csat-suneung

Q1 2 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

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

Q2 2 marks Vectors Introduction & 2D Magnitude of Vector Expression View

In coordinate space, the distance between point $\mathrm { P } ( 0,3,0 )$ and point $\mathrm { A } ( - 1,1 , a )$ is 2 times the distance between point P and point $\mathrm { B } ( 1,2 , - 1 )$. What is the value of the positive number $a$? [2 points]
(1) $\sqrt { 7 }$
(2) $\sqrt { 6 }$
(3) $\sqrt { 5 }$
(4) 2
(5) $\sqrt { 3 }$

Q3 2 marks Matrices Matrix Algebra and Product Properties View

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

Q4 3 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View

For the irrational equation $$\sqrt { 4 x ^ { 2 } - 5 x + 7 } - 4 x ^ { 2 } + 5 x = 1$$ What is the product of all real roots? [3 points]
(1) $- \frac { 1 } { 2 }$
(2) $- \frac { 3 } { 2 }$
(3) $- \frac { 5 } { 2 }$
(4) $- \frac { 7 } { 2 }$
(5) $- \frac { 9 } { 2 }$

Q5 3 marks Conic sections Locus and Trajectory Derivation View

In the coordinate plane, for point $\mathrm { A } ( 0,4 )$ and point P on the ellipse $\frac { x ^ { 2 } } { 5 } + y ^ { 2 } = 1$, let Q be the point other than A among the two points where the line passing through A and P meets the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 1$. When point P passes through all points on the ellipse, what is the length of the figure traced by point Q? [3 points]
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 2 } { 3 } \pi$
(5) $\frac { 3 } { 4 } \pi$

Q6 3 marks Combinations & Selection Distribution of Objects to Positions or Containers View

At a certain event venue, there are 5 locations where one banner can be installed at each location. There are three types of banners: A, B, and C, with 1 banner of type A, 4 banners of type B, and 2 banners of type C. When selecting and installing 5 banners at the 5 locations to satisfy the following conditions, how many possible cases are there? (Note: Banners of the same type are not distinguished from each other.) [3 points] (가) Banner A must be installed. (나) Banner B must be installed in at least 2 locations.
(1) 55
(2) 65
(3) 75
(4) 85
(5) 95

Q7 3 marks Probability Definitions Probability Using Set/Event Algebra View

Chulsu participated in a certain design competition. Participants receive scores in two categories, and the possible scores in each category are one of three types shown in the table. The probability that Chulsu receives score A in each category is $\frac { 1 } { 2 }$, the probability of receiving score B is $\frac { 1 } { 3 }$, and the probability of receiving score C is $\frac { 1 } { 6 }$. When the event of receiving an audience vote score and the event of receiving a judge score are independent, what is the probability that the sum of the two scores Chulsu receives is 70? [3 points]

Category	Score A	Score B	Score C
Audience Vote	40	30	20
Judge	50	40	30

(1) $\frac { 1 } { 3 }$
(2) $\frac { 11 } { 36 }$
(3) $\frac { 5 } { 18 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 2 } { 9 }$

Q8 3 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View

For the function $$f ( x ) = \begin{cases} x + 2 & ( x < - 1 ) \\ 0 & ( x = - 1 ) \\ x ^ { 2 } & ( - 1 < x < 1 ) \\ x - 2 & ( x \geqq 1 ) \end{cases}$$ which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [3 points]
$\langle$Remarks$\rangle$ ㄱ. $\lim _ { x \rightarrow 1 + 0 } \{ f ( x ) + f ( - x ) \} = 0$ ㄴ. The function $f ( x ) - | f ( x ) |$ is discontinuous at 1 point. ㄷ. There is no constant $a$ such that the function $f ( x ) f ( x - a )$ is continuous on the entire set of real numbers.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q9 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View

