2} 0& sqrt{2}& sqrt{2} 0 & 0 &- 1/sqrt{2} end{bmatrix}
H. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} &-1/sqrt{2} 0& -sqrt{2}& sqrt{2} 0 & 0 & 1/sqrt{2} end{bmatrix}
I. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} &-1/sqrt{2} 0& sqrt{2}& -sqrt{2} 0 & 0 & -1/sqrt{2} end{bmatrix}
J. `f(y_1,y_2,y_3)=y_1^2+y_2^2+y_3^2`, begin{bmatrix} 1/sqrt{2} & -1/sqrt{2} & 1/sqrt{2} 0& sqrt{2}& sqrt{2} 0 & 0 & 1/sqrt{2} end{bmatrix}
Answer: C
$int {{{dx} over {x{{left( {x - 1} right)}^2}}}} = left( {} right)$

Options:
A. $ln left| x right| + {1 over {x + 1}} + ln left| {x - 1} right| + C$
B. $ln left| x right| - {1 over {x - 1}} + ln left| {x + 1} right| + C$
C. $ln left| x right| + {1 over x} - ln left| {x - 1} right| + C$
D. $ln left| x right| + {1 over {x + 1}} - ln left| {x - 1} right| + C$
E. $ln left| x right| - {1 over {x + 1}} - ln left| {x - 1} right| + C$
F. $ln left| x right| - {1 over {x - 1}} + ln left| {x - 1} right| + C$
G. $ln left| x right| - {1 over {x - 1}} - ln left| {x - 1} right| + C$
H. $ln left| x right| + {1 over {x - 1}} - ln left| {x + 1} right| + C$
I. $ln left| x right| + {1 over {x - 1}} - ln left| {x - 1} right| + C$
J. $ln left| x right| - {1 over x} + ln left| {x + 1} right| + C$
Answer: G
The main cause of 'Bo Jue' discussed in the 'Plain Questions: Discussion on the Connection between Vitality and Heaven' is:

Options:
A. Yang Qi, when reversed, causes the blood and Qi to become disordered
B. Yang Qi, when weak, causes Yin Qi to rise
C. Yang Qi, when strong, cannot decline and be dense
D. Yang Qi, when greatly angered, causes the form and Qi to be depleted
E. Yang Qi, when greatly angered, causes the form and Qi to be severed
F. Yang Qi, when greatly angered, causes the form and Qi to be weakened
G. Yang Qi, when greatly angered, causes the form and Qi to be disrupted
H. Yang Qi, when greatly angered, causes the form and Qi to be cut off
I. Yang Qi, when greatly angered, causes the form and Qi to be obstructed
J. Yang Qi, when overworked, becomes excessive
Answer: H
China unwaveringly pursues an independent foreign policy of peace and follows a path of peaceful development, which is determined by the country's ( ).

Options:
A. socialist nature and international status
B. socialist nature and global influence
C. national strategy
D. socialist nature and national strategy
E. socialist principles and international strategy
F. socialist principles and national conditions
G. national conditions
H. ideology
I. socialist principles and international status
J. socialist principles and global standing
Answer: A
For a linear time-invariant continuous-time system, the free response to an excitation signal is \((e^{-3t}+e^{-t})u(t)\), and the forced response is \((1-e^{-2t})u(t)\). Which of the following statements is correct?

Options:
A. The system must be a marginally stable system
B. The system must be a second-order system
C. The system must be an unstable system
D. The system must be a first-order system
E. The zero-state response must include \((1-e^{-2t})u(t)\)
F. The system must be a marginally unstable system
G. The zero-input response must include \((e^{-3t}+e^{-t})u(t)\)
H. The system must be a stable system
I. The system must be a third-order system
Answer: H
"With the acquisition of new productive forces... people also change all their social relations. The hand-mill gives you society with the feudal lord; the steam-mill, society with the industrial capitalist." This passage indicates

Options:
A. the nature of production relations determines the state of productive forces
B. the development of production relations determines the state of productive forces
C. scientific and technological revolutions are powerful levers for promoting economic and social development
D. the state of production relations influences the development of productive forces
E. the development of productive forces determines the transformation of production relations
F. productive forces determine production relations
G. the development of production relations influences the state of productive forces
H. the state of productive forces determines the nature of production relations
I. productive forces influence production relations
J. the state of production relations determines the development of productive forces
Answer: F
Differentiability of a function at a point is the ( ) for the existence of partial derivatives of the function at that point.

Options:
A. sufficient condition and not always a necessary condition
B. sufficient condition and sometimes a necessary condition
C. neither a sufficient condition nor a necessary condition
D. sufficient and necessary condition
E. necessary condition and not always a sufficient condition
F. necessary condition and always a sufficient condition
G. sufficient condition but not a necessary condition
H. necessary condition but not a sufficient condition
I. necessary condition and not a sufficient condition
J. necessary condition and sometimes a sufficient condition
Answer: G
Atrioventricular valve closure occurs at ( )

Options:
A. end of ventricular systole
B. beginning of isovolumetric contraction
C. beginning of ventricular ejection
D. end of isovolumetric contraction
E. end of ventricular diastole
F. beginning of ventricular systole
G. beginning of isovolumetric relaxation
H. end of ventricular ejection
I. end of isovolumetric relaxation
J. beginning of ventricular diastole
Answer: B
The construction of ecological civilization in our country must be based on the current specific realities such as ( ) and so on, and follow a path of socialist ecological civilization construction that fits the national conditions.

