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  1. \documentclass[15pt]{exam}
  2. \usepackage[english]{babel}
  3. \usepackage[a4paper,top=1cm,bottom=1cm,left=2cm,right=2cm,marginparwidth=1.25cm]{geometry}
  4. \usepackage{amsmath}
  5. \usepackage{amsfonts}
  6. \usepackage{amssymb}
  7. \usepackage{amsthm}
  8. \usepackage{graphicx}
  9. \usepackage{pgfplots}
  10. \usepackage{tikz,tkz-tab}
  11. \usepackage[colorlinks=true, allcolors=blue]{hyperref}
  12. \pgfplotsset{compat=1.18}
  13. \renewcommand{\thechoice}{\bfseries\Alph{choice}}
  14. \everymath{\displaystyle}
  15. \title{2024 Vietnamese High School Graduation Exam\\Subject: Mathematics}
  16. \author{Code 101}
  17. \date{Date: June 27, 2024}
  18. \begin{document}
  19.     \maketitle
  20.     \thispagestyle{empty}
  21.     This exam consists of 50 multiple choice questions, 0.2 pts each. Total time given to complete the exam is 90 minutes. Calculators are allowed on this exam.
  22.     \begin{questions}
  23.         \question Given a complex number $z$ so that $\overline{z} = -5 + 6i$. What is the imaginary part of $z$?
  24.        
  25.         \begin{oneparchoices}
  26.             \choice $-5$
  27.             \choice $-6$
  28.             \choice $5$
  29.             \choice $6$
  30.         \end{oneparchoices}
  31.        
  32.         \question Which statement below is true?
  33.        
  34.         \begin{oneparchoices}
  35.             \choice $\int (2x+3) dx = \frac{1}{2} x^2 + 3x + C$
  36.             \choice $\int (2x+3) dx = x^2 + C$
  37.             \choice $\int (2x+3) dx = 2x^2 + 3x + C$
  38.             \choice $\int (2x+3) dx = x^2 + 3x + C$
  39.         \end{oneparchoices}
  40.        
  41.         \question In the 3-dimensional Cartesian coordinate space, consider a line $d: \frac{x+1}{1} = \frac{y-2}{-1} = \frac{z}{-3}$. Which vector below is a direction vector of $d$?
  42.        
  43.         \begin{oneparchoices}
  44.             \choice $\overrightarrow{u}_1 = (1; 2; 0)$
  45.             \choice $\overrightarrow{u}_2 = (-1; 2; 0)$
  46.             \choice $\overrightarrow{u}_3 = (1; -1; -3)$
  47.             \choice $\overrightarrow{u}_4 = (1; 1; 3)$
  48.         \end{oneparchoices}
  49.        
  50.         \question Given a cylinder with lateral surface area $S_{lateral} = 36 \pi$ and height $h = 6$. The radius of the cylinder is
  51.        
  52.         \begin{oneparchoices}
  53.             \choice $6$
  54.             \choice $9$
  55.             \choice $3$
  56.             \choice $12$
  57.         \end{oneparchoices}
  58.        
  59.         \question Which sequence below is an arithmetic sequence?
  60.        
  61.         \begin{oneparchoices}
  62.             \choice $1, 3, 5, 7$
  63.             \choice $1, 0, 2, 4$
  64.             \choice $1, 3, 5, 10$
  65.             \choice $1, 2, 3, -4$
  66.         \end{oneparchoices}
  67.        
  68.         \question Let $a$ and $b$ be arbitrary real numbers so that $a \neq 1$. What is $\log_{a^2} b^2$?
  69.        
  70.         \begin{oneparchoices}
  71.             \choice $\log_a b$
  72.             \choice $\log_{a^4} b$
  73.             \choice $(\log_a b)^2$
  74.             \choice $\log_a b^4$
  75.         \end{oneparchoices}
  76.        
  77.         \question Let $y = f(x)$ be a quartic function whose graph is the following curve:
  78.        
  79.         \begin{center}
  80.             \begin{tikzpicture}
  81.                 \begin{axis}[
  82.                     axis lines=middle,
  83.                     xlabel={$x$},
  84.                     ylabel={$y$},
  85.                     xtick={-1, 1},
  86.                     ytick={1, 2},
  87.                     yticklabel style={above right},
  88.                     axis equal,
  89.                     axis line style={->},
  90.                     xmin=-1.25, xmax=1.25,
  91.                     ymin=-1, ymax=2.5,
  92.                     enlargelimits=true
  93.                     ]
  94.                     \addplot[black, thick, domain=-2:2, samples=200] {(-1) * x^4 + 2 * x^2 + 1};
  95.                     \node at (axis cs:0,0) [anchor=south west] {$O$};
  96.                    