To determine the relative density of soil, a method of inserting a test device into the soil for investigation is used. When the effective vertical stress of the soil is $S$ and the resistance force received by the test device as it enters the soil is $R$, the relative density $D ( \% )$ of the soil can be calculated as follows. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (where the units of $S$ and $R$ are metric ton $/ \mathrm { m } ^ { 2 }$.) The effective vertical stress of soil A is 1.44 times the effective vertical stress of soil B, and the resistance force received by the test device as it enters soil A is 1.5 times the resistance force received as it enters soil B. When the relative density of soil B is $65 ( \% )$, what is the relative density of soil A (in $\%$)? (Use $\log 2 = 0.3$ for calculation.) [3 points]
(1) 81.5
(2) 78.2
(3) 74.9
(4) 71.6
(5) 68.3

Q10 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View

There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, determine two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.
Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$, and determine two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, determine two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process to obtain $S _ { n }$, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? [4 points]
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$

Q11 4 marks Circles Area and Geometric Measurement Involving Circles View

As shown in the figure, there are two circular disks with distance between centers $\sqrt { 3 }$ and radius 1, and a plane $\alpha$. The line $l$ passing through the centers of each disk is perpendicular to the planes of the two disks and makes an angle of $60 ^ { \circ }$ with plane $\alpha$. When sunlight shines perpendicular to plane $\alpha$ as shown in the figure, what is the area of the shadow cast by the two disks on plane $\alpha$? (Note: the thickness of the disks is negligible.) [4 points]
(1) $\frac { \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 8 }$
(2) $\frac { 2 } { 3 } \pi + \frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 1 } { 8 }$
(4) $\frac { 4 } { 3 } \pi + \frac { \sqrt { 3 } } { 16 }$
(5) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 4 }$

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

Set $S$ has $1 \times 2$ matrices as elements and set $T$ has $2 \times 1$ matrices as elements, as follows. $$S = \{ ( a \; b ) \mid a + b \neq 0 \} , \quad T = \left\{ \left. \binom { p } { q } \right\rvert \, p q \neq 0 \right\}$$ Which of the following are correct for element $A$ of set $S$? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. For element $P$ of set $T$, $PA$ does not have an inverse matrix. ㄴ. For element $B$ of set $S$ and element $P$ of set $T$, if $PA = PB$ then $A = B$. ㄷ. Among the elements of set $T$, there exists $P$ satisfying $PA \binom { 1 } { 1 } = \binom { 1 } { 1 }$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q13 3 marks Normal Distribution Normal Distribution Combined with Total Probability or Bayes' Theorem View

The distance from home to a traditional market for customers using a certain traditional market follows a normal distribution with mean 1740 m and standard deviation 500 m. Among customers whose distance from home to the market is 2000 m or more, 15\% use personal vehicles to come to the market, and among customers whose distance is less than 2000 m, 5\% use personal vehicles. When one customer who came to the market using a personal vehicle is randomly selected, what is the probability that the distance from this customer's home to the market is less than 2000 m? (Note: When $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( 0 \leqq Z \leqq 0.52 ) = 0.2$.) [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 7 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$

Q14 4 marks Circles Area and Geometric Measurement Involving Circles View

As shown in the figure, in the coordinate plane, for two points $\mathrm { A } , \mathrm { B }$ on the $x$-axis, the parabola $p _ { 1 }$ with vertex at A and the parabola $p _ { 2 }$ with vertex at B satisfy the following conditions. What is the area of triangle ABC? [4 points] (가) The focus of $p _ { 1 }$ is B, and the focus of $p _ { 2 }$ is the origin O. (나) $p _ { 1 }$ and $p _ { 2 }$ meet at two points $\mathrm { C } , \mathrm { D }$ on the $y$-axis. (다) $\overline { \mathrm { AB } } = 2$
(1) $4 ( \sqrt { 2 } - 1 )$
(2) $3 ( \sqrt { 3 } - 1 )$
(3) $2 ( \sqrt { 5 } - 1 )$
(4) $\sqrt { 3 } + 1$
(5) $\sqrt { 5 } + 1$

