csat-suneung

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

Find the value of $\sqrt[3]{24} \times 3^{\frac{2}{3}}$. [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

Q2 3 marks Trig Graphs & Exact Values View

For $\theta$ satisfying $\frac{3}{2}\pi < \theta < 2\pi$ and $\sin(-\theta) = \frac{1}{3}$, find the value of $\tan\theta$. [3 points]
(1) $-\frac{\sqrt{2}}{2}$
(2) $-\frac{\sqrt{2}}{4}$
(3) $-\frac{1}{4}$
(4) $\frac{1}{4}$
(5) $\frac{\sqrt{2}}{4}$

Q3 2 marks Differentiation from First Principles View

For the function $f(x) = 2x^3 - 5x^2 + 3$, find the value of $\lim_{h \rightarrow 0} \frac{f(2+h) - f(2)}{h}$. [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Q4 3 marks Curve Sketching Finding Parameters for Continuity View

Consider the function $$f(x) = \begin{cases} 3x - a & (x < 2) \\ x^2 + a & (x \geq 2) \end{cases}$$ If $f$ is continuous on the set of all real numbers, find the value of the constant $a$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Q5 3 marks Indefinite & Definite Integrals Recovering Function Values from Derivative Information View

A polynomial function $f(x)$ satisfies $$f'(x) = 3x(x-2), \quad f(1) = 6$$ Find the value of $f(2)$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Q6 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View

Let $S_n$ denote the sum of the first $n$ terms of a geometric sequence $\{a_n\}$. $$S_4 - S_2 = 3a_4, \quad a_5 = \frac{3}{4}$$ Find the value of $a_1 + a_2$. [3 points]
(1) 27
(2) 24
(3) 21
(4) 18
(5) 15

Q7 3 marks Stationary points and optimisation Find critical points and classify extrema of a given function View

For the function $f(x) = \frac{1}{3}x^3 - 2x^2 - 12x + 4$, if $f$ has a local maximum at $x = \alpha$ and a local minimum at $x = \beta$, find the value of $\beta - \alpha$. (Here, $\alpha$ and $\beta$ are constants.) [3 points]
(1) $-4$
(2) $-1$
(3) 2
(4) 5
(5) 8

Q8 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

A cubic function $f(x)$ satisfies $$xf(x) - f(x) = 3x^4 - 3x$$ for all real numbers $x$. Find the value of $\int_{-2}^{2} f(x)\,dx$. [3 points]
(1) 12
(2) 16
(3) 20
(4) 24
(5) 28

Q9 4 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View

For two points $\mathrm{P}(\log_5 3)$ and $\mathrm{Q}(\log_5 12)$ on a number line, the point that divides the line segment PQ internally in the ratio $m:(1-m)$ has coordinate 1. Find the value of $4^m$. (Here, $m$ is a constant with $0 < m < 1$.) [4 points]
(1) $\frac{7}{6}$
(2) $\frac{4}{3}$
(3) $\frac{3}{2}$
(4) $\frac{5}{3}$
(5) $\frac{11}{6}$

Q10 4 marks Variable acceleration (1D) Two-particle comparison problem View

At time $t = 0$, two points P and Q simultaneously depart from the origin and move on a number line. Their velocities at time $t$ ($t \geq 0$) are respectively $$v_1(t) = t^2 - 6t + 5, \quad v_2(t) = 2t - 7$$ Let $f(t)$ denote the distance between points P and Q at time $t$. The function $f(t)$ increases on the interval $[0, a]$, decreases on the interval $[a, b]$, and increases on the interval $[b, \infty)$. Find the distance traveled by point Q from time $t = a$ to time $t = b$. (Here, $0 < a < b$) [4 points]
(1) $\frac{15}{2}$
(2) $\frac{17}{2}$
(3) $\frac{19}{2}$
(4) $\frac{21}{2}$
(5) $\frac{23}{2}$

Q11 4 marks Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View

For an arithmetic sequence $\{a_n\}$ with nonzero common difference, $$|a_6| = a_8, \quad \sum_{k=1}^{5} \frac{1}{a_k a_{k+1}} = \frac{5}{96}$$ Find the value of $\sum_{k=1}^{15} a_k$. [4 points]
(1) 60
(2) 65
(3) 70
(4) 75
(5) 80

