MikaStars39/nano-eval huggingface dataset:
Find the gradient of the function $f(x,y) = \frac{ \sqrt{x} + y^2 }{ x \cdot y }$.

Options:
A. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x^2\cdot y},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x^2\cdot y^2} \right\rangle
B. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x\cdot y^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y}\right\rangle
C. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x\cdot y^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2} \right\rangle
D. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x^2\cdot y},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x^2\cdot y} \right\rangle
E. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{y\cdot x^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2}\right\rangle
F. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x\cdot y^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x^2\cdot y^2}\right\rangle
G. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x\cdot y^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x^2\cdot y} \right\rangle
H. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x^2\cdot y},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2}\right\rangle
I. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x^2\cdot y^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2} \right\rangle
J. \left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{x^2\cdot y},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2} \right\rangle
Answer: E
Solve the exponential equation exactly: $3^{\frac{ x }{ 14 }} = \frac{ 1 }{ 10 }$

Options:
A. -14 / (log10(3) * 2)
B. -14 / log3(10)
C. -14 / (log10(3) * 0.5)
D. -14 / (log10(3) + 1)
E. -14 / (log10(3) * 1.5)
F. -14 / (log10(3) - 1)
G. -14 / (log10(3) * 1.2)
H. -14 / (log10(3) * 1.1)
I. -14 / log10(3)
J. -14 / log3(10) + 1
Answer: I
Compute the integral:
$$
\int \frac{ -2 }{ e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} } \, dx
$$

Options:
A. C - \frac{1}{3} \cdot \left( \frac{2}{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}}} + 2 \cdot \ln\left( \frac{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} + 1}{1 + e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}}} \right) \right)
B. C - \frac{1}{3} \cdot \left( \frac{2}{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}}} + 2 \cdot \ln\left( \frac{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} - 1}{1 + e^{3 \cdot x} - \sqrt{1 + e^{6 \cdot x}}} \right) \right)
C. C - \frac{1}{3} \cdot \left( \frac{2}{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}}} + 2 \cdot \ln\left( \frac{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} - 1}{1 + e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}}} \right) \right)
D. C - \frac{1}{3} \cdot \left( \frac{2}{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}}} + 2 \cdot \ln\left( \frac{e^{3 \cdot x} + \sqrt{1 + e^{6 \cdot x}} + 1}{1 + e^{3 \cdot x} - \sqrt{1 + e^{6 \cdot x}}} \right) \right)
Answer: C
A triangle has the property that its sides form an arithmetic progression, and that the angle opposite the longest side is three times the angle opposite the shortest side. The ratio of the longest side to the shortest side can be expressed as $\frac{a+\sqrt{b}} {c}$ , where $a,b,c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a+b+c$ .

Options:
A. 47
B. 50
C. 46
D. 48
E. 49
Answer: B
Evaluate $\int_{0}^\pi \sin(5 \cdot x) \cdot \sqrt{\cos(5 \cdot x)} \, dx$ using substitution.

Options:
A. 2/15 - 2*sqrt(-1)/3
B. 2/15 - 2*sqrt(1)/15
C. 2/15 - 2*sqrt(-1)/15
D. 2/15 - 2*sqrt(-1)/5
E. 2/15 + 2*sqrt(-1)/15
Answer: E
Compute the integral:
$$
\int \frac{ 2 \cdot x+1 }{ (x-1) \cdot \sqrt{x^2-4 \cdot x+2} } \, dx
$$

Options:
A. 2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x-1}{2\cdot\sqrt{2}}\right)+C
B. 2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x-2}{(x-1)\cdot\sqrt{2}}\right)+C
C. 2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x-1}{\sqrt{2}\cdot(x-1)}\right)+C
D. 2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x}{(x-1)\cdot\sqrt{2}}\right)+C
E. 2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x-1}{(x-1)\cdot\sqrt{2}}\right)+C
F. 2\cdot\ln\left(\left|x-2+\sqrt{x^2-4\cdot x+2}\right|\right)-3\cdot\arcsin\left(\frac{x-1}{\sqrt{2}}\right)+C
Answer: D
A spinner has five sectors numbered ${-}1.25$ , ${-}1$ , $0$ , $1$ , and $1.25$ , each of which are equally likely to be spun. Ryan starts by writing the number $1$ on a blank piece of paper. Each minute, Ryan spins the spinner randomly and overwrites the number currently on the paper with the number multiplied by the number the spinner lands on. The expected value of the largest number Ryan ever writes on the paper can be written as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Options:
A. 16
B. 17
C. 14
D. 18
E. 15
F. 13
G. 19
Answer: B
A secret spy organization needs to spread some secret knowledge to all of its members. In the beginning, only $1$ member is informed. Every informed spy will call an uninformed spy such that every informed spy is calling a different uninformed spy. After being called, an uninformed spy becomes informed. The call takes $1$ minute, but since the spies are running low on time, they call the next spy directly afterward. However, to avoid being caught, after the third call an informed spy makes, the spy stops calling. How many minutes will it take for every spy to be informed, provided that the organization has $600$ spies?

Options:
A. 17
B. 10
C. 13
D. 12
E. 8
F. 9
G. 11
H. 16
I. 15
J. 14
Answer: B
Compute the limit using L'Hopital's Rule:
$$
\lim_{x \to 0} \left( \frac{ 1 }{ x^2 } - \cot(x)^2 \right)
$$

Options:
A. 1/9
B. 1/5
C. 1/3
D. 2/3
E. 1/4
F. 1/6
G. 1/2
Answer: D
The sequence $a_1, a_2,...$ is defined by $a_1 = 1$ and
$$$a_{k+1} = (-1)^{a_1+a_2+...+a_k}$$$
for all integer $k \ge 1$ . Find the integer $x$ , with $0 \le x \le 999$ , such that $a_1 + 2a_2 + ... + 2017a_{2017} -x$ is a
multiple of $1000$ .

Options:
A. 59
B. 9
C. 19
D. 39
E. 109
F. 199
G. 29
H. 49
Answer: B
Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Find the least number of distinct real zeros $f + 6f' + 12f'' + 8f'''$ must have.

