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int64 1
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|---|---|---|---|---|---|
1 | 1 | 1. $\left(2^{\sqrt{3}} \times 4\right)^{\sqrt{3} - 2}$ μ κ°μ? [2μ ] \begin{itemize} \item[1] \frac{1}{4} \item[2] \frac{1}{2} \item[3] 1 \item[4] 2 \item[5] 4 \end{itemize} | 2 | 2 | null |
2 | 2 | 2. ν¨μ $f(x) = x^3 + 3x^2 + x - 1$ μ λνμ¬ $f'(1)$μ κ°μ? [2μ ] \begin{itemize} \item[1] 6 \item[2] 7 \item[3] 8 \item[4] 9 \item[5] 10 \end{itemize} | 5 | 2 | null |
3 | 3 | 3. λ±μ°¨μμ΄ $\{a_n\}$μ λνμ¬ \[ a_2 = 6, \quad a_4 + a_6 = 36 \] μΌ λ, $a_{10}$μ κ°μ? [3μ ] \begin{itemize} \item[1] 30 \item[2] 32 \item[3] 34 \item[4] 36 \item[5] 38 \end{itemize} | 5 | 3 | null |
4 | 4 | 4. ν¨μ $( y = f(x) )$κ° λ€μκ³Ό κ°μ΄ μ μλμ΄ μλ€.
\[
f(x) =
\begin{cases}
-x+2, & x < -1, \\
2, & x = -1, \\
(3*x+3)/2, & -1 < x < 1, \\
1, & 1 \leq x < 2, \\
3, & x = 2, \\
1, & x \geq 2.
\end{cases}
\]
\[ \lim_{x \to -1-} f(x) + \lim_{x \to 2} f(x) \text{μ κ°μ? [3μ ]} \]
\begin{itemize} \item[1] 1 \item[2] 2 \item[3] 3 \item[4] 4 \item[5] 5 \end{itemize} | 4 | 3 | Removed figure and the statement referring to the figure. The figure is needed to solve the problem, so we paraphrased the figure into text. |
5 | 5 | 5. 첫째νμ΄ 1μΈ μμ΄ $\{a_n\}$μ΄ λͺ¨λ μμ°μ $n$μ λνμ¬ \[ a_{n+1} = \begin{cases} 2a_n & (a_n < 7) \\ a_n - 7 & (a_n \geq 7) \end{cases} \] μΌ λ, $\sum_{k=1}^{8} a_k$μ κ°μ? [3μ ] \begin{itemize} \item[1] 30 \item[2] 32 \item[3] 34 \item[4] 36 \item[5] 38 \end{itemize} | 1 | 3 | null |
6 | 6 | 6. λ°©μ μ $( 2x^3 - 3x^2 - 12x + k = 0 )$μ΄ μλ‘ λ€λ₯Έ μΈ μ€κ·Όμ κ°λλ‘ νλ μ μ $k$μ κ°μλ? [3μ ] \begin{itemize} \item[1] 20 \item[2] 23 \item[3] 26 \item[4] 29 \item[5] 32 \end{itemize} | 3 | 3 | null |
7 | 7 | 7. $( \pi < \theta < \frac{3}{2}\pi )$μΈ $\theta$μ λνμ¬ $\tan \theta - \frac{6}{\tan \theta} = 1$μΌ λ, $ \sin \theta + \cos \theta $μ κ°μ? [3μ ] \begin{itemize} \item[1] -\frac{2\sqrt{10}}{5} \item[2] -\frac{\sqrt{10}}{5} \item[3] 0 \item[4] \frac{\sqrt{10}}{5} \item[5] \frac{2\sqrt{10}}{5} \end{itemize} | 1 | 3 | null |
8 | 8 | 8. 곑μ $( y = x^2 - 5x )$μ μ§μ $( y = x )$λ‘ λλ¬μΈμΈ λΆλΆμ λμ΄λ₯Ό μ§μ $( x = k )$κ° μ΄λ±λΆν λ, μμ $k$μ κ°μ? [3μ ] \begin{itemize} \item[1] 3 \item[2] \frac{13}{4} \item[3] \frac{7}{2} \item[4] \frac{15}{4} \item[5] 4 \end{itemize} | 1 | 3 | null |
9 | 9 | 9. μ§μ $( y = 2x + k )$ κ° λ ν¨μ \[ y = \left( \frac{2}{3} \right)^{x+3} + 1, \quad y = \left( \frac{2}{3} \right)^{x+1} + \frac{8}{3} \] μ κ·Έλνμ λ§λλ μ μ κ°κ° $( \mathrm{P} )$, $( \mathrm{Q} )$λΌ νμ. $\overline{\mathrm{PQ}} = \sqrt{5}$μΌ λ, μμ $k$μ κ°μ? [4μ ] \begin{itemize} \item[1] \frac{31}{6} \item[2] \frac{16}{3} \item[3] \frac{11}{2} \item[4] \frac{17}{3} \item[5] \frac{35}{6} \end{itemize} | 4 | 4 | Removed figure. |
