csat-suneung

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View

What is the value of $9 ^ { \frac { 3 } { 2 } } \times 27 ^ { - \frac { 2 } { 3 } }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) 1
(3) $\sqrt { 3 }$
(4) 3
(5) $3 \sqrt { 3 }$

Q2 2 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

When the function $f ( x ) = 6 x ^ { 2 } + 2 a x$ satisfies $\int _ { 0 } ^ { 1 } f ( x ) d x = f ( 1 )$, what is the value of the constant $a$? [2 points]
(1) $- 4$
(2) $- 2$
(3) 0
(4) 2
(5) 4

Q3 3 marks Inequalities Solving an equation via substitution to reduce to quadratic form View

What is the sum of all real roots of the equation $\sqrt { x ^ { 2 } - 2 x + 1 } - \sqrt { x ^ { 2 } - 2 x } = \frac { 1 } { 2 }$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1

Q4 2 marks Matrices Matrix Algebra and Product Properties View

For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & - 2 \\ 1 & 0 \end{array} \right)$, what is the sum of all entries of the matrix $( A + B ) A$? [2 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

Q5 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View

The figure on the right shows a circle with center at the origin O and radius 1, and the graph of a quadratic function $y = f ( x )$ passing through the point $( 0 , - 1 )$ on the coordinate plane. The equation $$\frac { 1 } { f ( x ) + 1 } - \frac { 1 } { f ( x ) - 1 } = \frac { 2 } { x ^ { 2 } }$$ has how many distinct real roots $x$? [3 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

Q6 3 marks Composite & Inverse Functions Finding Parameters for Continuity View

For the function $f ( x ) = x ^ { 2 } - 4 x + a$ and the function $g ( x ) = \lim _ { n \rightarrow \infty } \frac { 2 | x - b | ^ { n } + 1 } { | x - b | ^ { n } + 1 }$, let $h ( x ) = f ( x ) g ( x )$. What is the value of $a + b$, the sum of the two constants $a , b$ such that the function $h ( x )$ is continuous for all real numbers $x$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

Q7 3 marks Exponential Functions Parameter Determination from Conditions View

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of the function $y = f ( x )$ and the function $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of $a + b$, the sum of the two constants $a , b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

Q8 3 marks Normal Distribution Sampling Distribution of the Mean View

A company produces women's general handball balls certified by the International Handball Federation. The weight of handball balls produced by this company follows a normal distribution with mean 350 g and standard deviation 16 g. The company determines that there is a problem in the production process if the average weight of 64 randomly selected handball balls is 346 g or less or 355 g or more. Using the standard normal distribution table on the right, what is the probability that the company determines there is a problem in the production process? [3 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.00	0.4772
2.25	0.4878
2.50	0.4938
2.75	0.4970

(1) 0.0290
(2) 0.0258
(3) 0.0184
(4) 0.0152
(5) 0.0092

Q9 4 marks Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

For a function $y = f ( x )$ defined on the closed interval $[ 0,5 ]$, define the function $g ( x )$ as $$g ( x ) = \begin{cases} \{ f ( x ) \} ^ { 2 } & ( 0 \leqq x \leqq 3 ) \\ ( f \circ f ) ( x ) & ( 3 < x \leqq 5 ) \end{cases}$$ Which of the following graphs of the function $y = f ( x )$ make the function $g ( x )$ continuous on the closed interval $[ 0,5 ]$? Select all that apply from . [4 points] ㄱ. [graph] ㄴ. [graph] ㄷ. [graph]
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

Q10 3 marks Proof by induction Fill in missing steps of a given induction proof View