Options:
A. current state of natural resource depletion
B. current state of industrial pollution
C. current state of natural ecological environment
D. economic development level
E. cultural development status
F. population quality
G. current state of natural resource management
H. current state of natural resource utilization
I. current state of urban infrastructure
J. current state of environmental policies
Answer: C
The 'NPCSC Interpretation' refers to the legislative interpretation made by the Standing Committee of the National People's Congress, the permanent legislative body of the People's Republic of China, on the Basic Law of the Hong Kong Special Administrative Region, which is the constitutional document of the Hong Kong Special Administrative Region formulated by the National People's Congress, the highest state power organ. Which of the following cases is a judgment made by the courts of the Hong Kong Special Administrative Region after actively requesting the 'NPCSC Interpretation'?

Options:
A. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2001 (20 July 2001)
B. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2000 (21 July 2000)
C. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2000 (23 July 2001)
D. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2000 (19 July 2001)
E. Sixtus Leung Chung Hang v. Chief Executive of The Hong Kong Special Administrative Region, Secretary for Justice [2017] HKCFA 55; (2017) 20 HKCFAR 390; FAMV 9/2017 (1 September 2017)
F. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2000 (21 July 2001)
G. Ng Ka Ling and Another v. The Director of Immigration [1999] HKCFA 72; [1999] 1 HKLRD 315; (1999) 2 HKCFAR 4; [1999] 1 HKC 291 ; FACV 14/1998 (29 January 1999)
H. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2000 (20 July 2001)
I. Democratic Republic of the Congo and Others v. FG Hemisphere Associates LLC [2011] HKCFA 41; (2011) 14 HKCFAR 95; [2011] 4 HKC 151; FACV 5/2010 (8 June 2011)
J. The Director of Immigration v. Chong Fung Yuen [2001] HKCFA 48; [2001] 2 HKLRD 533; (2001) 4 HKCFAR 211; FACV 26/2000 (22 July 2001)
Answer: H
If investment temporarily increases by 150 billion yuan, and the marginal propensity to consume is 0.8, then the level of income will increase ( )

Options:
A. 750 billion yuan and then stabilize at a lower level
B. 750 billion yuan and then return to the initial level
C. 720 billion yuan and then return to the initial level
D. 750 billion yuan and remain at this level
E. 150 billion yuan, and finally return to the original level
F. 190 billion yuan and remain at this level
G. 600 billion yuan and remain at this level
H. 675 billion yuan and remain at this level
I. 720 billion yuan and remain at this level
J. 600 billion yuan and then return to the initial level
Answer: D
A 70-year-old male patient has experienced chest pain and shortness of breath during physical exertion for the past six months, with activities such as sweeping and walking on flat ground being restricted. He has a history of hypertension for over ten years and had tuberculosis at the age of 20. The preferred further examination for this patient is ()

Options:
A. Chest X-ray
B. Stress Test (Treadmill Test)
C. Electrocardiogram (ECG)
D. Pulmonary Artery CT
E. Echocardiogram
F. Pulmonary Function Test
G. Cardiac CT
H. Stress Echocardiogram
I. Holter Monitor (Ambulatory ECG)
J. Cardiac MRI
Answer: E
The condition of hemiplegia, with pain on one side of the body but no change in speech or mental state, is due to a disease located between the skin and muscles. Treatment with large needles should be used to supplement what is deficient and reduce what is excessive, thus restoring balance." This quote is from ( ).

Options:
A. Zhen Jiu Yi Xue
B. Huang Di Nei Jing
C. Qian Jin Yao Fang
D. Zhen Jiu Zhi Nan
E. Zhen Jiu Da Cheng
F. Zhen Jiu Jia Yi Jing
G. Qian Jin Fang
Answer: B
After completing a project, renovation company A sold the remaining floorboards and other items to property management company B. However, the approved business scope on A's business license does not include the sale of floorboards. How is the legal nature of the transaction between A and B?

Options:
A. It is a civil legal act subject to regulatory approval
B. It is a partially valid civil legal act
C. It is a civil legal act with limited validity
D. It is a civil legal act with conditional validity
E. It is a civil legal act subject to judicial review
F. It is a valid civil legal act
G. It is an invalid civil legal act
H. It is a revocable civil legal act
I. It is a civil legal act subject to administrative review
J. It is a civil legal act of uncertain validity
Answer: F
Given non-negative functions f(n), g(h) and h(n), which of the following statements about $O,Theta,Omega$ is correct?

Options:
A. If $f(n)=O(h(n))$ and $g(n)=O(h(n))$, then $f(n)=g(n)$
B. $Theta(n)+Theta(n)>Theta(n)$
C. if $f(n)=O(h(n))$ and $g(n)=O(h(n))$, then $f(n)+g(n)=O(h(n))$
D. if $f(n)=O(h(n))$ and $g(n)=O(h(n))$, then $f(n)+g(n)=Theta(h(n))$
E. $Theta(n)+Theta(n/2)+Theta(n/4)+ldots+Theta(1)=Theta(n^2)$
F. if $f(n)=O(h(n))$ and $f(n)=Ω(h(n))$, then $f(n)=Theta(h(n))$
Answer: F
In the causes of postpartum hemorrhage, the first consideration for hysterectomy to achieve hemostasis is

Options:
A. placenta previa (critical type)
B. coagulopathy
C. uterine atony
D. placental abruption (severe type)
E. placental entrapment
F. placenta previa (severe type)
G. placenta previa (moderate type)
H. placental adhesion
I. placenta previa (mild type)
Answer: F
Gifts and gratuitous entrustment contracts belong to:

Options:
A. bilateral legal act, consensual legal act
B. bilateral legal act, gratuitous legal act, real legal act
C. bilateral legal act, gratuitous legal act, real legal act, consensual legal act
D. unilateral legal act, gratuitous legal act, real legal act
E. unilateral legal act, gratuitous legal act, consensual legal act, real legal act
F. bilateral legal act, gratuitous legal act, consensual legal act
G. unilateral legal act, real legal act
H. unilateral legal act, consensual legal act
I. bilateral legal act, real legal act
J. unilateral legal act, gratuitous legal act, consensual legal act
Answer: F
Given the recurrence relation of Fibonacci sequence is F(n)=F(n-1)+F(n-2), set G(n)=F(2n), then the recurrence relation of G(n) is: G(n)=___G(n-1)+___G(n-2)+____G(n-3) Please choose the coefficient value:

Options:
A. 3,0,0
B. 3,1,1
C. 3,0,-1
D. 1,1,1
E. 3,-1,0
F. 1,1,0
G. 3,1,0
H. 3,-1,-1
I. 3,0,1
J. 3,-1,1
Answer: E
Given that `beta ,alpha _1,alpha _2` are linearly dependent, and `beta ,alpha _2,alpha _3` are linearly independent, then ( )

Options:
A. `alpha _2` can be linearly represented by `beta ,alpha _1` with a zero scalar multiple.
B. ` beta` can be linearly represented by `alpha _1,alpha _2`.
C. `alpha _2` can be linearly represented by `beta ,alpha _1,alpha _3`
D. `alpha _1` can be linearly represented by `beta ,alpha _2,alpha _3`
E. `alpha _1` can be linearly represented by `beta ,alpha _2` with a zero scalar multiple.
F. `alpha _1,alpha _2,alpha _3` are linearly dependent;
G. `alpha _3` can be linearly represented by `beta ,alpha _1,alpha _2`
H. `alpha _1,alpha _2,alpha _3` are linearly independent;
I. `alpha _2` can be linearly represented by `beta ,alpha _1` with a non-zero scalar multiple.
J. `alpha _1` can be linearly represented by `beta ,alpha _2,alpha _3` with a non-zero scalar multiple.
Answer: B
The New York Times published an advertisement accusing the police authorities in Montgomery, Alabama, of brutally suppressing civil rights leader Martin Luther King Jr. and his supporters. Sullivan, as the head of the local police department, believed that the advertisement's report was inaccurate and filed a civil lawsuit with the Alabama court, demanding $500,000 in damages. Both the trial court and the state supreme court supported Sullivan's request, but the Supreme Court overturned the Alabama Supreme Court's decision. The reason is?

Options:
A. The New York Times' speech, while critical, should be protected under the First Amendment, but only if it does not cause substantial public harm
B. The New York Times' speech, while critical, did not constitute a significant threat to public officials, and thus should be protected by law
C. The New York Times' speech, while controversial, did not incite immediate violence or lawlessness
D. The New York Times' speech, while critical, did not constitute a significant threat to public officials, and thus should be protected under the First Amendment, but only if it does not cause substantial public harm
E. Alabama's racism violates fundamental moral principles and should not be protected by law
F. Debate on public issues should be uninhibited, robust, and wide-open
G. The New York Times' speech, while critical, did not constitute a direct threat to public officials, and thus should be protected by law
H. The New York Times' report was not significantly inaccurate to constitute defamation
I. The New York Times' speech, while critical, did not constitute a significant threat to public officials, and thus should be protected under the First Amendment
J. The New York Times' speech did not present a 'clear and present danger' to public order, and therefore should be protected by law
Answer: F
Assume that whether it rains tomorrow depends only on today's weather and not on past weather. Also, assume the probability of rain tomorrow given it rains today is a, and the probability of rain tomorrow given it does not rain today is b. Define rainy weather as state 0 and non-rainy weather as state 1. Given a=0.7, b=0.4, find the probability that it rains today and also rains on the fourth day.

Options:
A. 0.4332
B. 0.4248
C. 0.5678
D. 0.5752
E. 0.4251
F. 0.4252
G. 0.5750
H. 0.5748
I. 0.5751
J. 0.5749
Answer: J
The test of significant importance for distinguishing Mycobacterium tuberculosis from Mycobacterium bovis is

Options:
A. reduction of nitrate
B. synthesis of nicotinic acid
C. fermentation of glucose
D. reduction of nitrate to nitrite
E. fermentation of lactose
F. heat-stable nuclease test
G. reduction of nitrate to ammonia
H. reduction of nitrate to nitric oxide
I. reduction of nitrite
J. reduction of nitrate to nitrogen gas
Answer: F
The cells of the crescents in diffuse crescentic glomerulonephritis are

Options:
A. visceral epithelial cells of the glomerular capsule and macrophages
B. visceral epithelial cells of the glomerular capsule and fibroblasts
C. visceral epithelial cells of the glomerular capsule and plasma cells
D. mesangial cells
E. mesangial cells and endothelial cells
F. visceral epithelial cells of the glomerular capsule and monocytes
G. visceral epithelial cells of the glomerular capsule and neutrophils
H. visceral epithelial cells of the glomerular capsule and eosinophils
I. parietal epithelial cells of the glomerular capsule
J. visceral epithelial cells of the glomerular capsule and lymphocytes
Answer: F
Let the rank of matrix \( A_{m \times n} \) be \( R(A) = m < n \), and \( E_m \) be the \( m \)-order identity matrix. Which of the following conclusions is correct?

Options:
A. Matrix \( A \) must have a set of \( m \) column vectors that are linearly independent, but it may or may not have a set of \( m \) row vectors that are linearly dependent;
B. Matrix \( A \) must have a set of \( m \) column vectors that are linearly independent, but it does not necessarily have a set of \( m \) row vectors that are linearly independent;
C. Matrix \( A \) must have a set of \( m \) column vectors that are linearly independent, but it does not necessarily have a set of \( m \) row vectors that are linearly dependent;
D. Matrix \( A \) must have a set of \( m \) column vectors that are linearly dependent, and it must also have a set of \( m \) row vectors that are linearly dependent;
E. Any \( m \)-order minor of matrix \( A \) is not equal to zero;
F. Any \( m \) column vectors of matrix \( A \) must be linearly independent;
G. Matrix \( A \) must have a set of \( m \) column vectors that are linearly independent, and it must also have a set of \( m \) row vectors that are linearly independent;
H. Matrix \( A \) must have a set of \( m \) column vectors that are linearly independent, and it may or may not have a set of \( m \) row vectors that are linearly dependent;
I. Matrix \( A \) can certainly be transformed into the form \( (E_m, 0) \) through elementary row operations.
J. Matrix \( A \) must have a set of \( m \) column vectors that are linearly independent, and it may or may not have a set of \( m \) row vectors that are linearly independent;
Answer: G
`alpha _j` is the j-th column of the fourth-order determinant D (j=1, 2, 3, 4), and D=-5, then which of the following determinants equals -10?