Options:
A. 2
B. 5
C. 1
D. 8
E. 3
F. 7
G. 6
H. 4
Answer: A
Evaluate the limit:
$$
\lim_{x \to 0}\left(\frac{ \ln\left(1+x+x^2\right)+\ln\left(1-x+x^2\right) }{ x^2 }\right)
$$

Options:
A. 2
B. 0
C. -1
D. 1/3
E. 1
F. 3
G. 1/2
H. 2/3
Answer: E
Regular tetrahedron $ABCD$ has side length $6$ . Let $\ell$ be the line passing through the midpoints of $\overline{AB}$ and $\overline{CD}$ . The tetrahedron is rotated about $\ell$ ; let $\mathcal{S}$ be the intersection of all the rotated tetrahedra. The square of the volume of $\mathcal{S}$ can be expressed as $\tfrac{m}{n} \pi^2$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Options:
A. 89
B. 101
C. 103
D. 97
E. 95
F. 99
G. 91
H. 107
I. 93
Answer: A
Solve the integral:
$$
\int \sqrt{\frac{ 4 \cdot \sin(4 \cdot x) }{ 9 \cdot \cos(4 \cdot x)^9 }} \, dx
$$

Options:
A. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{1}{2}}
B. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{13}{2}}
C. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{11}{2}}
D. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{5}{2}}
E. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{19}{2}}
F. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{7}{2}}
G. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{17}{2}}
H. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{15}{2}}
I. C + \frac{1}{9} \cdot \left(\tan(4 \cdot x)\right)^{\frac{3}{2}} + \frac{1}{21} \cdot \left(\tan(4 \cdot x)\right)^{\frac{9}{2}}
Answer: F
Find the largest prime factor of the smallest positive integer $n$ such that $r_1, r_2, \ldots , r_{2006}$ are distinct integers such that the polynomial $(x-r_{1})(x-r_{2})\cdots (x-r_{2006})$ has exactly $n$ nonzero coefficients.

Options:
A. 249
B. 251
C. 247
D. 259
E. 255
F. 257
G. 243
H. 254
I. 253
J. 245
Answer: B
Let $\mathcal{F}$ be a collection of eight-element subsets of $S = \{1, 2, \dots, 24\}$ such that any five-element subset of $S$ is contained in a unique element of $\mathcal{F}$ . Let $N$ be the number of unordered pairs $\{A, B\}$ such that $A, B \in \mathcal{F}$ and $A \cap B = \emptyset$ . Find the remainder when $N$ is divided by $1000$ .

Options:
A. 385
B. 375
C. 390
D. 355
E. 397
F. 395
G. 380
H. 365
I. 370
J. 377
Answer: A
Find the last three digits of the remainder when $2^{96}-2^{72}+2^{28}+2^{14}$ is divided by $2017 \cdot 2081.$

Options:
A. 644
B. 648
C. 641
D. 643
E. 647
F. 639
G. 642
H. 646
I. 645
J. 640
Answer: J
Let $N$ denote the number of $8$ -tuples $(a_1,a_2,... a_8)$ of real numbers such that $a_1 = 10$ and
$$$\left|a^2_1 - a^2_2 \right|= 10$$$
$$$\left|a^2_2 - a^2_3 \right|= 20$$$
$$$...$$$
$$$\left|a^2_7 - a^2_8 \right|= 70$$$
$$$\left|a^1_8 - a^2_1 \right|= 80$$$
Determine the remainder obtained when $N$ is divided by $1000$ .

Options:
A. 477
B. 479
C. 470
D. 473
E. 474
F. 471
G. 478
H. 476
I. 475
J. 472
Answer: J
Let $x$ and $n$ be integers such that $n$ is positive and
$$$x^2=(10n+12)x+(4n^3+4n^2+6n+9).$$$
In addition, $x$ is the larger solution to the quadratic in $x$ above. Compute the smallest possible value of $x-2n$ .

Options:
A. 20
B. 27
C. 23
D. 22
E. 24
F. 21
G. 25
H. 26
I. 29
J. 28
Answer: F
Given a parallelogram $ABCD$ , let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$ , the distance from vertex $B$ to $\mathcal{P}$ is $25$ , and the distance from vertex $C$ to $\mathcal{P}$ is $36$ . Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$ .

Options:
A. 218
B. 228
C. 216
D. 220
E. 224
F. 232
G. 212
H. 204
I. 208
J. 196
Answer: D
Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2}$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2}$ . If $1 \le a \le 251$ , how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $5$ ?

Options:
A. 101
B. 102
C. 105
D. 98
E. 100
F. 99
G. 106
H. 103
I. 104
J. 107
Answer: A
Let $N$ denote the number of permutations of the $15$ -character string $AAAABBBBBCCCCCC$ such that
$\bullet$ None of the first four letters is an $A$ . (1)
$\bullet$ None of the next five letters is a $B$ . (2)
$\bullet$ None of the last six letters is a $C$ . (3)
Find the remainder when $N$ is divided by $1000$ .

Options:
A. 320
B. 340
C. 390
D. 380
E. 330
F. 310
G. 360
H. 300
I. 370
J. 350
Answer: A
Let $ABC$ be a triangle with $AB = 12$ , $AC = 16$ , and incenter $I$ . Lines $BI$ and $CI$ intersect the circumcircle of $\vartriangle ABC$ at $X \ne B$ and $Y \ne C$ , respectively. If $\overline{XY}$ is tangent to the incircle of $\vartriangle ABC$ , then find $BC^2$ .

Options:
A. 200
B. 210
C. 208
D. 204
E. 218
F. 224
G. 220
H. 216
I. 214
J. 212
Answer: C
There are precisely $N$ distinct non-constant polynomials $P(x)$ with nonnegative integer coefficients such that $P(10) = 2023$ . Find the remainder when $N$ is divided by $1000$ .

Options:
A. 753
B. 746
C. 748
D. 754
E. 752
F. 751
G. 749
H. 747
I. 750
J. 745
Answer: C
Let $S$ be the set of positive integers with only odd digits satisfying the following condition: any $x \in S$ with $n$ digits must be divisible by $5^n$ . Let $A$ be the sum of the $20$ smallest elements of $S$ . Find the remainder upon dividing $A$ by $1000$ .

Options:
A. 810
B. 825
C. 830
D. 815
E. 832
F. 828
G. 820
H. 835
I. 822
J. 840
Answer: C
In $\triangle{ABC},$ let $P$ be on $AC$ such that $BP$ bisects $\angle{ABC}.$ Similarly, let $Q$ be on $AB$ such that $CQ$ bisects $\angle{ACB}.$ Let $BP$ intersect $CQ$ at $R$ . It is known $APRQ$ can be inscribed in a circle. Given that $AP=23$ and $AQ=13,$ find the square of the length of $AR.$

Options:
A. 435
B. 436
C. 433
D. 430
E. 437
F. 431
G. 432
H. 434
I. 439
J. 438
Answer: G
Let $P(x)=x^3-2x+3$ and $Q(x)=x^7-6x^5+9x^4+12x^3-36x^2+17x+36$ . Let $z$ be a randomly selected root of $P(x)$ . Compute the expected value of $Q(z)$ .