The sequence $\left\{ a _ { n } \right\}$ satisfies $$\left\{ \begin{array} { l } a _ { 1 } = \frac { 1 } { 2 } \\ ( n + 1 ) ( n + 2 ) a _ { n + 1 } = n ^ { 2 } a _ { n } \quad ( n = 1,2,3 , \cdots ) \end{array} \right.$$ The following is a proof by mathematical induction that for all natural numbers $n$ $$\sum _ { k = 1 } ^ { n } a _ { k } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k ^ { 2 } } - \frac { n } { n + 1 }$$ holds.
(1) When $n = 1$, (left side) $= \frac { 1 } { 2 }$, (right side) $= 1 - \frac { 1 } { 2 } = \frac { 1 } { 2 }$, so (*) holds.
(2) Assume that (*) holds when $n = m$: $\sum _ { k = 1 } ^ { m } a _ { k } = \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 }$
Now show that (*) holds when $n = m + 1$. $$\begin{aligned} & \sum _ { k = 1 } ^ { m + 1 } a _ { k } = \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 } + a _ { m + 1 } \\ = & \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 } + \square \text{ (a) } a _ { m } \\ = & \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 } \\ & \quad + \frac { m ^ { 2 } } { ( m + 1 ) ( m + 2 ) } \cdot \frac { ( m - 1 ) ^ { 2 } } { m ( m + 1 ) } \cdot \cdots \cdot \frac { 1 ^ { 2 } } { 2 \cdot 3 } a _ { 1 } \end{aligned}$$ Therefore, (*) also holds when $n = m + 1$. Thus, for all natural numbers $n$, (*) holds.
What are the expressions that should be filled in for (a), (b), and (c) in the above proof? [3 points]
(1) $\dfrac{\text{(a)}}{\dfrac{m}{(m+1)(m+2)}} \quad \dfrac{\text{(b)}}{1} \quad \dfrac{\text{(c)}}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)^2}$
(2) $\dfrac{m}{(m+1)(m+2)} \quad \dfrac{m}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)}$
(3) $\dfrac{m^2}{(m+1)(m+2)} \quad \dfrac{1}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)^2}$
(4) $\dfrac{m^2}{(m+1)(m+2)} \quad \dfrac{1}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)}$
(5) $\dfrac{m^2}{(m+1)(m+2)} \quad \dfrac{m}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)^2}$

Q11 4 marks Chain Rule Limit Involving Derivative Definition of Composed Functions View

A polynomial function $f ( x )$ and two natural numbers $m , n$ satisfy $$\begin{array} { l l } \lim _ { x \rightarrow \infty } \frac { f ( x ) } { x ^ { m } } = 1 , & \lim _ { x \rightarrow \infty } \frac { f ^ { \prime } ( x ) } { x ^ { m - 1 } } = a \\ \lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { n } } = b , & \lim _ { x \rightarrow 0 } \frac { f ^ { \prime } ( x ) } { x ^ { n - 1 } } = 9 \end{array}$$ Which of the following are correct? Select all that apply from . (where $a , b$ are real numbers.) [4 points]
ㄱ. $m \geqq n$ ㄴ. $a b \geqq 9$ ㄷ. If $f ( x )$ is a cubic function, then $a m = b n$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q12 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View

Let the set $U$ be $$U = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, a , b , c , d \text{ are positive numbers not equal to } 1 \right\}$$ Let the subset $S$ of $U$ be $$S = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$ Which of the following are correct? Select all that apply from . [4 points]
ㄱ. If $A = \left( \begin{array} { c c } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$. ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$. ㄷ. If $A \in S$, then $A$ has an inverse matrix.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q13 3 marks Stationary points and optimisation Geometric or applied optimisation problem View

For a natural number $n$, when two points $\mathrm { P } _ { n - 1 } , \mathrm { P } _ { n }$ are on the graph of the function $y = x ^ { 2 }$, the point $\mathrm { P } _ { n + 1 }$ is determined according to the following rule.
(a) The coordinates of the two points $\mathrm { P } _ { 0 } , \mathrm { P } _ { 1 }$ are $(0,0)$ and $(1,1)$, respectively.
(b) The point $\mathrm { P } _ { n + 1 }$ is the intersection of the line passing through point $\mathrm { P } _ { n }$ and perpendicular to the line $\mathrm { P } _ { n - 1 } \mathrm { P } _ { n }$ and the graph of the function $y = x ^ { 2 }$. (where $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ are distinct points.) Let $l _ { n } = \overline { \mathrm { P } _ { n - 1 } \mathrm { P } _ { n } }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { l _ { n } } { n }$? [3 points]
(1) $2 \sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) $\sqrt { 3 }$
(5) $\sqrt { 2 }$