Q12 4 marks Areas by integration View

For the function $f(x) = \frac{1}{9}x(x-6)(x-9)$ and a real number $t$ with $0 < t < 6$, the function $g(x)$ is defined as $$g(x) = \begin{cases} f(x) & (x < t) \\ -(x-t) + f(t) & (x \geq t) \end{cases}$$ Find the maximum area of the region enclosed by the graph of $y = g(x)$ and the $x$-axis. [4 points]
(1) $\frac{125}{4}$
(2) $\frac{127}{4}$
(3) $\frac{129}{4}$
(4) $\frac{131}{4}$
(5) $\frac{133}{4}$

Q13 4 marks Sine and Cosine Rules Multi-step composite figure problem View

As shown in the figure, $$\overline{\mathrm{AB}} = 3, \quad \overline{\mathrm{BC}} = \sqrt{13}, \quad \overline{\mathrm{AD}} \times \overline{\mathrm{CD}} = 9, \quad \angle\mathrm{BAC} = \frac{\pi}{3}$$ For quadrilateral ABCD, let $S_1$ denote the area of triangle ABC, $S_2$ denote the area of triangle ACD, and $R$ denote the circumradius of triangle ACD. If $S_2 = \frac{5}{6}S_1$, find the value of $\frac{R}{\sin(\angle\mathrm{ADC})}$. [4 points]
(1) $\frac{54}{25}$
(2) $\frac{117}{50}$
(3) $\frac{63}{25}$
(4) $\frac{27}{10}$
(5) $\frac{72}{25}$

Q14 4 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View

For two natural numbers $a$ and $b$, the function $f(x)$ is defined as $$f(x) = \begin{cases} 2x^3 - 6x + 1 & (x \leq 2) \\ a(x-2)(x-b) + 9 & (x > 2) \end{cases}$$ For a real number $t$, let $g(t)$ denote the number of intersection points of the graph of $y = f(x)$ and the line $y = t$. $$g(k) + \lim_{t \rightarrow k-} g(t) + \lim_{t \rightarrow k+} g(t) = 9$$ If the number of real numbers $k$ satisfying this condition is 1, find the maximum value of $a + b$ for the ordered pair $(a, b)$ of two natural numbers. [4 points]
(1) 51
(2) 52
(3) 53
(4) 54
(5) 55

Q15 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View

A sequence $\{a_n\}$ with a natural number as its first term satisfies $$a_{n+1} = \begin{cases} 2^{a_n} & (\text{if } a_n \text{ is odd}) \\ \frac{1}{2}a_n & (\text{if } a_n \text{ is even}) \end{cases}$$ for all natural numbers $n$. Find the sum of all values of $a_1$ such that $a_6 + a_7 = 3$. [4 points]
(1) 139
(2) 146
(3) 153
(4) 160
(5) 167

Q16 3 marks Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View

Solve the equation $3^{x-8} = \left(\frac{1}{27}\right)^x$ for the real number $x$. [3 points]

Q17 3 marks Product & Quotient Rules View

For the function $f(x) = (x+1)(x^2+3)$, find the value of $f'(1)$. [3 points]

Q18 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View

For two sequences $\{a_n\}$ and $\{b_n\}$, $$\sum_{k=1}^{10} a_k = \sum_{k=1}^{10} (2b_k - 1), \quad \sum_{k=1}^{10} (3a_k + b_k) = 33$$ Find the value of $\sum_{k=1}^{10} b_k$. [3 points]

Q19 3 marks Standard trigonometric equations Solve trigonometric inequality View

For the function $f(x) = \sin\frac{\pi}{4}x$, find the sum of all natural numbers $x$ satisfying the inequality $$f(2+x)f(2-x) < \frac{1}{4}$$ for $0 < x < 16$. [3 points]

Q20 4 marks Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View

For a real number $a > \sqrt{2}$, define the function $f(x)$ as $$f(x) = -x^3 + ax^2 + 2x$$ The tangent line to the curve $y = f(x)$ at the point $\mathrm{O}(0,0)$ intersects the curve $y = f(x)$ at another point A. The tangent line to the curve $y = f(x)$ at point A intersects the $x$-axis at point B. If point A lies on the circle with diameter OB, find the value of $\overline{\mathrm{OA}} \times \overline{\mathrm{AB}}$. [4 points]

Q21 4 marks Stationary points and optimisation Composite or piecewise function extremum analysis View

For a positive number $a$, the function $f(x)$ defined on $x \geq -1$ is $$f(x) = \begin{cases} -x^2 + 6x & (-1 \leq x < 6) \\ a\log_4(x-5) & (x \geq 6) \end{cases}$$ For a real number $t \geq 0$, let $g(t)$ denote the maximum value of $f(x)$ on the closed interval $[t-1, t+1]$. If the minimum value of the function $g(t)$ on the interval $[0, \infty)$ is 5, find the minimum value of the positive number $a$. [4 points]

