csat-suneung

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

What is the value of $27 ^ { \frac { 1 } { 3 } } + \log _ { 2 } 4$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Q2 2 marks Matrices Matrix Algebra and Product Properties View

For two matrices $A = \left( \begin{array} { l l } 3 & 0 \\ 0 & 3 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & 1 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of the matrix $A B + 2 B$? [2 points]
(1) 10
(2) 8
(3) 6
(4) 4
(5) 2

Q3 2 marks Chain Rule Continuity Conditions via Composition View

For two constants $a , b$, if $\lim _ { x \rightarrow 3 } \frac { \sqrt { x + a } - b } { x - 3 } = \frac { 1 } { 4 }$, what is the value of $a + b$? [2 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11

Q4 3 marks Conic sections Tangent and Normal Line Problems View

Let Q be the point where the tangent line at point $\mathrm { P } ( a , b )$ on the parabola $y ^ { 2 } = 4 x$ meets the $x$-axis. When $\overline { \mathrm { PQ } } = 4 \sqrt { 5 }$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) 21
(2) 32
(3) 45
(4) 60
(5) 77

Q5 3 marks Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View

On plane $\alpha$, there is a right isosceles triangle ABC with $\angle \mathrm { A } = 90 ^ { \circ }$ and $\overline { \mathrm { BC } } = 6$. A point P outside plane $\alpha$ is at a distance of 4 from the plane, and the foot of the perpendicular from P to plane $\alpha$ is point A. What is the distance from point P to line BC? [3 points]
(1) $3 \sqrt { 2 }$
(2) 5
(3) $3 \sqrt { 3 }$
(4) $4 \sqrt { 2 }$
(5) 6

Q6 3 marks Conditional Probability Bayes' Theorem with Production/Source Identification View

10\% of the emails Chulsu received contain the word ``travel''. 50\% of emails containing ``travel'' are advertisements, and 20\% of emails not containing ``travel'' are advertisements. Given that an email Chulsu received is an advertisement, what is the probability that it contains the word ``travel''? [3 points]
(1) $\frac { 5 } { 23 }$
(2) $\frac { 6 } { 23 }$
(3) $\frac { 7 } { 23 }$
(4) $\frac { 8 } { 23 }$
(5) $\frac { 9 } { 23 }$

Q7 3 marks Permutations & Arrangements Selection and Task Assignment View

A company employee has 6 tasks to handle in total, including A and B. The employee wants to handle 4 tasks including A and B today, and A must be handled before B. What is the number of ways to select the tasks to handle today and determine the order of handling the selected tasks? [3 points]
(1) 60
(2) 66
(3) 72
(4) 78
(5) 84

Q8 3 marks Solving quadratics and applications Counting solutions or configurations satisfying a quadratic system View

For a real number $a$, let $f ( a )$ be the number of elements in the set $$\left\{ x \mid a x ^ { 2 } + 2 ( a - 2 ) x - ( a - 2 ) = 0 , x \text { is a real number } \right\}$$ Which of the following statements in are correct? [3 points]
Remarks ᄀ. $\lim _ { a \rightarrow 0 } f ( a ) = f ( 0 )$ ㄴ. There are 2 real numbers $c$ such that $\lim _ { a \rightarrow c + 0 } f ( a ) \neq \lim _ { a \rightarrow c - 0 } f ( a )$. ㄷ. The function $f ( a )$ is discontinuous at 3 points.
(1) ᄂ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

Q9 4 marks Normal Distribution Process Capability or Quality Compliance Assessment View

The internal pressure strength of bottles produced at a certain factory follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$, and bottles with internal pressure strength less than 40 are classified as defective. The process capability index $G$ for evaluating the process capability of this factory is calculated as $$G = \frac { m - 40 } { 3 \sigma }$$ When $G = 0.8$, what is the probability that a randomly selected bottle is defective, using the standard normal distribution table below? [4 points]