Options:
A. 6
B. 10
C. 9
D. 7.4
E. 8
F. 7.5
G. 7.6
H. 7.8
I. 7
J. 7.2
Answer: E
Let $\lfloor x \rfloor$ be the largest integer such that $\lfloor x \rfloor \le x$ , and let $\{x\} = x -\lfloor x \rfloor$ . How many values of $x $ satisfy $x + \lfloor x \rfloor \cdot \{x\} = 23$ ?

Options:
A. 14
B. 15
C. 11
D. 16
E. 12
F. 18
G. 13
H. 19
I. 10
J. 17
Answer: E
Two distinct roots of the equation $f(x) = 2x^2-8nx+10x-n^2+35n-76$ , where the constant $n$ is an integer, are positive primes. Find the sum of the roots of $f(x)$ .

Options:
A. 9
B. 13
C. 10
D. 12
E. 8
F. 16
G. 14
H. 7
I. 15
J. 11
Answer: H
Define $a_n=\displaystyle\sum_{i=0}^{n}f(i)$ for $n\ge 0$ and $a_0=0$ . Given that $f(x)$ is a polynomial, and $a_1, a_2+1, a_3+8, a_4+27, a_5+64, a_6+125, \cdots$ is an arithmetic sequence, find the smallest positive integer value of $x$ such that $f(x)<-2012$ .

Options:
A. 27
B. 29
C. 21
D. 26
E. 28
F. 30
G. 23
H. 22
I. 24
J. 25
Answer: E
$p$ , $q$ , and $r$ are positive real numbers such that
$$$p^2 + pq + q^2 = 211$$$
$$$q^2 + qr + r^2 = 259$$$
$$$r^2 + rp + p^2 = 307$$$
Compute the value of $pq + qr + rp$ .

Options:
A. 251
B. 247
C. 255
D. 250
E. 252
F. 254
G. 249
H. 253
I. 248
J. 245
Answer: H
$C_1$ is a circle with radius $164$ and $C_2$ is a circle internally tangent to $C_1$ that passes through the center of $C_1$ . $\overline{AB}$ is a chord in $C_1$ of length $320$ tangent to $C_2$ at $D$ where $AD>BD$ . Given that $BD=a-b\sqrt{c}$ where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime, what is $a+b+c$ ?

Options:
A. 201
B. 209
C. 204
D. 210
E. 208
F. 202
G. 203
H. 206
I. 205
J. 207
Answer: D
Shanille O'Keal is practicing free throws. She makes the first shot and misses the second one. For each subsequent shot, the likelihood of making it is equal to the ratio of successful shots to total shots taken so far. What is the probability that she makes exactly 702 out of her first 4684 shots?

Options:
A. \frac{1}{4688}
B. \frac{1}{4686}
C. \frac{1}{4687}
D. \frac{1}{4684}
E. \frac{1}{4680}
F. \frac{1}{4685}
G. \frac{1}{4681}
H. \frac{1}{4682}
I. \frac{1}{4683}
J. \frac{1}{4689}
Answer: I
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots?

Options:
A. \frac{1}{100}
B. \frac{1}{104}
C. \frac{1}{107}
D. \frac{1}{103}
E. \frac{1}{101}
F. \frac{1}{106}
G. \frac{1}{99}
H. \frac{1}{102}
I. \frac{1}{108}
J. \frac{1}{105}
Answer: G
Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?

Options:
A. 2 - \frac{10}{\pi}
B. 2 - \frac{13}{\pi}
C. 2 - \frac{9}{\pi}
D. 2 - \frac{5}{\pi}
E. 2 - \frac{7}{\pi}
F. 2 - \frac{11}{\pi}
G. 2 - \frac{8}{\pi}
H. 2 - \frac{12}{\pi}
I. 2 - \frac{6}{\pi}
J. 2 - \frac{4}{\pi}
Answer: I
A game involves jumping to the right on the real number line. If $a$ and $b$ are real numbers and $b > a$, the cost of jumping from $a$ to $b$ is $b^3-ab^2$. For what maximum value of $c$ can one travel from $0$ to $1$ in a finite number of jumps with total cost exactly $c$?

Options:
A. \boxed{\frac{3}{4}}
B. \boxed{\frac{15}{16}}
C. \boxed{\frac{5}{6}}
D. \boxed{1}
E. \boxed{\frac{1}{2}}
F. \boxed{\frac{9}{10}}
G. \boxed{\frac{17}{18}}
H. \boxed{\frac{11}{12}}
I. \boxed{\frac{13}{14}}
J. \boxed{\frac{7}{8}}
Answer: D
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.

Options:
A. \frac{p-3}{2}
B. \frac{p-1}{3}
C. \frac{p-7}{2}
D. \frac{p-1}{4} + 1
E. \frac{p-1}{2} - 1
F. \frac{p-1}{4}
G. \frac{p-1}{2}
H. \frac{p-9}{2}
I. \frac{p-1}{2} + 1
J. \frac{p-5}{2}
Answer: G
Determine the smallest positive integer $y$ such that for any polynomial $t(x)$ with integer coefficients and any integer $k$, the value \[ t^{(y)}(k) = \left. \frac{d^y}{dx^y} t(x) \right|_{x=k} \] (the $y$-th derivative of $t(x)$ evaluated at $k$) is divisible by 2016.

Options:
A. 11
B. 12
C. 8
D. 4
E. 9
F. 5
G. 10
H. 7
I. 14
J. 6
Answer: C
Find the product of all values of $\alpha$ for which the curves $y = \alpha x^2 + \alpha x + \frac{1}{24}$ and $x = \alpha y^2 + \alpha y + \frac{1}{24}$ are tangent to each other.

Options:
A. -8
B. -3
C. -7
D. -6
E. -4
F. -3.5
G. -1
H. -2.5
I. -2
J. -5
Answer: B
Given positive integers $v$, define the sequence $o(v)$ by the following rules: $o(1) = 1$, $o(2v) = o(v)$, and $o(2v+1) = (-1)^v o(v)$. Find the value of \[ \sum_{v=1}^{1565} o(v) o(v+2). \]

Options:
A. 0
B. -3
C. 2
D. -7
E. 1
F. -5
G. -4
H. -6
I. -1
J. -2
Answer: I
Solve the integral:
$$
\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$

Options:
A. C + sqrt((x+4)/(x-4)) * (x-20) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
B. C + sqrt((x+4)/(x-4)) * (x-21) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
C. C + sqrt((x+4)/(x-4)) * (x-18) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
D. C + sqrt((x+4)/(x-4)) * (x-16) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
E. C + sqrt((x+4)/(x-4)) * (x-25) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
F. C + sqrt((x+4)/(x-4)) * (x-22) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
G. C + sqrt((x+4)/(x-4)) * (x-24) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
H. C + sqrt((x+4)/(x-4)) * (x-19) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
I. C + sqrt((x+4)/(x-4)) * (x-17) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
J. C + sqrt((x+4)/(x-4)) * (x-23) - 12 * ln(abs((sqrt(x-4) - sqrt(x+4)) / (sqrt(x-4) + sqrt(x+4))))
Answer: A
Find the range of $f(x) = \frac{ x }{ 1+x^2 }$.

