MikaStars39/nano-eval huggingface dataset:
Consider the set $\{1,2,\dots,i\}$ and all its nonempty subsets, which we'll denote as $N_1, N_2, \dots, N_{2^i-1}$ in any arbitrary order. Now, construct a matrix $J$ of size $(2^i-1) \times (2^i-1)$ where each element $m_{ij}$ is defined as follows: \[ m_{ij} = \begin{cases} 1 & \text{if $N_i$ and $N_j$ have at least one element in common,} \\ 0 & \text{if $N_i$ and $N_j$ are disjoint.} \end{cases} \]Your task is to evaluate the determinant of $J$. In case multiple determinant values are possible, compute the sum of all distinct determinant values.

Options:
A. 1
B. -4
C. -1
D. -2
E. 0
F. 2
G. 1/2
H. -3
I. 3
J. 4
Answer: E
Fix an integer $b \geq 2$. Let $f(1) = 1$, $f(2) = 2$, and for each $n \geq 3$, define $f(n) = n f(d)$, where $d$ is the number of base-$b$ digits of $n$. For which values of $b$ does \[ \sum_{n=1}^\infty \frac{1}{f(n)} \] converge?

Options:
A. 7
B. 9
C. 8
D. 4
E. 10
F. 2
G. 3
H. 11
I. 6
J. 5
Answer: F
Consider the set $\{1,2,\dots,z\}$ and all its nonempty subsets, which we'll denote as $L_1, L_2, \dots, L_{2^z-1}$ in any arbitrary order. Now, construct a matrix $W$ of size $(2^z-1) \times (2^z-1)$ where each element $m_{ij}$ is defined as follows: \[ m_{ij} = \begin{cases} 1 & \text{if $L_i$ and $L_j$ have at least one element in common,} \\ 0 & \text{if $L_i$ and $L_j$ are disjoint.} \end{cases} \]Your task is to evaluate the determinant of $W$. In case multiple determinant values are possible, compute the sum of all distinct determinant values.

Options:
A. 5
B. 1
C. -4
D. 2
E. 4
F. -2
G. -1
H. 0
I. -3
J. 3
Answer: H
Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[4]{15}$ with an accuracy of 0.0001.

Options:
A. 1.967
B. 1.9682
C. 1.970
D. 1.9675
E. 1.9685
F. 1.9683
G. 1.968
H. 1.966
I. 1.969
J. 1.965
Answer: G
A doctor injects a patient with $13$ milligrams of radioactive dye that decays exponentially. After $12$ minutes, there are $4.75$ milligrams of dye remaining in the patient's system. Which is an appropriate model for this situation?

Options:
A. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{11}\cdot t}
B. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{15}\cdot t}
C. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{12.5}\cdot t}
D. f(t)=13\cdot e^{\frac{\ln\left(\frac{13}{4.75}\right)}{12}\cdot t}
E. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{16}\cdot t}
F. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{12}\cdot t}
G. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{10}\cdot t}
H. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{11.5}\cdot t}
I. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{14}\cdot t}
J. f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{13}\cdot t}
Answer: F
Evaluate the integral:
$$
I = \int \left(x^3 + 3\right) \cdot \cos(2 \cdot x) \, dx
$$

Options:
A. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 32\cdot x^2\cdot\cos(2\cdot x) + C
B. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 32\cdot x\cdot\cos(2\cdot x) + C
C. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 32\cdot x^4\cdot\sin(2\cdot x) + C
D. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 32\cdot x\cdot\sin(2\cdot x) + C
E. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) - 32\cdot x^4\cdot\cos(2\cdot x) + C
F. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + C
G. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 64\cdot x^2\cdot\sin(2\cdot x) + C
H. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\cos(2\cdot x) + C
I. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 32\cdot x\cdot\cos(2\cdot x) - 16\cdot x^2\cdot\sin(2\cdot x) + C
J. 384\cdot\sin(2\cdot x)+128\cdot x^3\cdot\sin(2\cdot x)+192\cdot x^2\cdot\cos(2\cdot x)-96\cdot\cos(2\cdot x)-192\cdot x\cdot\sin(2\cdot x) + 32\cdot x^2\cdot\sin(2\cdot x) + C
Answer: F
Find the volume of the solid whose boundaries are given in the rectangular coordinates:
$$
\sqrt{x^2+y^2} \le z \le \sqrt{16-x^2-y^2}, \quad x \ge 0, \quad y \ge 0
$$

Options:
A. 64\cdot\pi-16\cdot\pi\cdot\sqrt{2}/3
B. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/8
C. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/4
D. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/6
E. 64\cdot\pi-16\cdot\pi\cdot\sqrt{2}/6
F. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/5
G. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/9
H. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/3
I. 64\cdot\pi-32\cdot\pi\cdot\sqrt{2}/7
J. 64\cdot\pi-16\cdot\pi\cdot\sqrt{2}/2
Answer: D
Calculate the derivative of the function $r = \ln\left(\sqrt[4]{\frac{ 1+\tan(\varphi) }{ 1-\tan(\varphi) }}\right)$.

Options:
A. 1/2 * sec(φ) * tan(φ) * sec(φ)
B. 1/2 * sec(φ) * tan(φ)
C. 1/2 * sec(φ) * tan(φ/2)
D. 1/2 * sec(φ) * tan(2 * φ)
E. 1/2 * sec(2 * φ)
F. 1/2 * sec(φ) * sec(φ)
G. 1/2 * sec(φ/2)
H. 1/2 * sec(φ) * sec(φ/2)
I. 1/2 * sec(φ) * sec(2 * φ)
J. 1/2 * sec(φ)
Answer: E
Compute the integral:
$$
\int \frac{ 1 }{ (x+4) \cdot \sqrt{x^2+2 \cdot x+5} } \, dx
$$