  97.                     \coordinate (A) at (axis cs:-1, 0);
  98.                     \coordinate (B) at (axis cs:-1, 2);
  99.                     \coordinate (C) at (axis cs:0, 2);
  100.                     \coordinate (D) at (axis cs:1, 0);
  101.                     \coordinate (E) at (axis cs:1, 2);
  102.                    
  103.                     \draw[dashed, black] (A) -- (B);
  104.                     \draw[dashed, black] (C) -- (B);
  105.                     \draw[dashed, black] (D) -- (E);
  106.                     \draw[dashed, black] (C) -- (E);
  107.                 \end{axis}
  108.             \end{tikzpicture}
  109.         \end{center}
  110.        
  111.         How many real solutions does the equation $f(x) = \frac{3}{2}$ have?
  112.        
  113.         \begin{oneparchoices}
  114.             \choice $3$
  115.             \choice $4$
  116.             \choice $0$
  117.             \choice $2$
  118.         \end{oneparchoices}
  119.        
  120.         \question Consider a triangular prism with a base area $B = 6$ and height $h = 3$. The volume of the triangular prism is
  121.        
  122.         \begin{oneparchoices}
  123.             \choice $24$
  124.             \choice $6$
  125.             \choice $12$
  126.             \choice $18$
  127.         \end{oneparchoices}
  128.        
  129.         \question Let $f(x)$ be a function whose derivative is continuous on the real numbers, so that $f(1) = 3$ and $f(2) = 1$. What is the value of $\int_{1}^{2} f'(x) dx$?
  130.        
  131.         \begin{oneparchoices}
  132.             \choice $4$
  133.             \choice $2$
  134.             \choice $-2$
  135.             \choice $-4$
  136.         \end{oneparchoices}
  137.        
  138.         \question The vertical asymptote of the function $y = \frac{4x-1}{3x+2}$ is
  139.        
  140.         \begin{oneparchoices}
  141.             \choice $x = \frac{-2}{3}$
  142.             \choice $x = \frac{4}{3}$
  143.             \choice $y = \frac{4}{3}$
  144.             \choice $y = \frac{-2}{3}$
  145.         \end{oneparchoices}
  146.        
  147.         \question $z = i + i^2 + i^3 = ?$
  148.        
  149.         \begin{oneparchoices}
  150.             \choice $-1$
  151.             \choice $-1 + 2i$
  152.             \choice $1$
  153.             \choice $i$
  154.         \end{oneparchoices}
  155.        
  156.         \question On the interval $(-\infty; +\infty)$, which function below is the function $F(x) = \frac{1}{2} \sin 2x$ an antiderivative of?
  157.        
  158.         \begin{oneparchoices}
  159.             \choice $f_3(x) = \frac{-1}{2} \cos 2x$
  160.             \choice $f_4(x) = \frac{-1}{4} \cos 2x$
  161.             \choice $f_2(x) = \cos 2x$
  162.             \choice $f_1(x) = -\cos 2x$
  163.         \end{oneparchoices}
  164.        
  165.         \question If $\int_{-2}^{1} f(x) dx = -1$ and $\int_{1}^{7} f(x) dx = -5$, what is $\int_{-2}^{7} f(x) dx$?
  166.        
  167.         \begin{oneparchoices}
  168.             \choice $-4$
  169.             \choice $5$
  170.             \choice $-6$
  171.             \choice $4$
  172.         \end{oneparchoices}
  173.        
  174.         \question Let $y = f(x)$ be a function with the following table of variations:
  175.        
  176.         \begin{tikzpicture}
  177.             \tkzTab
  178.             [lgt=3,espcl=2.5]
  179.             {$x$/1, $f’(x)$/1, $f(x)$/2.5}
  180.             {$-\infty$, $-1$, $2$, $+\infty$}
  181.             {,-,0,+,0,-,}
  182.             {+/ $+\infty$, -/ $-2$, +/ $1$ , -/ $-\infty$}
  183.         \end{tikzpicture}
  184.        
