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 3 marks Exponential Equations & Modelling Exponential Inequality Solving View

What is the sum of all natural numbers $x$ that satisfy the exponential inequality $\left( 3 ^ { x } - 5 \right) \left( 3 ^ { x } - 100 \right) < 0$? [3 points]
(1) 5
(2) 7
(3) 9
(4) 11
(5) 13

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 2 marks Geometric Sequences and Series Determine the Limit of a Sequence via Geometric Series View

When $\lim _ { n \rightarrow \infty } \frac { a \times 6 ^ { n + 1 } - 5 ^ { n } } { 6 ^ { n } + 5 ^ { n } } = 4$, what is the value of the constant $a$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 3 } { 2 }$

Q5 3 marks Independent Events View

Two events $A$ and $B$ are mutually independent, and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \mathrm { P } ( A \cap B ) = \mathrm { P } ( A ) - \mathrm { P } ( B )$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { 2 } { 5 }$
(5) $\frac { 1 } { 2 }$

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 outcomes are there? (Note: banners of the same type are not distinguished from each other.) [3 points]
(a) Banner A must be installed.
(b) Banner B is 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

A student named Chulsu participated in a 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 audience voting scores and the event of receiving judge scores are mutually 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 Voting	40	30	20
Judges	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 Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View

The probability distribution table of the random variable $X$ is as follows.

$X$	- 1	0	1	2	Total
$\mathrm { P } ( X = x )$	$\frac { 3 - a } { 8 }$	$\frac { 1 } { 8 }$	$\frac { 3 + a } { 8 }$	$\frac { 1 } { 8 }$	1

When $\mathrm { P } ( 0 \leqq X \leqq 2 ) = \frac { 7 } { 8 }$, what is the value of the expected value $\mathrm { E } ( X )$ of the random variable $X$? [3 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 3 } { 8 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 5 } { 8 }$
(5) $\frac { 3 } { 4 }$

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

To determine the relative density of soil, a test device is inserted into the soil for investigation. 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. $$D = - 98 + 66 \log \frac { R } { \sqrt { S } }$$ (Here, 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 Radians, Arc Length and Sector Area View

There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1$ and $\overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ are determined 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 $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$ such that two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ are determined as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ are determined 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, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$ for the obtained $S _ { n }$? [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 3 marks Exponential Functions Graph Transformations and Symmetry View

On the coordinate plane, the graph of the exponential function $y = a ^ { x }$ is reflected about the $y$-axis, then translated 3 units in the $x$-direction and 2 units in the $y$-direction. The resulting graph passes through the point $( 1,4 )$. What is the value of the positive number $a$? [3 points]
(1) $\sqrt { 2 }$
(2) 2
(3) $2 \sqrt { 2 }$
(4) 4
(5) $4 \sqrt { 2 }$

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

Sets $S$ and $T$ with $1 \times 2$ matrices and $2 \times 1$ matrices as elements, respectively, are as follows. $$S = \{ ( a \; b ) \mid a + b \neq 0 \} , \quad T = \left\{ \left. \binom { p } { q } \right\rvert \, p q \neq 0 \right\}$$ For an element $A$ of set $S$, which of the following statements in are correct? [4 points]
ㄱ. For an element $P$ of set $T$, $PA$ does not have an inverse matrix. ㄴ. For an element $B$ of set $S$ and an 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 a customer's home to a traditional market follows a normal distribution with mean 1740 m and standard deviation 500 m. Among customers whose distance from home to market is 2000 m or more, 15\% use private vehicles to come to the market, and among customers whose distance is less than 2000 m, 5\% use private vehicles. When one customer who came to the market using a private vehicle is randomly selected, what is the probability that the distance from this customer's home to the market is less than 2000 m? (Here, when $Z$ is a random variable following the standard normal distribution, use $\mathrm { P } ( 0 \leqq Z \leqq 0.52 ) = 0.2$ for calculation.) [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 Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

On the coordinate plane, for a natural number $n$, let $\mathrm { A } _ { n }$ be the point where the two lines $y = \frac { 1 } { n } x$ and $x = n$ meet, and let $\mathrm { B } _ { n }$ be the point where the line $x = n$ and the $x$-axis meet. Let $\mathrm { C } _ { n }$ be the center of the circle inscribed in triangle $\mathrm { A } _ { n } \mathrm { OB } _ { n }$, and let $S _ { n }$ be the area of triangle $\mathrm { A } _ { n } \mathrm { OC } _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } } { n }$? [4 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

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 $$b _ { n + 1 } = b _ { n } + ( \text{(a)} )$$ The general term of the sequence $\left\{ b _ { n } \right\}$ is $b _ { n } =$ (b) so $\frac { a _ { 1 } a _ { 2 } \cdots a _ { n } } { n ! } =$ (b). $\vdots$ Therefore, $a _ { 1 } = 1$ and $a _ { n } = \frac { ( n - 1 ) ( 2 n - 1 ) } { 2 n - 3 }$ for $n \geqq 2$.
When the expression that fits (a) is $f ( n )$ and the expression that fits (b) 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

