csat-suneung

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

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

Q2 3 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View

For an arithmetic sequence $\left\{ a _ { n } \right\}$, $$a _ { 2 } = 6 , \quad a _ { 4 } + a _ { 6 } = 36$$ what is the value of $a _ { 10 }$? [3 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

Q3 2 marks Chain Rule Straightforward Polynomial or Basic Differentiation View

For the function $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + x - 1$, what is the value of $f ^ { \prime } ( 1 )$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

Q4 3 marks Curve Sketching Limit Reading from Graph View

The graph of the function $y = f ( x )$ is shown in the figure.
What is the value of $\lim _ { x \rightarrow - 1 - } f ( x ) + \lim _ { x \rightarrow 2 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Q5 3 marks Sequences and series, recurrence and convergence Summation of sequence terms View

For a sequence $\left\{ a _ { n } \right\}$ with first term 1, satisfying for all natural numbers $n$: $$a _ { n + 1 } = \begin{cases} 2 a _ { n } & \left( a _ { n } < 7 \right) \\ a _ { n } - 7 & \left( a _ { n } \geq 7 \right) \end{cases}$$ what is the value of $\sum _ { k = 1 } ^ { 8 } a _ { k }$? [3 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

Q6 3 marks Stationary points and optimisation Count or characterize roots using extremum values View

How many integers $k$ are there such that the equation $2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + k = 0$ has three distinct real roots? [3 points]
(1) 20
(2) 23
(3) 26
(4) 29
(5) 32

Q7 3 marks Reciprocal Trig & Identities View

For $\theta$ satisfying $\pi < \theta < \frac { 3 } { 2 } \pi$ and $\tan \theta - \frac { 6 } { \tan \theta } = 1$, what is the value of $\sin \theta + \cos \theta$? [3 points]
(1) $- \frac { 2 \sqrt { 10 } } { 5 }$
(2) $- \frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $\frac { \sqrt { 10 } } { 5 }$
(5) $\frac { 2 \sqrt { 10 } } { 5 }$

Q8 3 marks Areas by integration View

When the line $x = k$ bisects the area enclosed by the curve $y = x ^ { 2 } - 5 x$ and the line $y = x$, what is the value of the constant $k$? [3 points]
(1) 3
(2) $\frac { 13 } { 4 }$
(3) $\frac { 7 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 4

Q9 4 marks Exponential Equations & Modelling Geometric Properties of Exponential/Logarithmic Curves View

The line $y = 2 x + k$ meets the graphs of the two functions $$y = \left( \frac { 2 } { 3 } \right) ^ { x + 3 } + 1 , \quad y = \left( \frac { 2 } { 3 } \right) ^ { x + 1 } + \frac { 8 } { 3 }$$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When $\overline { \mathrm { PQ } } = \sqrt { 5 }$, what is the value of the constant $k$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 16 } { 3 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 17 } { 3 }$
(5) $\frac { 35 } { 6 }$

Q10 4 marks Tangents, normals and gradients Determine unknown parameters from tangent conditions View

For a cubic function $f ( x )$, the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = x f ( x )$ at the point $( 1,2 )$ coincide. What is the value of $f ^ { \prime } ( 2 )$? [4 points]
(1) $-18$
(2) $-17$
(3) $-16$
(4) $-15$
(5) $-14$

Q11 4 marks Standard trigonometric equations Geometric problem using trigonometric function graphs View

For a positive number $a$, there is a function $$f ( x ) = \tan \frac { \pi x } { a }$$ defined on the set $\left\{ x \left\lvert \, - \frac { a } { 2 } < x \leq a \right. , x \neq \frac { a } { 2 } \right\}$. As shown in the figure, there is a line passing through three points $\mathrm { O } , \mathrm { A } , \mathrm { B }$ on the graph of $y = f ( x )$. Let $\mathrm { C }$ be the point other than $\mathrm { A }$ where the line parallel to the $x$-axis passing through point $\mathrm { A }$ meets the graph of $y = f ( x )$. When triangle $\mathrm { ABC }$ is equilateral, what is the area of triangle $\mathrm { ABC }$? (Here, $\mathrm { O }$ is the origin.) [4 points]
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 17 \sqrt { 3 } } { 12 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 4 }$
(5) $\frac { 7 \sqrt { 3 } } { 6 }$