Q14 4 marks Circles Infinite Series or Sequences Involving Circles View

On the coordinate plane, there is a circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + y ^ { 2 } = 1$. As shown in the figure, when a tangent line $l$ with positive slope is drawn from the origin to the circle $C _ { 1 }$, let the point of tangency be $\mathrm { P } _ { 1 }$. Let $C _ { 2 }$ be a circle with center on the line $l$, passing through the point $\mathrm { P } _ { 1 }$, and tangent to the $x$-axis, and let $\mathrm { P } _ { 2 }$ be the point of tangency with the $x$-axis. Let $C _ { 3 }$ be a circle with center on the $x$-axis, passing through the point $\mathrm { P } _ { 2 }$, and tangent to the line $l$, and let $\mathrm { P } _ { 3 }$ be the point of tangency with the line $l$. Let $C _ { 4 }$ be a circle with center on the line $l$, passing through the point $\mathrm { P } _ { 3 }$, and tangent to the $x$-axis, and let $\mathrm { P } _ { 4 }$ be the point of tangency with the $x$-axis. Continuing this process, let $S _ { n }$ be the area of the circle $C _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? (Note: the radius of circle $C _ { n + 1 }$ is smaller than the radius of circle $C _ { n }$.) [4 points]
(1) $\frac { 3 } { 2 } \pi$
(2) $2 \pi$
(3) $\frac { 5 } { 2 } \pi$
(4) $3 \pi$
(5) $\frac { 7 } { 2 } \pi$

Q15 4 marks Combinations & Selection Counting Integer Solutions to Equations View

A certain volunteer service center operates the following four volunteer activity programs every day.

Program	A	B	C	D
Volunteer Activity Hours	1 hour	2 hours	3 hours	4 hours

Chulsu wants to participate in one program each day for 5 days at this volunteer service center and create a volunteer activity plan so that the total volunteer activity hours is 8 hours. How many different volunteer activity plans can be created? [4 points]
(1) 47
(2) 44
(3) 41
(4) 38
(5) 35

Q16 4 marks Probability Definitions Conditional Probability and Bayes' Theorem View

Pouches A and B each contain five marbles with the numbers $1,2,3,4,5$ written on them, one number per marble. Chulsu draws one marble from pouch A, and Younghee draws one marble from pouch B. They check the numbers on the two marbles and do not put them back. This trial is repeated. What is the probability that the numbers on the two marbles drawn the first time are different, and the numbers on the two marbles drawn the second time are the same? [4 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$

Q17 4 marks Probability Definitions Verifying Statements About Probability Properties View

In information theory, when an event $E$ occurs, the information content $I ( E )$ of the event $E$ is defined as follows. $$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$ Which of the following are correct? Select all that apply from . (Note: the probability that event $E$ occurs, $\mathrm { P } ( E )$, is positive, and the unit of information content is bits.) [4 points]
ㄱ. If event $E$ is rolling one die and getting an odd number, then $I ( E ) = 1$. ㄴ. If two events $A , B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A , B$ with $\mathrm { P } ( A ) > 0 , \mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q18 3 marks Chain Rule Limit Involving Derivative Definition of Composed Functions View

For a polynomial function $f ( x )$, if $\lim _ { x \rightarrow 2 } \frac { f ( x + 1 ) - 8 } { x ^ { 2 } - 4 } = 5$, find the value of $f ( 3 ) + f ^ { \prime } ( 3 )$. [3 points]