10 | 10 | 10. μΌμ°¨ν¨μ $( f(x) )$μ λνμ¬ κ³‘μ $( y = f(x) )$ μμ μ $( 0, 0 )$μμμ μ μ κ³Ό 곑μ $( y = x f(x) )$ μμ μ $( 1, 2 )$μμμ μ μ μ΄ μΌμΉν λ, $f'(2)$μ κ°μ? [4μ ] \begin{itemize} \item[1] -18 \item[2] -17 \item[3] -16 \item[4] -15 \item[5] -14 \end{itemize} | 5 | 4 | null |
11 | 11 | 11. μμ $a$μ λνμ¬ μ§ν© $\left\{ x \ \middle| \ -\frac{a}{2} < x \leq a, \ x \neq \frac{a}{2} \right\}$ μμ μ μλ ν¨μ \[ f(x) = \tan \frac{\pi x}{a} \] κ° μλ€. κ·Έλ¦Όκ³Ό κ°μ΄ ν¨μ $y = f(x)$μ κ·Έλν μμ μΈ μ $( \mathrm{O, A, B} )$λ₯Ό μ§λλ μ§μ μ΄ μλ€. μ $( \mathrm{A} )$λ₯Ό μ§λκ³ $x$μΆμ ννν μ§μ μ΄ ν¨μ $y = f(x)$μ κ·Έλνμ λ§λλ μ μ€ $( \mathrm{A} )$κ° μλ μ μ $( \mathrm{C} )$λΌ νμ. μΌκ°ν $( \mathrm{ABC} )$κ° μ μΌκ°νμΌ λ, μΌκ°ν $( \mathrm{ABC} )$μ λμ΄λ? (λ¨, $( \mathrm{O} )$λ μμ μ΄λ€.) [4μ ] \begin{itemize} \item[1] \frac{3\sqrt{3}}{2} \item[2] \frac{17\sqrt{3}}{12} \item[3] \frac{4\sqrt{3}}{3} \item[4] \frac{5\sqrt{3}}{4} \item[5] \frac{7\sqrt{3}}{6} \end{itemize} | 3 | 4 | Removed figure and the statement referring to the figure. |
12 | 12 | 12. μ€μ μ 체μ μ§ν©μμ μ°μμΈ ν¨μ $f(x)$κ° λͺ¨λ μ€μ $x$μ λνμ¬ \[ \{f(x)\}^3 - \{f(x)\}^2 - x^2 f(x) + x^2 = 0 \] μ λ§μ‘±μν¨λ€. ν¨μ $f(x)$μ μ΅λκ°μ΄ 1μ΄κ³ μ΅μκ°μ΄ 0μΌ λ, \[ f\left( -\frac{4}{3} \right) + f(0) + f\left( \frac{1}{2} \right) \] μ κ°μ? [4μ ] \begin{itemize} \item[1] \frac{1}{2} \item[2] 1 \item[3] \frac{3}{2} \item[4] 2 \item[5] \frac{5}{2} \end{itemize} | 3 | 4 | null |
13 | 13 | 13. λ μμ $( a, b \ (1 < a < b) )$μ λνμ¬ μ’ννλ©΄ μμ λ μ $(a, \log_2 a), \ (b, \log_2 b)$λ₯Ό μ§λλ μ§μ μ $y$μ νΈκ³Ό λ μ $(a, \log_4 a), \ (b, \log_4 b)$λ₯Ό μ§λλ μ§μ μ $y$μ νΈμ΄ κ°λ€. ν¨μ $f(x) = a^{bx} + b^{ax}$μ λνμ¬ $f(1) = 40$μΌ λ, $f(2)$μ κ°μ? [4μ ] \begin{itemize} \item[1] 760 \item[2] 800 \item[3] 840 \item[4] 880 \item[5] 920 \end{itemize} | 2 | 4 | null |
14 | 14 | 14. μμ§μ μλ₯Ό μμ§μ΄λ μ $\mathrm{P}$μ μκ° $t$μμμ μμΉ $x(t)$κ° λ μμ $a$, $b$μ λνμ¬ \[ x(t) = t(t - 1)(at + b) \quad (a \neq 0) \] μ΄λ€. μ $\mathrm{P}$μ μκ° $t$μμμ μλ $v(t)$κ° $\int_0^1 |v(t)| \, dt = 2$λ₯Ό λ§μ‘±μν¬ λ, μλ γ±, γ΄, γ· μ€μμ μ³μ κ²λ§μ μλ λλ‘ κ³ λ₯Έ κ²μ? [4μ ]
\begin{itemize} \item[γ±.] $\int_0^1 v(t) \, dt = 0$ \item[γ΄.] $|x(t_1)| > 1$μΈ $t_1$μ΄ μ΄λ¦°κ΅¬κ° $(0, 1)$μ μ‘΄μ¬νλ€. \item[γ·.] $0 \leq t \leq 1$μΈ λͺ¨λ $t$μ λνμ¬ $|x(t)| < 1$μ΄λ©΄ $x(t_2) = 0$μΈ $t_2$κ° μ΄λ¦°κ΅¬κ° $(0, 1)$μ μ‘΄μ¬νλ€. \end{itemize}
\begin{itemize} \item[1] γ± \item[2] γ±, γ΄ \item[3] γ±, γ· \item[4] γ΄, γ· \item[5] γ±, γ΄, γ· \end{itemize} | 3 | 4 | <보기> changed to 'μλ γ±,γ΄,γ·, μ€'. |
15 | 15 | 15. λ μ $( \mathrm{O}_1, \mathrm{O}_2 )$λ₯Ό κ°κ° μ€μ¬μΌλ‘ νκ³ λ°μ§λ¦μ κΈΈμ΄κ° $(\overline{\mathrm{O}_1\mathrm{O}_2} )$μΈ λ μ $( C_1, C_2 )$κ° μλ€. κ·Έλ¦Όκ³Ό κ°μ΄ μ $( C_1 )$ μμ μλ‘ λ€λ₯Έ μΈ μ $( \mathrm{A}, \mathrm{B}, \mathrm{C} )$μ μ $( C_2 )$ μμ μ $( \mathrm{D} )$κ° μ£Όμ΄μ Έ μκ³ , μΈ μ $( \mathrm{A}, \mathrm{O}_1, \mathrm{O}_2 )$μ μΈ μ $( \mathrm{C}, \mathrm{O}_2, \mathrm{D} )$κ° κ°κ° ν μ§μ μμ μλ€.