Q15 4 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View

The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$ and $$a _ { n + 1 } = n + 1 + \frac { ( n - 1 ) ! } { a _ { 1 } a _ { 2 } \cdots a _ { n } } \quad ( n \geqq 1 )$$ The following is part of the process of finding the general term $a _ { n }$.
For all natural numbers $n$, $$a _ { 1 } a _ { 2 } \cdots a _ { n } a _ { n + 1 } = a _ { 1 } a _ { 2 } \cdots a _ { n } \times ( n + 1 ) + ( n - 1 ) !$$ If $b _ { n } = \frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! }$, then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + ( \text{(가)} )$$ The general term of the sequence $\left\{ b _ { n } \right\}$ is $b _ { n } =$ (나) so $\frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! } =$ (나). $\vdots$ Therefore, $a _ { 1 } = 1$ and $a _ { n } = \frac { ( n - 1 ) ( 2 n - 1 ) } { 2 n - 3 } ( n \geqq 2 )$.
When the expression that fits (가) is $f ( n )$ and the expression that fits (나) is $g ( n )$, what is the value of $f ( 13 ) \times g ( 7 )$? [4 points]
(1) $\frac { 1 } { 70 }$
(2) $\frac { 1 } { 77 }$
(3) $\frac { 1 } { 84 }$
(4) $\frac { 1 } { 91 }$
(5) $\frac { 1 } { 98 }$

Q16 4 marks Exponential Functions True/False or Multiple-Statement Verification View

In the coordinate plane, let the two points where the curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet be $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and let the point where the curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet be $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q17 4 marks Variable acceleration (1D) True/false or multiple-statement verification View

Point P starts from the origin and moves on a number line. The velocity $v ( t )$ of point P at time $t ( 0 \leqq t \leqq 5 )$ is as follows. $$v ( t ) = \begin{cases} 4 t & ( 0 \leqq t < 1 ) \\ - 2 t + 6 & ( 1 \leqq t < 3 ) \\ t - 3 & ( 3 \leqq t \leqq 5 ) \end{cases}$$ For a real number $x$ with $0 < x < 3$, let $f ( x )$ be the minimum among:

the distance traveled from time $t = 0$ to $t = x$,
the distance traveled from time $t = x$ to $t = x + 2$,
the distance traveled from time $t = x + 2$ to $t = 5$.

Which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. $f ( 1 ) = 2$ ㄴ. $f ( 2 ) - f ( 1 ) = \int _ { 1 } ^ { 2 } v ( t ) d t$ ㄷ. The function $f ( x )$ is differentiable at $x = 1$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ

Q18 3 marks Stationary points and optimisation Determine parameters from given extremum conditions View

When the local minimum value of the function $f ( x ) = ( x - 1 ) ^ { 2 } ( x - 4 ) + a$ is 10, find the value of the constant $a$. [3 points]

Q19 3 marks Inequalities Integer Solutions of an Inequality View

For the fractional inequality in $x$ $$1 + \frac { k } { x - k } \leqq \frac { 1 } { x - 1 }$$ Find the value of the natural number $k$ such that the number of integers $x$ satisfying the inequality is 3. [3 points]

Q20 3 marks Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View

The volume of the solid of revolution created by rotating the region enclosed by the two curves $y = \sqrt { x } , y = \sqrt { - x + 10 }$ and the $x$-axis around the $x$-axis is $a \pi$. Find the value of $a$. [3 points]

Q21 3 marks Vectors 3D & Lines Line-Plane Intersection View

In coordinate space, let A be the intersection point of the line $\frac { x } { 2 } = y = z + 3$ and the plane $\alpha : x + 2 y + 2 z = 6$. A sphere with center at point $( 1 , - 1,5 )$ passing through point A intersects plane $\alpha$ to form a figure with area $k \pi$. Find the value of $k$. [3 points]

Q22 4 marks Addition & Double Angle Formulae Geometric Configuration with Trigonometric Identities View

As shown in the figure, there is an equilateral triangle ABC and a circle O with diameter AC on a plane. Point D on segment BC is determined such that $\angle \mathrm { DAB } = \frac { \pi } { 15 }$. When point X moves on circle O, let P be the point where the dot product $\overrightarrow { \mathrm { AD } } \cdot \overrightarrow { \mathrm { CX } }$ of the two vectors $\overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { CX } }$ is minimized. If $\angle \mathrm { ACP } = \frac { q } { p } \pi$, find the value of $p + q$. (Note: $p$ and $q$ are coprime natural numbers.) [4 points]

Q23 Sequences and Series Evaluation of a Finite or Infinite Sum View

For a natural number $n \geq 2$, consider the set $$\left\{ 3 ^ { 2 k - 1 } \mid k \text { is a natural number, } 1 \leqq k \leqq n \right\}$$ Let $S$ be the set containing only all possible values obtained by multiplying two distinct elements of this set, and let $f ( n )$ be the number of elements in $S$. For example, $f ( 4 ) = 5$. Find the value of $\sum _ { n = 2 } ^ { 11 } f ( n )$.