Q19 3 marks Vectors Introduction & 2D Dot Product Computation View

As shown in the figure, in a rectangular parallelepiped $\mathrm { ABCD } - \mathrm { EFGH }$ with $\overline { \mathrm { AB } } = \overline { \mathrm { AD } } = 4$ and $\overline { \mathrm { AE } } = 8$, let P be the point that divides the edge AE in the ratio $1 : 3$, and let Q, R, S be the midpoints of edges $\mathrm { AB }$, $\mathrm { AD }$, and $\mathrm { FG }$, respectively. Let T be the midpoint of segment QR. Find the value of the dot product $\overrightarrow { \mathrm { TP } } \cdot \overrightarrow { \mathrm { QS } }$ of vectors $\overrightarrow { \mathrm { TP } }$ and $\overrightarrow { \mathrm { QS } }$. [3 points]

Q20 3 marks Conic sections Equation Determination from Geometric Conditions View

An ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is inscribed in a quadrilateral formed by connecting the four vertices of the ellipse $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1$. When the two foci of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ are $\mathrm { F } ( b , 0 ) , \mathrm { F } ^ { \prime } ( - b , 0 )$, find the value of $a^2 + b^2$ (or the relevant quantity as stated in the problem). [3 points]

Q21 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View

Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]

Q22 4 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View

As shown in the figure, three cylinders with radius $\sqrt{3}$ and different heights are mutually externally tangent and placed on a plane $\alpha$. Let $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ be the centers of the bases of the three cylinders that do not meet plane $\alpha$. Triangle $\mathrm{QPR}$ is an isosceles triangle, and the angle between plane $\mathrm{QPR}$ and plane $\alpha$ is $60°$. If the heights of the three cylinders are $8$, $a$, and $b$ respectively, find the value of $a + b$. (Given: $8 < a < b$) [4 points]

Q23 4 marks Vectors: Lines & Planes MCQ: Cross-Section or Surface Area of a Solid View

In coordinate space, let $C$ be the circle formed by the intersection of the sphere $S : x^2 + y^2 + z^2 = 4$ and the plane $\alpha : y - \sqrt{3}z = 2$. For point $\mathrm{A}(0, 2, 0)$ on circle $C$, let $\mathrm{P}$ and $\mathrm{Q}$ be the endpoints of a diameter of circle $C$ such that $\overline{\mathrm{AP}} = \overline{\mathrm{AQ}}$. Let $\mathrm{R}$ be another point where the line passing through $\mathrm{P}$ and perpendicular to plane $\alpha$ meets sphere $S$. If the area of triangle $\mathrm{ARQ}$ is $s$, find the value of $s^2$. [4 points]

Q26 3 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

(Calculus) For $0 \leqq x < 2\pi$, the sum of all distinct values of $x$ satisfying the equation $\sin 2x = 2\cos x - 2\cos^2 x$ is? [3 points]
(1) $\pi$
(2) $\frac{5}{4}\pi$
(3) $\frac{3}{2}\pi$
(4) $\frac{7}{4}\pi$
(5) $2\pi$

Q26b 3 marks Measures of Location and Spread View

(Probability and Statistics) A certain basketball player practices free throws 40 times every day. The stem-and-leaf plot below shows the number of successful free throws each day for the first 10 days, with the tens digit as the stem and the units digit as the leaf. On the 11th day, the number of successful free throws was $n$, and the average number of successful free throws for the first 11 days was equal to the mode in the stem-and-leaf plot below. What is the value of $n$? [3 points]

Stem	\multicolumn{4}{\|c\|}{Leaf}
0	9
1	7	9
2	1	4	4	6
3	0	1	3

(1) 24
(2) 26
(3) 28
(4) 30
(5) 32

Q27 3 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

(Calculus) A continuous function $f(x)$ defined on the closed interval $[0, 1]$ satisfies $f(0) = 0$, $f(1) = 1$, has a second derivative on the open interval $(0, 1)$, and $f'(x) > 0$, $f''(x) > 0$. Which of the following is equal to $\int_0^1 \{f^{-1}(x) - f(x)\} dx$? [3 points]
(1) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{2n}$
(2) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{2}{n}$
(3) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(4) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{2n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(5) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{2k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$