Q12 4 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View

A function $f ( x )$ continuous on the entire set of real numbers satisfies $$\{ f ( x ) \} ^ { 3 } - \{ f ( x ) \} ^ { 2 } - x ^ { 2 } f ( x ) + x ^ { 2 } = 0$$ for all real numbers $x$. When the maximum value of $f ( x )$ is 1 and the minimum value is 0, what is the value of $f \left( - \frac { 4 } { 3 } \right) + f ( 0 ) + f \left( \frac { 1 } { 2 } \right)$? [4 points]
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 3 } { 2 }$
(4) 2
(5) $\frac { 5 } { 2 }$

Q13 4 marks Laws of Logarithms Determine Parameters of a Logarithmic Function View

For two constants $a , b$ with $1 < a < b$, the $y$-intercept of the line passing through the two points $\left( a , \log _ { 2 } a \right) , \left( b , \log _ { 2 } b \right)$ and the $y$-intercept of the line passing through the two points $\left( a , \log _ { 4 } a \right) , \left( b , \log _ { 4 } b \right)$ are equal.
For the function $f ( x ) = a ^ { b x } + b ^ { a x }$ with $f ( 1 ) = 40$, what is the value of $f ( 2 )$? [4 points]
(1) 760
(2) 800
(3) 840
(4) 880
(5) 920

Q14 4 marks Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View

The position $x ( t )$ of a point P moving on a number line at time $t$ is given by $$x ( t ) = t ( t - 1 ) ( a t + b ) \quad ( a \neq 0 )$$ for two constants $a , b$. The velocity $v ( t )$ of point P at time $t$ satisfies $\int _ { 0 } ^ { 1 } | v ( t ) | d t = 2$. Which of the following statements in the given options are correct? [4 points]
Given statements: ᄀ. $\int _ { 0 } ^ { 1 } v ( t ) d t = 0$ ㄴ. There exists $t _ { 1 }$ in the open interval $( 0,1 )$ such that $\left| x \left( t _ { 1 } \right) \right| > 1$. ㄷ. If $| x ( t ) | < 1$ for all $t$ with $0 \leq t \leq 1$, then there exists $t _ { 2 }$ in the open interval $( 0,1 )$ such that $x \left( t _ { 2 } \right) = 0$.
(1) ᄀ
(2) ᄀ, ㄴ
(3) ᄀ, ㄷ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ

