A geometric sequence $\left\{ a_{n} \right\}$ with first term and common ratio both equal to a positive number $k$ satisfies $$\frac{a_{4}}{a_{2}} + \frac{a_{2}}{a_{1}} = 30$$ What is the value of $k$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
Q3
2 marksDifferentiation from First PrinciplesView
For the function $f(x) = x^{3} - 8x + 7$, what is the value of $\lim_{h \rightarrow 0} \frac{f(2+h) - f(2)}{h}$? [2 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
The function $$f(x) = \left\{ \begin{array}{cc} 5x + a & (x < -2) \\ x^{2} - a & (x \geq -2) \end{array} \right.$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points] (1) 6 (2) 7 (3) 8 (4) 9 (5) 10
When $\cos\left(\frac{\pi}{2} + \theta\right) = -\frac{1}{5}$, what is the value of $\frac{\sin\theta}{1 - \cos^{2}\theta}$? [3 points] (1) $-5$ (2) $-\sqrt{5}$ (3) $0$ (4) $\sqrt{5}$ (5) $5$
A polynomial function $f(x)$ satisfies $$\int_{0}^{x} f(t)\, dt = 3x^{3} + 2x$$ for all real numbers $x$. What is the value of $f(1)$? [3 points] (1) 7 (2) 9 (3) 11 (4) 13 (5) 15
For two real numbers $a = 2\log\frac{1}{\sqrt{10}} + \log_{2}20$ and $b = \log 2$, what is the value of $a \times b$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
For the function $f(x) = 3x^{2} - 16x - 20$, $$\int_{-2}^{a} f(x)\, dx = \int_{-2}^{0} f(x)\, dx$$ When this condition is satisfied, what is the value of the positive number $a$? [4 points] (1) 16 (2) 14 (3) 12 (4) 10 (5) 8
The function $f(x) = a\cos bx + 3$ is defined on the closed interval $[0, 2\pi]$ and has a maximum value of 13 at $x = \frac{\pi}{3}$. For the ordered pair $(a, b)$ of two natural numbers $a$ and $b$ satisfying this condition, what is the minimum value of $a + b$? [4 points] (1) 12 (2) 14 (3) 16 (4) 18 (5) 20
A point P starts at time $t = 0$ and moves on a number line. At time $t$ ($t \geq 0$), its position $x$ is given by $$x = t^{3} - \frac{3}{2}t^{2} - 6t$$ What is the acceleration of point P at the time when its direction of motion changes after starting? [4 points] (1) 6 (2) 9 (3) 12 (4) 15 (5) 18
A sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and an arithmetic sequence $\left\{ b_{n} \right\}$ with $b_{1} = 2$ satisfy $$\sum_{k=1}^{n} \frac{a_{k}}{b_{k+1}} = \frac{1}{2}n^{2}$$ for all natural numbers $n$. What is the value of $\sum_{k=1}^{5} a_{k}$? [4 points] (1) 120 (2) 125 (3) 130 (4) 135 (5) 140
A cubic function $f(x)$ with leading coefficient 1 satisfies $$f(1) = f(2) = 0, \quad f'(0) = -7$$ Let Q be the point where the line segment OP intersects the curve $y = f(x)$ other than P, where O is the origin and $\mathrm{P}(3, f(3))$. Let $A$ be the area enclosed by the curve $y = f(x)$, the $y$-axis, and the line segment OQ, and let $B$ be the area enclosed by the curve $y = f(x)$ and the line segment PQ. What is the value of $B - A$? [4 points] (1) $\frac{37}{4}$ (2) $\frac{39}{4}$ (3) $\frac{41}{4}$ (4) $\frac{43}{4}$ (5) $\frac{45}{4}$