  185.         The minimum point of the above function is
  186.        
  187.         \begin{oneparchoices}
  188.             \choice $x=2$
  189.             \choice $x=-1$
  190.             \choice $x=1$
  191.             \choice $x=-2$
  192.         \end{oneparchoices}
  193.        
  194.         \question The solution set of the inequality $\log_{\frac{1}{2}} (x+2) > -1$ is
  195.        
  196.         \begin{oneparchoices}
  197.             \choice $(-2;1)$
  198.             \choice $(0;+\infty)$
  199.             \choice $(-2;0)$
  200.             \choice $(-\infty;0)$
  201.         \end{oneparchoices}
  202.        
  203.         \question Among the functions below, which one is the function whose graph resembles the following curve:
  204.        
  205.         \begin{center}
  206.             \begin{tikzpicture}
  207.                 \begin{axis}[
  208.                     axis lines=middle,
  209.                     xlabel={$x$},
  210.                     ylabel={$y$},
  211.                     ticks=none,
  212.                     axis equal,
  213.                     axis line style={->},
  214.                     xmin=-1, xmax=0.5,
  215.                     ymin=-2, ymax=3,
  216.                     enlargelimits=true
  217.                     ]
  218.                     \addplot[black, thick, domain=-4:2, samples=200] {x^3 + 3 * x^2 - 1};
  219.                     \node at (axis cs:0,0) [anchor=south east] {$O$};
  220.                 \end{axis}
  221.             \end{tikzpicture}
  222.         \end{center}
  223.        
  224.         \begin{oneparchoices}
  225.             \choice $y = -x^3 + 3x^2 + 3$
  226.             \choice $y = x^4 - 2x^2 - 4$
  227.             \choice $y = \frac{x-2}{2x+1}$
  228.             \choice $y = x^3 + 3x^2 - 1$
  229.         \end{oneparchoices}
  230.        
  231.         \question In the 3-dimensional Cartesian coordinate space, consider two points $A(1; -2; 3)$ and $B(3; 0; 1)$. Let $(S)$ be the sphere whose diameter is $AB$, the center of $(S)$ is
  232.        
  233.         \begin{oneparchoices}
  234.             \choice $(2; -1; 2)$
  235.             \choice $(-1; -1; 1)$
  236.             \choice $(4; -2; 4)$
  237.             \choice $(1; 1; -1)$
  238.         \end{oneparchoices}
  239.        
  240.         \question The solution of the equation $2^{2x} = 2^{x+6}$ is
  241.        
  242.         \begin{oneparchoices}
  243.             \choice $x=-6$
  244.             \choice $x=2$
  245.             \choice $x=6$
  246.             \choice $x=-2$
  247.         \end{oneparchoices}
  248.        
  249.         \question Let $y = f(x)$ be a function whose derivative is $f'(x) = 2x + 4, \forall x \in \mathbb{R}$. On which interval below is the function decreasing?
  250.        
  251.         \begin{oneparchoices}
  252.             \choice $(-\infty; -2)$
  253.             \choice $(2; 4)$
  254.             \choice $(-2; +\infty)$
  255.             \choice $(2; +\infty)$
  256.         \end{oneparchoices}
  257.        
  258.         \question Which function below is an exponential function?
  259.        
  260.         \begin{oneparchoices}
  261.             \choice $y = x^{2024}$
  262.             \choice $y = 2024^x$
  263.             \choice $y = \log_3 x$
  264.             \choice $y = x^{-4}$
  265.         \end{oneparchoices}
  266.        
  267.         \question On the interval $(0; +\infty)$, the derivative of $y = x^{\frac{1}{7}}$ is
  268.        
  269.         \begin{oneparchoices}
  270.             \choice $y' = \frac{1}{7} x^{\frac{-6}{7}}$
  271.             \choice $y' = \frac{1}{7} x^{\frac{6}{7}}$
  272.             \choice $y' = x^{\frac{-6}{7}}$
  273.             \choice $y' = \frac{7}{8} x^{\frac{8}{7}}$
  274.         \end{oneparchoices}
  275.        
  276.         \question Given a cone with base radius $r = 3$ and slant height $l = 5$, what is the altitude of the cone?
  277.        
  278.         \begin{oneparchoices}
  279.             \choice $4$
  280.             \choice $5$
  281.             \choice $\sqrt{34}$
  282.             \choice $2$
  283.         \end{oneparchoices}
  284.        