μ΄λ $(\angle \mathrm{B}\mathrm{O}_1\mathrm{A} = \theta_1)$, $(\angle \mathrm{O}_2\mathrm{O}_1\mathrm{C} = \theta_2)$, $(\angle \mathrm{O}_1\mathrm{O}_2\mathrm{D} = \theta_3)$μ΄λΌ νμ.
λ€μμ $( \overline{\mathrm{A}\mathrm{B}} : \overline{\mathrm{O}_1\mathrm{D}} = 1 : 2\sqrt{2} )$μ΄κ³ $( \theta_3 = \theta_1 + \theta_2 )$μΌ λ, μ λΆ $( \mathrm{A}\mathrm{B} )$μ μ λΆ $( \mathrm{C}\mathrm{D} )$μ κΈΈμ΄μ λΉλ₯Ό ꡬνλ κ³Όμ μ΄λ€.
\[ \begin{aligned} &\angle \mathrm{C}\mathrm{O}_2\mathrm{O}_1 + \angle \mathrm{O}_1\mathrm{O}_2\mathrm{D} = \pi \text{μ΄λ―λ‘ } \theta_3 = \frac{\pi}{2} + \frac{\theta_2}{2} \text{μ΄κ³ } \\ &\theta_3 = \theta_1 + \theta_2 \text{μμ } 2\theta_1 + \theta_2 = \pi \text{μ΄λ―λ‘ } \angle \mathrm{C}\mathrm{O}_1\mathrm{B} = \theta_1 \text{μ΄λ€.} \\ &\text{μ΄λ } \angle \mathrm{O}_2\mathrm{O}_1\mathrm{B} = \theta_1 + \theta_2 = \theta_3 \text{μ΄λ―λ‘ μΌκ°ν } \mathrm{O}_1\mathrm{O}_2\mathrm{B} \text{μ μΌκ°ν } \mathrm{O}_2\mathrm{O}_1\mathrm{D} \text{λ ν©λμ΄λ€.} \\ &\overline{\mathrm{A}\mathrm{B}} = k \text{λΌ ν λ} \\ &\overline{\mathrm{B}\mathrm{O}_2} = \overline{\mathrm{O}_1\mathrm{D}}= 2\sqrt{2}k \text{μ΄λ―λ‘ } \overline{\mathrm{A}\mathrm{O}_2} = \text{(κ°)μ΄κ³ ,} \\ &\angle \mathrm{B}\mathrm{O}_2\mathrm{A} = \frac{\theta_1}{2} \text{μ΄λ―λ‘ } \cos \frac{\theta_1}{2} = \text{(λ) μ΄λ€.} \\ &\text{μΌκ°ν } \mathrm{O}_2\mathrm{B}\mathrm{C} \text{μμ} \\ &\overline{\mathrm{B}\mathrm{C}} = k, \overline{\mathrm{B}\mathrm{O}_2} = 2\sqrt{2}k, \angle \mathrm{C}\mathrm{O}_2\mathrm{B} = \frac{\theta_1}{2} \text{μ΄λ―λ‘} \\ &\text{μ½μ¬μΈλ²μΉμ μνμ¬ } \overline{\mathrm{O}_2\mathrm{C}} = \text{(λ€) μ΄λ€.} \\ &\overline{\mathrm{C}\mathrm{D}} = \overline{\mathrm{O}_2\mathrm{D}} + \overline{\mathrm{O}_2\mathrm{C}} = \overline{\mathrm{O}_1\mathrm{O}_2} + \overline{\mathrm{O}_2\mathrm{C}} \text{μ΄λ―λ‘} \\ &\overline{\mathrm{A}\mathrm{B}} : \overline{\mathrm{C}\mathrm{D}} = k : \left(\frac{\text{(κ°)}}{2} + \text{(λ€)}\right) \text{μ΄λ€.} \end{aligned} \]
μμ (κ°), (λ€)μ μλ§μ μμ κ°κ° $( f(k), g(k) )$λΌ νκ³ , (λ)μ μλ§μ μλ₯Ό $( p )$λΌ ν λ, $( f(p) \times g(p) )$μ κ°μ? [4μ ]
\begin{itemize} \item[1] \frac{169}{27} \item[2] \frac{56}{9} \item[3] \frac{167}{27} \item[4] \frac{166}{27} \item[5] \frac{55}{9} \end{itemize} | 2 | 4 | Removed figure and the statement referring to the figure. |
16 | 16 | 16. $\log_2 120 - \frac{1}{\log_{15} 2}$ μ κ°μ ꡬνμμ€. [3μ ] | 3 | 3 | null |
17 | 17 | 17. ν¨μ $f(x)$μ λνμ¬ $f'(x) = 3x^2 + 2x$μ΄κ³ $f(0) = 2$μΌ λ, $f(1)$μ κ°μ ꡬνμμ€. [3μ ] | 4 | 3 | null |
18 | 18 | 18. μμ΄ $\{a_n\}$μ λνμ¬ \[ \sum_{k=1}^{10} a_k - \sum_{k=1}^{7} \frac{a_k}{2} = 56, \quad \sum_{k=1}^{10} 2a_k - \sum_{k=1}^{8} a_k = 100 \] μΌ λ, $a_8$μ κ°μ ꡬνμμ€. [3μ ] | 12 | 3 | null |
19 | 19 | 19. ν¨μ $f(x) = x^3 + ax^2 - (a^2 - 8a)x + 3$μ΄ μ€μ μ 체μ μ§ν©μμ μ¦κ°νλλ‘ νλ μ€μ $a$μ μ΅λκ°μ ꡬνμμ€. [3μ ] | 6 | 3 | null |
20 | 20 | 20. μ€μ μ 체μ μ§ν©μμ λ―ΈλΆκ°λ₯ν ν¨μ $( f(x) )$κ° λ€μ 쑰건μ λ§μ‘±μν¨λ€.