$z$	$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.2	0.4861
2.3	0.4893
2.4	0.4918
2.5	0.4938

(1) 0.0139
(2) 0.0107
(3) 0.0082
(4) 0.0062
(5) 0.0038

Q10 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View

Shellfish filter suspended matter. When the water temperature is $t \left( { } ^ { \circ } \mathrm { C } \right)$ and the individual weight is $w ( \mathrm {~g} )$, the amounts filtered in 1 hour by shellfish A and B (in L) are $Q _ { \mathrm { A } }$ and $Q _ { \mathrm { B } }$ respectively, and the following relational equations hold. $$\begin{aligned} Q _ { \mathrm { A } } & = 0.01 t ^ { 1.25 } w ^ { 0.25 } \\ Q _ { \mathrm { B } } & = 0.05 t ^ { 0.75 } w ^ { 0.30 } \end{aligned}$$ When the water temperature is $20 { } ^ { \circ } \mathrm { C }$ and the individual weights of shellfish A and B are each 8 g, the value of $\frac { Q _ { \mathrm { A } } } { Q _ { \mathrm { B } } }$ is $2 ^ { a } \times 5 ^ { b }$. What is the value of $a + b$? (Here, $a$ and $b$ are rational numbers.) [3 points]
(1) 0.15
(2) 0.35
(3) 0.55
(4) 0.75
(5) 0.95

Q11 4 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View

As shown in the figure, the graph of the cubic function $y = f ( x )$ is tangent to the $x$-axis at point $\mathrm { P } ( 2,0 )$ and meets the graph of the linear function $y = g ( x )$ only at point P. When $1 < f ( 0 ) < g ( 0 )$, what is the number of real roots of the equation $$f ( x ) + g ( x ) = \frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) }$$ ? [4 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

Q12 3 marks Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

The following is a proof by mathematical induction that the equality $$\sum _ { k = 0 } ^ { n } \frac { { } _ { n } \mathrm { C } _ { k } } { { } _ { n + 4 } \mathrm { C } _ { k } } = \frac { n + 5 } { 5 }$$ holds for all natural numbers $n$.

(1) When $n = 1$, $$( \text { Left side } ) = \frac { { } _ { 1 } \mathrm { C } _ { 0 } } { { } _ { 5 } \mathrm { C } _ { 0 } } + \frac { { } _ { 1 } \mathrm { C } _ { 1 } } { { } _ { 5 } \mathrm { C } _ { 1 } } = \frac { 6 } { 5 } , ( \text { Right side } ) = \frac { 1 + 5 } { 5 } = \frac { 6 } { 5 }$$ so the given equality holds.
(2) Assume that when $n = m$, the equality $$\sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } } = \frac { m + 5 } { 5 }$$ holds. When $n = m + 1$, $$\sum _ { k = 0 } ^ { m + 1 } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k } } = \text { (가) } + \sum _ { k = 0 } ^ { m } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k + 1 } }$$ For a natural number $l$, $${ } _ { l + 1 } \mathrm { C } _ { k + 1 } = \text { (나) } \cdot { } _ { l } \mathrm { C } _ { k } \quad ( 0 \leqq k \leqq l )$$ so $$\sum _ { k = 0 } ^ { m } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k + 1 } } = \text { (다) } \cdot \sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } }$$ Therefore, $$\begin{aligned} \sum _ { k = 0 } ^ { m + 1 } \frac { m + 1 } { { } _ { m + 5 } \mathrm { C } _ { k } } & = \text { (가) } + \text { (다) } \cdot \sum _ { k = 0 } ^ { m } \frac { { } _ { m } \mathrm { C } _ { k } } { { } _ { m + 4 } \mathrm { C } _ { k } } \\ & = \frac { m + 6 } { 5 } \end{aligned}$$ Thus, the given equality holds for all natural numbers $n$.
Which of the following are correct for (가), (나), and (다)? [3 points]