Q24 4 marks Stationary points and optimisation Composite or piecewise function extremum analysis View

There is a quartic function $f ( x )$ with leading coefficient 1, $f ( 0 ) = 3$, and $f ^ { \prime } ( 3 ) < 0$. For a real number $t$, define the set $S$ as $$S = \{ a \mid \text{the function } | f ( x ) - t | \text{ is not differentiable at } x = a \}$$ and let $g ( t )$ be the number of elements in set $S$. When the function $g ( t )$ is discontinuous only at $t = 3$ and $t = 19$, find the value of $f ( - 2 )$. [4 points]

Q25 4 marks Sequences and Series Limit Evaluation Involving Sequences View

For a natural number $m$, there are blocks in the shape of identical cubes stacked with 1 block in column 1, 2 blocks in column 2, 3 blocks in column 3, $\cdots$, and $m$ blocks in column $m$. The following procedure is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove $\frac { 1 } { 2 }$ of the blocks in that column from the column.
After completing all block removal procedures, let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$. For example, $f ( 2 ) = 2 , f ( 3 ) = 5 , f ( 4 ) = 6$. $$\lim _ { n \rightarrow \infty } \frac { f \left( 2 ^ { n + 1 } \right) - f \left( 2 ^ { n } \right) } { f \left( 2 ^ { n + 2 } \right) } = \frac { q } { p }$$ Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]

Q26 (Calculus) 3 marks Reciprocal Trig & Identities View

When $\tan \frac { \theta } { 2 } = \frac { \sqrt { 2 } } { 2 }$, what is the value of $\sec \theta$? (where $0 < \theta < \frac { \pi } { 2 }$ ) [3 points]
(1) 3
(2) $\frac { 10 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) 4
(5) $\frac { 13 } { 3 }$

Q26 (Probability and Statistics) 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View

The probability mass function of a discrete random variable $X$ is $$\mathrm { P } ( X = x ) = \frac { a x + 2 } { 10 } ( x = - 1,0,1,2 )$$ What is the value of the variance $\mathrm { V } ( 3 X + 2 )$ of the random variable $3 X + 2$? (where $a$ is a constant.) [3 points]
(1) 9
(2) 18
(3) 27
(4) 36
(5) 45

Q26 (Discrete Mathematics) 3 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View

Among the partitions of the natural number 7, how many distinct partitions can be expressed as the sum of natural numbers not exceeding 3? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

Q27 (Calculus) 3 marks Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View

On the coordinate plane, what is the slope of the tangent line to the curve $y ^ { 3 } = \ln \left( 5 - x ^ { 2 } \right) + x y + 4$ at the point $( 2,2 )$? [3 points]
(1) $- \frac { 3 } { 5 }$
(2) $- \frac { 1 } { 2 }$
(3) $- \frac { 2 } { 5 }$
(4) $- \frac { 3 } { 10 }$
(5) $- \frac { 1 } { 5 }$

Q27 (Probability and Statistics) 3 marks Combinations & Selection Combinatorial Probability View

In a table tennis competition with 4 male table tennis players and 4 female table tennis players, when 2 people are randomly selected to form 4 teams, what is the probability that exactly 2 teams consist of 1 male and 1 female? [3 points]
(1) $\frac { 3 } { 7 }$
(2) $\frac { 18 } { 35 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 24 } { 35 }$
(5) $\frac { 27 } { 35 }$

Q28 (Calculus) 3 marks Integration by Substitution Determine J−I or Compare Related Integrals via Substitution View