As shown in the figure, in triangle ABC, point D is taken on segment AB such that $\overline{\mathrm{AD}} : \overline{\mathrm{DB}} = 3 : 2$, and a circle $O$ centered at A passing through D intersects segment AC at point E. $\sin A : \sin C = 8 : 5$, and the ratio of the areas of triangles ADE and ABC is $9 : 35$. When the circumradius of triangle ABC is 7, what is the maximum area of triangle PBC for a point P on circle $O$? (Given: $\overline{\mathrm{AB}} < \overline{\mathrm{AC}}$) [4 points] (1) $18 + 15\sqrt{3}$ (2) $24 + 20\sqrt{3}$ (3) $30 + 25\sqrt{3}$ (4) $36 + 30\sqrt{3}$ (5) $42 + 35\sqrt{3}$
For a constant $a$ ($a \neq 3\sqrt{5}$) and a quadratic function $f(x)$ with negative leading coefficient, the function $$g(x) = \begin{cases} x^{3} + ax^{2} + 15x + 7 & (x \leq 0) \\ f(x) & (x > 0) \end{cases}$$ satisfies the following conditions. (가) The function $g(x)$ is differentiable on the set of all real numbers. (나) The equation $g'(x) \times g'(x - 4) = 0$ has exactly 4 distinct real roots. What is the value of $g(-2) + g(2)$? [4 points] (1) 30 (2) 32 (3) 34 (4) 36 (5) 38
A sequence $\left\{ a_{n} \right\}$ satisfies $$a_{n} + a_{n+4} = 12$$ for all natural numbers $n$. What is the value of $\sum_{n=1}^{16} a_{n}$? [3 points]
For a positive number $a$, let the function $f(x)$ be $$f(x) = 2x^{3} - 3ax^{2} - 12a^{2}x$$ When the local maximum value of $f(x)$ is $\frac{7}{27}$, what is the value of $f(3)$? [3 points]
Let $k$ be the $x$-coordinate of the intersection point of the curve $y = \left(\frac{1}{5}\right)^{x-3}$ and the line $y = x$. A function $f(x)$ defined on the set of all real numbers satisfies the following conditions. For all real numbers $x > k$, $f(x) = \left(\frac{1}{5}\right)^{x-3}$ and $f(f(x)) = 3x$. What is the value of $f\left(\frac{1}{k^{3} \times 5^{3k}}\right)$? [4 points]
A function $f(x) = x^{3} + ax^{2} + bx + 4$ satisfies the following condition for two integers $a$ and $b$. What is the maximum value of $f(1)$? [4 points] For all real numbers $\alpha$, the limit $\lim_{x \rightarrow \alpha} \frac{f(2x+1)}{f(x)}$ exists.
All terms of a sequence $\left\{ a_{n} \right\}$ are integers and satisfy the following conditions. What is the sum of the values of $\left| a_{1} \right|$? [4 points] (가) For all natural numbers $n$, $$a_{n+1} = \begin{cases} a_{n} - 3 & \left(\left| a_{n} \right| \text{ is odd}\right) \\ \frac{1}{2}a_{n} & \left(a_{n} = 0 \text{ or } \left| a_{n} \right| \text{ is even}\right) \end{cases}$$ (나) The minimum value of the natural number $m$ such that $\left| a_{m} \right| = \left| a_{m+2} \right|$ is 3.