(가)	(나)	(다)
(1) 1	$\frac { l + 2 } { k + 2 }$	$\frac { m + 1 } { m + 4 }$
(2) 1	$\frac { l + 1 } { k + 1 }$	$\frac { m + 1 } { m + 5 }$
(3) 1	$\frac { l + 1 } { k + 1 }$	$\frac { m + 1 } { m + 4 }$
(4) $m + 1$	$\frac { l + 1 } { k + 1 }$	$\frac { m + 1 } { m + 5 }$
(5) $m + 1$	$\frac { l + 2 } { k + 2 }$	$\frac { m + 1 } { m + 4 }$

Q13 4 marks Matrices Matrix Algebra and Product Properties View

For a $2 \times 2$ matrix $A$ and matrix $B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$ satisfying $( B A ) ^ { 2 } = \left( \begin{array} { l l } 1 & 1 \\ 1 & 2 \end{array} \right)$, what is the matrix $( A B ) ^ { 2 }$? [4 points]
(1) $\left( \begin{array} { l l } 1 & 1 \\ 1 & 2 \end{array} \right)$
(2) $\left( \begin{array} { l l } 2 & 1 \\ 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right)$
(4) $\left( \begin{array} { l l } 1 & 2 \\ 2 & 1 \end{array} \right)$
(5) $\left( \begin{array} { l l } 1 & 1 \\ 2 & 1 \end{array} \right)$

Q14 4 marks Vectors Introduction & 2D True/False or Multiple-Statement Verification View

In the plane, the pentagon ABCDE satisfies $$\overline { \mathrm { AB } } = \overline { \mathrm { BC } } , \overline { \mathrm { AE } } = \overline { \mathrm { ED } } , \angle \mathrm {~B} = \angle \mathrm { E } = 90 ^ { \circ }$$ Which of the following statements in are correct? [4 points]
Remarks ㄱ. For the midpoint M of segment BE, $\overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { AE } }$ and $\overrightarrow { \mathrm { AM } }$ are parallel to each other. ㄴ. $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AE } } = - \overrightarrow { \mathrm { BC } } \cdot \overrightarrow { \mathrm { ED } }$ ㄷ. $| \overrightarrow { \mathrm { BC } } + \overrightarrow { \mathrm { ED } } | = | \overrightarrow { \mathrm { BE } } |$
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

Q15 4 marks Circles Area and Geometric Measurement Involving Circles View

As shown in the figure, draw a circle $\mathrm { O } _ { 1 }$ centered at the origin with radius 3, and let the four points where circle $\mathrm { O } _ { 1 }$ meets the coordinate axes be $\mathrm { A } _ { 1 } ( 0,3 )$, $\mathrm { B } _ { 1 } ( - 3,0 ) , \mathrm { C } _ { 1 } ( 0 , - 3 ) , \mathrm { D } _ { 1 } ( 3,0 )$ respectively. Two circles passing through both points $\mathrm { B } _ { 1 } , \mathrm { D } _ { 1 }$ and centered at points $\mathrm { A } _ { 1 } , \mathrm { C } _ { 1 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 1 }$ at points $\mathrm { C } _ { 2 } , \mathrm {~A} _ { 2 }$ respectively.
Let $S _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm {~A} _ { 2 } \mathrm { D } _ { 1 }$, and let $T _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 2 } \mathrm { D } _ { 1 }$.
Draw circle $\mathrm { O } _ { 2 }$ with segment $\mathrm { A } _ { 2 } \mathrm { C } _ { 2 }$ as diameter, and let the two points where circle $\mathrm { O } _ { 2 }$ meets the $x$-axis be $\mathrm { B } _ { 2 } , \mathrm { D } _ { 2 }$ respectively. Two circles passing through both points $\mathrm { B } _ { 2 } , \mathrm { D } _ { 2 }$ and centered at points $\mathrm { A } _ { 2 } , \mathrm { C } _ { 2 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 2 }$ at points $\mathrm { C } _ { 3 } , \mathrm {~A} _ { 3 }$ respectively.
Let $S _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm {~A} _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm {~A} _ { 3 } \mathrm { D } _ { 2 }$, and let $T _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 3 } \mathrm { D } _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm {~A} _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm {~A} _ { n + 1 } \mathrm { D } _ { n }$ obtained in the $n$-th step, and let $T _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { C } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { C } _ { n + 1 } \mathrm { D } _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } \left( S _ { n } + T _ { n } \right)$? [4 points]
(1) $6 ( \sqrt { 2 } + 1 )$
(2) $6 ( \sqrt { 3 } + 1 )$
(3) $6 ( \sqrt { 5 } + 1 )$
(4) $9 ( \sqrt { 2 } + 1 )$
(5) $9 ( \sqrt { 3 } + 1 )$