Options:
A. -\frac{1}{3} \le y \le \frac{1}{3}
B. -\frac{1}{\sqrt{3}} \le y \le \frac{1}{\sqrt{3}}
C. -\frac{1}{4} \le y \le \frac{1}{4}
D. -\frac{1}{\sqrt{2}} \le y \le \frac{1}{\sqrt{2}}
E. -\frac{1}{\sqrt{6}} \le y \le \frac{1}{\sqrt{6}}
F. -\frac{1}{2}\le y\le\frac{1}{2}
G. -\frac{1}{\sqrt{5}} \le y \le \frac{1}{\sqrt{5}}
H. -1 \le y \le 1
I. -\frac{1}{\sqrt{7}} \le y \le \frac{1}{\sqrt{7}}
J. -\frac{1}{\sqrt{4}} \le y \le \frac{1}{\sqrt{4}}
Answer: F
Compute the limit:
$$
\lim_{x \to 0} \left( \frac{ x-1 }{ 2 \cdot x^2 } + \frac{ 1 }{ x \cdot \left( e^{2 \cdot x} - 1 \right) } \right)
$$

Options:
A. 1/9
B. 1/3
C. 1/4
D. 1/8
E. 1/10
F. 1/12
G. 1/5
H. 1/6
I. 1/15
J. 1/7
Answer: H
Find the area of the figure enclosed between the curves $y = 3 \cdot x^2$, $y = \frac{ x^2 }{ 6 }$, and $y = 2$.

Options:
A. \frac{48-8\cdot\sqrt{4}}{3\cdot\sqrt{3}}
B. \frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{5}}
C. \frac{48-8\cdot\sqrt{6}}{3\cdot\sqrt{3}}
D. \frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{4}}
E. \frac{48-8\cdot\sqrt{3}}{3\cdot\sqrt{2}}
F. \frac{48-8\cdot\sqrt{3}}{3\cdot\sqrt{3}}
G. \frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{3}}
H. \frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{6}}
I. \frac{48-8\cdot\sqrt{3}}{3\cdot\sqrt{6}}
J. \frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{2}}
Answer: G
Compute the integral:
$$
4 \cdot \int \cos(4 \cdot x)^4 \, dx
$$

Options:
A. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{10}\right)+C
B. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{4}\right)+C
C. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{5}\right)+C
D. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{7}\right)+C
E. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{6}\right)+C
F. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{8}\right)+C
G. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)-\frac{x}{2}\right)+C
H. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{2}\right)+C
I. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{3}\right)+C
J. 4 \cdot \int \cos(4 \cdot x)^4 \, dx = \frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{12}\right)+C
Answer: H
Evaluate $\lim_{x \to 0}\left(\left(\frac{ \sin(2 \cdot x) }{ 2 \cdot x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}\right)$.

Options:
A. e^{-\frac{1}{10}}
B. e^{-\frac{1}{14}}
C. e^{-\frac{1}{7}}
D. e^{-\frac{1}{5}}
E. e^{-\frac{1}{6}}
F. e^{-\frac{1}{8}}
G. e^{-\frac{1}{4}}
H. e^{-\frac{1}{3}}
I. e^{-\frac{1}{12}}
J. e^{-\frac{1}{9}}
Answer: E
Consider a limousine that gets $m(v) = \frac{ 120-2 \cdot v }{ 5 }$ mi/gal at speed $v$, the chauffer costs $\$15$/h, and gas is $\$3.5$/gal. Find the cheapest driving speed.

Options:
A. 34.42
B. 34.62
C. 34.22
D. 33.02
E. 34.32
F. 34.52
G. 35.02
H. 34.12
I. 34.72
J. 34.02
Answer: J
Find the 3rd order Taylor polynomial $P_{3}(x)$ of $f(x) = \sqrt[3]{2 \cdot x + 1}$ about $a = 0$.

Options:
A. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{32}{81}\cdot x^3
B. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{80}{81}\cdot x^3
C. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{50}{81}\cdot x^3
D. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{40}{81}\cdot x^3
E. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{60}{81}\cdot x^3
F. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{20}{81}\cdot x^3
G. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{10}{27}\cdot x^3
H. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{36}{81}\cdot x^3
I. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{48}{81}\cdot x^3
J. 1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{44}{81}\cdot x^3
Answer: D
Compute the limit:
$$
\lim_{x \to 0}\left(\frac{ -\sin(x) }{ x }\right)^{\frac{ 1 }{ 4 \cdot x^2 }}
$$

Options:
A. e^{-\frac{1}{20}}
B. e^{-\frac{1}{32}}
C. e^{-\frac{1}{30}}
D. e^{-\frac{1}{26}}
E. e^{-\frac{1}{16}}
F. e^{-\frac{1}{24}}
G. e^{-\frac{1}{12}}
H. e^{-\frac{1}{48}}
I. e^{-\frac{1}{28}}
J. e^{-\frac{1}{36}}
Answer: F
Compute the integral:
$$
3 \cdot \int \cos(2 \cdot x)^6 \, dx
$$

Options:
A. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/64*(sin(4*x))^3 + C
B. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/16*(sin(4*x))^2 + C
C. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/128*(sin(4*x))^3 + C
D. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/64*(sin(4*x))^2 + C
E. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/96*(sin(4*x))^3 + C
F. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/24*(sin(4*x))^2 + C
G. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/24*(sin(4*x))^3 + C
H. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/48*(sin(4*x))^3 + C
I. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/16*(sin(4*x))^3 + C
J. 3/8*sin(4*x) + 9/128*sin(8*x) + 15/16*x - 1/32*(sin(4*x))^3 + C
Answer: J
Find the sum of the series $\sum_{k=2}^\infty \left(\frac{ 3^{k-1} }{ 4^{3 \cdot k+1} }\right)$.

Options:
A. 3/15616
B. 3/15619
C. 3/15624
D. 3/15628
E. 3/15626
F. 3/15620
G. 3/15625
H. 3/15618
I. 3/15617
J. 3/15632
Answer: A
Calculate $\sqrt[3]{30}$ with an estimate error of $0.001$, using series expansion.