  285.         \question Let $y = f(x)$ be a function with the following table of variations:
  286.        
  287.         \begin{tikzpicture}
  288.             \tkzTab
  289.             [lgt=3,espcl=2.5]
  290.             {$x$/1, $y'$/1, $y$/2.5}
  291.             {$-\infty$, $-2$, $0$, $2$, $+\infty$}
  292.             {,-,0,+,0,-,0,+,}
  293.             {+/ $+\infty$, -/ $-3$, +/ $0$ , -/ $-3$, +/ $+\infty$}
  294.         \end{tikzpicture}
  295.        
  296.         How many extrema points does the function have?
  297.        
  298.         \begin{oneparchoices}
  299.             \choice $3$
  300.             \choice $2$
  301.             \choice $4$
  302.             \choice $1$
  303.         \end{oneparchoices}
  304.        
  305.         \question In the 3-dimensional Cartesian coordinate space, consider two vectors $\overrightarrow{a} = (2; 3; -1)$ and $\overrightarrow{b} = (-3; 2; -4)$. What is $\overrightarrow{a} + \overrightarrow{b}$?
  306.        
  307.         \begin{oneparchoices}
  308.             \choice $(-1; -5; 5)$
  309.             \choice $(-5; -1; -3)$
  310.             \choice $(-1; 5; -5)$
  311.             \choice $(1; -5; 5)$
  312.         \end{oneparchoices}
  313.        
  314.         \question In the 3-dimensional Cartesian coordinate space, the equation of a plane passing through $M(3; 4; -2)$ and perpendicular to the axis $Oz$ is
  315.        
  316.         \begin{oneparchoices}
  317.             \choice $y-4=0$
  318.             \choice $z+2=0$
  319.             \choice $x+y+z-5=0$
  320.             \choice $x-3=0$
  321.         \end{oneparchoices}
  322.        
  323.         \question Consider a quadrilateral pyramid whose volume is $V = 3a^3$ and base area is $B = a^2$. The height of the pyramid is
  324.        
  325.         \begin{oneparchoices}
  326.             \choice $a$
  327.             \choice $6a$
  328.             \choice $3a$
  329.             \choice $9a$
  330.         \end{oneparchoices}
  331.        
  332.         \question How many ways are there to arrange 6 people into a line?
  333.        
  334.         \begin{oneparchoices}
  335.             \choice $36$
  336.             \choice $720$
  337.             \choice $1$
  338.             \choice $6$
  339.         \end{oneparchoices}
  340.        
  341.         \question On the Cartesian coordinate plane, $M(2; -5)$ is the point representing the complex number $z$. The real part of $z$ is
  342.        
  343.         \begin{oneparchoices}
  344.             \choice $-5$
  345.             \choice $-2$
  346.             \choice $2$
  347.             \choice $5$
  348.         \end{oneparchoices}
  349.        
  350.         \question Consider a pyramid $S.ABCD$ whose base is a square of length $a$, side $SA$ is perpendicular to the base plane and $SA = \sqrt{2} a$. The distance from $C$ to the plane $(SBD)$ is
  351.        
  352.         \begin{oneparchoices}
  353.             \choice $\frac{2\sqrt{10}}{5} a$
  354.             \choice $\frac{\sqrt{6}}{3} a$
  355.             \choice $\frac{\sqrt{10}}{10} a$
  356.             \choice $\frac{\sqrt{10}}{5} a$
  357.         \end{oneparchoices}
  358.        
  359.         \question In the 3-dimensional Cartesian coordinate space, consider a point $A(1; 2; -1)$ and a plane $(P): 2x-z+1=0$. The line passing through $A$ and perpendicular to $(P)$ is
  360.        
  361.         \begin{oneparchoices}
  362.             \choice $\begin{cases}x=2+t\\y=2t\\z=-1-t\end{cases}$
  363.             \choice $\begin{cases}x=1+2t\\y=2-t\\z=-1+t\end{cases}$
  364.             \choice $\begin{cases}x=-1+2t\\y=-2\\z=1-t\end{cases}$
  365.             \choice $\begin{cases}x=1+2t\\y=2\\z=-1-t\end{cases}$
  366.         \end{oneparchoices}
  367.        
  368.         \question Consider a complex number $z=3+4i$. The modulus of $iz$ is
  369.        