Q27b 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View

(Probability and Statistics) A coin is tossed three times, and based on the results, a score is obtained as a random variable $X$ according to the following rules. (가) If the same face does not appear consecutively, the score is 0 points. (나) If the same face appears consecutively exactly twice, the score is 1 point. (다) If the same face appears consecutively three times, the score is 3 points.
What is the variance $\mathrm{V}(X)$ of the random variable $X$? [3 points]
(1) $\frac{9}{8}$
(2) $\frac{19}{16}$
(3) $\frac{5}{4}$
(4) $\frac{21}{16}$
(5) $\frac{11}{8}$

Q28 3 marks Differentiating Transcendental Functions Full function study with transcendental functions View

(Calculus) For the function $f(x) = 4\ln x + \ln(10 - x)$, which of the following statements in are correct? [3 points]
ㄱ. The maximum value of function $f(x)$ is $13\ln 2$. ㄴ. The equation $f(x) = 0$ has two distinct real roots. ㄷ. The graph of function $y = e^{f(x)}$ is concave downward on the interval $(4, 8)$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q28b 3 marks Combinations & Selection Combinatorial Probability View

(Probability and Statistics) There are 9 balls in a bag, each labeled with a natural number from 1 to 9. When 4 balls are randomly drawn simultaneously from the bag, what is the probability that the sum of the largest and smallest numbers on the drawn balls is at least 7 and at most 9? [3 points]
(1) $\frac{5}{9}$
(2) $\frac{1}{2}$
(3) $\frac{4}{9}$
(4) $\frac{7}{18}$
(5) $\frac{1}{3}$

Q29 4 marks Chain Rule Derivative of Inverse Functions View

(Calculus) Let the function $f(x)$ be defined as $$f(x) = \int_a^x \{2 + \sin(t^2)\} dt$$ If $f''(a) = \sqrt{3}a$, find the value of $(f^{-1})'(0)$. (Given: $a$ is a constant satisfying $0 < a < \sqrt{\frac{\pi}{2}}$) [4 points]
(1) $\frac{1}{10}$
(2) $\frac{1}{5}$
(3) $\frac{3}{10}$
(4) $\frac{2}{5}$
(5) $\frac{1}{2}$

Q29b 4 marks Normal Distribution Multiple-Choice Conceptual Question on Normal Distribution Properties View

(Probability and Statistics) Random variables $X$ and $Y$ follow normal distributions with mean 0 and variances $\sigma^2$ and $\frac{\sigma^2}{4}$ respectively, and random variable $Z$ follows the standard normal distribution. For two positive numbers $a$ and $b$, $$\mathrm{P}(|X| \leqq a) = \mathrm{P}(|Y| \leqq b)$$ Which of the following statements in are correct? [4 points] ㄱ. $a > b$ ㄴ. $\mathrm{P}\left(Z > \frac{2b}{\sigma}\right) = \mathrm{P}\left(Y > \frac{a}{2}\right)$ ㄷ. If $\mathrm{P}(Y \leqq b) = 0.7$, then $\mathrm{P}(|X| \leqq a) = 0.3$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q30 4 marks Connected Rates of Change Geometric Related Rates with Distance or Angle View

(Calculus) Point $\mathrm{A}$ is on circle $\mathrm{O}$ with radius 1. As shown in the figure, for a positive angle $\theta$, two points $\mathrm{B}$ and $\mathrm{C}$ on circle $\mathrm{O}$ are chosen such that $\angle \mathrm{BAC} = \theta$ and $\overline{\mathrm{AB}} = \overline{\mathrm{AC}}$. Let $r(\theta)$ denote the radius of the inscribed circle of triangle $\mathrm{ABC}$. If $\lim_{\theta \rightarrow \pi - 0} \frac{r(\theta)}{(\pi - \theta)^2} = \frac{q}{p}$, find the value of $p^2 + q^2$. (Given: $p$ and $q$ are coprime natural numbers.) [4 points]

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