Q15 4 marks Sine and Cosine Rules Multi-step composite figure problem View

There are two circles $C _ { 1 } , C _ { 2 }$ with centers $\mathrm { O } _ { 1 } , \mathrm { O } _ { 2 }$ respectively and radii equal to $\overline { \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } }$. As shown in the figure, three distinct points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ on circle $C _ { 1 }$ and a point $\mathrm { D }$ on circle $C _ { 2 }$ are given, with three points $\mathrm { A } , \mathrm { O } _ { 1 } , \mathrm { O } _ { 2 }$ and three points $\mathrm { C } , \mathrm { O } _ { 2 } , \mathrm { D }$ each on a line.
Let $\angle \mathrm { BO } _ { 1 } \mathrm {~A} = \theta _ { 1 } , \angle \mathrm { O } _ { 2 } \mathrm { O } _ { 1 } \mathrm { C } = \theta _ { 2 } , \angle \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } \mathrm { D } = \theta _ { 3 }$.
The following is the process of finding the ratio of the lengths of segments AB and CD when $\overline { \mathrm { AB } } : \overline { \mathrm { O } _ { 1 } \mathrm { D } } = 1 : 2 \sqrt { 2 }$ and $\theta _ { 3 } = \theta _ { 1 } + \theta _ { 2 }$.
Since $\angle \mathrm { CO } _ { 2 } \mathrm { O } _ { 1 } + \angle \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } \mathrm { D } = \pi$, we have $\theta _ { 3 } = \frac { \pi } { 2 } + \frac { \theta _ { 2 } } { 2 }$, and from $\theta _ { 3 } = \theta _ { 1 } + \theta _ { 2 }$, we get $2 \theta _ { 1 } + \theta _ { 2 } = \pi$, so $\angle \mathrm { CO } _ { 1 } \mathrm {~B} = \theta _ { 1 }$. Since $\angle \mathrm { O } _ { 2 } \mathrm { O } _ { 1 } \mathrm {~B} = \theta _ { 1 } + \theta _ { 2 } = \theta _ { 3 }$, triangles $\mathrm { O } _ { 1 } \mathrm { O } _ { 2 } \mathrm {~B}$ and $\mathrm { O } _ { 2 } \mathrm { O } _ { 1 } \mathrm { D }$ are congruent. Let $\overline { \mathrm { AB } } = k$. Since $\overline { \mathrm { BO } _ { 2 } } = \overline { \mathrm { O } _ { 1 } \mathrm { D } } = 2 \sqrt { 2 } k$, we have $\overline { \mathrm { AO } _ { 2 } } =$ (a) and since $\angle \mathrm { BO } _ { 2 } \mathrm {~A} = \frac { \theta _ { 1 } } { 2 }$, we have $\cos \frac { \theta _ { 1 } } { 2 } =$ (b). In triangle $\mathrm { O } _ { 2 } \mathrm { BC }$, with $\overline { \mathrm { BC } } = k , \overline { \mathrm { BO } _ { 2 } } = 2 \sqrt { 2 } k , \angle \mathrm { CO } _ { 2 } \mathrm {~B} = \frac { \theta _ { 1 } } { 2 }$, by the law of cosines, $\overline { \mathrm { O } _ { 2 } \mathrm { C } } =$ (c). Since $\overline { \mathrm { CD } } = \overline { \mathrm { O } _ { 2 } \mathrm { D } } + \overline { \mathrm { O } _ { 2 } \mathrm { C } } = \overline { \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } } + \overline { \mathrm { O } _ { 2 } \mathrm { C } }$, $\overline { \mathrm { AB } } : \overline { \mathrm { CD } } = k : \left( \frac { \text{(a)} } { 2 } + \text{(c)} \right)$.
Let the expressions for (a) and (c) be $f ( k )$ and $g ( k )$ respectively, and let the number for (b) be $p$. What is the value of $f ( p ) \times g ( p )$? [4 points]
(1) $\frac { 169 } { 27 }$
(2) $\frac { 56 } { 9 }$
(3) $\frac { 167 } { 27 }$
(4) $\frac { 166 } { 27 }$
(5) $\frac { 55 } { 9 }$

Q16 3 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

Find the value of $\log _ { 2 } 120 - \frac { 1 } { \log _ { 15 } 2 }$. [3 points]

Q17 3 marks Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

For a function $f ( x )$ with $f ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x$ and $f ( 0 ) = 2$, find the value of $f ( 1 )$. [3 points]

Q18 3 marks Sequences and Series Evaluation of a Finite or Infinite Sum View

For a sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } a _ { k } - \sum _ { k = 1 } ^ { 7 } \frac { a _ { k } } { 2 } = 56 , \quad \sum _ { k = 1 } ^ { 10 } 2 a _ { k } - \sum _ { k = 1 } ^ { 8 } a _ { k } = 100$$ find the value of $a _ { 8 }$. [3 points]

Q19 3 marks Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Find the maximum value of the real number $a$ such that the function $f ( x ) = x ^ { 3 } + a x ^ { 2 } - \left( a ^ { 2 } - 8 a \right) x + 3$ is increasing on the entire set of real numbers. [3 points]

Q20 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

A function $f ( x )$ differentiable on the entire set of real numbers satisfies the following conditions.
(a) On the closed interval $[ 0,1 ]$, $f ( x ) = x$.
(b) For some constants $a , b$, on the interval $[ 0 , \infty )$, $f ( x + 1 ) - x f ( x ) = a x + b$. Find the value of $60 \times \int _ { 1 } ^ { 2 } f ( x ) d x$. [4 points]

Q21 4 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View

A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.
(a) $\left| a _ { 1 } \right| = 2$
(b) For all natural numbers $n$, $\left| a _ { n + 1 } \right| = 2 \left| a _ { n } \right|$.
(c) $\sum _ { n = 1 } ^ { 10 } a _ { n } = - 14$ Find the value of $a _ { 1 } + a _ { 3 } + a _ { 5 } + a _ { 7 } + a _ { 9 }$. [4 points]