  370.         \begin{oneparchoices}
  371.             \choice $7$
  372.             \choice $49$
  373.             \choice $25$
  374.             \choice $5$
  375.         \end{oneparchoices}
  376.        
  377.         \question Consider an acute angle $xOy$ formed by two rays $Ox$ and $Oy$. Consider 5 (\textit{resp.} 6) distinct points on rays $Ox$ (\textit{resp.} $Oy$), so that all these points are not coincident with $O$. We choose three random points out of the given 12 points (including $O$ and the 11 points given above). What is the probability that the three points chosen form a triangle?
  378.        
  379.         \begin{oneparchoices}
  380.             \choice $\frac{19}{22}$
  381.             \choice $\frac{27}{44}$
  382.             \choice $\frac{3}{4}$
  383.             \choice $\frac{39}{44}$
  384.         \end{oneparchoices}
  385.        
  386.         \question A car is in motion with a velocity of $20 m/s$, when the driver brakes. From that point onwards, the car is in constant deceleration, with the velocity changing as a function of time $v(t) = -4t + 20 (m/s)$ where $t$ is the time elapsed in seconds from the point when the driver starts braking. The total distance travelled by the car from the point when the driver starts braking to the point when the car stops is:
  387.        
  388.         \begin{oneparchoices}
  389.             \choice $32m$
  390.             \choice $50m$
  391.             \choice $48m$
  392.             \choice $30m$
  393.         \end{oneparchoices}
  394.        
  395.         \question In the 3-dimensional Cartesian coordinate space, consider two points $A(1;2;3)$ and $B(3;2;5)$. Let $M$ be the point satisfying $\overrightarrow{MB} = 3\overrightarrow{MA}$, the norm of $\overrightarrow{OM}$ is
  396.        
  397.         \begin{oneparchoices}
  398.             \choice $\frac{\sqrt{74}}{2}$
  399.             \choice $2\sqrt{2}$
  400.             \choice $8$
  401.             \choice $2\sqrt{14}$
  402.         \end{oneparchoices}
  403.        
  404.         \question Consider a pyramid $S.ABC$ with the base $ABC$ being an isosceles triangle with $\angle A = 90^{\circ}$, $BC = 2a$, $SA \perp (ABC)$ and $SA = \sqrt{3} a$. The angle between planes $(SBC)$ and $(ABC)$ is
  405.        
  406.         \begin{oneparchoices}
  407.             \choice $60^{\circ}$
  408.             \choice $90^{\circ}$
  409.             \choice $30^{\circ}$
  410.             \choice $45^{\circ}$
  411.         \end{oneparchoices}
  412.        
  413.         \question The maximum value of the function $f(x) = -6x^3 + 27x^2 - 16x + 1$ on the interval $[1;5]$ is
  414.        
  415.         \begin{oneparchoices}
  416.             \choice $6$
  417.             \choice $\frac{329}{9}$
  418.             \choice $\frac{-14}{9}$
  419.             \choice $-154$
  420.         \end{oneparchoices}
  421.        
  422.         \question Consider a quartic function $y = f(x)$. The graph of its derivative $f'(x)$ is the curve below. On which interval below is $f(x)$ increasing?
  423.        
  424.         \begin{center}
  425.             \begin{tikzpicture}
  426.                 \begin{axis}[
  427.                     axis lines=middle,
  428.                     xlabel={$x$},
  429.                     ylabel={$y$},
  430.                     xtick={-1, 1, 2},
  431.                     ymajorticks=false,
  432.                     axis equal,
  433.                     axis line style={->},
  434.                     xmin=-1.25, xmax=3,
  435.                     ymin=-1, ymax=2.5,
  436.                     enlargelimits=true
  437.                     ]
  438.                     \addplot[black, thick, domain=-2:4, samples=200] {0.592593 * x^3 - 1.18519 * x^2 - 0.592593 * x + 1.18519};
  439.                     \node at (axis cs:0,0) [anchor=south west] {$O$};
  440.                 \end{axis}
  441.             \end{tikzpicture}
  442.         \end{center}
  443.        
  444.         \begin{oneparchoices}
  445.             \choice $(-\infty; -1)$
  446.             \choice $(-1; 2)$
  447.             \choice $(1; 2)$
  448.             \choice $(-1; 1)$
  449.         \end{oneparchoices}
  450.        