Options:
A. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} - 4 - x - \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 + \sqrt{13} + x + \sqrt{x^2 + 2 \cdot x + 5}\right)
B. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} + 4 - x + \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 - \sqrt{13} - x + \sqrt{x^2 + 2 \cdot x + 5}\right)
C. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} - 4 + x - \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 + \sqrt{13} - x - \sqrt{x^2 + 2 \cdot x + 5}\right)
D. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} + 4 + x - \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 - \sqrt{13} - x + \sqrt{x^2 + 2 \cdot x + 5}\right)
E. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} + 4 - x - \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 - \sqrt{13} + x + \sqrt{x^2 + 2 \cdot x + 5}\right)
F. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} - 4 - x + \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 + \sqrt{13} - x - \sqrt{x^2 + 2 \cdot x + 5}\right)
G. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} - 4 + x - \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 + \sqrt{13} - x + \sqrt{x^2 + 2 \cdot x + 5}\right)
H. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} - 4 - x + \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 + \sqrt{13} + x - \sqrt{x^2 + 2 \cdot x + 5}\right)
I. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} + 4 + x + \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 - \sqrt{13} - x - \sqrt{x^2 + 2 \cdot x + 5}\right)
J. C + \frac{1}{\sqrt{13}} \cdot \ln\left(\sqrt{13} - 4 + x + \sqrt{x^2 + 2 \cdot x + 5}\right) - \frac{1}{\sqrt{13}} \cdot \ln\left(4 + \sqrt{13} - x - \sqrt{x^2 + 2 \cdot x + 5}\right)
Answer: A
Compute the integral:
$$
\int \frac{ 2 \cdot x-1 }{ \sqrt{8 \cdot x^2+4 \cdot x-10} } \, dx
$$

Options:
A. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+11\cdot x\right|\right)\right)+C
B. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+8\cdot x\right|\right)\right)+C
C. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+6\cdot x\right|\right)\right)+C
D. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+14\cdot x\right|\right)\right)+C
E. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+4\cdot x\right|\right)\right)+C
F. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+18\cdot x\right|\right)\right)+C
G. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+12\cdot x\right|\right)\right)+C
H. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+16\cdot x\right|\right)\right)+C
I. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+9\cdot x\right|\right)\right)+C
J. frac{1}{32\cdot\sqrt{2}}\cdot\left(16\cdot\sqrt{4\cdot x^2+2\cdot x-5}-24\cdot\ln\left(\left|2+4\cdot\sqrt{4\cdot x^2+2\cdot x-5}+10\cdot x\right|\right)\right)+C
Answer: B
Compute the integral:
$$
\int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x)^3 } \, dx
$$

Options:
A. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{1}{4} \cdot \cos(2 \cdot x)
B. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{11}{8} \cdot \cos(2 \cdot x)
C. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{15}{8} \cdot \cos(2 \cdot x)
D. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{13}{8} \cdot \cos(2 \cdot x)
E. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{17}{8} \cdot \cos(2 \cdot x)
F. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{5}{4} \cdot \cos(2 \cdot x)
G. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{9}{8} \cdot \cos(2 \cdot x)
H. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{1}{2} \cdot \cos(2 \cdot x)
I. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{7}{4} \cdot \cos(2 \cdot x)
J. C + \frac{3}{8} \cdot \ln\left(\frac{1 + \cos(2 \cdot x)}{1 - \cos(2 \cdot x)}\right) - \frac{\left(\cos(2 \cdot x)\right)^3}{4 - 4 \cdot \left(\cos(2 \cdot x)\right)^2} - \frac{3}{4} \cdot \cos(2 \cdot x)
Answer: J
Find the Taylor series of the given function $f(x) = \frac{ 1 }{ (x-1)^2 }$ centered at the indicated point: $a=0$ . (Hint: Differentiate $\frac{ 1 }{ 1-x }$.)

Options:
A. ∑_{n=0}^∞ (n·x^n)
B. ∑_{n=0}^∞ (n·x^n + (n+1)·x^n)
C. ∑_{n=0}^∞ (n·x^(n+2))
D. ∑_{n=0}^∞ (n·x^(n+1))
E. ∑_{n=0}^∞ (n·x^(n-1))
F. ∑_{n=0}^∞ (n·x^n + x^(n-1))
G. ∑_{n=0}^∞ (n·x^n + x^(n+1))
H. ∑_{n=0}^∞ ((n+1)·x^n)
I. ∑_{n=0}^∞ (n·x^n + x^n)
J. ∑_{n=0}^∞ (n·x^n + (n+1)·x^(n-1))
Answer: H
Evaluate $L=\lim_{P(x,y) \to P(a,2 \cdot b)}\left(\frac{ a^2 \cdot x \cdot y-2 \cdot a \cdot b \cdot x^2+a \cdot b \cdot y^2-2 \cdot b^2 \cdot x \cdot y }{ a^2 \cdot x \cdot y-2 \cdot a \cdot b \cdot x^2-a \cdot b \cdot y^2+2 \cdot b^2 \cdot x \cdot y }\right)$

Options:
A. L=\frac{a^2+2\cdot b^2}{a^2-2\cdot b^2}
B. L=\frac{a^2-2\cdot b^2}{a^2-2\cdot b^2}
C. L=\frac{a^2+4\cdot b^2}{a^2-4\cdot b^2}
D. L=\frac{a^2-4\cdot b^2}{a^2-2\cdot b^2}
E. L=\frac{a^2+2\cdot b^2}{a^2+4\cdot b^2}
F. L=\frac{a^2-4\cdot b^2}{a^2+4\cdot b^2}
G. L=\frac{a^2+2\cdot b^2}{a^2+2\cdot b^2-1}
H. L=\frac{a^2-2\cdot b^2}{a^2+2\cdot b^2}
I. L=\frac{a^2+2\cdot b^2}{a^2+2\cdot b^2}
J. L=\frac{a^2+4\cdot b^2}{a^2-2\cdot b^2}
Answer: A
Simplify the expression $\left(1+\tan\left(\theta\right)\right)^2-2 \cdot \tan\left(\theta\right)$ by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.