Q16 4 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View

For a natural number $n ( n \geqq 2 )$, let $a _ { n }$ and $b _ { n } \left( a _ { n } < b _ { n } \right)$ be the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$
(1) ᄀ
(2) ᄂ
(3) ᄃ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ

Q17 4 marks Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View

For a quartic function $f ( x )$ with leading coefficient 1, the function $g ( x )$ satisfies the following conditions. (가) When $- 1 \leqq x < 1$, $g ( x ) = f ( x )$. (나) For all real numbers $x$, $g ( x + 2 ) = g ( x )$.
Which of the following statements in are correct? [4 points]
Remarks ㄱ. If $f ( - 1 ) = f ( 1 )$ and $f ^ { \prime } ( - 1 ) = f ^ { \prime } ( 1 )$, then $g ( x )$ is differentiable on the entire set of real numbers. ㄴ. If $g ( x )$ is differentiable on the entire set of real numbers, then $f ^ { \prime } ( 0 ) f ^ { \prime } ( 1 ) < 0$. ㄷ. If $g ( x )$ is differentiable on the entire set of real numbers and $f ^ { \prime } ( 1 ) > 0$, then there exists $c$ in the interval $( - \infty , - 1 )$ such that $f ^ { \prime } ( c ) = 0$.
(1) ᄀ
(2) ᄂ
(3) ᄀ, ᄃ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

Q18 3 marks Product & Quotient Rules View

For the function $f ( x ) = \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + x - 2 \right)$, find the value of $f ^ { \prime } ( 2 )$. [3 points]

Q19 3 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View

Find the product of all real roots of the irrational equation $\sqrt { x ^ { 2 } - 7 x + 15 } = x ^ { 2 } - 7 x + 9$. [3 points]

Q20 3 marks Vectors: Lines & Planes Find Cartesian Equation of a Plane View

In coordinate space, the equation of the plane that is perpendicular to the line $\frac { x - 2 } { 2 } = \frac { y - 2 } { 3 } = z - 1$ and passes through the point $( 1 , - 5,2 )$ is $2 x + a y + b z + c = 0$. Find the value of $a + b + c$. [3 points]

Q21 4 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

There is a function $f ( x ) = x ^ { 2 } + a x + b \quad ( a \geqq 0 , b > 0 )$. For a natural number $n \geq 2$, divide the closed interval $[ 0,1 ]$ into $n$ equal parts, and let the division points (including both endpoints) be $$0 = x _ { 0 } , x _ { 1 } , x _ { 2 } , \cdots , x _ { n - 1 } , x _ { n } = 1$$ respectively. Let $A _ { k }$ be the area of the rectangle with base $\left[ x _ { k - 1 } , x _ { k } \right]$ and height $f \left( x _ { k } \right)$. $( k = 1,2 , \cdots , n )$
Given that the sum of the areas of the two rectangles at the ends is $$A _ { 1 } + A _ { n } = \frac { 7 n ^ { 2 } + 1 } { n ^ { 3 } }$$ find the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 8 k } { n } A _ { k }$. [4 points]