Options:
A. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} + {alpha _4}, {alpha _4} - {alpha _2}|
B. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} - {alpha _4}, {alpha _4} + {alpha _2}|
C. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} + {alpha _4}, {alpha _4} + {alpha _1} - {alpha _2}|
D. |{alpha _1}, {alpha _1} + {alpha _2}, {alpha _1} + {alpha _2} + {alpha _3}, {alpha _1} + {alpha _2} + {alpha _3} + {alpha _4}|
E. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} - {alpha _4}, {alpha _4} - {alpha _1}|
F. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} - {alpha _4}, {alpha _4} + {alpha _1}|
G. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} + {alpha _4}, {alpha _4} - 2{alpha _1}|
H. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} + {alpha _4}, {alpha _4} + {alpha _1}|
I. |2{alpha _1}, 2{alpha _2}, 2{alpha _3}, 2{alpha _4}|
J. |{alpha _1} + {alpha _2}, {alpha _2} + {alpha _3}, {alpha _3} + {alpha _4}, {alpha _4} - {alpha _1}|
Answer: J
$int_{-infty}^{+infty} frac{a^{3/2}}{s^2+a^2} ds = $

Options:
A. $0$
B. $frac{1}{a}$
C. $frac{3}{a}$
D. $frac{1}{2a^2}$
E. $frac{1}{2a}$
F. $frac{2}{a}$
G. $2a$
H. $frac{3}{2a}$
I. $frac{3}{2a^2}$
J. $frac{1}{a^2}$
Answer: F
The zone position code of a Chinese character is 3721, its international code is ()

Options:
A. 4543H
B. 4554H
C. 4535H
D. 4533H
E. 6554H
F. 4545H
G. 4555H
H. 3555H
I. 4553H
J. 5445H
Answer: C
When the variance \( D(X) = \sigma^2 \) of a normal population \( X \) is unknown, the statistic used to test the expectation \( EX = \mu_0 \) is ( ).

Options:
A. \frac{(\bar{x} - \mu_0)\sqrt{n-1}}{\left( \sum\limits_{k=1}^n (\bar{x} - x_k)^2 \right)^{\frac{1}{2}}}
B. \frac{(\bar{x} - \mu_0)\sqrt{n(n-2)}}{\left( \sum\limits_{k=1}^n (\bar{x} - x_k)^2 \right)^{\frac{1}{2}}}
C. \( rac{(ar{x} - \mu_0)n}{\left( \sum\limits_{k=1}^n (ar{x} - x_k)^2 ight)^{rac{1}{2}}} \)
D. \( \frac{(\bar{x} - \mu_0)\sqrt{n(n-1)}}{\left( \sum\limits_{k=1}^n (\bar{x} - x_k)^2 \right)^{\frac{1}{2}}} \)
E. \frac{(\bar{x} - \mu_0)\sqrt{n(n+1)}}{\left( \sum\limits_{k=1}^n (\bar{x} - x_k)^2 \right)^{\frac{1}{2}}}
F. \( rac{ar{x} - \mu_0}{\left( \sum\limits_{k=1}^n (ar{x} - x_k)^2 ight)^{rac{1}{2}}} \)
G. \( rac{(ar{x} - \mu_0)(n-1)}{\left( \sum\limits_{k=1}^n (ar{x} - x_k)^2 ight)^{rac{1}{2}}} \)
H. \frac{(\bar{x} - \mu_0)\sqrt{n}}{\left( \sum\limits_{k=1}^n (\bar{x} - x_k)^2 \right)^{\frac{1}{2}}}
Answer: D
The ancient saying 'Luxury is the beginning of peril and decline' embodies the philosophical principle of

Options:
A. The dialectical relationship between qualitative change and quantitative change
B. The dialectical relationship between qualitative change and external conditions
C. The law of negation of negation
D. The dialectical relationship between internal causes and external effects
E. The dialectical relationship between cause and effect
F. The opposition and unity of the universality and particularity of contradictions
G. The dialectical relationship between quantitative change and qualitative change
H. Phenomenon is the external manifestation of essence
I. The dialectical relationship between internal causes and external conditions
J. The dialectical relationship between quantitative change and external conditions
Answer: G
The Dao is nothing greater than non-action, and conduct is nothing greater than being cautious. When a gentleman governs, he appears as if there is nothing to do, as silent as if there is no sound, the government offices seem as if there are no officials, and the villages seem as if there are no people. There are no disputes in the neighborhoods, and the elderly and young are not worried in their courtyards. Those nearby have nothing to discuss, and those far away have nothing to hear. There are no soldiers traveling at night on the postal roads, and no night calls for conscription in the villages. Dogs do not bark at night, and roosters do not crow at night. The elderly enjoy their food in the hall, and the young men till the fields. The above two paragraphs are selected from "Xinyu" (New Discourses) written by Lu Jia for Liu Bang during the early Han Dynasty. Which of the following statements about this passage is correct?