Options:
A. 1 + sec(θ) * tan(θ)^2
B. 1 + sec(θ) * tan(θ)
C. 1 + 2 * sec(θ)
D. 1 + tan(θ)^2 + 2 * tan(θ)
E. 1 + sec(θ)^2
F. 1 + 2 * tan(θ)
G. 1 + tan(θ)^2
H. 1 + sec(θ) * (1 + tan(θ))
I. 1 + sec(θ)
J. sec(θ)^2
Answer: J
Find the radius of convergence of the series:
$$
\sum_{k=1}^\infty \left(\frac{ \left(k!\right)^3 \cdot x^k }{ (2 \cdot k)! }\right)
$$

Options:
A. 1/8
B. 1/2
C. 1/16
D. 1/128
E. 1/64
F. 1
G. 1/256
H. 1/32
I. 1/4
J. 0
Answer: J
Compute $\lim_{x \to 0}\left(\frac{ 2 \cdot \cos(x)+4 }{ 5 \cdot x^3 \cdot \sin(x) }-\frac{ 6 }{ 5 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series.

Options:
A. 1/155
B. 1/170
C. 1/120
D. 1/180
E. 1/150
F. 1/135
G. 1/190
H. 1/140
I. 1/160
J. 1/145
Answer: E
Solve the integral:
$$
\int \frac{ 4 \cdot \cos(6 \cdot x)^3 }{ 9 \cdot \sin(6 \cdot x)^9 } \, dx
$$

Options:
A. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{7} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
B. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{14} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
C. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{5} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
D. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{8} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
E. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{10} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
F. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{6} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
G. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{9} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
H. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{11} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
I. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{16} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
J. C - \frac{2}{27} \cdot \left( \frac{1}{3} \cdot \left( \cot(6 \cdot x) \right)^6 + \frac{1}{4} \cdot \left( \cot(6 \cdot x) \right)^4 + \frac{1}{12} \cdot \left( \cot(6 \cdot x) \right)^8 \right)
Answer: D
Let $x$ and $y$ be real numbers with $0 < x < y < 1$ such that
$$$x^1y^0 + x^1y^2 + x^1y^4 + x^1y^6 + \ldots = x^0y^1 + x^2y^1 + x^4y^1 + x^6y^1 + \ldots$$$
The smallest value of $c$ such that we must have
$$$c > x^{1/2}y^{1/2} + x^{3/2}y^{3/2} + x^{5/2}y^{5/2} + x^{7/2}y^{7/2} + \ldots$$$
can be expressed in form $\tfrac{\sqrt{a}}{b}$ for relatively prime positive integers $a$ and $b$ . Find $a + b$ .

Options:
A. 7
B. 6
C. 8
D. 5
Answer: D
Solve the integral:
$$
\int 3 \cdot \sin(2 \cdot x)^4 \cdot \cos(2 \cdot x)^2 \, dx
$$

Options:
A. \frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}+\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C
B. \frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}-\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C
C. \frac{3}{16}\cdot\left(x+\frac{\sin(8\cdot x)}{8}-\frac{\sin(4\cdot x)}{8}+\frac{\sin(12\cdot x)}{24}\right)+C
D. \frac{3}{16}\cdot\left(x-\frac{\sin(8\cdot x)}{8}+\frac{\sin(4\cdot x)}{8}-\frac{\sin(12\cdot x)}{24}\right)+C
Answer: B
Find the first derivative $y_{x}'$ of the function:
$$
x = \arcsin\left(\frac{ t }{ \sqrt{2+2 \cdot t^2} }\right), \quad y = \arccos\left(\frac{ 1 }{ \sqrt{2+2 \cdot t^2} }\right), \quad t \ge 0
$$

Options:
A. t\cdot\sqrt{t^2+2}/\sqrt{2\cdot t^2+1}
B. t\cdot\sqrt{t^2+1}/\sqrt{2\cdot t^2+3}
C. t\cdot\sqrt{t^2+1}/\sqrt{2\cdot t^2+1}
D. t\cdot\sqrt{t^2+1}/\sqrt{2\cdot t^2+2}
E. t\cdot\sqrt{t^2+2}/\sqrt{2\cdot t^2+2}
Answer: A
Alvin, Simon, and Theodore are running around a $1000$ -meter circular track starting at different positions. Alvin is running in the opposite direction of Simon and Theodore. He is also the fastest, running twice as fast as Simon and three times as fast as Theodore. If Alvin meets Simon for the first time after running $312$ meters, and Simon meets Theodore for the first time after running $2526$ meters, how far apart along the track (shorter distance) did Alvin and Theodore meet?

Options:
A. 626
B. 627
C. 628
D. 374
E. 624
F. 625
Answer: D
If $\tan 15^\circ \tan 25^\circ \tan 35^\circ =\tan \theta$ and $0^\circ \le \theta \le 180^\circ,$ find $\theta.$

Options:
A. 10
B. 35
C. 15
D. 45
E. 25
F. 5
G. 20
Answer: F
Compute the integral:
$$
\int_{0}^3 \frac{ 1 }{ x^2 + 2 \cdot x - 8 } \, dx
$$

Options:
A. ∞
B. -3
C. -2
D. -1
E. -∞
F. 2
G. -4
Answer: A
For a positive integer $N$, let $f_N$\footnote{Corrected from $F_N$ in the source.} be the function defined by \[ f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin((2n+1)x). \] Determine the smallest constant $M$ such that $f_N(x) \leq M$ for all $N$ and all real $x$.

Options:
A. \frac{3\pi}{10}
B. \frac{\pi}{2}
C. \frac{5\pi}{16}
D. \frac{3\pi}{16}
E. \frac{3\pi}{8}
F. \frac{\pi}{4}
G. \frac{7\pi}{24}
H. \frac{\pi}{3}
Answer: F
Let $\star (x)$ be the sum of the digits of a positive integer $x$ . $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$ , we have that $\star (n)=12$ and $0\le n< 10^{7}$ . If $m$ is the number of elements in $\mathcal{S}$ , compute $\star(m)$ .

Options:
A. 27
B. 25
C. 26
D. 22
E. 28
F. 23
G. 24
H. 21
I. 29
Answer: C
Consider a function $i(o)$ defined for all positive real numbers $o$ as follows: \[ i(o) = \lim_{s \to 0} \left((o+1)^{s+1} - o^{s+1}\right)^{{1}/{s}}. \] Determine the value of $\lim_{o \to \infty} \frac{i(o)}{o}$.