Q22 Differential equations Qualitative Analysis of DE Solutions View

For a natural number $n$, point $\mathrm { A } _ { n }$ is on the $x$-axis. Point $\mathrm { A } _ { n + 1 }$ is determined according to the following rule. (가) The coordinates of point $\mathrm { A } _ { 1 }$ are $( 2,0 )$. (나) (1) Let $\mathrm { P } _ { n }$ be the point where the line passing through point $\mathrm { A } _ { n }$ and parallel to the $y$-axis meets the curve $y = \frac { 1 } { x } ( x > 0 )$.
(2) Let $\mathrm { Q } _ { n }$ be the point obtained by reflecting $\mathrm { P } _ { n }$ about the line $y = x$.
(3) Let $\mathrm { R } _ { n }$ be the point where the line passing through $\mathrm { Q } _ { n }$ and parallel to the $y$-axis meets the $x$-axis.

Q26 3 marks Addition & Double Angle Formulae Direct Double Angle Evaluation View

[Calculus] When $\tan \theta = - \sqrt { 2 }$, what is the value of $\sin \theta \tan 2 \theta$? (where $\frac { \pi } { 2 } < \theta < \pi$ ) [3 points]
(1) $\frac { 2 \sqrt { 3 } } { 3 }$
(2) $\sqrt { 3 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 3 }$
(5) $2 \sqrt { 3 }$

Q26b 3 marks Measures of Location and Spread View

[Probability and Statistics] Which of the following is the correct shape of the graph of the function representing the median according to the value of $x$ in the following data? (where $x \geqq 0$ ) [3 points] $10 , \quad 28 , \quad x , \quad 20 , \quad 8 , \quad 2 , \quad 25 , \quad 7 , \quad 17$

Q26c 3 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View

[Discrete Mathematics] A sequence $\left\{ a _ { n } \right\}$ satisfies $$\left\{ \begin{array} { l } a _ { 1 } = 2 , a _ { 2 } = 5 \\ a _ { n } = 2 a _ { n - 1 } + a _ { n - 2 } \end{array} \quad ( n \geqq 3 ) \right.$$ What is the value of $a _ { 5 }$? [3 points]
(1) 70
(2) 72
(3) 74
(4) 76
(5) 78

Q27 3 marks Chain Rule Geometric Limit with Parametric Chain Rule View

[Calculus] As shown in the figure, let Q be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point P on the circle meets the $x$-axis. For point $\mathrm { A } ( - 1,0 )$ and the origin O, let $\angle \mathrm { PAO } = \theta$. Find the value of $\lim _ { \theta \rightarrow \frac { \pi } { 4 } - 0 } \frac { \overline { \mathrm { PQ } } - \overline { \mathrm { OQ } } } { \theta - \frac { \pi } { 4 } }$. (where point P is in the first quadrant) [3 points]
(1) 2
(2) $\sqrt { 3 }$
(3) $\frac { 3 } { 2 }$
(4) 1
(5) $\frac { \sqrt { 2 } } { 2 }$

Q27b 3 marks Binomial Distribution Compute Expectation, Variance, or Standard Deviation View

[Probability and Statistics] A certain math class has 10 groups, each consisting of 3 male students and 2 female students. When 2 people are randomly selected from each group, let $X$ be the random variable representing the number of groups in which only male students are selected. What is the expected value $\mathrm { E } ( X )$ of $X$? (Note: No student belongs to more than one group.) [3 points]
(1) 6
(2) 5
(3) 4
(4) 3
(5) 2

Q28 3 marks Tangents, normals and gradients Common tangent line to two curves View

[Calculus] The tangent line to the curve $y = e ^ { x }$ at the point $( 1 , e )$ is tangent to the curve $y = 2 \sqrt { x - k }$. What is the value of the real number $k$? [3 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e ^ { 4 } }$
(4) $\frac { 1 } { 1 + e }$
(5) $\frac { 1 } { 1 + e ^ { 2 } }$

Q28b 3 marks Conditional Probability Bayes' Theorem with Production/Source Identification View

[Probability and Statistics] A training facility has three courses A, B, and C to be experienced in order, with the entrance and exit being the same. There are 30 envelopes at course A, 60 envelopes at course B, and 90 envelopes at course C. Each envelope contains 1, 2, or 3 coupons. The following table shows the number of envelopes by the number of coupons for each course.