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

A cubic function $f(x)$ with leading coefficient 1 satisfies the following condition.
For the function $f(x)$, $$f(k-1)f(k+1) < 0$$ has no integer solutions for $k$.
If $f'\left(-\frac{1}{4}\right) = -\frac{1}{4}$ and $f'\left(\frac{1}{4}\right) < 0$, find the value of $f(8)$. [4 points]

Q23 2 marks Permutations & Arrangements Word Permutations with Repeated Letters View

The number of ways to arrange all 5 letters $x, x, y, y, z$ in a row is? [2 points]
(1) 10
(2) 20
(3) 30
(4) 40
(5) 50

Q23_calculus 2 marks Differentiating Transcendental Functions Limit involving transcendental functions View

Find the value of $\lim_{x \rightarrow 0} \frac{\ln(1+3x)}{\ln(1+5x)}$. [2 points]
(1) $\frac{1}{5}$
(2) $\frac{2}{5}$
(3) $\frac{3}{5}$
(4) $\frac{4}{5}$
(5) 1

Q23_geometry 2 marks Vectors 3D & Lines Section Division and Coordinate Computation View

For two points $\mathrm{A}(a, -2, 6)$ and $\mathrm{B}(9, 2, b)$ in coordinate space, the midpoint of segment AB has coordinates $(4, 0, 7)$. What is the value of $a + b$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

Q24 3 marks Independent Events View

Two events $A$ and $B$ are independent, and $$\mathrm{P}(A \cap B) = \frac{1}{4}, \quad \mathrm{P}(A^C) = 2\mathrm{P}(A)$$ Find the value of $\mathrm{P}(B)$. (Here, $A^C$ is the complement of $A$.) [3 points]
(1) $\frac{3}{8}$
(2) $\frac{1}{2}$
(3) $\frac{5}{8}$
(4) $\frac{3}{4}$
(5) $\frac{7}{8}$

Q24_calculus 3 marks Parametric differentiation View

For the curve represented parametrically by $$x = \ln(t^3 + 1), \quad y = \sin\pi t$$ where $t > 0$, find the value of $\frac{dy}{dx}$ at $t = 1$. [3 points]
(1) $-\frac{1}{3}\pi$
(2) $-\frac{2}{3}\pi$
(3) $-\pi$
(4) $-\frac{4}{3}\pi$
(5) $-\frac{5}{3}\pi$

Q24_geometry 3 marks Circles Tangent Lines and Tangent Lengths View

For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{6} = 1$, what is the slope of the tangent line at the point $(\sqrt{3}, -2)$ on the ellipse? (where $a$ is a positive number) [3 points]
(1) $\sqrt{3}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{4}$
(5) $\frac{\sqrt{3}}{5}$

Q25 3 marks Permutations & Arrangements Probability via Permutation Counting View

There are 6 cards with the numbers $1, 2, 3, 4, 5, 6$ written on them, one number per card. When all 6 cards are arranged in a row in random order using each card exactly once, find the probability that the sum of the two numbers on the cards at both ends is at most 10. [3 points]
(1) $\frac{8}{15}$
(2) $\frac{19}{30}$
(3) $\frac{11}{15}$
(4) $\frac{5}{6}$
(5) $\frac{14}{15}$

Q25_calculus 3 marks Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View

Two functions $f(x)$ and $g(x)$ are defined and differentiable on the set of all positive real numbers. $g(x)$ is the inverse function of $f(x)$, and $g'(x)$ is continuous on the set of all positive real numbers. For all positive numbers $a$, $$\int_1^a \frac{1}{g'(f(x))f(x)}\,dx = 2\ln a + \ln(a+1) - \ln 2$$ and $f(1) = 8$. Find the value of $f(2)$. [3 points]
(1) 36
(2) 40
(3) 44
(4) 48
(5) 52

Q25_geometry 3 marks Vectors Introduction & 2D Magnitude of Vector Expression View

For two vectors $\vec{a}$ and $\vec{b}$, $$|\vec{a}| = \sqrt{11}, \quad |\vec{b}| = 3, \quad |2\vec{a} - \vec{b}| = \sqrt{17}$$ What is the value of $|\vec{a} - \vec{b}|$? [3 points]
(1) $\frac{\sqrt{2}}{2}$
(2) $\sqrt{2}$
(3) $\frac{3\sqrt{2}}{2}$
(4) $2\sqrt{2}$
(5) $\frac{5\sqrt{2}}{2}$