For a natural number $n$ ($n \geq 2$), let the line $x = \frac{1}{n}$ meet the two ellipses $$C_{1} : \frac{x^{2}}{2} + y^{2} = 1, \quad C_{2} : 2x^{2} + \frac{y^{2}}{2} = 1$$ at points P and Q respectively in the first quadrant. Let $\alpha$ be the $x$-intercept of the tangent line to ellipse $C_{1}$ at point P, and let $\beta$ be the $x$-intercept of the tangent line to ellipse $C_{2}$ at point Q. How many values of $n$ satisfy $6 \leq \alpha - \beta \leq 15$? [3 points] (1) 7 (2) 9 (3) 11 (4) 13 (5) 15
For two events $A$ and $B$, $$\mathrm{P}(A \mid B) = \mathrm{P}(A) = \frac{1}{2}, \quad \mathrm{P}(A \cap B) = \frac{1}{5}$$ What is the value of $\mathrm{P}(A \cup B)$? [3 points] (1) $\frac{1}{2}$ (2) $\frac{3}{5}$ (3) $\frac{7}{10}$ (4) $\frac{4}{5}$ (5) $\frac{9}{10}$
A sample of size 256 is randomly extracted from a population following a normal distribution $\mathrm{N}\left(m, 2^{2}\right)$. The 95\% confidence interval for $m$ obtained using the sample mean is $a \leq m \leq b$. What is the value of $b - a$? (Given: When $Z$ is a random variable following the standard normal distribution, $\mathrm{P}(|Z| \leq 1.96) = 0.95$.) [3 points] (1) 0.49 (2) 0.52 (3) 0.55 (4) 0.58 (5) 0.61
A survey was conducted on the preferences for Subject A and Subject B among 16 students in a class. Each student who participated in the survey chose one of the two subjects. 9 students chose Subject A and 7 students chose Subject B. When 3 students are randomly selected from the 16 students who participated in the survey, what is the probability that at least one of the 3 selected students chose Subject B? [3 points] (1) $\frac{3}{4}$ (2) $\frac{4}{5}$ (3) $\frac{17}{20}$ (4) $\frac{9}{10}$ (5) $\frac{19}{20}$
As shown in the figure, a solid figure has as its base the region enclosed by the curve $y = \sqrt{\frac{x+1}{x(x + \ln x)}}$, the $x$-axis, and the two lines $x = 1$ and $x = e$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is a square, what is the volume of this solid figure? [3 points] (1) $\ln(e+1)$ (2) $\ln(e+2)$ (3) $\ln(e+3)$ (4) $\ln(2e+1)$ (5) $\ln(2e+2)$
There is a bag containing 5 cards with the numbers $1, 3, 5, 7, 9$ written on them, one number per card. A trial is performed by randomly drawing one card from the bag, confirming the number on the card, and putting it back. This trial is repeated 3 times, and let $\bar{X}$ be the average of the three numbers confirmed. When $\mathrm{V}(a\bar{X} + 6) = 24$, what is the value of the positive number $a$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be $$g(x) = f\left(e^{x}\right) + e^{x}$$ The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points] (1) $\frac{1}{36}$ (2) $\frac{1}{18}$ (3) $\frac{1}{12}$ (4) $\frac{1}{9}$ (5) $\frac{5}{36}$
As shown in the figure, for a tetrahedron ABCD with $\overline{\mathrm{AB}} = 6$, $\overline{\mathrm{BC}} = 4\sqrt{5}$, let M be the midpoint of segment BC. Triangle AMD is equilateral and line BC is perpendicular to plane AMD. Find the area of the orthogonal projection of the circle inscribed in triangle ACD onto plane BCD. [3 points] (1) $\frac{\sqrt{10}}{4}\pi$ (2) $\frac{\sqrt{10}}{6}\pi$ (3) $\frac{\sqrt{10}}{8}\pi$ (4) $\frac{\sqrt{10}}{10}\pi$ (5) $\frac{\sqrt{10}}{12}\pi$
For the set $X = \{1, 2, 3, 4, 5, 6\}$, how many functions $f : X \rightarrow X$ satisfy the following conditions? [4 points] (가) The value of $f(1) \times f(6)$ is a divisor of 6. (나) $2f(1) \leq f(2) \leq f(3) \leq f(4) \leq f(5) \leq 2f(6)$ (1) 166 (2) 171 (3) 176 (4) 181 (5) 186