  451.         \question Let $a, b > 1$ be real numbers, $\log_{ab} b$ is equal to
  452.        
  453.         \begin{oneparchoices}
  454.             \choice $\frac{1}{1+\log_{b} a}$
  455.             \choice $\frac{1}{\log_{b} a}$
  456.             \choice $1-\log_{b} a$
  457.             \choice $1+\log_{b} a$
  458.         \end{oneparchoices}
  459.        
  460.         \question Let $y=f(x)$ be a function satisfying $f(e) = \frac{1}{5}$ and $f'(x) = \frac{1}{3} \ln x, \forall x \in (0;+\infty)$. Given that $\int_{e}^{e^3} \frac{f(x)}{x^2} dx = ae^{-3} + be^{-1} + c$, where $a$, $b$, $c$ are rational numbers, on which interval among the following does the value of $a-b+c$ belong to?
  461.        
  462.         \begin{oneparchoices}
  463.             \choice $\left(\frac{1}{2}; \frac{3}{4}\right)$
  464.             \choice $\left(\frac{1}{4}; \frac{1}{2}\right)$
  465.             \choice $\left(\frac{3}{4}; 1\right)$
  466.             \choice $\left(0; \frac{1}{4}\right)$
  467.         \end{oneparchoices}
  468.        
  469.         \question How many integers $a>1$ exist so that for each $a$, there exists no more than 4 integers $b$ such that $5^{b^2} < 25^{-b} . a^{b+2}$?
  470.        
  471.         \begin{oneparchoices}
  472.             \choice $125$
  473.             \choice $100$
  474.             \choice $99$
  475.             \choice $124$
  476.         \end{oneparchoices}
  477.        
  478.         \question Let $f(x)$ be a quartic function with three extrema points $\frac{-3}{2}$, $2$, $\frac{11}{2}$ and has a minimum value. The inequality $f(x) \leq m$ has solutions belonging to the interval $[0;3]$ if and only if
  479.        
  480.         \begin{oneparchoices}
  481.             \choice $m \geq f(3)$
  482.             \choice $f(2) \geq m \geq f(3)$
  483.             \choice $m \geq f(0)$
  484.             \choice $m \geq f(2)$
  485.         \end{oneparchoices}
  486.        
  487.         \question How many integer values of $m$ exist so that for each $m$, there exists exactly two complex numbers $z$ satisfying $|z-1-5i|+|z-1+5i|=10$ and $|z-2-i|=m$?
  488.        
  489.         \begin{oneparchoices}
  490.             \choice $5$
  491.             \choice $4$
  492.             \choice $2$
  493.             \choice $3$
  494.         \end{oneparchoices}
  495.        
  496.         \question Consider the function $f(x) = ax^3 + bx^2 + cx + d$ so that it has two extrema points $x_1$, $x_2$ ($x_1 < x_2$) satisfying $x_1 + x_2 = 0$. The area of the closed region bounded by the curve $y = f'(x).f''(x)$ and the horizontal axis is $\frac{9}{4}$. Given that $\int_{x_1}^{x_2} \frac{f'(x)}{3^x + 1} dx = \frac{-7}{2}$, on which interval among the following does the value of $\int_{0}^{x_2} (x+2)f''(x) dx$ belong to?
  497.        
  498.         \begin{oneparchoices}
  499.             \choice $(6;7)$
  500.             \choice $(-1;0)$
  501.             \choice $(0;1)$
  502.             \choice $(-7;-6)$
  503.         \end{oneparchoices}
  504.        
  505.         \question Consider a right prism $ABC.A'B'C'$ where the base $ABC$ is an isosceles right triangle with $\angle A = 90^{\circ}$ and $AB=a$. Given that the angle between planes $(A'BC)$ and $(ABC)$ is $30^{\circ}$, the volume of the prism is
  506.        
  507.         \begin{oneparchoices}
  508.             \choice $\frac{\sqrt{6} a^3}{12}$
  509.             \choice $\frac{\sqrt{6} a^3}{36}$
  510.             \choice $\frac{\sqrt{6}}{4} a^3$
  511.             \choice $\frac{3\sqrt{6}}{4} a^3$
  512.         \end{oneparchoices}
  513.        