On the coordinate plane, the two points where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet are $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and the point where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet is $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following statements in are correct? [4 points]
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 Combinations & Selection Combinatorial Probability View

There are 2 students each from Korea, China, and Japan. When these 6 students each randomly select and sit in one of 6 seats with assigned seat numbers as shown in the figure, what is the probability that the two students from the same country sit such that the difference in their seat numbers is 1 or 10? [4 points]

11

12

13

21

22

23

(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$

Q18 3 marks Permutations & Arrangements Factorial and Combinatorial Expression Simplification View

Find the natural number $n$ that satisfies the equation $2 \times {}_{n}\mathrm{C}_{3} = 3 \times {}_{n}\mathrm{P}_{2}$. [3 points]

Q19 3 marks Laws of Logarithms Solve a Logarithmic Equation View

When $\alpha$ is the root of the logarithmic equation $\log _ { 3 } ( x - 4 ) = \log _ { 9 } ( 5 x + 4 )$, find the value of $\alpha$. [3 points]

Q20 3 marks Combinations & Selection Partitioning into Teams or Groups View

When 6 different balls are placed 3 each in two baskets A and B, how many possible outcomes are there? [3 points]

Q21 3 marks Binomial Distribution Compute Expectation, Variance, or Standard Deviation View

When the trial of simultaneously tossing 2 coins is repeated 10 times, let $X$ be the random variable representing the number of times both coins show heads. Find the variance $\mathrm { V } ( 4 X + 1 )$ of the random variable $4 X + 1$. [3 points]

Q22 4 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with nonzero common difference, the three terms $a _ { 2 } , a _ { 4 } , a _ { 9 }$ form a geometric sequence with common ratio $r$ in this order. Find the value of $6r$. [4 points]

Q23 4 marks 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 $\sum _ { n = 2 } ^ { 11 } f ( n )$. [4 points]

Q24 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View

For a natural number $A$, let the characteristic of $\log A$ be $n$ and the mantissa be $\alpha$. Find the number of values of $A$ such that $n \leqq 2\alpha$ holds. (Given: $3.1 < \sqrt { 10 } < 3.2$) [4 points]

Q25 4 marks Sequences and Series Limit Evaluation Involving Sequences View

For a natural number $m$, blocks in the shape of identical cubes are 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 trial is repeated until there are no columns with an even number of blocks remaining.
For each column with an even number of blocks, remove from that column a number of blocks equal to $\frac { 1 } { 2 }$ of the number of blocks in that column.
Let $f ( m )$ be the sum of the number of blocks remaining in columns 1 through $m$ after all block removal trials are completed. 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 $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

Q26 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

The sequence $\left\{ a _ { n } \right\}$ satisfies $$2 a _ { n + 1 } = a _ { n } + a _ { n + 2 }$$ for all natural numbers $n$. When $a _ { 2 } = - 1 , a _ { 3 } = 2$, what is the sum of the first 10 terms of the sequence $\left\{ a _ { n } \right\}$? [3 points]
(1) 95
(2) 90
(3) 85
(4) 80
(5) 75

Q27 3 marks Normal Distribution Sampling Distribution of the Mean View

The duration of one use of a public bicycle in a certain city follows a normal distribution with mean 60 minutes and standard deviation 10 minutes. When 25 uses of the public bicycle are randomly sampled and surveyed, what is the probability that the total duration of the 25 uses is 1450 minutes or more, using the standard normal distribution table on the right? [3 points]

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

(1) 0.8351
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938

Q28 4 marks Matrices Linear System and Inverse Existence View

A certain company converts the raw scores of applicants' reasoning ability test and spatial perception ability test for use. When the raw score of the reasoning ability test is $x$ and the raw score of the spatial perception ability test is $y$, the two converted scores $p$ and $q$ are as follows. $$\binom { p } { q } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 3 \end{array} \right) \binom { x } { y }$$ When the converted scores of applicants A, B, and C are as shown in the table, let the raw scores of applicants A, B, and C on the reasoning ability test be $a , b , c$, respectively. Which of the following correctly represents the order of magnitude of $a , b , c$? [4 points]

Converted Score / Test Taker	A	B	C
$p$	45	50	45
$q$	40	50	50

(1) $a > b > c$
(2) $a > c > b$
(3) $b > a > c$
(4) $b > c > a$
(5) $c > b > a$

Q29 4 marks Matrices Matrix Power Computation and Application View

For a $2 \times 2$ square matrix $A$, the $(i,j)$ component $a_{ij}$ is $$a_{ij} = i - j \quad (i = 1,2,\ j = 1,2)$$ What is the $(2,1)$ component of the matrix $A + A^2 + A^3 + \cdots + A^{2010}$? [4 points]
(1) $-2010$
(2) $-1$
(3) $0$
(4) $1$
(5) $2010$

Q30 4 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

The sequence $\{a_n\}$ satisfies the following for all natural numbers $n$: $$\sum_{k=1}^{n} a_k = \log \frac{(n+1)(n+2)}{2}$$ Let $\sum_{k=1}^{20} a_{2k} = p$. Find the value of $10^p$. [4 points]

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