The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is $$f'(x) = -x + e^{1 - x^{2}}$$ For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points] (1) $\frac{1}{2}e + \frac{1}{2}$ (2) $\frac{1}{2}e + \frac{2}{3}$ (3) $\frac{1}{2}e + \frac{5}{6}$ (4) $\frac{2}{3}e + \frac{1}{2}$ (5) $\frac{2}{3}e + \frac{2}{3}$
In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points] (1) $\sqrt{43}$ (2) $\sqrt{47}$ (3) $\sqrt{51}$ (4) $\sqrt{55}$ (5) $\sqrt{59}$
A random variable $X$ follows a normal distribution $\mathrm{N}\left(m_{1}, \sigma_{1}^{2}\right)$ and a random variable $Y$ follows a normal distribution $\mathrm{N}\left(m_{2}, \sigma_{2}^{2}\right)$, satisfying the following conditions. For all real numbers $x$, $\mathrm{P}(X \leq x) = \mathrm{P}(X \geq 40 - x)$ and $\mathrm{P}(Y \leq x) = \mathrm{P}(X \leq x + 10)$. When $\mathrm{P}(15 \leq X \leq 20) + \mathrm{P}(15 \leq Y \leq 20) = 0.4772$ using the standard normal distribution table below, what is the value of $m_{1} + \sigma_{2}$? (Given: $\sigma_{1}$ and $\sigma_{2}$ are positive.) [4 points]
There is a hyperbola $x^{2} - \frac{y^{2}}{35} = 1$ with foci at $\mathrm{F}(c, 0)$, $\mathrm{F}'(-c, 0)$ ($c > 0$). For a point P on this hyperbola in the first quadrant, let Q be a point on line $\mathrm{PF}'$ such that $\overline{\mathrm{PQ}} = \overline{\mathrm{PF}}$. When triangle $\mathrm{QF'F}$ and triangle $\mathrm{FF'P}$ are similar, the area of triangle PFQ is $\frac{q}{p}\sqrt{5}$. Find the value of $p + q$. (Here, $\overline{\mathrm{PF}'} < \overline{\mathrm{QF}'}$ and $p$ and $q$ are coprime natural numbers.) [4 points]
Five coins are placed in a line on a table. At the start, the coins in the 1st and 2nd positions show heads, and the coins in the remaining 3 positions show tails. Using these 5 coins and one die, the following trial is performed. Roll the die once. If the result is $k$, if $k \leq 5$, flip the coin in the $k$-th position once and place it back, if $k = 6$, flip all coins once and place them back. After repeating this trial 3 times, what is the probability that all 5 coins show heads? Express the answer as $\frac{q}{p}$. What is the value of $p + q$? (Given: $p$ and $q$ are coprime natural numbers.) [4 points]
For two constants $a$ ($1 \leq a \leq 2$) and $b$, the function $f(x) = \sin(ax + b + \sin x)$ satisfies the following conditions. (가) $f(0) = 0$ and $f(2\pi) = 2\pi a + b$ (나) The minimum positive value of $t$ such that $f'(0) = f'(t)$ is $4\pi$. Let $A$ be the set of all values of $\alpha$ in the open interval $(0, 4\pi)$ where the function $f(x)$ has a local maximum. If $n$ is the number of elements in set $A$ and $\alpha_{1}$ is the smallest element in set $A$, then $n\alpha_{1} - ab = \frac{q}{p}\pi$. What is the value of $p + q$? [4 points]
In the coordinate plane, there is a square ABCD with side length 4. $$|\overrightarrow{\mathrm{XB}} + \overrightarrow{\mathrm{XC}}| = |\overrightarrow{\mathrm{XB}} - \overrightarrow{\mathrm{XC}}|$$ Let $S$ be the figure formed by points X satisfying this condition. For a point P on figure $S$, $$4\overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{PB}} + 2\overrightarrow{\mathrm{PD}}$$ Let Q be the point satisfying this condition. If the maximum and minimum values of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{AQ}}$ are $M$ and $m$ respectively, find the value of $M \times m$. [4 points]