Options:
A. 759/243
B. 758/243
C. 755/243
D. 757/243
E. 756/243
F. 753/243
G. 750/243
H. 751/243
I. 752/243
J. 754/243
Answer: C
Find the directional derivative of $f(x,y,z) = x^2 + y \cdot z$ at $P(1,-3,2)$ in the direction of increasing $t$ along the path
$\vec{r}(t) = t^2 \cdot \vec{i} + 3 \cdot t \cdot \vec{j} + \left(1-t^3\right) \cdot \vec{k}$.

Options:
A. 11/sqrt(198)
B. 11/sqrt(55)
C. 11/sqrt(176)
D. 11/sqrt(88)
E. 11/sqrt(44)
F. 11/sqrt(22)
G. 11/sqrt(11)
H. 11/sqrt(132)
I. 11/sqrt(66)
J. 11/sqrt(33)
Answer: F
Let $k$ be a positive integer with a first digit four such that after removing the first digit, you get another positive integer, $m$ , that satisfies $14m+1=k$ . Find the number of possible values of $m$ between $0$ and $10^{2007}$ .

Options:
A. 334
B. 335
C. 333
D. 330
E. 339
F. 336
G. 337
H. 331
I. 338
J. 332
Answer: B
Compute the integral:
$$
\int \sin(5 \cdot x)^6 \cdot \cos(5 \cdot x)^2 \, dx
$$

Options:
A. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{20}\cdot\sin(20\cdot x)-\frac{1}{8}\cdot\frac{1}{10}\cdot\frac{1}{3}\cdot\sin(10\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{40}\cdot\sin(40\cdot x)+C
B. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{20}\cdot\sin(20\cdot x)-\frac{1}{8}\cdot\frac{1}{10}\cdot\frac{1}{3}\cdot\sin(10\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{40}\cdot\sin(40\cdot x)^2+C
C. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{20}\cdot\sin(20\cdot x)-\frac{1}{8}\cdot\frac{1}{10}\cdot\frac{1}{3}\cdot\sin(10\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{40}\cdot\sin(40\cdot x)^3+C
D. \frac{1}{32}\cdot x-\frac{1}{32}\cdot\frac{1}{20}\cdot\sin(20\cdot x)-\frac{1}{8}\cdot\frac{1}{10}\cdot\frac{1}{3}\cdot\sin(10\cdot x)^3+\frac{1}{128}\cdot x-\frac{1}{128}\cdot\frac{1}{40}\cdot\sin(40\cdot x)^4+C
Answer: A
Consider the sequence defined by $y_0=1$, $y_1=2$, and $y_a=4y_{a-1}-y_{a-2}$ for $a\geq 2$. Determine the smallest odd prime factor of $y_{25}$.

Options:
A. 199
B. 197
C. 191
D. 193
E. 181
Answer: E
Find the inverse of the function $f(x) = \log_{a}\left(x + \sqrt{x^2 + 1}\right)$, where $a > 1$.

Options:
A. \frac{a^{2\cdot x}+1}{2\cdot a^{2x}}
B. frac{a^{2\cdot x}+1}{2\cdot a^x}
C. \frac{a^{2\cdot x}-1}{2\cdot a^{2x}}
D. \frac{a^{2\cdot x}+1}{2\cdot a^{x}}
E. frac{a^{2\cdot x}-1}{2\cdot a^x}
Answer: E
Find the zeros of the function $f(x) = \left(1 - \tan(x)\right) \cdot \left(1 + \sin(2 \cdot x)\right) - \tan(x) + \cos(2 \cdot x) - 1$.

Options:
A. x_1=\frac{\pi}{4}+\pi\cdot n, x_2=\arctan\left(\frac{1}{4}\right)+\pi\cdot n
B. x_1=\frac{\pi}{4}+\pi\cdot n, x_2=\arctan\left(-\frac{1}{2}\right)+\pi\cdot n
C. x_1=\frac{\pi}{4}+\pi\cdot n, x_2=\arctan\left(-\frac{1}{3}\right)+\pi\cdot n
D. x_1=-\frac{\pi}{4}+\pi\cdot n, x_2=\arctan\left(\frac{1}{2}\right)+\pi\cdot n
E. x_1=\frac{\pi}{4}+\pi\cdot n, x_2=\arctan\left(\frac{1}{3}\right)+\pi\cdot n
F. x_1=\frac{\pi}{4}+\pi\cdot n, x_2=\arctan\left(\frac{1}{2}\right)+\pi\cdot n
Answer: D
Consider a sequence of independent random variables $H_1, H_2, \dots$ defined for $0 \leq p \leq 1/2$, where each $H_i$ follows the distribution: \[ H_i = \begin{cases} 1 & \text{with probability $p$,} \\ -1 & \text{with probability $p$,} \\ 0 & \text{with probability $1-2p$.} \end{cases} \] Let $y$ be a positive integer and $i, u_1, \dots, u_y$ be integers. Define $P(i, u_1, \dots, u_y)$ as the probability of the event $u_1 H_1 + \cdots + u_y H_y = i$. What is the largest value of $p$ that ensures \[ P(0, u_1, \dots, u_y) \geq P(i, u_1, \dots, u_y) \] holds true for any choice of positive integer $y$ and integers $i, u_1, \dots, u_y$?

Options:
A. \frac{1}{2}
B. \frac{1}{6}
C. \frac{1}{4}
D. \frac{1}{7}
E. \frac{1}{8}
F. \frac{1}{5}
G. \frac{1}{3}
Answer: C
Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$ . (Repeated digits are allowed.)

Options:
A. 435
B. 425
C. 420
D. 445
E. 455
F. 430
G. 415
H. 440
Answer: A
Find $L=\lim_{(x,y) \to (2,3)}\left(\frac{ x^2-y^2+10 \cdot y-25 }{ x^2-y^2-10 \cdot x+25 }\right)$.

Options:
A. -4/3
B. -3/2
C. -2/3
D. -3/4
E. -3/5
F. -1/3
G. -1/2
H. -1/4
Answer: C
Let $U$ be the positive integer consisting of 1998 decimal digits, all of which are 1; that is, \[U=1111\cdots 11.\] Determine the $619$th digit after the decimal point in the decimal expansion of $\sqrt{U}$.

Options:
A. 9
B. 5
C. 4
D. 6
E. 7
F. 2
G. 8
H. 3
I. 1
Answer: H
Solve the equation:
$$
\tan\left(\frac{ \pi }{ 4 }+x\right)+\tan\left(\frac{ \pi }{ 4 }-x\right)=2
$$

Options:
A. x=π⋅n - π/4
B. x=π⋅n + π/8
C. x=π⋅n
D. x=π⋅n + π/6
E. x=π⋅n + π/12
F. x=π⋅n + π/16
G. x=π⋅n + π/2
H. x=π⋅n + π/4
I. x=π⋅n + π/3
Answer: C
Which combination correctly identifies the dance that originated in Harlem, New York City, in the 1920s? Consider the following dances:
I. Swing Dance
II. Jitterbug
III. Charleston
IV. Lindy Hop
Select the option that correctly describes the dance.