There is a function $f ( x )$ that is differentiable on the set of all real numbers. For all real numbers $x$, $f ( 2 x ) = 2 f ( x ) f ^ { \prime } ( x )$, and $$f ( a ) = 0 , \quad \int _ { 2 a } ^ { 4 a } \frac { f ( x ) } { x } d x = k \quad ( a > 0,0 < k < 1 )$$ When this holds, what is the value of $\int _ { a } ^ { 2 a } \frac { \{ f ( x ) \} ^ { 2 } } { x ^ { 2 } } d x$ expressed in terms of $k$? [3 points]
(1) $\frac { k ^ { 2 } } { 4 }$
(2) $\frac { k ^ { 2 } } { 2 }$
(3) $k ^ { 2 }$
(4) $k$
(5) $2 k$

Q28 (Probability and Statistics) 3 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View

The daily production of employees at a certain company varies depending on their length of service. The daily production of an employee with $n$ months of service ( $1 \leqq n \leqq 100$ ) follows a normal distribution with mean $a n + 100$ ( $a$ is a constant) and standard deviation 12. When the probability that the daily production of an employee with 16 months of service is 84 or less is 0.0228, find the probability that the daily production of an employee with 36 months of service is at least 100 and at most 142 using the standard normal distribution table.

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
1.0	0.3413
1.5	0.4332
2.0	0.4772
2.5	0.4938

[3 points]
(1) 0.7745
(2) 0.8185
(3) 0.9104
(4) 0.9270
(5) 0.9710

Q29 (Calculus) 4 marks Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View

For all functions $f ( x )$ that are differentiable on the set of all real numbers and satisfy the following conditions, what is the minimum value of $\int _ { 0 } ^ { 2 } f ( x ) d x$? [4 points] (가) $f ( 0 ) = 1 , f ^ { \prime } ( 0 ) = 1$ (나) If $0 < a < b < 2$, then $f ^ { \prime } ( a ) \leqq f ^ { \prime } ( b )$. (다) On the interval $( 0,1 )$, $f ^ { \prime \prime } ( x ) = e ^ { x }$.
(1) $\frac { 1 } { 2 } e - 1$
(2) $\frac { 3 } { 2 } e - 1$
(3) $\frac { 5 } { 2 } e - 1$
(4) $\frac { 7 } { 2 } e - 2$
(5) $\frac { 9 } { 2 } e - 2$

Q29 (Probability and Statistics) 4 marks Measures of Location and Spread View

There are two datasets A and B. The mean and median of dataset A, which consists of 5 distinct numbers, are both 25. Dataset B consists of 7 numbers, where 5 of them match the data in A, and the remaining 2 are $x$ and $y$. Which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q30 (Calculus) 4 marks Parametric differentiation View

On the coordinate plane, as shown in the figure, for a point P on the circle $x ^ { 2 } + y ^ { 2 } = 1$, let $\theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$ be the angle that the line segment OP makes with the positive direction of the $x$-axis. Let Q be the point where the line passing through P and parallel to the $x$-axis meets the curve $y = e ^ { x } - 1$, and let R be the foot of the perpendicular from Q to the $x$-axis. Let T be the intersection point of the line segment OP and the line segment QR, and let $S ( \theta )$ be the area of triangle ORT. When $\lim _ { \theta \rightarrow + 0 } \frac { S ( \theta ) } { \theta ^ { 3 } } = a$, find the value of $60 a$. [4 points]

Q30 (Probability and Statistics) 4 marks Confidence intervals Determine minimum sample size for a desired interval width View

We want to investigate the proportion of antibody possession for a specific disease among Korean adults. Let $p$ be the proportion of antibody possession in the population, and let $\hat { p }$ be the proportion of antibody possession in a sample of $n$ people randomly selected from the population. Find the minimum value of $n$ such that the probability that $| \hat { p } - p | \leq 0.16 \sqrt { \hat { p } ( 1 - \hat { p } ) }$ is at least 0.9544. (where $Z$ is a random variable following the standard normal distribution and $\mathrm { P } ( 0 \leq Z \leq 2 ) = 0.4772$.) [4 points]

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2011 csat__math-science