Options:
A. e
B. e^{2/3}
C. e^{3/4}
D. e^{1/3}
E. e^{1/4}
F. e^{5/4}
G. e^{1/2}
H. e^2
I. e^{3/2}
Answer: A
Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function
$$
x = \frac{ 1 }{ 2 } \cdot \left(\cos(t)\right)^3, \quad y = 6 \cdot \left(\sin(t)\right)^3
$$

Options:
A. 8 / (cos(t)^5 * sin(t))
B. 8 / (cos(t)^3 * sin(t)^2)
C. 8 / (cos(t)^5 * sin(t)^2)
D. 8 / (cos(t)^3 * sin(t)^4)
E. 8 / (cos(t)^4 * sin(t))
F. 8 / (cos(t)^4 * sin(t)^3)
G. 8 / (cos(t)^3 * sin(t))
H. 8 / (cos(t)^4 * sin(t)^2)
I. 8 / (cos(t)^3 * sin(t)^3)
Answer: E
Which combination correctly identifies the dance style influenced by African and Caribbean dance traditions? Consider the following dance styles:
I. Hip Hop
II. Ballet
III. Salsa
IV. Tap Dance
Select the option that correctly describes the dance style.

Options:
A. III
B. II
C. II and III
D. II and IV
E. I and II
F. IV
G. I and III
H. I and IV
I. I
J. III and IV
Answer: I
Poisson d'or est un roman de _______. Consider the following authors:
I. J.M.G. Le Clézio
II. Richard Millet
III. Emile Zola
IV. Guy de Maupassant
Select the correct author.

Options:
A. I and IV
B. II and III
C. I
D. I and II
E. II and IV
F. IV
G. III and IV
H. III
I. I and III
J. II
Answer: C
The only restriction we place on the initial solution of a transportation problem is that: we must have nonzero quantities in a majority of the boxes.

Options:
A. demand must equal supply.
B. None of the above
C. we must have a number (equal to the number of rows plus the number of columns minus one) of boxes which contain nonzero quantities.
D. demand need not equal supply.
E. we must have a number (equal to the number of rows plus the number of columns minus one) of boxes which contain zero quantities.
F. all constraints need not be satisfied.
G. demand may equal supply.
H. all constraints must be satisfied.
I. all constraints may be satisfied.
J. demand must greater than supply.
Answer: H
A company has built a new 200-square-meter factory building, and is now ready to arrange equipment for the production of a certain product. The company is now idle to produce 3 sets of equipment of four types of products A, B, C, D, and the daily production capacity of each type of equipment is given in the following table. A, B, C, D Covers an area of 40 20 10 5 Daily production capacity 100 60 20 8 In the case of limited plant size, the plant can produce up to one ( ) product per day.

Options:
A. 500
B. 540
C. 550
D. 524
E. 520
F. 516
G. 528
H. 518
I. 530
J. 526
Answer: E
The river city bus company is responsible for the operation and maintenance management of the city's public transport vehicles, and is known to be required at all times of the day
The number of people is shown in the table,
Time Hours: 5:00-7:00, 7:00-9:00, 9:00-11:00, 11:00-13:00, 13:00-15:00
Number required: 26 50 34 40 38
Time Hours: 15:00-17:00, 17:00-19:00, 19:00-21:00, 21:00-23:00
Number required: 50 46 36 26
All the personnel of the company work for 6 hours continuously every day from Bu Yan, and arrange one day off every week
The company needs to be equipped with () personnel to meet the normal operation of the company.

Options:
A. 147
B. 136
C. 158
D. 145
E. 139
F. 103
G. 149
H. 130
I. 128
J. 126
Answer: A
Consider Linear Programming Issues:
$max\ z=(3+2\lambda )x_1+(5+\lambda)x_2+2x_3$
$s.t.-2x_1+x_2+x_3\leq 5+6\lambda
\\
3x_1+x_2-x_3\le 10-8\lambda
\\
x_1,x_2,x_3,\lambda\ge0$
When the objective function reaches its maximum,$\lambda$is

Options:
A. 4.25
B. 2.5
C. 2.25
D. 3.5
E. 3.25
F. 3.75
G. 3
H. 1
I. 1.25
J. 1.5
Answer: D
An enterprise produces the same product for two types of customers, and the demand of customer A is$x_1$,each piece is sold at:$70-4x_1$,B Customer needs are:$x_2$,each piece is sold at:$150-15x_2$.The cost per product produced by the enterprise is:$100+15(x_1+x_2)$The maximum profit of the enterprise is:

Options:
A. 380.81
B. 384.81
C. 392.81
D. 386.81
E. 378.81
F. 394.81
G. 382.81
H. 390.81
I. 388.81
J. 376.81
Answer: C
Due to the influence of the propagation distance of Dingwen, the distribution pattern of forest plants is mainly

Options:
A. Uniform Distribution
B. Chi-square Distribution
C. Jump Distribution
D. Clustered Distribution
E. Random Distribution
F. Symmetrical Distribution
G. Free Distribution
H. Distributed Distribution
I. Mosaic Distribution
Answer: D
There are different types of ecological pyramids, but only the Always Positive Pyramid

Options:
A. Plant Pyramid
B. Mixed Pyramid
C. Energy Pyramid
D. Pyramid of Quantities
E. Animalia Pyramid
F. Biomass Pyramid
G. Health Pyramid
H. Body Pyramid
I. Statistics Pyramid
J. Khufu Pyramid
Answer: C
Among the following minerals, they are secondary minerals

Options:
A. Calcite
B. Quartz
C. Plaster
D. Hematite
E. Magnetite
F. Amphibole
G. Orthoclase
H. Pyroxene
I. Mica
J. Peridot
Answer: A
The land area on the earth's surface is relatively fixed, and the soil has been formed for a long time; The gradual expansion of the occupied area and the increasing degradation of the land are indicative of soil resources

Options:
A. Relative non-renewability
B. Mass variability
C. Differences in distribution patterns
D. Sustainable use
E. Spatial distribution is fixed
F. Temporal evolutionary dynamics
G. Variability in development difficulty
H. Quantity is finite
I. Limited efficiency of utilization
J. Scarcity of resources
Answer: A
Once a neutral state has expressed its neutrality, it becomes applicable between a belligerent and a neutral state ( ).