Options:
A. This passage suggests that the government should maintain a visible presence to ensure that people are aware of its authority and influence.
B. This passage emphasizes that the government should not overly interfere in the lives of ordinary people.
C. After Liu Bang's death, the Huang-Lao thought of non-action represented by 'Xinyu' was abandoned
D. This passage emphasizes that the government should focus on maintaining strict laws and regulations to ensure social order.
E. This is an embodiment of Shusun Tong's Confucian thought
F. This passage suggests that the government should maintain a minimal presence while ensuring economic prosperity.
G. This passage suggests that the government should maintain a balance between non-interference and active engagement in the lives of the people.
H. This passage emphasizes that the government should maintain a minimal presence while ensuring basic social order.
I. Although Liu Bang liked Lu Jia's 'Xinyu', he did not appreciate this passage
J. This passage suggests that the government should actively engage in the daily lives of the people to ensure order and harmony.
Answer: B
A 68-year-old male. He has had recurrent chest pain for 2 weeks, which is related to labor and can be relieved by rest. Three hours ago, he developed persistent chest pain, which progressively worsened, accompanied by dyspnea and inability to lie flat. His blood pressure is 110/70 mmHg, heart rate is 120 beats per minute, with regular rhythm. A grade 3/6 systolic murmur can be heard at the apex of the heart, and scattered wheezes and moist rales are heard in both lungs. The preferred examination is

Options:
A. Echocardiogram with Doppler
B. Echocardiography with Doppler
C. Cardiac CT
D. Holter monitoring
E. Echocardiogram with stress test
F. Echocardiogram
G. Electrocardiogram
H. Echocardiography
I. Troponin
J. Cardiac MRI
Answer: G
Consider a call option on a non-dividend-paying stock. Assume the stock price is $49, the option strike price is $50, the risk-free rate is 5% per annum, the annual volatility of the stock price is 20%, and the option maturity is 20 weeks (0.3846 years). The Gamma value of the option is:

Options:
A. 0.059
B. 0.068
C. 0.053
D. 0.064
E. 0.024
F. 0.069
G. 0.060
H. 0.066
I. 0.043
J. 0.065
Answer: H
The following set of functions is linearly dependent in the domain ( ).

Options:
A. (sin x cos x, 2 sin x cos x + sin 2x)
B. (sin x, cos x)
C. (tan x, cot x)
D. (sin x cos x, 2 sin x cos^2 x)
E. (sin x cos x, sin 2x)
F. (sin x cos x, 2 sin x cos x)
G. (sec x, csc x)
H. (sin x cos x, 2 sin x cos x + sin x)
I. (sin x cos x, 2 sin x cos x - sin 2x)
J. (sin x cos x, 2 sin^2 x cos x)
Answer: E
The one who praised the revolution as a 'good medicine to enlighten the people and remove the old to make way for the new', and advocated using violent revolution to overthrow the Qing government and establish a bourgeois democratic republic is ( ).

Options:
A. The Revolutionary Army
B. The Revolutionary Critique of Kang Youwei
C. The Revolutionary Critique of Kang Youwei's Discussion of Revolution
D. Rebuttal to Kang Youwei's Discussion of Revolution
E. Rebuttal to Kang Youwei's Discussion of Reform
F. Rebuttal to Kang Youwei's Discussion of Reform and Revolution
G. The World's Constant Warning
H. The Revolutionary Critique of Kang Youwei's Discussion
I. The Revolutionary Manifesto
J. The Warning Bell
Answer: D
A 43-year-old female presented with severe right upper quadrant pain that worsened intermittently over 3 days, accompanied by chills, high fever, nausea, and vomiting. Physical examination revealed jaundice, a temperature of 39°C, a pulse rate of 120 beats per minute, and a blood pressure of 80/60 mmHg. The patient was delirious and confused, with muscle tension in the left of the epigastrium, the liver palpable 2 cm below the costal margin, a white blood cell count of 15×10^9/L, 80% neutrophils, and a total serum bilirubin level of 50 μmol/L. The preferred treatment for this case is

Options:
A. cholecystectomy and common bile duct decompression
B. cholecystectomy and abdominal drainage
C. cholecystectomy and common bile duct exploration
D. common bile duct decompression and cholecystostomy
E. common bile duct decompression and cholecystectomy
F. anti-shock and anti-infection treatment
G. common bile duct decompression and drainage
H. cholecystostomy
I. common bile duct decompression and anti-infection treatment
J. common bile duct exploration and drainage
Answer: G
A patient diagnosed with Japanese encephalitis, on the third day of hospitalization, shows a significant increase in blood pressure, unequal pupils, neck stiffness, and apnea. What is the first emergency measure that should be taken?

Options:
A. sedation
B. 20% mannitol
C. 10% mannitol
D. corticosteroids
E. 18% mannitol
F. 22% mannitol
G. furosemide
H. 25% mannitol
I. oxygen inhalation
J. 15% mannitol
Answer: B
If a firm is in a perfectly competitive product market and factor A is its only variable factor, then the firm's demand curve for factor A is given by which of the following?

Options:
A. MFCa curve
B. MRPa + MFCa curve
C. All of the above
D. VMPa * MPPa curve
E. MRPa curve
F. MRPa * VMPa curve
G. MRPa - MFCa curve
H. VMPa curve
I. MRPa * MFCa curve
J. MPPa curve
Answer: H
The main immune response causing immune damage in Hemorrhagic Fever with Renal Syndrome is?

Options:
A. Type II hypersensitivity reaction with immune complex deposition
B. Type IV hypersensitivity reaction with immune complex deposition
C. Type IV hypersensitivity reaction
D. Type III hypersensitivity reaction with immune cell activation
E. Type II hypersensitivity reaction with immune cell activation
F. Type II hypersensitivity reaction
G. Type III hypersensitivity reaction
H. Type V hypersensitivity reaction
I. Type I hypersensitivity reaction
J. Type III hypersensitivity reaction with immune complex deposition
Answer: G
If $x$ is an odd number, then find the largest integer that always divides the expression
$$(10x+2)(10x+6)(5x+5)$$

Options:
A. 720
B. 840
C. 1200
D. 1920
E. 480
F. 1680
G. 120
H. 1120
I. 960
J. 1440
Answer: I
On square $ABCD,$ points $E$ and $F$ lie on side $AD$ such that $AE=EF=FD.$ The circumcircle of triangle $ACF$ intersects ray $AB$ again at a point $G,$ such that $GF=20.$ Find the area of $ABCD.$
Author: Jason Tang

Options:
A. 172
B. 200
C. 140
D. 150
E. 190
F. 182
G. 170
H. 162
I. 160
J. 180
Answer: J
Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$ .