\begin{itemize} \item[(κ°)] λ«νκ΅¬κ° $[0, 1]$μμ $f(x) = x$μ΄λ€. \item[(λ)] μ΄λ€ μμ $a, b$μ λνμ¬ κ΅¬κ° $[0, \infty)$μμ $f(x+1) - x f(x) = ax + b$μ΄λ€. \end{itemize}
\[ 60 \times \int_1^2 f(x) \, dx \] μ κ°μ ꡬνμμ€. [4μ ] | 110 | 4 | null |
21 | 21 | 21. μμ΄ $\{a_n\}$μ΄ λ€μ 쑰건μ λ§μ‘±μν¨λ€.
\begin{itemize} \item[(κ°)] $( |a_1| = 2 )$ \item[(λ)] λͺ¨λ μμ°μ $( n )$μ λνμ¬ $( |a_{n+1}| = 2|a_n| )$μ΄λ€. \item[(λ€)] $\sum_{n=1}^{10} a_n = -14$ \end{itemize}
$a_1 + a_3 + a_5 + a_7 + a_9$μ κ°μ ꡬνμμ€. [4μ ] | 678 | 4 | null |
22 | 22 | 22. μ΅κ³ μ°¨νμ κ³μκ° $\frac{1}{2}$ μΈ μΌμ°¨ν¨μ $f(x)$μ μ€μ $t$μ λνμ¬ λ°©μ μ $f'(x) = 0$μ΄ λ«νκ΅¬κ° $[t, t+2]$μμ κ°λ μ€κ·Όμ κ°μλ₯Ό $g(t)$λΌ ν λ, ν¨μ $g(t)$λ λ€μ 쑰건μ λ§μ‘±μν¨λ€.
\begin{itemize} \item[(κ°)] λͺ¨λ μ€μ $( a )$μ λνμ¬ $( \lim_{t \to a+} g(t) + \lim_{t \to a-} g(t) \leq 2 )$μ΄λ€. \item[(λ)] $( g(f(1)) = g(f(4)) = 2, \ g(f(0)) = 1 )$ \end{itemize}
$f(5)$μ κ°μ ꡬνμμ€. [4μ ] | 9 | 4 | null |
23 | 23_prob | 23. λ€νμ $(x+2)^7$μ μ κ°μμμ $x^5$μ κ³μλ? [2μ ] \begin{itemize} \item[1] 42 \item[2] 56 \item[3] 70 \item[4] 84 \item[5] 98 \end{itemize} | 4 | 2 | null |
24 | 24_prob | 24. νλ₯ λ³μ $X$κ° μ΄νλΆν¬ $\mathrm{B}\left(n, \frac{1}{3}\right)$μ λ°λ₯΄κ³ $\mathrm{V}(2X) = 40$μΌ λ, $n$μ κ°μ? [3μ ] \begin{itemize} \item[1] 30 \item[2] 35 \item[3] 40 \item[4] 45 \item[5] 50 \end{itemize} | 4 | 3 | null |
25 | 25_prob | 25. λ€μ 쑰건μ λ§μ‘±μν€λ μμ°μ $a, \ b, \ c, \ d, \ e$μ λͺ¨λ μμμ $(a, b, c, d, e)$μ κ°μλ? [3μ ]
\begin{itemize} \item[(κ°)] $a + b + c + d + e = 12$ \item[(λ)] $\left| a^2 - b^2 \right| = 5$ \end{itemize}
\begin{itemize} \item[1] 30 \item[2] 32 \item[3] 34 \item[4] 36 \item[5] 38 \end{itemize} | 1 | 3 | null |
26 | 26_prob | 26. $( 1 )$λΆν° $( 10 )$κΉμ§ μμ°μκ° νλμ© μ ν μλ $( 10 )$μ₯μ μΉ΄λκ° λ€μ΄ μλ μ£Όλ¨Έλκ° μλ€. μ΄ μ£Όλ¨Έλμμ μμλ‘ μΉ΄λ $( 3 )$μ₯μ λμμ κΊΌλΌ λ, κΊΌλΈ μΉ΄λμ μ ν μλ μΈ μμ°μ μ€μμ κ°μ₯ μμ μκ° $( 4 )$ μ΄νμ΄κ±°λ $( 7 )$ μ΄μμΌ νλ₯ μ? [3μ ]
\begin{itemize} \item[1] \frac{4}{5} \item[2] \frac{5}{6} \item[3] \frac{13}{15} \item[4] \frac{9}{10} \item[5] \frac{14}{15} \end{itemize} | 3 | 3 | Removed figure. |
27 | 27_prob | 27. μ΄λ μλμ°¨ νμ¬μμ μμ°νλ μ κΈ° μλμ°¨μ 1ν μΆ©μ μ£Όν 거리λ νκ· μ΄ $m$μ΄κ³ νμ€νΈμ°¨κ° $\sigma$μΈ μ κ·λΆν¬λ₯Ό λ°λ₯Έλ€κ³ νλ€.