Q22 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View

For a cubic function $f ( x )$ with leading coefficient $\frac { 1 } { 2 }$ and a real number $t$, let $g ( t )$ be the number of real roots of the equation $f ^ { \prime } ( x ) = 0$ in the closed interval $[ t , t + 2 ]$. The function $g ( t )$ satisfies the following conditions.
(a) For all real numbers $a$, $\lim _ { t \rightarrow a + } g ( t ) + \lim _ { t \rightarrow a - } g ( t ) \leq 2$.
(b) $g ( f ( 1 ) ) = g ( f ( 4 ) ) = 2 , g ( f ( 0 ) ) = 1$ Find the value of $f ( 5 )$. [4 points]

Q23 (Probability and Statistics) 2 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View

In the expansion of the polynomial $( x + 2 ) ^ { 7 }$, what is the coefficient of $x ^ { 5 }$? [2 points]
(1) 42
(2) 56
(3) 70
(4) 84
(5) 98

Q23 (Calculus) Sequences and series, recurrence and convergence Multiple-choice on sequence properties View

What is the value of $\lim _ { n \rightarrow \infty } \frac { \frac { 5 } { n } + \frac { 3 } { n ^ { 2 } } } { \frac { 1 } { n } - \frac { 2 } { n ^ { 3 } } }$?

Q23 (Geometry) 2 marks Vectors 3D & Lines Section Division and Coordinate Computation View

In coordinate space, let P be the point obtained by reflecting point $\mathrm { A } ( 2,1,3 )$ across the xy-plane, and let Q be the point obtained by reflecting point A across the yz-plane. What is the length of segment PQ? [2 points]
(1) $5 \sqrt { 2 }$
(2) $2 \sqrt { 13 }$
(3) $3 \sqrt { 6 }$
(4) $2 \sqrt { 14 }$
(5) $2 \sqrt { 15 }$

Q24 (Probability and Statistics) 3 marks Binomial Distribution Find Parameters from Moment Conditions View

A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 2 X ) = 40$. What is the value of $n$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50

Q24 (Geometry) 3 marks Conic sections Eccentricity or Asymptote Computation View

For a hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 6 } = 1$ with one focus at $( 3 \sqrt { 2 } , 0 )$, what is the length of the major axis? (Given that $a$ is a positive number.) [3 points]
(1) $3 \sqrt { 3 }$
(2) $\frac { 7 \sqrt { 3 } } { 2 }$
(3) $4 \sqrt { 3 }$
(4) $\frac { 9 \sqrt { 3 } } { 2 }$
(5) $5 \sqrt { 3 }$

Q25 (Probability and Statistics) 3 marks Combinations & Selection Counting Integer Solutions to Equations View

How many ordered pairs $( a , b , c , d , e )$ of natural numbers satisfy the following conditions? [3 points]
(a) $a + b + c + d + e = 12$
(b) $\left| a ^ { 2 } - b ^ { 2 } \right| = 5$
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

Q25 (Geometry) 3 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View

In the coordinate plane, consider two lines $$\frac { x + 1 } { 2 } = y - 3 , \quad x - 2 = \frac { y - 5 } { 3 }$$ If $\theta$ is the acute angle between these lines, what is the value of $\cos \theta$? [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { \sqrt { 5 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 4 }$
(4) $\frac { \sqrt { 7 } } { 4 }$
(5) $\frac { \sqrt { 2 } } { 2 }$

Q26 (Probability and Statistics) 3 marks Principle of Inclusion/Exclusion View

A bag contains 10 cards with natural numbers from 1 to 10 written on them, one number per card. When drawing 3 cards simultaneously at random from the bag, what is the probability that the smallest of the three natural numbers on the drawn cards is at most 4 or at least 7? [3 points]
(1) $\frac { 4 } { 5 }$
(2) $\frac { 5 } { 6 }$
(3) $\frac { 13 } { 15 }$
(4) $\frac { 9 } { 10 }$
(5) $\frac { 14 } { 15 }$