Options:
A. 678
B. 660
C. 688
D. 675
E. 680
F. 685
G. 682
H. 670
I. 672
J. 690
Answer: E
How many ways can Elena place $1$ s inside the nine boxes of a $3 \times 3$ square such that each row and column of the $3 \times 3$ square contains at least one $1$ ? (Assume that each box either has a $1$ placed by Elena or nothing inside it; rotations and reflections are considered distinct.)
Problem Five,

Options:
A. 256
B. 240
C. 270
D. 260
E. 279
F. 264
G. 243
H. 258
I. 252
J. 265
Answer: J
$\{A_n\}_{n\ge1}$ is a sequence of positive integers such that
$$$a_n = 2a_{n-1} + n^2$$$
for all integers $n > 1$ . Compute the remainder obtained when $a_{2004}$ is divided by $1000$ if $a_1 = 1$ .

Options:
A. 57
B. 56
C. 60
D. 52
E. 54
F. 59
G. 55
H. 58
I. 53
J. 61
Answer: H
Let $p$ , $q$ , $r$ , and $s$ be positive real numbers such that
$$\begin{align*}p+q+r+s&=15\\p^2+q^2+r^2+s^2&=n\end{align*}$$
and $p\ge q\ge r\ge s$ . Given that the minimum possible value of $pq-rs$ is $12$ for a choice of constant $n$ , the sum of the possible values of $n$ is $x$ . If $x$ is an integer, compute $x$ . If $x$ can be expressed as $\frac mn$ for relatively prime positive integers $m$ and $n\ne1$ , compute $m+n$ . If $x$ can be expressed as $\frac{a+b\sqrt c}d$ for positive integers $a$ , $b$ , $c$ , and $d$ such that $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime, compute $a+b+c+d$ .

Options:
A. 46
B. 39
C. 42
D. 44
E. 41
F. 40
G. 45
H. 47
I. 38
J. 43
Answer: B
The sum of all positive integers $n$ such that the base $2$ representation of $n$ has exactly $8$ $1$ 's and $8$ $0$ 's (the first digit cannot be $0$ ) is $S$ . Compute the remainder when $S$ is divided by $1000$ .

Options:
A. 389
B. 388
C. 382
D. 384
E. 385
F. 381
G. 380
H. 386
I. 387
J. 383
Answer: F
Let $S(n)$ be the sum of the squares of the digits of $n$ . How many positive integers $n>2007$ satisfy the inequality
$$$n-S(n)\le 2007?$$$

Options:
A. 66
B. 65
C. 67
D. 62
E. 68
F. 69
G. 61
H. 60
I. 64
J. 63
Answer: J
An AIME has $15$ questions, $5$ of each of three difficulties: easy, medium, and hard. Let $e(X)$ denote the number of easy questions up to question $X$ (including question $X$ ). Similarly define $m(X)$ and $h(X)$ . Let $N$ be the number of ways to arrange the questions in the AIME such that, for any $X$ , $e(X) \ge m(X) \ge h(X)$ and if a easy and hard problem are consecutive, the easy always comes first. Find the remainder when $N$ is divided by $1000$ .

Options:
A. 760
B. 765
C. 769
D. 766
E. 761
F. 763
G. 762
H. 767
I. 768
J. 764
Answer: J
A sequence of polynomials is defined by the recursion $P_1(x) = x+1$ and
$$$P_{n}(x) = \frac{(P_{n-1}(x)+1)^5 - (P_{n-1}(-x)+1)^5}{2}$$$
for all $n \geq 2$ . Find the remainder when $P_{2022}(1)$ is divided by $1000$ .

Options:
A. 610
B. 611
C. 617
D. 615
E. 612
F. 616
G. 618
H. 614
I. 619
J. 613
Answer: F
Find the sum of all positive integers $x$ such that there exists integers $a$ and $b$ that satisfy
$$$|x^2 - 92x + 2099| = 2^a3^b - 8.$$$

Options:
A. 837
B. 835
C. 836
D. 830
E. 839
F. 834
G. 833
H. 831
I. 838
J. 832
Answer: G
In $\triangle{ABC}$ , let $D$ and $E$ be points on sides $AC$ and $BC$ , respectively, so that $DE$ is parallel to $AB$ . Let $P$ be the intersection of $AE$ and $BD$ , and assume that the area of $\triangle{ABP}=36$ and the area of $\triangle{EDP}=25$ . Find the area of $\triangle{ABC}$

Options:
A. 384
B. 376
C. 369
D. 385
E. 390
F. 375
G. 391
H. 396
I. 361
J. 380
Answer: H
A sequence $\{R_n\}_{n \ge 0}$ obeys the recurrence $7R_n = 64-2R_{n-1} +9R_{n-2}$ for any integers $n \ge 2$ . Additionally, $R_0 = 10 $ and $R_1 = -2$ . Let
$$$S = \sum^{\infty}_{i=0} \frac{R_i}{2^i}.$$$
$S$ can be expressed as $\frac{m}{n}$ for two relatively prime positive integers $m$ and $n$ . Determine the value of $m + n$ .