μ΄ μλμ°¨ νμ¬μμ μμ°ν μ κΈ° μλμ°¨ 100λλ₯Ό μμμΆμΆνμ¬ μ»μ 1ν μΆ©μ μ£Όν 거리μ νλ³Ένκ· μ΄ $\overline{x_1}$μΌ λ, λͺ¨νκ· $m$μ λν μ λ’°λ 95\%μ μ 뒰ꡬκ°μ΄ $a \le m \le b$μ΄λ€.
μ΄ μλμ°¨ νμ¬μμ μμ°ν μ κΈ° μλμ°¨ 400λλ₯Ό μμμΆμΆνμ¬ μ»μ 1ν μΆ©μ μ£Όν 거리μ νλ³Ένκ· μ΄ $\overline{x_2}$μΌ λ, λͺ¨νκ· $m$μ λν μ λ’°λ 99\%μ μ 뒰ꡬκ°μ΄ $c \le m \le d$μ΄λ€.
$\overline{x_1} - \overline{x_2} = 1.34$μ΄κ³ $a = c$μΌ λ, $b - a$μ κ°μ? (λ¨, μ£Όν 거리μ λ¨μλ kmμ΄κ³ , $Z$κ° νμ€μ κ·λΆν¬λ₯Ό λ°λ₯΄λ νλ₯ λ³μμΌ λ $\mathrm{P}(|Z| \le 1.96) = 0.95$, $\mathrm{P}(|Z| \le 2.58) = 0.99$λ‘ κ³μ°νλ€.) [3μ ]
\begin{itemize} \item[1] 5.88 \item[2] 7.84 \item[3] 9.80 \item[4] 11.76 \item[5] 13.72 \end{itemize} | 2 | 3 | null |
28 | 28_prob | 28. λ μ§ν© $X = \{1, 2, 3, 4, 5\}$, $Y = \{1, 2, 3, 4\}$μ λνμ¬ λ€μ 쑰건μ λ§μ‘±μν€λ $X$μμ $Y$λ‘μ ν¨μ $f$μ κ°μλ? [4μ ]
\begin{itemize} \item[(κ°)] μ§ν© $X$μ λͺ¨λ μμ $x$μ λνμ¬ $f(x) \geq \sqrt{x}$μ΄λ€. \item[(λ)] ν¨μ $f$μ μΉμμ μμμ κ°μλ 3μ΄λ€. \end{itemize}
\begin{itemize} \item[1] 128 \item[2] 138 \item[3] 148 \item[4] 158 \item[5] 168 \end{itemize} | 1 | 4 | null |
29 | 29_prob | 29. λ μ°μνλ₯ λ³μ $( X )$μ $( Y )$κ° κ°λ κ°μ λ²μλ $( 0 \leq X \leq 6 )$, $( 0 \leq Y \leq 6 )$μ΄κ³ , $( X )$μ $( Y )$μ νλ₯ λ°λν¨μλ κ°κ° $( f(x), g(x) )$μ΄λ€. νλ₯ λ³μ $( X )$μ νλ₯ λ°λν¨μ $( f(x) )$κ° λ€μκ³Ό κ°μ΄ μ μλμ΄ μλ€.
\[
f(x) =
\begin{cases}
0, & x < 0, \\
\frac{1}{12}x, & 0 \leq x < 3, \\
\frac{1}{4}, & 3 \leq x \leq 5, \\
\frac{1}{4}(6-x), & 5 < x \leq 6, \\
0, & x > 6.
\end{cases}
\]
\[ 0 \leq x \leq 6\ \text{μΈ λͺ¨λ } x \text{μ λνμ¬} \]
\[ f(x) + g(x) = k \quad (k \text{λ μμ}) \]
λ₯Ό λ§μ‘±μν¬ λ, $( \mathrm{P}(6k \leq Y \leq 15k) = \frac{q}{p} )$μ΄λ€. $( p + q )$μ κ°μ ꡬνμμ€. (λ¨, $( p )$μ $( q )$λ μλ‘μμΈ μμ°μμ΄λ€.) [4μ ] | 31 | 4 | Removed figure and the statement referring to the figure. The figure is needed to solve the problem, so we paraphrased the figure into text. |
30 | 30_prob | 30. ν° κ³΅κ³Ό κ²μ κ³΅μ΄ κ°κ° 10κ° μ΄μ λ€μ΄ μλ λ°κ΅¬λμ λΉμ΄ μλ μ£Όλ¨Έλκ° μλ€. ν κ°μ μ£Όμ¬μλ₯Ό μ¬μ©νμ¬ λ€μ μνμ νλ€.