Q26 (Geometry) 3 marks Conic sections Circle-Conic Interaction with Tangency or Intersection View

For an ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a point A in the first quadrant on the ellipse. Among circles that are tangent to both lines $\mathrm { AF } , \mathrm { AF } ^ { \prime }$ and have their center on the y-axis, let C be the circle whose center has a negative y-coordinate. When the center of circle C is B and the area of quadrilateral $\mathrm { AFBF } ^ { \prime }$ is 72, what is the radius of circle C? [3 points]
(1) $\frac { 17 } { 2 }$
(2) 9
(3) $\frac { 19 } { 2 }$
(4) 10
(5) $\frac { 21 } { 2 }$

Q27 (Probability and Statistics) 3 marks Confidence intervals Algebraic problem using two confidence intervals View

A certain automobile company produces electric vehicles whose driving range on a single charge follows a normal distribution with mean $m$ and standard deviation $\sigma$.
When a sample of 100 electric vehicles produced by this company is randomly selected and the sample mean of the driving range is $\overline { x _ { 1 } }$, the 95\% confidence interval for the population mean $m$ is $a \leq m \leq b$.
When a sample of 400 electric vehicles produced by this company is randomly selected and the sample mean of the driving range is $\overline { x _ { 2 } }$, the 99\% confidence interval for the population mean $m$ is $c \leq m \leq d$.
If $\overline { x _ { 1 } } - \overline { x _ { 2 } } = 1.34$ and $a = c$, what is the value of $b - a$? (Here, the unit of driving range is km, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95 , \mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points]
(1) 5.88
(2) 7.84
(3) 9.80
(4) 11.76
(5) 13.72

Q27 (Geometry) 3 marks Vectors: Lines & Planes Volume of Pyramid/Tetrahedron Using Planes and Lines View

As shown in the figure, there is a cube $\mathrm { ABCD } - \mathrm { EFGH }$ with edge length 4. Let M be the midpoint of segment AD. What is the area of triangle MEG? [3 points]
(1) $\frac { 21 } { 2 }$
(2) 11
(3) $\frac { 23 } { 2 }$
(4) 12
(5) $\frac { 25 } { 2 }$

Q28 (Probability and Statistics) 4 marks Combinations & Selection Counting Functions or Mappings with Constraints View

For two sets $X = \{ 1,2,3,4,5 \} , Y = \{ 1,2,3,4 \}$, how many functions $f$ from $X$ to $Y$ satisfy the following conditions? [4 points]
(a) For all elements $x$ in set $X$, $f ( x ) \geq \sqrt { x }$.
(b) The range of function $f$ has exactly 3 elements.
(1) 128
(2) 138
(3) 148
(4) 158
(5) 168

Q28 (Geometry) 4 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View

For two positive numbers $a , p$, let $\mathrm { F } _ { 1 }$ be the focus of the parabola $( y - a ) ^ { 2 } = 4 p x$, and let $\mathrm { F } _ { 2 }$ be the focus of the parabola $y ^ { 2 } = - 4 x$.
When segment $\mathrm { F } _ { 1 } \mathrm {~F} _ { 2 }$ meets the two parabolas at points $\mathrm { P } , \mathrm { Q }$ respectively, $\overline { \mathrm { F } _ { 1 } \mathrm {~F} _ { 2 } } = 3$ and $\overline { \mathrm { PQ } } = 1$. What is the value of $a ^ { 2 } + p ^ { 2 }$? [4 points]
(1) 6
(2) $\frac { 25 } { 4 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 27 } { 4 }$
(5) 7

Q29 (Probability and Statistics) 4 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View

Two continuous random variables $X$ and $Y$ have ranges $0 \leq X \leq 6$ and $0 \leq Y \leq 6$, with probability density functions $f ( x )$ and $g ( x )$ respectively. The graph of the probability density function $f ( x )$ of random variable $X$ is shown in the figure.
For all $x$ with $0 \leq x \leq 6$, $$f ( x ) + g ( x ) = k \text{ (where } k \text{ is a constant)}$$
When $\mathrm { P } ( 6 k \leq Y \leq 15 k ) = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