Options:
A. 446
B. 449
C. 443
D. 448
E. 447
F. 442
G. 440
H. 445
I. 441
J. 444
Answer: C
For each arrangement of the ten numbers $1,4,7,10,13,16,19,22,25,28$ around the circle, let $N$ denote the largest of the ten sums obtained by adding each number to the sum of its neighbors. What is the smallest value of $N$ that can be obtained?

Options:
A. 49
B. 45
C. 43
D. 47
E. 44
F. 47.5
G. 48
H. 50
I. 46
J. 47.3
Answer: G
Compute, to the nearest integer, the area of the region enclosed by the graph of
$$$13x^2-20xy+ 52y^2 - 10x + 52y = 563.$$$

Options:
A. 70
B. 76
C. 78
D. 71
E. 74
F. 75
G. 77
H. 73
I. 79
J. 72
Answer: F
On a test with $30$ problems, each correct answer is worth $5$ points, while each wrong answer deducts $4$ points. There is no credit/deduction for any problem left blank. Find the smallest positive integer $N$ such that it is impossible to get $N$ points on such a test.

Options:
A. 114
B. 116
C. 120
D. 113
E. 119
F. 117
G. 121
H. 115
I. 118
J. 112
Answer: E
Point $P$ is located inside $\triangle{ABC}$ so that $\angle{PAB}=\angle{PBC}=\angle{PCA}$ . The sides of the triangle are $AB=13$ , $BC=14$ , $CA=15$ . The tangent of $\angle{PAB}$ may be expressed in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ .

Options:
A. 469
B. 466
C. 461
D. 462
E. 465
F. 467
G. 460
H. 464
I. 463
J. 468
Answer: I
In parallelogram $ABCD$ , let a circle passing through $A$ intersect sides $AB, AC, AD$ in points $P,Q,R$ , respectively. Given that $AD=2013(AB)$ and $ AP=2013 (AR) $ , find the remainder when $ \frac{(AQ) (AC)}{(AB) (AR)} $ is divided by $1000$ .

Options:
A. 25
B. 29
C. 28
D. 20
E. 22
F. 21
G. 24
H. 27
I. 26
J. 23
Answer: I
Let $S$ be a subset of $[1,2,3,....,1000]$ so that if $a$ and $b$ are any (not necessarily distinct) elements of $S$ , then their product $a \cdot b$ , is not an element of $S$ . What is the maximum number of elements of $S$ .

Options:
A. 970
B. 972
C. 978
D. 976
E. 971
F. 975
G. 977
H. 969
I. 973
J. 974
Answer: H
Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^3 - x$. Let \[ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) = \frac{f(x)}{g(x)} \] for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.

Options:
A. 3986
B. 3983
C. 3985
D. 3989
E. 3987
F. 3988
G. 3984
H. 3982
I. 3980
J. 3981
Answer: G
Determine the smallest positive integer $d$ such that for any polynomial $p(x)$ with integer coefficients and any integer $k$, the value \[ p^{(d)}(k) = \left. \frac{d^d}{dx^d} p(x) \right|_{x=k} \] (the $d$-th derivative of $p(x)$ evaluated at $k$) is divisible by 2016.

Options:
A. 14
B. 8
C. 7
D. 15
E. 9
F. 10
G. 12
H. 6
I. 16
J. 11
Answer: B
For all real numbers $x$ satisfying $x^4+173056\leq 2768x^2$, what is the maximum value of $f(x)=x^3-3x$?

Options:
A. 140450
B. 140455
C. 140453
D. 140456
E. 140452
F. 140457
G. 140451
H. 140459
I. 140458
J. 140454
Answer: E
Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).\]

Options:
A. 2
B. 6
C. 1
D. 3.75
E. 3.25
F. 5
G. 3.5
H. 4
I. 2.5
J. 3
Answer: J
Determine the count of positive integers $X$ that fulfill all of these criteria: \begin{{enumerate}} \item[(i)] $X$ is a multiple of 505000. \item[(ii)] The number of decimal digits in $X$ does not exceed 505000. \item[(iii)] When written in decimal notation, $X$ consists of a sequence of ones immediately followed by a sequence of zeros (either sequence may be empty). \end{{enumerate}}

Options:
A. \boxed{31877746252}
B. \boxed{31877746254}
C. \boxed{31877746253}
D. \boxed{31877746258}
E. \boxed{31877746259}
F. \boxed{31877746255}
G. \boxed{31877746256}
H. \boxed{31877746250}
I. \boxed{31877746251}
J. \boxed{31877746257}
Answer: I
Evaluate \[ \int_0^1 \left( C(-s-1) \sum_{k=1}^12456 \frac{1}{s+k} \right)\,ds, \] where $C(\alpha)$ is the coefficient of $x^12456$ in the expansion of the power series of $(1 + g)^\alpha$ centered at $g=0$.

Options:
A. 12458
B. 12450
C. 12453
D. 12457
E. 12455
F. 12459
G. 12454
H. 12456
I. 12451
J. 12452
Answer: H
Let $C$ be the unit circle $x^2+y^2=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$ and $y$-axes with diagonal $pq$. What is the probability that no point of $R$ lies outside of $C$?

Options:
A. \frac{4}{\pi^2}
B. \frac{1}{\pi^2}
C. \frac{3}{\pi^2}
D. \frac{12}{\pi^2}
E. \frac{7}{\pi^2}
F. \frac{6}{\pi^2}
G. \frac{8}{\pi^2}
H. \frac{2}{\pi^2}
I. \frac{5}{\pi^2}
J. \frac{10}{\pi^2}
Answer: A
Given $g(x) = \frac{ 1 }{ 3 } \cdot (a+b) \cdot x^3 + \frac{ 1 }{ 2 } \cdot (a+b+c) \cdot x^2 - (a+b+c+d) \cdot x + a \cdot b \cdot c \cdot d$, simplify the derivative of $g(x)$ if $x^2 + x = a + b$.