\[ \begin{array}{|c|} \hline \text{μ£Όμ¬μλ₯Ό ν λ² λμ Έ} \\ \text{λμ¨ λμ μκ° 5 μ΄μμ΄λ©΄} \\ \text{λ°κ΅¬λμ μλ ν° κ³΅ 2κ°λ₯Ό μ£Όλ¨Έλμ λ£κ³ ,} \\ \text{λμ¨ λμ μκ° 4 μ΄νμ΄λ©΄} \\ \text{λ°κ΅¬λμ μλ κ²μ 곡 1κ°λ₯Ό μ£Όλ¨Έλμ λ£λλ€.} \\ \hline \end{array} \]
μμ μνμ 5λ² λ°λ³΅ν λ, $( n(1 \leq n \leq 5) )$λ²μ§Έ μν ν μ£Όλ¨Έλμ λ€μ΄ μλ ν° κ³΅κ³Ό κ²μ 곡μ κ°μλ₯Ό κ°κ° $( a_n )$, $( b_n )$μ΄λΌ νμ. $( a_5 + b_5 \geq 7 )$μΌ λ, $( a_k = b_k )$μΈ μμ°μ $( k(1 \leq k \leq 5) )$κ° μ‘΄μ¬ν νλ₯ μ $( \frac{q}{p} )$μ΄λ€. $( p + q )$μ κ°μ ꡬνμμ€. (λ¨, $(p)$μ $(q)$λ μλ‘μμΈ μμ°μμ΄λ€.) [4μ ] | 191 | 4 | null |
31 | 23_calc | 23. \[ \lim_{n \to \infty} \frac{\frac{5}{n} + \frac{3}{n^2}}{\frac{1}{n} - \frac{2}{n^3}} \text{μ κ°μ? [2μ ]} \] \begin{itemize} \item[1] 1 \item[2] 2 \item[3] 3 \item[4] 4 \item[5] 5 \end{itemize} | 5 | 2 | null |
32 | 24_calc | 24. μ€μ μ 체μ μ§ν©μμ λ―ΈλΆκ°λ₯ν ν¨μ $f(x)$κ° λͺ¨λ μ€μ $x$μ λνμ¬ \[ f(x^3 + x) = e^x \] μ λ§μ‘±μν¬ λ, $f'(2)$μ κ°μ? [3μ ] \begin{itemize} \item[1] e \item[2] \frac{e}{2} \item[3] \frac{e}{3} \item[4] \frac{e}{4} \item[5] \frac{e}{5} \end{itemize} | 4 | 3 | null |
33 | 25_calc | 25. λ±λΉμμ΄ $\{a_n\}$μ λνμ¬ \[ \sum_{n=1}^{\infty} (a_{2n-1} - a_{2n}) = 3, \quad \sum_{n=1}^{\infty} a_n^2 = 6 \] μΌ λ, $\sum_{n=1}^{\infty} a_n$ μ κ°μ? [3μ ] \begin{itemize} \item[1] 1 \item[2] 2 \item[3] 3 \item[4] 4 \item[5] 5 \end{itemize} | 2 | 3 | null |
34 | 26_calc | 26. \[ \lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^2 + 2kn}{k^3 + 3k^2 n + n^3} \text{μ κ°μ?} \quad [3 \text{μ }] \] \begin{itemize} \item[1] \ln 5 \item[2] \frac{\ln 5}{2} \item[3] \frac{\ln 5}{3} \item[4] \frac{\ln 5}{4} \item[5] \frac{\ln 5}{5} \end{itemize} | 3 | 3 | null |
35 | 27_calc | 27. μ’ννλ©΄ μλ₯Ό μμ§μ΄λ μ $\mathrm{P}$μ μκ° $t \ (t>0)$μμμ μμΉκ° 곑μ $y = x^2$κ³Ό μ§μ $y = t^2 x - \frac{\ln t}{8}$κ° λ§λλ μλ‘ λ€λ₯Έ λ μ μ μ€μ μΌ λ, μκ° $t=1$μμ $t=e$κΉμ§ μ $\mathrm{P}$κ° μμ§μΈ 거리λ? [3μ ] \begin{itemize} \item[1] \frac{e^4}{2} - \frac{3}{8} \item[2] \frac{e^4}{2} - \frac{5}{16} \item[3] \frac{e^4}{2} - \frac{1}{4} \item[4] \frac{e^4}{2} - \frac{3}{16} \item[5] \frac{e^4}{2} - \frac{1}{8} \end{itemize} | 1 | 3 | null |
36 | 28_calc | 28. ν¨μ $( f(x) = 6\pi (x - 1)^2 )$μ λνμ¬ ν¨μ $( g(x) )$λ₯Ό \[ g(x) = 3f(x) + 4\cos f(x) \] λΌ νμ. $( 0 < x < 2 )$μμ ν¨μ $( g(x) )$κ° κ·Ήμκ° λλ $( x )$μ κ°μλ? [4μ ] \begin{itemize} \item[1] 6 \item[2] 7 \item[3] 8 \item[4] 9 \item[5] 10 \end{itemize} | 2 | 4 | null |
37 | 29_calc | 29. κ·Έλ¦Όκ³Ό κ°μ΄ κΈΈμ΄κ° 2μΈ μ λΆ $(\mathrm{AB})$λ₯Ό μ§λ¦μΌλ‘ νλ λ°μμ΄ μλ€. νΈ $(\mathrm{AB})$ μμ λ μ $(\mathrm{P})$, $(\mathrm{Q})$λ₯Ό $(\angle \mathrm{PAB} = \theta)$, $(\angle \mathrm{QBA} = 2\theta)$κ° λλλ‘ μ‘κ³ , λ μ λΆ $(\mathrm{AP})$, $(\mathrm{BQ})$μ κ΅μ μ $(\mathrm{R})$λΌ νμ. μ λΆ $(\mathrm{AB})$ μμ μ $(\mathrm{S})$, μ λΆ $(\mathrm{BR})$ μμ μ $(\mathrm{T})$, μ λΆ $(\mathrm{AR})$ μμ μ $(\mathrm{U})$λ₯Ό μ λΆ $(\mathrm{UT})$κ° μ λΆ $(\mathrm{AB})$μ νννκ³ μΌκ°ν $(\mathrm{STU})$κ° μ μΌκ°νμ΄ λλλ‘ μ‘λλ€. λ μ λΆ $(\mathrm{AR})$, $(\mathrm{QR})$μ νΈ $(\mathrm{AQ})$λ‘ λλ¬μΈμΈ λΆλΆμ λμ΄λ₯Ό $(f(\theta))$, μΌκ°ν $(\mathrm{STU})$μ λμ΄λ₯Ό $(g(\theta))$λΌ ν λ,
\[ \lim_{\theta \to 0+} \frac{g(\theta)}{\theta \times f(\theta)} = \frac{q}{p} \sqrt{3} \]
μ΄λ€. $(p + q)$μ κ°μ ꡬνμμ€.