Options:
A. Trade Law
B. Neutral country law
C. Environmental law
D. International law
E. Humanitarian law
F. Laws of belligerents
G. Law of the sea
H. Air Law
I. Third Country Laws
J. Neutrality Legislation
Answer: J
Guo and his 10-year-old son went to a restaurant to eat, and when they went to the toilet, they left their handbags on the seat and asked their sons to take care of them, and when they came back, they found that their handbags were missing. Guo's request for compensation from the restaurant was denied, so he filed a civil lawsuit. Which of the following statements is true under the right to consumer safety?

Options:
A. The restaurant shall give a clear reminder to the customer to keep his belongings properly, otherwise he shall be liable for the loss of the customer's belongings
B. The Hotel shall be liable only for property damage caused by its mismanagement, and shall not be liable for any loss of property due to personal reasons of the Guest
C. The restaurant shall ensure the safety of the customer's property when receiving the service, and bear the risk of loss of the customer's belongings
D. The Hotel shall assume unlimited liability for all losses incurred by the Customer due to unforeseen events (regardless of fault).
E. The Hotel shall be liable for any damage to the property of all guests, regardless of whether or not they are at fault
F. The hotel shall make a notice and reminder about the custody of property in a conspicuous position, otherwise it will bear part of the responsibility for the damage to the guest's property
G. The restaurant shall ensure that its service environment is absolutely safe, and shall be liable for all losses suffered by customers in the restaurant
H. The Hotel shall be responsible for the proper custody of valuables deposited by the Guest, and shall be liable for compensation for any loss of the items due to improper management
I. The restaurant shall ensure that the catering services provided by it meet the requirements of protecting personal and property safety, but shall not assume the obligation to keep the customer's belongings, nor shall it bear the risk of loss of the customer's belongings
J. The Hotel shall not be liable for property damage caused by force majeure (e.g. natural disasters).
Answer: I
Country A is a Member State of the United Nations. In 2006, a United Nations agency in country A, on behalf of the United Nations, signed an agreement with the Government of country A to purchase a batch of office supplies. Due to the delay in the delivery of country A, a dispute arose between the two parties. Which of the following options is correct under the Charter of the United Nations and the relevant rules of international law?

Options:
A. The above-mentioned acts on behalf of the United Nations shall be considered to be common to all States Members of the United Nations
B. The General Assembly of the United Nations shall have the right to request an advisory opinion from the International Court of Justice in respect of the dispute, whether or not State A agrees to it
C. The Security Council may impose a binding obligation on any State to submit to the jurisdiction of the International Court of Justice without the consent of that State
D. Purchases by the United Nations are automatically deemed to be the same obligations of all Member States
E. As a political international organization, the above-mentioned purchase by the United Nations organization shall be invalid ab initio
F. The General Assembly of the United Nations shall have the right to institute proceedings against State A before the International Court of Justice in respect of the dispute, with or without the consent of State A
G. The Secretary-General of the United Nations may bring proceedings concerning disputes directly before the International Court of Justice without the authorization of any body
H. The judgment of the International Court of Justice shall be binding on either party to the dispute without prior confirmation of its jurisdiction
I. As an international organization, all acts of the United Nations are not subject to the laws of any State
J. All decisions of the United Nations shall be considered legally binding without procedural review
Answer: A
Which of the following statements about the physical properties of wood is correct? Consider the following statements:
I. Wood is a porous material composed only of the cell wall substance of wood cells.
II. Wood density includes only green density and air-dry density.
III. Air-dry density = air-dry volume / air-dry mass.
IV. Green density = green mass / green volume.
Select the correct statement.

Options:
A. II and III
B. I
C. IV
D. III
E. III and IV
F. II
G. I and III
H. I and II
I. II and IV
J. I and IV
Answer: C
A 72-year-old man with a history of diabetes, dyslipidemia, and hypertension presents to the emergency department with sudden-onset right-sided facial paresthesia, word-finding difficulty, and lightheadedness. He was at a restaurant, in general good health, enjoying a dinner with friends when the 1st symptoms started. He takes atorvastatin, lisinopril, hydrochlorothiazide, and a multivitamin every day. A review of symptoms was negative. His symptoms spontaneously resolved after 5 minutes. In the emergency department, the vital signs included: blood pressure of 132/80 mm Hg, heart rate of 86/min, respiratory rate of 20/min, and oxygen saturation of 99% on room air. The laboratory results included:
Hemoglobin 11.3 g/dL
Hematocrit 33%
WBC count 9,800/mm3
Platelet count 247,000/mm3
International normalized ratio (INR) 1.1
Prothrombin time (PT) 11.5 seconds
Partial thromboplastin time (PTT) 32 seconds
His electrocardiogram (ECG) and his head computed tomography (CT) scan were normal. A duplex ultrasound scan shows 70-89% stenosis in the left carotid artery and 50% in the right. In the course of evaluation, his aphasia recurred for 3 minutes. A brain magnetic resonance imaging (MRI) scan was negative for areas of ischemia or intracranial hemorrhage. His symptom recurrence prompted his hospital admission for inpatient care. What is the next step in management?

Options:
A. Dabigatran
B. Carotid endarterectomy
C. Carotid artery angioplasty
D. Enoxaparin
E. Extracranial-intracranial bypass surgery
F. Thrombolysis
G. Carotid artery stenting
H. Carotid artery angiography
I. IV unfractionated heparin
J. Carotid artery angioplasty with stenting
Answer: B
A 57-year-old woman presents to an outpatient clinic with lower extremity weakness and lower back pain. The patient says that her symptoms began 2 weeks ago when she was working in her garden and have progressively worsened to the extent she currently is unable to walk on her own. She describes the pain as sharp, severe and descending bilaterally from her lower back to her lateral ankles along the posterior surface of her thighs and legs. She also states that she has had several episodes of urinary incontinence for the past couple of days. The patient denies having any similar pain or incontinence in the past. No other significant past medical history. Current medications are alendronate 5 mg orally daily and a daily multivitamin. Her temperature is 37.0°C (98.6℉), the blood pressure is 110/70 mm Hg, the pulse is 72/min, the respiratory rate is 15/min, and oxygen saturation is 99% on room air. On physical examination, the patient appears to be in significant distress. Strength is ⅗ in her thighs bilaterally and ⅖ in the legs bilaterally left greater than right. Muscle tone is decreased in the lower extremities. The patellar reflex is 1+ bilaterally and plantar reflex is 0+ bilaterally. Fine touch and pain and temperature sensation are decreased in the lower extremities bilaterally, left greater than right. Saddle anesthesia is present. Which of the following is the next, best step in the management of this patient?