Options:
A. (a+b)^2+c*x-(a+b+c+d+6)
B. (a+b)^2+c*x-(a+b+c+d+3)
C. (a+b)^2+c*x-(a+b+c+d+2)
D. (a+b)^2+c*x-(a+b+c+d+4)
E. (a+b)^2+c*x-(a+b+c+d-2)
F. (a+b)^2+c*x-(a+b+c+d+5)
G. (a+b)^2+c*x-(a+b+c+d)
H. (a+b)^2+c*x-(a+b+c+d-1)
I. (a+b)^2+c*x-(a+b+c)
J. (a+b)^2+c*x-(a+b+c+d+1)
Answer: G
Calculate $E = \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(22) }$.

Options:
A. \\frac{ \\cos(70) \\cdot \\cos(10) + \\cos(80) \\cdot \\cos(20) }{ \\cos(68) \\cdot \\cos(8) + \\cos(82) \\cdot \\cos(29) }
B. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(27) }
C. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(28) }
D. \\frac{ \\cos(70) \\cdot \\cos(10) + \\cos(80) \\cdot \\cos(20) }{ \\cos(68) \\cdot \\cos(8) + \\cos(82) \\cdot \\cos(21) }
E. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(23) }
F. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(26) }
G. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(24) }
H. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(25) }
I. \frac{ \cos(70) \cdot \cos(10) + \cos(80) \cdot \cos(20) }{ \cos(68) \cdot \cos(8) + \cos(82) \cdot \cos(22) + 0.01 }
J. 1
Answer: J
Compute the integral:
$$
\int \frac{ x^3 }{ \sqrt{4 \cdot x^2+4 \cdot x+5} } \, dx
$$

Options:
A. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2+3}\right)+C
B. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2-1}\right)+C
C. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2+1}\right)+C
D. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2+2}\right)+C
E. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2-2}\right)+C
F. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2+4}\right)+C
G. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2+5}\right)+C
H. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{4\cdot x^2+4\cdot x+5}\right)+C
I. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2}\right)+C
J. \left(\frac{1}{12}\cdot x^2-\frac{5}{48}\cdot x-\frac{5}{96}\right)\cdot\sqrt{4\cdot x^2+4\cdot x+5}+\frac{5}{16}\cdot\ln\left(x+\frac{1}{2}+\sqrt{1+\left(x+\frac{1}{2}\right)^2-3}\right)+C
Answer: I
Find the equation of an ellipse with the following conditions:
1. The ellipse is tangent to the y-axis at $(0,3)$.
2. The ellipse intersects the x-axis at $(3,0)$ and $(7,0)$.
3. The axes of the ellipse are parallel to the cartesian axes.

Options:
A. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{85}=1
B. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{50}=1
C. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{120}=1
D. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{90}=1
E. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{130}=1
F. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{80}=1
G. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{75}=1
H. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{110}=1
I. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{60}=1
J. \frac{(x-5)^2}{25}+\frac{7\cdot(y-3)^2}{100}=1
Answer: G
Compute $\int_{0}^{\frac{ 1 }{ 3 }} e^{-\frac{ x^2 }{ 3 }} \, dx$ with accuracy $0.00001$.

Options:
A. 0.32926
B. 0.32922
C. 0.32921
D. 0.32920
E. 0.32927
F. 0.32924
G. 0.32929
H. 0.32923
I. 0.32928
J. 0.32925
Answer: A
Evaluate $\lim_{x \to 0}\left(\left(\frac{ \sin(4 \cdot x) }{ 4 \cdot x }\right)^{\frac{ 1 }{ 3 \cdot x^2 }}\right)$.

Options:
A. e^{-\frac{14}{9}}
B. e^{-\frac{7}{9}}
C. e^{-\frac{8}{9}}
D. e^{-\frac{4}{9}}
E. e^{-\frac{2}{9}}
F. e^{-\frac{12}{9}}
G. e^{-\frac{10}{9}}
H. e^{-\frac{18}{9}}
I. e^{-\frac{1}{9}}
J. e^{-\frac{16}{9}}
Answer: C
Calculate $e$ with an estimated error of $0.001$, using the series expansion.

Options:
A. 2.7187
B. 2.7186
C. 2.7188
D. 2.7181
E. 2.7185
F. 2.7189
G. 2.7183
H. 2.7182
I. 2.7184
J. 2.7180
Answer: D
Calculate the integral:
$$
\int_{-\sqrt{2}}^{\sqrt{2}} \frac{ 23 \cdot x^7+7 \cdot x^6-130 \cdot x^5-72 \cdot x^3-112 \cdot x^2+4 \cdot x+7 }{ x^2+4 } \, dx
$$

Options:
A. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{19}
B. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{16}
C. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{20}
D. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{18}
E. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{15}
F. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{14}
G. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{11}
H. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{12}
I. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{17}
J. 7\cdot\arctan\left(\frac{1}{\sqrt{2}}\right)-\frac{392\cdot\sqrt{2}}{13}
Answer: E
Solve the integral:
$$
\int 2 \cdot \cot(14 \cdot x)^6 \, dx
$$

Options:
A. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 - \frac{1}{7} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
B. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 + \arctan\left( \cot(14 \cdot x) \right) \right)
C. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 + \frac{1}{4} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
D. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 - \arctan\left( \cot(14 \cdot x) \right) \right)
E. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 + \frac{1}{6} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
F. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 + \frac{1}{9} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
G. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 - \frac{1}{2} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
H. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 + \frac{1}{8} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
I. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 - \frac{1}{4} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
J. C - \frac{1}{7} \cdot \left( \frac{1}{5} \cdot \left( \cot(14 \cdot x) \right)^5 + \cot(14 \cdot x) - \frac{1}{3} \cdot \left( \cot(14 \cdot x) \right)^3 + \frac{1}{2} \cdot \arctan\left( \cot(14 \cdot x) \right) \right)
Answer: D