(λ¨, $(0 < \theta < \frac{\pi}{6})$μ΄κ³ , $(p)$μ $(q)$λ μλ‘μμΈ μμ°μμ΄λ€.) [4μ ] | 11 | 4 | Removed figure and the statement referring to the figure. |
38 | 30_calc | 30. μ€μ μ 체μ μ§ν©μμ μ¦κ°νκ³ λ―ΈλΆκ°λ₯ν ν¨μ $f(x)$κ° λ€μ 쑰건μ λ§μ‘±μν¨λ€.
\begin{itemize} \item[(κ°)] $f(1) = 1$, \quad $\int_{1}^{2} f(x) \, dx = \frac{5}{4}$ \item[(λ)] ν¨μ $f(x)$μ μν¨μλ₯Ό $g(x)$λΌ ν λ, $x \geq 1$μΈ λͺ¨λ μ€μ $x$μ λνμ¬ $g(2x) = 2f(x)$μ΄λ€. \end{itemize}
\[ \int_{1}^{8} x f'(x) \, dx = \frac{q}{p} \text{μΌ λ, } p+q \text{μ κ°μ ꡬνμμ€.} \]
(λ¨, $p$μ $q$λ μλ‘μμΈ μμ°μμ΄λ€.) [4μ ] | 143 | 4 | null |
39 | 23_geom | 23. μ’ν곡κ°μ μ $\mathrm{A}(2, 1, 3)$μ $xy$ νλ©΄μ λνμ¬ λμΉμ΄λν μ μ $\mathrm{P}$λΌ νκ³ , μ $\mathrm{A}$λ₯Ό $yz$ νλ©΄μ λνμ¬ λμΉμ΄λν μ μ $\mathrm{Q}$λΌ ν λ, μ λΆ $\mathrm{PQ}$μ κΈΈμ΄λ? [2μ ]
\begin{itemize} \item[1] 5 \sqrt{2} \item[2] 2 \sqrt{13} \item[3] 3 \sqrt{6} \item[4] 2 \sqrt{14} \item[5] 2 \sqrt{15} \end{itemize} | 2 | 2 | null |
40 | 24_geom | 24. ν μ΄μ μ μ’νκ° $\left( 3\sqrt{2}, 0 \right)$ μΈ μ곑μ $\frac{x^2}{a^2} - \frac{y^2}{6} = 1$ μ μ£ΌμΆμ κΈΈμ΄λ? (λ¨, $a$ λ μμμ΄λ€.) [3μ ]
\begin{itemize} \item[1] 3\sqrt{3} \item[2] \frac{7\sqrt{3}}{2} \item[3] 4\sqrt{3} \item[4] \frac{9\sqrt{3}}{2} \item[5] 5\sqrt{3} \end{itemize} | 3 | 3 | null |
41 | 25_geom | 25. μ’ννλ©΄μμ λ μ§μ \[ \frac{x+1}{2} = y - 3, \quad x - 2 = \frac{y - 5}{3} \] κ° μ΄λ£¨λ μκ°μ ν¬κΈ°λ₯Ό $\theta$λΌ ν λ, $\cos \theta$μ κ°μ? [3μ ]
\begin{itemize} \item[1] \frac{1}{2} \item[2] \frac{\sqrt{5}}{4} \item[3] \frac{\sqrt{6}}{4} \item[4] \frac{\sqrt{7}}{4} \item[5] \frac{\sqrt{2}}{2} \end{itemize} | 5 | 3 | null |
42 | 26_geom | 26. λ μ΄μ μ΄ $( \mathrm{F}, \mathrm{F'} )$μΈ νμ $\frac{x^2}{64} + \frac{y^2}{16} = 1$ μμ μ μ€ μ 1μ¬λΆλ©΄μ μλ μ $( \mathrm{A} )$κ° μλ€. λ μ§μ $( \mathrm{AF}, \mathrm{AF'} )$μ λμμ μ νκ³ μ€μ¬μ΄ $y$μΆ μμ μλ μ μ€ μ€μ¬μ $y$μ’νκ° μμμΈ κ²μ $( C )$λΌ νμ. μ $( C )$μ μ€μ¬μ $( \mathrm{B} )$λΌ ν λ μ¬κ°ν $( \mathrm{AFBF'} )$μ λμ΄κ° 72μ΄λ€. μ $( C )$μ λ°μ§λ¦μ κΈΈμ΄λ? [3μ ]
\begin{itemize} \item[1] \frac{17}{2} \item[2] 9 \item[3] \frac{19}{2} \item[4] 10 \item[5] \frac{21}{2} \end{itemize} | 2 | 3 | Removed figure. |
43 | 27_geom | 27. κ·Έλ¦Όκ³Ό κ°μ΄ ν λͺ¨μ리μ κΈΈμ΄κ° 4μΈ μ μ‘면체 $\mathrm{ABCD - EFGH}$ κ° μλ€. μ λΆ $\mathrm{AD}$ μ μ€μ μ $\mathrm{M}$μ΄λΌ ν λ, μΌκ°ν $\mathrm{MEG}$ μ λμ΄λ? [3μ ]