Options:
A. Immediate transfer to the emergency department for imaging studies and initiation of muscle relaxants and analgesics
B. Outpatient management with a 3-day course of meloxicam and tolperisone and reassess
C. Immediate transfer to the emergency department for imaging studies and outpatient follow-up in one week
D. Immediate transfer to the emergency department for imaging studies and follow-up in one week
E. Immediate transfer to the emergency department for imaging studies and initiation of steroid therapy
F. Outpatient management with a 3-day course of diclofenac and gabapentin and reassess
G. Immediate transfer to the emergency department for imaging studies and inpatient observation for 48 hours
H. Recommend non-emergent inpatient spinal manipulation program
I. Outpatient management with 3 days of strict bed rest and reassess
J. Immediate transfer to the emergency department for management
Answer: J
Male patient with EHL weakness

Options:
A. S2-S3 slip disc
B. L5-L6 slip disc
C. L1-L2 slip disc
D. L3-L4 slip disc
E. L4-L5 slip disc
F. L5-L4 slip disc
G. L2-L3 slip disc
H. L5-S2 slip disc
I. L4-L6 slip disc
J. L5-S1 slip disc
Answer: E
Denominator of infant mortality rate is?

Options:
A. Per live birth
B. Per 1000 live births
C. Per 10000 live births
D. Per 1,000 live births (annual)
E. Per 500 live births
F. per 100 live births
G. Per 1500 live births
H. Per 2000 live births
I. Per lakh live births
J. Per 10,000 live births
Answer: B
What is the numerator in perinatal mortality:-

Options:
A. Late neonatal death of fetus >1000 g
B. Still birth <500 g
C. Early neonatal death of fetus >750 g
D. Early neonatal death of fetus >1000 g
E. Early neonatal death of fetus >900 g
F. Early neonatal death of fetus >1250 g
G. Abortion after 16 weeks
H. Post neonatal period death
I. Early neonatal death of fetus >500 g
J. Early neonatal death of fetus >1100 g
Answer: D
Two farmers were brought and they died soon. Smell of bitter almonds was coming from their mouth. They died of which poison

Options:
A. Potassium Cyanide
B. Aconite
C. Carbon Monoxide
D. Arsenic
E. Cyanide Gas
F. Hydrogen Cyanide
G. Organophosphorus
H. Sodium Cyanide
I. Calcium Cyanide
J. Opium
Answer: F
Maximum chances of injury in a child patient occurs during:

Options:
A. 12-14 months
B. 10-12 months
C. 12-15 months
D. 13-16 months
E. 14-16 months
F. 16 months
G. 11-13 months
H. 11-14 months
I. 8 months
J. 9-12 months
Answer: C
Tardy ulnar nerve palsy is seen in:

Options:
A. Cubitus varus deformity
B. Cubitus varus
C. Ulnar deviation
D. Cubitus valgus
E. Genu varus
F. Cubitus valgus deformity with ulnar deviation
G. Genu valgus
H. Cubitus valgus deformity
I. Cubitus valgus with ulnar deviation
J. Ulnar deviation with cubitus varus
Answer: D
The acceptable ambient temperature of summer in terms of thermal comfo is

Options:
A. 28 to 32degC
B. 23 to 27degC
C. 23 to 26degC
D. 21 to 25degC
E. 24 to 29degC
F. 24 to 28degC
G. 25 to 29degC
H. 22 to 26degC
I. 27 to 28degC
J. 20 to 23degC
Answer: B
Global cerebral dysfunction characterized by alteration in cognitive function and consciousness is known as -

Options:
A. Confusion
B. Acute confusion
C. Cognitive decline
D. Encephalopathy
E. Disorientation
F. Dementia
G. Acute delirium
H. Delirium
I. Depression
J. Acute anxiety
Answer: H
Shanti, aged 27 yrs presented with complaints of difficulty in reading near print. There is ptosis and diplopia in all directions. She is having-

Options:
A. III CN palsy with ptosis but without diplopia
B. III CN palsy with ptosis and intermittent diplopia
C. III CN palsy
D. III CN palsy with ptosis and diplopia
E. Presbyopia
F. IV CN palsy
G. III CN palsy with ptosis and intermittent diplopia in some directions
H. III CN palsy with ptosis and constant diplopia
I. VI CN palsy
J. Myasthenia gravis
Answer: J
Medical Termination of Pregnancy Act 1971, was amended in 2002 to include

Options:
A. 'Mentally healthy' in place of lunatic
B. 'Emotionally unstable' in place of lunatic
C. Risk to mother's life as an indication
D. 'Mentally ill' in place of lunatic
E. 'Mentally ill' as an additional indication
F. 'Psychiatric disorders' in place of lunatic
G. 'Mental disorders' in place of lunatic
H. POG upto 20 weeks
I. Failure of contraception as an indication
J. 'Mentally disabled' in place of lunatic
Answer: D
30 year old female comes with visual defects, breast engorgement and Amenorrhea. Investigation of choice is

Options:
A. Serum prolactin concentration
B. Biopsy
C. Serum TSH
D. CT brain
E. Serum prolactin concentration level
F. Serum prolactin level concentration
G. Serum prolactin level
H. Mr angiography
I. Serum hCG
J. Serum prolactin
Answer: J
Preventive dose of anti D gamma globulin given to mother is-

Options:
A. 275 meg
B. 300 meg
C. 250 meg
D. 200 meg
E. 350 meg
F. 325 meg
G. 400 meg
H. 500 meg
I. 375 meg
J. 600 meg
Answer: B
For Contralateral weakness and sensory loss, worse in face and arm; homonymous hemianopsia; aphasia or neglect syndrome select the site of the lesion of cerebrovascular disease,

Options:
A. Internal carotid aery
B. Middle cerebral aery (damaged lateral branch)
C. Middle cerebral aery (damaged posterior branch)
D. Middle cerebral aery
E. Middle cerebral artery (damaged anterior branch)
F. Anterior cerebral aery
G. Midbasilar aery
H. Posterior cerebral aery
I. Penetrating branch, middle cerebral aery
Answer: D
Return of ulbocavernous reflex in spinal shock -

Options:
A. Temporary reversal of spinal shock symptoms
B. Partial lesion of spinal cord
C. Complete transection of spinal cord
D. Temporary improvement in spinal cord function
E. Early sign of spinal cord damage
F. Temporary return of neurological function during spinal shock
G. Incomplete transection of spinal cord
H. Sign of recovery from spinal shock
I. Early sign of spinal cord recovery
J. Indication of progressing spinal cord damage
Answer: H
Which of the following is a causative agent of Farmer's Lung?