Q29 (Calculus) 4 marks Parametric integration View

As shown in the figure, there is a semicircle with diameter AB of length 2. Two points $\mathrm { P } , \mathrm { Q }$ are taken on arc AB such that $\angle \mathrm { PAB } = \theta , \angle \mathrm { QBA } = 2 \theta$, and the intersection of two line segments $\mathrm { AP } , \mathrm { BQ }$ is denoted R. Points S on segment AB, point T on segment BR, and point U on segment AR are chosen such that segment UT is parallel to segment AB and triangle STU is equilateral. Let $f ( \theta )$ be the area of the region enclosed by two line segments $\mathrm { AR } , \mathrm { QR }$ and arc AQ, and let $g ( \theta )$ be the area of triangle STU. When $\lim _ { \theta \rightarrow 0 + } \frac { g ( \theta ) } { \theta \times f ( \theta ) } = \frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Given that $0 < \theta < \frac { \pi } { 6 }$ and $p$ and $q$ are coprime natural numbers.) [4 points]

Q29 (Geometry) 4 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View

In the coordinate plane, for a parallelogram OACB with $\overline { \mathrm { OA } } = \sqrt { 2 } , \overline { \mathrm { OB } } = 2 \sqrt { 2 }$ and $\cos ( \angle \mathrm { AOB } ) = \frac { 1 } { 4 }$, point P satisfies the following conditions. (가) $\overrightarrow { \mathrm { OP } } = s \overrightarrow { \mathrm { OA } } + t \overrightarrow { \mathrm { OB } } ( 0 \leq s \leq 1, 0 \leq t \leq 1 )$ (나) $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { BP } } \cdot \overrightarrow { \mathrm { BC } } = 2$
For a point X moving on a circle centered at O and passing through point A, let $M$ and $m$ be the maximum and minimum values of $| 3 \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OX } } |$ respectively. When $M \times m = a \sqrt { 6 } + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Given that $a$ and $b$ are rational numbers.) [4 points]

Q30 (Probability and Statistics) 4 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View

There is a basket containing at least 10 white balls and at least 10 black balls, and an empty bag. Using one die, the following trial is performed.
Roll the die once. If the result is 5 or more, put 2 white balls from the basket into the bag. If the result is 4 or less, put 1 black ball from the basket into the bag.
When the above trial is repeated 5 times, let $a _ { n }$ and $b _ { n }$ denote the number of white balls and black balls in the bag after the $n$-th trial ($1 \leq n \leq 5$) respectively. Given that $a _ { 5 } + b _ { 5 } \geq 7$, what is the probability that there exists a natural number $k$ ($1 \leq k \leq 5$) such that $a _ { k } = b _ { k }$? If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

Q30 (Calculus) 4 marks Integration by Parts Definite Integral Evaluation by Parts View

A function $f ( x )$ that is increasing and differentiable on the set of all real numbers satisfies the following conditions. (가) $f ( 1 ) = 1 , \int _ { 1 } ^ { 2 } f ( x ) d x = \frac { 5 } { 4 }$ (나) When the inverse function of $f ( x )$ is $g ( x )$, for all real numbers $x \geq 1$, $g ( 2 x ) = 2 f ( x )$. When $\int _ { 1 } ^ { 8 } x f ^ { \prime } ( x ) d x = \frac { q } { p }$, find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.) [4 points]

Q30 (Geometry) 4 marks Vectors 3D & Lines Volume of a 3D Solid View

In coordinate space, there is a sphere $$S : ( x - 2 ) ^ { 2 } + ( y - \sqrt { 5 } ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 25$$ with center $\mathrm { C } ( 2 , \sqrt { 5 } , 5 )$ passing through point $\mathrm { P } ( 0,0,1 )$. For a point Q moving on the circle formed by the intersection of sphere $S$ and plane OPC, and a point R moving on sphere $S$, let $\mathrm { Q } _ { 1 }$ and $\mathrm { R } _ { 1 }$ be the orthogonal projections of points $\mathrm { Q }$ and $\mathrm { R }$ onto the xy-plane respectively.
For two points $\mathrm { Q } , \mathrm { R }$ that maximize the area of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$, the area of the orthogonal projection of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$ onto plane PQR is $\frac { q } { p } \sqrt { 6 }$. Find the value of $p + q$. [4 points]

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