  514.         \question In the 3-dimensional Cartesian coordinate space, consider two lines $d_1: \frac{x-2}{1}=\frac{y-4}{3}=\frac{z+3}{-5}$ and $d_2:\frac{x+2}{1}=\frac{y+2}{-1}=\frac{z+1}{-1}$. Let $(S)$ be the sphere with the smallest radius among the set of all spheres tangent to both $d_1$ and $d_2$, the equation of $(S)$ is
  515.        
  516.         \begin{oneparchoices}
  517.             \choice $(x+1)^2 + y^2 + (z-1)^2 = 6$
  518.             \choice $x^2 + (y-3)^2 + (z+4)^2 = 6$
  519.             \choice $(x-1)^2 + (y+2)^2 + (z+1)^2 = 6$
  520.             \choice $x^2 + (y+1)^2 + z^2 = 6$
  521.         \end{oneparchoices}
  522.        
  523.         \question Consider the function $f(x) = \frac{2}{x^3} + \ln \frac{x+3}{x-3}$. How many integers $a \in (-\infty; 2100)$ satisfy $f(a-2024)+f(6a-27) \geq 0$?
  524.        
  525.         \begin{oneparchoices}
  526.             \choice $2096$
  527.             \choice $288$
  528.             \choice $1807$
  529.             \choice $360$
  530.         \end{oneparchoices}
  531.        
  532.         \question Consider the quadratic equation $az^2 + bz + c = 0$ ($a, b, c \in \mathbb{R}$, $a \neq 0$) with two complex solutions $z_1$ and $z_2$ whose imaginary parts are nonzero and $\left \lvert 2z_1 - \frac{1}{9} \right \rvert = |z_1 - z_2|$. Suppose that $z_1 = \frac{1}{\sqrt{k}}$ and $w$ is a complex number satisfying $cw^2 + bw + a = 0$, how many positive integers $k$ exist so for each $k$, there exists exactly 9 complex numbers $z_3$ so that $Im(z_3) \in \mathbb{Z}$, $z_3 - w$ is purely imaginary and $|z_3| \leq |w|$?
  533.        
  534.         \begin{oneparchoices}
  535.             \choice $23$
  536.             \choice $22$
  537.             \choice $11$
  538.             \choice $12$
  539.         \end{oneparchoices}
  540.        
  541.         \question Consider a pyramid $S.ABC$ whose base is triangle $ABC$ such that $\angle A = 90^{\circ}$, $AB=2a$. $SAB$ is an equilateral triangle belonging to the plane perpendicular with the base plane. The surface area of the circumscribed sphere of the pyramid is
  542.        
  543.         \begin{oneparchoices}
  544.             \choice $\frac{25\pi}{9} a^2$
  545.             \choice $\frac{25\pi}{3} a^2$
  546.             \choice $\frac{28\pi}{3} a^2$
  547.             \choice $\frac{28\pi}{9} a^2$
  548.         \end{oneparchoices}
  549.        
  550.         \question In the 3-dimensional Cartesian coordinate space, consider two points $A(1;6;-1)$, $B(2;-4;-1)$ and a sphere $(S)$ with center $I(1;2;-1)$ passing through $A$. The point $M(a;b;c)$ ($c>0$) belongs to the spherical surface $(S)$ so that $IAM$ is an obtuse triangle whose area is $2\sqrt{7}$ and the distance between lines $BM$ and $AI$ is maximized. On which interval among the following does the value of $a+b+c$ belong to?
  551.        
  552.         \begin{oneparchoices}
  553.             \choice $\left(1; \frac{3}{2} \right)$
  554.             \choice $\left(\frac{3}{2}; 2 \right)$
  555.             \choice $\left(\frac{5}{2}; 3 \right)$
  556.             \choice $\left(2; \frac{5}{2} \right)$
  557.         \end{oneparchoices}
  558.        
  559.         \question Consider a quartic function $y = f(x)$ so that $f(-1) = 5$. The function $y = f'(x)$ is increasing on $(-\infty; +\infty)$, $f'(4)=0$ and $f'(-1)=a$. How many integers $a \in (-100;0)$ exist so for each $a$, the function $y = \left\lvert f(x) + \frac{5}{x^2} \right\rvert$ has exactly 3 extrema points belonging to $(-1;+\infty)$?
  560.        
  561.         \begin{oneparchoices}
  562.             \choice $9$
  563.             \choice $89$
  564.             \choice $10$
  565.             \choice $90$
  566.         \end{oneparchoices}
  567.     \end{questions}
  568. \end{document}
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