Options:
A. Thermophilic Streptomycetes
B. Aspergillus Fumigatus
C. Thermophilic Bacteria
D. Thermophilic Actinobacter
E. Mycobacterium Avium
F. Aspergillus Flavus
G. Actinobacter
H. Thermophilic Streptomyces
I. Thermophilic Mycobacterium
J. Thermophilic Actinomycetes
Answer: J
Bilateral inter nuclear ophthalmoplegia is pathognomonic of-

Options:
A. Devic's disease
B. Optic neuritis
C. Pontine haemorrhage
D. Bitemporal hemianopia
E. Neuromyelitis optica
F. Multiple sclerosis
G. Pontine glioma
H. Multiple sclerosis (RRMS)
I. Brainstem encephalitis
J. Lateral medullary syndrome
Answer: F
Which of the following is true for ESI act?

Options:
A. Funeral benefit is up to Rs 50,000
B. The State Government's share of expenditure on medical care is 1/11; the ESI Corporation's share of expenditure on medical care is 10/11 of total cost
C. The State Government's share of expenditure on medical care is 1/6; the ESI Corporation's share of expenditure on medical care is 5/6 of total cost
D. The State Government's share of expenditure on medical care is 1/12; the ESI Corporation's share of expenditure on medical care is 11/12 of total cost
E. The State Government's share of expenditure on medical care is 1/9; the ESI Corporation's share of expenditure on medical care is 8/9 of total cost
F. Employee has to contribute 4.75% and employer contributes 8.75%
G. Person with daily wages of Rs 70 has to contribute Rs 300 towards ESI
H. The State Government's share of expenditure on medical care is 1/10; the ESI Corporation's share of expenditure on medical care is 9/10 of total cost
I. The State Government's share of expenditure on medical care is 1/8; the ESI Corporation's share of expenditure on medical care is 7/8 of total cost
J. The State Government's share of expenditure on medical care is 1/7; the ESI Corporation's share of expenditure on medical care is 6/7 of total cost
Answer: I
A patient of parkinsonism is managed with L-dopa. If Vit. B-complex is administered concurrently-

Options:
A. The action of L-dopa in brain will be weakened but not eliminated
B. The bioavailability of L-dopa in the central nervous system will be reduced
C. The bioavailability of L-dopa will be increased
D. Decarboxylation of L-dopa in brain will be decreased
E. The action of L-dopa in brain will be potentiated
F. The bioavailability of L-dopa in the peripheral system will be reduced
G. The action of L-dopa in brain will be diminished
H. Side effects will be ameliorated
I. Decreased efficacy will result
J. The bioavailability of L-dopa will be reduced
Answer: I
Under ICDS one Anganwadi is for:

Options:
A. 700
B. 10000
C. 1100
D. 3000
E. 1200
F. 800
G. 1500
H. 1000
I. 900
J. 5000
Answer: H
Frecieuncy of Rest tremor in Parkinson's disease is -

Options:
A. 3-4/sec
B. 3-5/sec
C. 4-6/sec
D. 2/sec
E. 5-7/sec
F. 8/sec
G. 3.5-5.5/sec
H. 14/sec
I. 6-8/sec
J. 7-9/sec
Answer: C
MSAFP levels should generally be done at what gestational age?

Options:
A. 14 weeks
B. 16 weeks
C. 20 weeks
D. 17 weeks
E. 9 weeks
F. 18 weeks
G. 11 weeks
H. 13 weeks
I. 12 weeks
J. 15 weeks
Answer: J
Fluent aphasia without weakness occurs due to which of the following culprit arteries?

Options:
A. PCA (nondominant lobe)
B. Superior division of MCA (nondominant lobe)
C. Middle division of MCA (dominant lobe)
D. Superior division of PCA (dominant lobe)
E. Superior division of MCA (dominant lobe)
F. Hubner's branch of ACA
G. Inferior division of MCA
H. Superior division of PCA (nondominant lobe)
I. Inferior division of MCA (nondominant lobe)
J. Middle division of MCA (nondominant lobe)
Answer: G
Which of the following is a feature of absence seizure

Options:
A. Seizure typically starts in adulthood
B. Seizure precipitated by emotional stress
C. EEG shows 4-5 / sec spike and wave pattern
D. Associated with post - ictal confusion
E. Seizure precipitated by visual stimuli
F. During seizure, automatism present
G. Seizure precipitated by physical stress
H. Seizure typically starts in adolescence
I. Seizure precipitated by hyperventilation
J. Seizure typically lasts for more than 5 minutes
Answer: I
A patient was surgically treated for bilateral pheochromocytoma. He has staed developing gait abnormality hence MRI head was done. What is the probable diagnosis?