\begin{itemize} \item[1] \frac{21}{2} \item[2] 11 \item[3] \frac{23}{2} \item[4] 12 \item[5] \frac{25}{2} \end{itemize} | 4 | 3 | Removed figure and the statement referring to the figure. |
44 | 28_geom | 28. λ μμ $( a )$, $( p )$μ λνμ¬ ν¬λ¬Όμ $( (y - a)^2 = 4px )$μ μ΄μ μ $( \mathrm{F}_1 )$μ΄λΌ νκ³ , ν¬λ¬Όμ $( y^2 = -4x )$μ μ΄μ μ $( \mathrm{F}_2 )$λΌ νμ. μ λΆ $( \mathrm{F}_1 \mathrm{F}_2 )$κ° λ ν¬λ¬Όμ κ³Ό λ§λλ μ μ κ°κ° $( \mathrm{P} )$, $( \mathrm{Q} )$λΌ ν λ, $( \overline{\mathrm{F}_1 \mathrm{F}_2} = 3 )$, $( \overline{\mathrm{P}\mathrm{Q}} = 1 )$μ΄λ€. $( a^2 + p^2 )$μ κ°μ? [4μ ]
\begin{itemize} \item[1] 6 \item[2] \frac{25}{4} \item[3] \frac{13}{2} \item[4] \frac{27}{4} \item[5] 7 \end{itemize} | 5 | 4 | Removed figure. |
45 | 29_geom | 29. μ’ννλ©΄μμ $\overline{\mathrm{OA}} = \sqrt{2}$, $\overline{\mathrm{OB}} = 2\sqrt{2}$μ΄κ³
\[ \cos(\angle \mathrm{AOB}) = \frac{1}{4} \]
μΈ ννμ¬λ³ν $\mathrm{OACB}$μ λνμ¬ μ $\mathrm{P}$κ° λ€μ 쑰건μ λ§μ‘±μν¨λ€.
\begin{itemize} \item[(κ°)] $\overrightarrow{\mathrm{OP}} = s \overrightarrow{\mathrm{OA}} + t \overrightarrow{\mathrm{OB}} \quad (0 \leq s \leq 1, \ 0 \leq t \leq 1)$ \item[(λ)] $\overrightarrow{\mathrm{OP}} \cdot \overrightarrow{\mathrm{OB}} + \overrightarrow{\mathrm{BP}} \cdot \overrightarrow{\mathrm{BC}} = 2$ \end{itemize}
μ $\mathrm{O}$λ₯Ό μ€μ¬μΌλ‘ νκ³ μ $\mathrm{A}$λ₯Ό μ§λλ μ μλ₯Ό μμ§μ΄λ μ $\mathrm{X}$μ λνμ¬ $|3\overrightarrow{\mathrm{OP}} - \overrightarrow{\mathrm{OX}}|$μ μ΅λκ°κ³Ό μ΅μκ°μ κ°κ° $M$, $m$μ΄λΌ νμ. $M \times m = a\sqrt{6} + b$μΌ λ, $a^2 + b^2$μ κ°μ ꡬνμμ€. (λ¨, $a$μ $b$λ μ 리μμ΄λ€.) [4μ ] | 100 | 4 | Removed figure. |
46 | 30_geom | 30. μ’ν곡κ°μ μ€μ¬μ΄ $\mathrm{C}(2, \sqrt{5}, 5)$μ΄κ³ μ $\mathrm{P}(0, 0, 1)$μ μ§λλ ꡬ \[ S: (x - 2)^2 + (y - \sqrt{5})^2 + (z - 5)^2 = 25 \] κ° μλ€. ꡬ $S$κ° νλ©΄ $\mathrm{OPC}$μ λ§λμ μκΈ°λ μ μλ₯Ό μμ§μ΄λ μ $\mathrm{Q}$, ꡬ $S$ μλ₯Ό μμ§μ΄λ μ $\mathrm{R}$μ λνμ¬ λ μ $\mathrm{Q}, \mathrm{R}$μ $xy$νλ©΄ μλ‘μ μ μ¬μμ κ°κ° $\mathrm{Q}_1, \mathrm{R}_1$μ΄λΌ νμ.
μΌκ°ν $\mathrm{O}\mathrm{Q}_1\mathrm{R}_1$μ λμ΄κ° μ΅λκ° λλλ‘ νλ λ μ $\mathrm{Q}, \mathrm{R}$μ λνμ¬ μΌκ°ν $\mathrm{O}\mathrm{Q}_1\mathrm{R}_1$μ νλ©΄ $\mathrm{PQR}$ μλ‘μ μ μ¬μμ λμ΄λ $\frac{q}{p} \sqrt{6}$μ΄λ€. $p+q$μ κ°μ ꡬνμμ€.
(λ¨, $\mathrm{O}$λ μμ μ΄κ³ μΈ μ $\mathrm{O}, \mathrm{Q}_1, \mathrm{R}_1$μ ν μ§μ μμ μμ§ μμΌλ©°, $p$μ $q$λ μλ‘μμΈ μμ°μμ΄λ€.) [4μ ] | 23 | 4 | Removed figure. |
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