Options:
A. III
B. IV
C. I and IV
D. III and IV
E. II and III
F. I and III
G. I and II
H. II and IV
I. I
J. II
Answer: B
Le petit fût est écrit par _______. Consider the following authors:
I. Emile Zola
II. Guy de Maupassant
III. Alain Robbe-Grillet
IV. Jean-François Billetter
Select the correct author.

Options:
A. I and IV
B. II and IV
C. II and III
D. IV
E. II
F. I and II
G. III and IV
H. I
I. I and III
J. III
Answer: E
In the traveling salesman problem, we need to calculate the possible number of all routes.For a salesman who has to visit n cities which of the following are the ways of his tour plan

Options:
A. (n+2)!
B. n-1
C. 3n
D. n!
E. n
F. (n-1)!
G. (n+1)!
H. n*n
I. (n-2)!
J. 2n
Answer: F
A factory plans to produce two kinds of products, A and B, and the equipment time, raw materials A and B, the profits that can be obtained, and the amount of resources available for each set of products are shown in the following table. The plan should be arranged according to the () scheme to make the most profit of the plant A B Available Resources Equipment (number) 2 3 14 Raw material A (kg) 8 0 16 Raw material B (kg) 0 3 12 Profit (10,000 yuan) 2 3

Options:
A. Production of 4 sets of A and 3 sets of B
B. Production of 3 sets of A and 4 sets of B
C. Production of 2 sets of A and 4 sets of B
D. Production of 2 sets of A and 5 sets of B
E. Production of 4 sets of A and 5 sets of B
F. Production of 1 set of A and 4 sets of B
G. Production of 4 sets of A and 2 sets of B
H. Production of 2 sets of A and 3 sets of B
I. Production of 3 sets of A and 5 sets of B
J. Production of 4 sets of A and 4 sets of B
Answer: F
An enterprise plans to build a communication network to provide services to the outside world. According to market forecasts, the probability of good business development in the future is 0.7, and the probability of poor business development is 0.3. There are three planning schemes to choose from: Option 1, a direct investment of 30 million yuan to build a large-scale network. If the business development is good, it can make a profit of 10 million yuan per year, if the business development is not good, the annual loss is 2 million yuan, and the service period is 10 years: plan 2, invest 14 million yuan to build a small-scale network. If the business development is good, it can make a profit of 4 million yuan per year, and if the business development is not good, it can still make a profit of 3 million yuan per year, and the service period is 10 years: plan 3, the first 3 years will be implemented according to plan 2, that is, 14 million yuan will be invested to build a small-scale network, and the income will be the same as plan 2. After 3 years, if the business development is not good, it will continue to be implemented according to plan 2; If the business development is good, an additional 20 million yuan will be invested in network expansion, and the service period after the expansion will be 7 years, and the annual profit will be 9.5 million yuan. According to the above conditions, it can be calculated that ( ).

Options:
A. The expected net income for option 3 is $32.95 million
B. The expected net income for Option 3 is $45.95 million
C. The expected net income of option 3 is $35.95 million
D. The expected net benefit of option 1 is $40 million
E. Option 1 is optimal
F. The expected net benefit for option 3 is $34.95 million
G. Option 2 is optimal
H. Option 3 is not optimal
I. The expected net income for Option 3 is $36.95 million
J. The expected net benefit of option 1 is $50 million
Answer: C
Known Linear Programming Issues:
$min \ z=2x_1-x_2+2x_3$
$s.t.-x_1+x_2+x_3=4
\\
-x_1+x_2-kx_3\le 6
\\
x_1\le0,x_2\ge0,x_3 \quad unconstrained$
The optimal solution is$x_1=-5,x_2=,x_3=-1$
Then the value of k is () and the optimal solution of the dual problem is ()

Options:
A. -1,-12
B. -1,-10
C. 1,-14
D. 0,-12
E. 1,-10
F. 1,12
G. 1,-12
H. -2,-12
I. -1,-8
J. 1,-8
Answer: G
According to the contract, the number and date of delivery of engines provided by a factory to users are: 40 units at the end of the first quarter, 60 units at the end of the second quarter, and 80 units at the end of the third quarter. The maximum production capacity of the plant is 100 units per quarter, and the production cost per quarter is $50x+0.2x^2$ (yuan), where x is the number of engines produced in the season. If the factory produces too much, the excess engine can be moved to the next season for delivery to the customer, so the factory needs to pay a storage fee, which is 4 yuan per engine per quarter. The plant was asked how many engines it should produce per quarter in order to meet the delivery contract and minimize the cost to the plant (assuming that engines were out of stock at the beginning of the first season). The minimum fee is:

Options:
A. 10840
B. 11280
C. 11460
D. 11320
E. 12020
F. 12380
G. 11240
H. 11200
I. 12040
Answer: B
Two players write any of the three numbers (0,1,2) on a piece of paper without letting the other know. Let the first person guess the sum of the numbers written by the two people, and then let the second person guess, but stipulate that the second person cannot guess the same number as the first person. The guesser wins 1 yuan from the opponent, and if anyone can't guess correctly, it will be a tie. Answer, how many pure strategies do each of the two players have?

Options:
A. 512
B. 2024
C. 2560
D. 3136
E. 3048
F. 3072
G. 1024
H. 2048
I. 2346
J. 4096
Answer: F
The most light-loving of the following tree species is:

Options:
A. Orchid
B. Hosta
C. Staghorn fern
D. Birch
E. Sugimoto
F. Camellia
G. Korean pine
H. Spruce
I. Evergreen
J. Clematis
Answer: D
Plants flower, grow, and dormant are generally adapted to their range.

Options:
A. River density
B. Photoperiod variation
C. Topography
D. Elevation
E. Geographical location
F. Floor area
G. Temperature period
H. Soil fertility
I. Vegetation coverage
J. Precipitation
Answer: B
The type of soil acidity by which soil acidity is determined is:

Options:
A. Active acids
B. Hydrolytic acids
C. Titratable acids
D. Binding acids
E. Free acids
F. Immobilized acids
G. Weak acid
H. Latent acids
I. Strong acids
J. Exchange acids
Answer: A
Crimes against humanity are the murder or extermination of ( ) before or during the war
and other inhumane acts.