Options:
A. Von Hippel Landau Disorder
B. Multiple endocrine neoplasia type 3
C. Multiple endocrine neoplasia type 2B
D. Multiple endocrine neoplasia type 2A
E. Von Hippel Landau disease
F. Von Recklinghausen Disease
G. Autoimmune polyglandular syndrome
H. Multiple endocrine neoplasia type 4
I. Von Hippel Landau Syndrome
J. Neurofibromatosis type 1
Answer: I
A middle aged female presents with prolonged history of back pain followed by slowly progressive weakness of both lower limbs, spasticity and recent onset difficulty in micturation. On neurological examination there is evidence of dorsal myelopathy. MRI scan of spine shows a well-defined mid-dorsal intradural homogenous contrast enhancing mass lesion. The likely diagnosis is:

Options:
A. Intramedullary schwannoma
B. Intramedullary neuroma
C. Intradural Lipoma
D. Intramedullary meningioma
E. Intradural schwannoma
F. Intramedullary ependymoma
G. Dermoid cyst
H. Intradural meningioma
I. Spinal meningioma
J. Epidermoid cyst
Answer: I
Pearl's index indicates:

Options:
A. Postpartum mortality
B. Maternal mortality
C. Contraceptive prevalence
D. LBW/1000
E. LBW
F. Population
G. Malnutrition
H. Postpartum hemorrhage
I. Contraceptive failure
J. Birth rate
Answer: I
Determining level of fluoride in community water fluoridation programme depends on:

Options:
A. Average annual humidity of the place
B. Economic factors
C. Median annual temperature of the place
D. Average weight of children of the area
E. Altitude of area above sea level
F. Mean annual temperature of the place
G. Average annual sunshine hours of the place
H. Mean monthly temperature of the place
I. Average annual temperature of the place
J. Average annual rainfall of the place
Answer: F
Indication of Fecal contamination of water is due to presence of -

Options:
A. Enterococci
B. Coliform
C. Klebsiella
D. Enterobacter
E. Shigella
F. E. Coli
G. Pseudomonas
H. Clostridium
I. Salmonella
J. Bacillus
Answer: F
If untreated, percentage of mother to child transmission of HIV during delivery without intervention in a non-breast fed child is:

Options:
A. 15-20%
B. 20-35%
C. 15-25%
D. 10-20%
E. 25-40%
F. 5%
G. 15-30%
H. 10-15%
I. 40-50%
J. 10-25%
Answer: G
Following endaerectomy on the right common carotid, a patient is found to be blind in the right eye. It is appears that a small thrombus embolized during surgery and lodged in the aery supplying the optic nerve. Which aery would be blocked?

Options:
A. Infraorbital aery
B. Anterior ciliary aery
C. Optic aery
D. Nasociliary aery
E. Lacrimal aery
F. Anterior ciliary aery (left)
G. Central aery of the retina
H. Posterior ciliary aery (right)
I. Nasociliary aretry
J. Posterior ciliary aery
Answer: G
Billing's method of contraception refers to:

Options:
A. Monitoring cervical position
B. Coitus interruptus method
C. Monitoring cervical mucus and spermatozoa
D. Monitoring cervical mucus and basal body temperature
E. Fertility awareness method
F. Monitoring cervical mucus and cervical position
G. Monitoring cervical mucus and BBT
H. Cervical mucus method
I. Rhythm method
J. Monitoring BBT
Answer: H
Home delivery of contraception is carried out by

Options:
A. ASHA worker
B. Community health assistant
C. Auxiliary health worker
D. Anganawadi worker
E. Auxiliary health midwife
F. Female health worker
G. Male health worker
H. Auxiliary nurse-midwife assistant
I. Auxiliary nurse-midwife
J. Community health worker
Answer: A
Traveller&;s diarrhoea is caused by

Options:
A. E. coli O157
B. Giardiasis
C. Campylobacter jejuni
D. E. coli O157:H7
E. E. coli O104:H7
F. E. coli O104:H4
G. Shigella
H. E. Coli
I. E. histolytica
J. Vibrio cholerae
Answer: H
Earliest sign of raised ICT is:

Options:
A. Silver beaten appearance
B. Thickening of dorsum sella
C. Copper beaten appearance
D. Displacement of dorsum sella
E. Erosion of dorsum sphenoidale
F. Displacement of pineal gland
G. Erosion of dorsum sella
H. Erosion of petrous ridge
I. Erosion of petrous pyramid
J. Erosion of dorsum petrosum
Answer: G
Nerve involved in Foot drop

Options:
A. Posterior Tibial Nerve
B. Common Peroneal Nerve
C. Lateral Branch of the Common Peroneal Nerve
D. Lateral Branch of the Peroneal Nerve
E. Superficial Branch of the Peroneal Nerve
F. Deep Peroneal Nerve
G. Anterior Tibial Nerve
H. Tibial pa of Sciatic Nerve
I. Medial Branch of the Common Peroneal Nerve
J. Superficial Peroneal Nerve
Answer: F
The only intact eye movement in one and a half syndrome is

Options:
A. Abduction of both eyes
B. Adduction of both contralateral and ipsilateral eyes
C. Adduction of both eyes
D. Abduction of both contralateral eyes
E. Abduction of both ipsilateral eyes
F. Abduction of both contralateral and ipsilateral eyes
G. Abduction of contralateral eye
H. Adduction of Ipsilateral eye
I. Adduction of contralateral eye
J. Abduction of Ipsilateral eye
Answer: G
A patient presents with ascending muscle weakness for 2 days. On examination, the limb is flaccid. What investigation should be done first?

Options:
A. Serum calcium
B. Serum phosphate
C. Serum creatinine
D. Serum creatine kinase
E. Serum magnesium
F. Serum sodium
G. Serum potassium
H. Serum glucose
I. Serum chloride
J. Serum potassium concentration
Answer: G
Which of the following is a favorable prognostic indicator in Bell's palsy?

Options:
A. Persistence of the stapedial reflex
B. Persistence of the acoustic reflex
C. Absence of the stapedial reflex
D. Incomplete paralysis within a week
E. Persistence of the acoustic reflex within a week
F. Complete paralysis within a week
G. Severe taste impairment
H. Absence of the acoustic reflex
I. Hyperacusis
J. Absence of the acoustic reflex within a week
Answer: A
A patient presents with ataxia, urinary incontinence and dementia. The likely diagnosis is:

Options:
A. Parkinson's disease
B. Idiopathic Normal Pressure Hydrocephalus with associated incontinence
C. Alzheimer's Disease
D. Idiopathic Normal Pressure Hydrocephalus without associated incontinence
E. Idiopathic Normal Pressure Hydrocephalus
F. Steel Richardson syndrome
G. Multi-infarct Dementia
H. Normal Pressure Hydrocephalus
I. Wilson's Disease
J. Chronic Subdural Hematoma
Answer: H