\multicolumn{1}{\|c\|}{Number of Coupons}	1	2	3	Total
A	20	10	0	30
B	30	20	10	60
C	40	30	20	90

After completing each course, a student randomly selects one envelope from that course and receives the coupons inside. A student who started first completed all three courses and received a total of 4 coupons. What is the probability that the student received 2 coupons at course B? [3 points]
(1) $\frac { 6 } { 23 }$
(2) $\frac { 8 } { 23 }$
(3) $\frac { 10 } { 23 }$
(4) $\frac { 12 } { 23 }$
(5) $\frac { 14 } { 23 }$

Q29 4 marks Integration by Parts Prove an Integral Identity or Equality View

[Calculus] For two functions $f ( x )$ and $g ( x )$ that have second derivatives on the set of all real numbers, consider the definite integral $$\int _ { 0 } ^ { 1 } \left\{ f ^ { \prime } ( x ) g ( 1 - x ) - g ^ { \prime } ( x ) f ( 1 - x ) \right\} d x$$ Let the value of this integral be $k$. Which of the following statements in are correct? [4 points]
ㄱ. $\int _ { 0 } ^ { 1 } \left\{ f ( x ) g ^ { \prime } ( 1 - x ) - g ( x ) f ^ { \prime } ( 1 - x ) \right\} d x = - k$ ㄴ. If $f ( 0 ) = f ( 1 )$ and $g ( 0 ) = g ( 1 )$, then $k = 0$. ㄷ. If $f ( x ) = \ln \left( 1 + x ^ { 4 } \right)$ and $g ( x ) = \sin \pi x$, then $k = 0$.
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

Q29b 4 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View

[Probability and Statistics] One method to determine whether a certain bone fossil belongs to animal A or animal B is to use the length of a specific part. The length of this part in animal A follows a normal distribution $\mathrm { N } \left( 10,0.4 ^ { 2 } \right)$, and the length of this part in animal B follows a normal distribution $\mathrm { N } \left( 12,0.6 ^ { 2 } \right)$. If the length of this part is less than $d$, it is judged to be a fossil of animal A, and if it is at least $d$, it is judged to be a fossil of animal B. Find the value of $d$ such that the probability of judging an animal A fossil as an animal A fossil equals the probability of judging an animal B fossil as an animal B fossil. (The unit of length is cm.) [4 points]
(1) 10.4
(2) 10.5
(3) 10.6
(4) 10.7
(5) 10.8

Q30 4 marks Parametric integration View

[Calculus] The position $( x , y )$ of a point P moving on the coordinate plane at time $t$ is given by $$\left\{ \begin{array} { l } x = 4 ( \cos t + \sin t ) \\ y = \cos 2 t \end{array} \quad ( 0 \leqq t \leqq 2 \pi ) \right.$$ When the distance traveled by point P from $t = 0$ to $t = 2 \pi$ is $a \pi$, find the value of $a ^ { 2 }$. [4 points]

Q30b 4 marks Confidence intervals Determine minimum sample size for a desired interval width View

[Probability and Statistics] A survey of 100 randomly selected people from city A regarding the safest mode of transportation found that 20 people chose express buses. Using this result, a 95\% confidence interval for the proportion of people who chose express buses was found to be $[ a , b ]$. For city B, a 95\% confidence interval is to be constructed for the proportion of people who think express buses are the safest mode of transportation based on a random sample of $n$ people. Find the minimum value of $n$ such that the maximum allowable sampling error of this confidence interval is at most $\frac { b - a } { 2 }$. [4 points]

csat-suneung

Papers (39)

2010 csat__math-science