Options:
A. Refugees
B. Evacuees
C. Irregular armed forces
D. Volunteer Army
E. Terrorist organizations
F. Soldiers
G. Regular armed forces
H. Peacekeeping forces
I. Prisoners of war
J. Civilians
Answer: J
Which of the following statements shows that human rights have a history in their origins?

Options:
A. "The exercise of rights is limited to non-infringement"
B. "Law is the embodiment of the will of the ruling class"
C. "Everyone should have the right to pursue happiness"
D. " There are no rights without obligations, and there are no obligations without rights
E. " Rights can never go beyond the economic structure of society and the cultural development that is conditioned by it"
F. "Justice is the paramount principle for reconciling the interests of the individual with the public interest"
G. "Freedom as a well-founded necessity"
H. " According to the law of nature, all men are born free, without knowing that there are slaves,
It doesn't matter if it's released."
I. "The law must be obeyed or it loses its relevance as a law"
J. " All men are created equal, and they are all endowed with certain non-alienable rights from their 'Creator'
Answer: E
Since piracy in country A seriously endangers the transportation security of major international shipping routes, at the request of country A, the United Nations Security Council passed a resolution authorizing warships of other countries to enter the territorial waters of country A to combat piracy within a specified period of time with the consent of country A. Pursuant to this resolution, the warship of country B entered the territorial waters of country A to rescue the merchant ship of country C that was being pursued by pirates. Which of the following options is correct for this?

Options:
A. Security Council resolutions, once adopted, automatically authorize all States to exercise unrestricted maritime jurisdiction
B. The entry of a warship of country B into the territorial waters of country A for the purpose of rescuing a merchant ship of country C is under the protective jurisdiction
C. Security Council resolutions do not make the entry of warships into territorial waters to combat piracy customary international law
D. The Security Council does not have the authority to make resolutions authorizing foreign warships to enter the territorial waters of State A to combat piracy
E. A warship of a State may enter the territorial waters of any State unconditionally by protecting its own merchant vessels
F. Foreign warships may enter the territorial waters of any State to combat piracy in accordance with the resolutions of the Security Council
G. In the event of an act of piracy, any State may enter the territorial waters of another State to enforce the law without regard to sovereignty
H. The consent of any State is not required to enter the territorial waters of any State, provided that the warship is on an international mission
I. Security Council resolutions allow access to territorial waters to combat piracy, which means that all States are ready to do the same
J. Warships of any State may freely enter the territorial waters of other States without any authorization to combat piracy
Answer: E
Which of the following tree species does not have normal axial resin canals? Consider the following tree species:
I. Pinus genus
II. Picea genus
III. Keteleeria genus
IV. Tsuga genus
Select the combination that includes the species without normal axial resin canals.

Options:
A. I and II
B. II and IV
C. I
D. III
E. II
F. II and III
G. III and IV
H. IV
I. I and IV
J. I and III
Answer: H
A 59-year-old man is brought to the emergency department by a coworker for right arm weakness and numbness. The symptoms started suddenly 2 hours ago. His coworker also noticed his face appears to droop on the right side and his speech is slurred. He has a history of hypertension, hyperlipidemia, type 2 diabetes, and peripheral arterial disease. He works as a partner at a law firm and has been under more stress than usual lately. His father died of a stroke at age 70. The patient has smoked a pack of cigarettes daily for the last 40 years. He drinks two pints (750 mL) of whiskey each week. He takes aspirin, atorvastatin, lisinopril, and metformin daily. He is 167.6 cm (5 ft 6 in) tall and weighs 104.3 kg (230 lb); BMI is 37 kg/m2. His temperature is 37.1°C (98.8°F), pulse is 92/min, respirations are 15/min, and blood pressure is 143/92 mm Hg. He is fully alert and oriented. Neurological examination shows asymmetry of the face with droop of the lips on the right. There is 3/5 strength in right wrist flexion and extension, and right finger abduction. Sensation to light touch and pinprick is reduced throughout the right arm. Which of the following is the strongest predisposing factor for this patient's condition?

Options:
A. Aspirin use
B. Hypertension
C. Increased stress
D. Excessive alcohol intake
E. Type 2 diabetes mellitus
F. Hyperlipidemia
G. Positive family history
H. Obesity
I. Positive family history of stroke
J. Tobacco use
Answer: B
A 44-year-old female is admitted to the neurological service. You examine her chart and note that after admission she was started on nimodipine. Which of the following pathologies would benefit from this pharmacologic therapy?

Options:
A. Subarachnoid hemorrhage
B. Ischemic stroke
C. Thromboembolic stroke
D. Epidural hematoma
E. Subdural hematoma
F. Intracranial aneurysm
G. Intracranial aneurysm with subarachnoid hemorrhage
H. Intracranial aneurysm with thromboembolic stroke
I. Transient ischemic attack
J. Pseudotumor cerebri
Answer: A
Early loss of bladder control is seen in-

Options:
A. Sacral plexus
B. Lumbar plexus
C. Gullain barre syndrome
D. Cauda equina
E. Conus medullaris
F. Thoracic plexus
G. Thoracic spinal cord
H. Lumbosacral plexus
I. Amyotrophic lateral sclerosis
J. Cervical spinal cord
Answer: E
Chemical waste disposal under Biomedical waste management guidelines is:

Options:
A. Recycling
B. Steam Sterilization
C. Landfilling
D. Plasma Treatment
E. Microwaving
F. Chemical Decontamination
G. Incineration
H. Autoclaving
I. High-Temperature Pyrolysis
J. Chemical Neutralization
Answer: G
Punch drunk syndrome is seen in?

Options:
A. Professional wrestlers
B. Professional fighters
C. Martial artists
D. Professional boxers
E. Head injury
F. Wrestlers
G. Alcoholic
H. Drug abuse
I. Boxers
J. Football players
Answer: I
Sheela 26-year-old female, currently taking phenytoin for idiopathic grand mal epilepsy, and is consulting you because she wants to avoid pregnancy. She is on oral contraceptive pills (OCPs) for the time being. She has searched the internet and read that phenytoin could decrease the effectiveness of her OCPs. Which one of the followings is a suitable alternative?

Options:
A. Mizolastine
B. Levetiracetam
C. Ethosuximide
D. Lamotrigine
E. Clobazam
F. Topiramate.
G. Gabapentin
H. Phenobarbitone.
I. Carbamazepine.
J. Valproate
Answer: D
Instrument having Shank with alternating cutting edges

Options:
A. RACB
B. RACH
C. K-Flex
D. Real world endo
E. K-5
F. K-6
G. Endotech endo. instrument
H. RACT
I. RACE
J. RASE
Answer: I
A farmer working in the sugar cane industry comes to you for advice on how to prevent fungal infection (bagassosis) of the lung. His elder brother was diagnosed to have bagassosis 1 month ago. You would advise him to spray the sugar cane with which of the following?

Options:
A. 2% propionic acid
B. 2% acetic acid
C. 10% hypochlorite solution
D. 2% formic acid
E. 2.2% propionic acid
F. 1% propionic acid
G. 2.5% propionic acid
H. 1.5% propionic acid
I. 1.8% propionic acid
J. 3% propionic acid
Answer: A
Charcot's aery is -

Options:
A. Fronto-polar aery
B. Striate branch of middle meningeal aery
C. Striate branch of middle cerebral aery
D. Striate branch of anterior communicating aery
E. Striate branch of posterior communicating aery
F. Striate branch of internal carotid aery
G. Striate branch of anterior cerebral aery
H. Calloso-marginal aery
I. Medial striate branch of anterior cerebral aery
J. Striate branch of posterior cerebral aery
Answer: C
According to International Health regulation (IHR) Act, a pregnant woman, with the following duration of pregnancy (in weeks), can't travel by air to other country

Options:
A. 32
B. 34
C. 36
D. 20
E. 28
F. 39
G. 35
H. 37
I. 33
J. 38
Answer: C
A 15- year -female patient develops loss of appetite, cough followed by neck rigidity. CSF examination shows- glucose 40 mg%,protein 150mg%. chloride 52 meq/lit. with full of lymphocytes. The likely diaganosis is -

Options:
A. Bacterial meningitis
B. Cryptococcal meningitis
C. Bacterial meningitis with low CSF glucose
D. Viral meningitis
E. Viral encephalitis
F. Tubercular meningitis
G. Bacterial meningitis with CSF pleocytosis
H. Viral meningitis with CSF abnormalities
I. Bacterial meningitis with atypical presentation
J. Brain abscess
Answer: F
Kata thermometer is used to measure: September 2007

Options:
A. Air density
B. Air heat exchange efficiency
C. Air temperature
D. Degree of air thermal conductivity
E. Degree of hotness in air
F. Air heat exchange rate
G. Degree of air humidity
H. Air pressure
I. Air cooling power
J. Air heat capacity
Answer: I
Definition of extremely low birth weight: (Repeat)

Options:
A. Wt <2 kg
B. Wt <0.8 kg
C. Wt <2.5 kg
D. Wt <1.1 kg
E. Wt <1 kg
F. Wt <0.9 kg
G. Wt <1.5 kg
H. Wt <1.2 kg
I. Wt <0.7 kg
J. Wt <0.95 kg
Answer: E
Lafora's disease presents with?

Options:
A. Juvenile myoclonic epilepsy
B. Myoclonic muscular dystrophy
C. Paial seizures
D. G.T.C.S
E. Juvenile myoclonic seizures
F. Juvenile onset myoclonic epilepsy
G. Myoclonic seizures
H. Myoclonic epilepsy
I. Absence seizures
J. Petit mal epilepsy
Answer: H
Pyknolepsy is seen in

Options:
A. Lennox gastout syndrome
B. Epilepsy with myoclonic absences
C. Adrenoleukodystrophy
D. Absence seizures
E. Typical absence syndrome
F. Temporal lobe epilepsy
G. Epilepsy with generalized tonic-clonic seizures
H. Marcolepsy
I. Juvenile myoclonic epilepsy
J. Atonic seizures
Answer: D
Intravenous PA is indicated in ischemic stroke within how many hours of onset of disease process

Options:
A. 5 hours
B. 6 hours
C. 1 hour
D. 4 hours
E. 8 hours
F. 9 hours
G. 12 hours
H. 3 hours
I. 2 hours
J. 7 hours
Answer: H
Initial screening test for newborn hearing disorder -

Options:
A. Auditory Steady State Response (ASSR)
B. Otoacoustic Emissions (OAE)
C. ABR - Auditory Brainstem Response
D. ABR - Auditory Brainstem Reflex
E. ABR - Auditory Brainstem Evoked Response
F. ABR - Auditory Evoked Response
G. Visual reinforcement audiometry
H. Conditioned Play Audiometry
I. Free Field Audiometry
J. Transient Evoked Otoacoustic Emissions (TEOAE)
Answer: B
Proposed guideline values for Radioactivity in Drinking water is

Options:
A. Gross a activity 1.0 Bq/L and Gross b activity 0.5 Bq/L
B. Gross a activity 0.5 Bq/L and Gross b activity 1.0 Bq/L
C. Gross a activity 10 Bq/L and Gross b activity 1.0 Bq/L
D. Gross a activity 1.0 Bq/L and Gross b activity 0.1 Bq/L
E. Gross a activity 0.1 Bq/L and Gross b activity 0.5 Bq/L
F. Gross a activity 0.5 Bq/L and Gross b activity 0.5 Bq/L
G. Gross a activity 0.5 Bq/L and Gross b activity 0.1 Bq/L
H. Gross a activity 0.1 Bq/L and Gross b activity 1.0 Bq/L
I. Gross a activity 1.0 Bq/L and Gross b activity 10.0 Bq/L
J. Gross a activity 1.0 Bq/L and Gross b activity 1.0 Bq/L
Answer: H
Epidemic dropsy results from -

Options:
A. Datura and Argemone mixed poisoning
B. Organophosphorus poisoning
C. Aluminum phosphide poisoning
D. Argemone and Belladonna poisoning
E. Dhatura and Belladonna poisoning
F. Belladonna poisoning
G. Arsenic poisoning
H. Argemone poisoning
I. Dhatura poisoning
J. Datura and Argemone poisoning
Answer: H
Hatter's shakes are seen in: AIIMS 09; AP 10

Options:
A. Thallium poisoning
B. Lead poisoning
C. Gold poisoning
D. Mercury vapor exposure
E. Arsenic poisoning
F. Mercury poisoning
G. Zinc poisoning
H. Silver poisoning
I. Copper poisoning
J. Tin poisoning
Answer: F
What is the content of Pediatric iron folic acid tablet supplied under the reproductive and child health programme?

Options:
A. Iron 20 mg, Folic Acid 150 mcg
B. Iron 60 mg, Folic Acid 500 mcg
C. Iron 100 mg, Folic Acid 500 mcg
D. Iron 20 mg, Folic Acid 50 mcg
E. Iron 500 mg, Folic Acid 100 mcg
F. Iron 30 mg, Folic Acid 100 mcg
G. Iron 10 mg, Folic Acid 500 mcg
H. Iron 20 mg, Folic Acid 100 mcg
I. Iron 20 mg, Folic Acid 200 mcg
J. Iron 10 mg, Folic Acid 100 mcg
Answer: H