Q26 3 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

When 4 coins are tossed simultaneously, let $X$ be the random variable representing the number of coins showing heads. Define the discrete random variable $Y$ as $$Y = \begin{cases} X & (\text{if } X \text{ takes the value } 0 \text{ or } 1) \\ 2 & (\text{if } X \text{ takes a value of } 2 \text{ or more}) \end{cases}$$ Find the value of $\mathrm{E}(Y)$. [3 points]
(1) $\frac{25}{16}$
(2) $\frac{13}{8}$
(3) $\frac{27}{16}$
(4) $\frac{7}{4}$
(5) $\frac{29}{16}$

Q26_calculus 3 marks Volumes of Revolution Volume by Cross Sections with Known Geometry View

As shown in the figure, there is a solid figure with base formed by the curve $y = \sqrt{(1-2x)\cos x}$ ($\frac{3}{4}\pi \leq x \leq \frac{5}{4}\pi$) and the $x$-axis and the two lines $x = \frac{3}{4}\pi$ and $x = \frac{5}{4}\pi$. When this solid figure is cut by a plane perpendicular to the $x$-axis, all cross-sections are squares. Find the volume of this solid figure. [3 points]
(1) $\sqrt{2}\pi - \sqrt{2}$
(2) $\sqrt{2}\pi - 1$
(3) $2\sqrt{2}\pi - \sqrt{2}$
(4) $2\sqrt{2}\pi - 1$
(5) $2\sqrt{2}\pi$

Q26_geometry 3 marks Vectors 3D & Lines Dihedral Angle Computation View

There is a plane $\alpha$ in coordinate space. Let $\mathrm{A}'$ and $\mathrm{B}'$ be the orthogonal projections of two distinct points $\mathrm{A}$ and $\mathrm{B}$ (not on plane $\alpha$) onto plane $\alpha$, respectively. $$\overline{\mathrm{AB}} = \overline{\mathrm{A'B'}} = 6$$ Let $\mathrm{M}'$ be the orthogonal projection of the midpoint M of segment AB onto plane $\alpha$. A point P is chosen on plane $\alpha$ such that $$\overline{\mathrm{PM'}} \perp \overline{\mathrm{A'B'}}, \quad \overline{\mathrm{PM'}} = 6$$ When the area of the orthogonal projection of triangle $\mathrm{A'B'P}$ onto plane ABP is $\frac{9}{2}$, what is the length of segment PM? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24

Q27 3 marks Confidence intervals Find a specific bound or margin of error from the CI formula View

From a population following a normal distribution $\mathrm{N}(m, 5^2)$, a sample of size 49 is randomly extracted, and the sample mean is $\bar{x}$. The 95\% confidence interval for the population mean $m$ is $a \leq m \leq \frac{6}{5}a$. Find the value of $\bar{x}$. (Here, if $Z$ is a random variable following the standard normal distribution, $\mathrm{P}(|Z| \leq 1.96) = 0.95$.) [3 points]
(1) 15.2
(2) 15.4
(3) 15.6
(4) 15.8
(5) 16.0

Q27_calculus 3 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View

For a real number $t$, let $f(t)$ denote the slope of the line passing through the origin and tangent to the curve $y = \frac{1}{e^x} + e^t$. For the constant $a$ satisfying $f(a) = -e\sqrt{e}$, find the value of $f'(a)$. [3 points]
(1) $-\frac{1}{3}e\sqrt{e}$
(2) $-\frac{1}{2}e\sqrt{e}$
(3) $-\frac{2}{3}e\sqrt{e}$
(4) $-\frac{5}{6}e\sqrt{e}$
(5) $-e\sqrt{e}$

Q27_geometry 3 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View

Let F be the focus of the parabola $y^2 = 8x$. From a point A on the parabola, drop a perpendicular to the directrix of the parabola, with the foot of the perpendicular being B. Let C and D be the two points where the line BF intersects the parabola. When $\overline{\mathrm{BC}} = \overline{\mathrm{CD}}$, what is the area of triangle ABD? (where $\overline{\mathrm{CF}} < \overline{\mathrm{DF}}$ and point A is not the origin) [3 points]
(1) $100\sqrt{2}$
(2) $104\sqrt{2}$
(3) $108\sqrt{2}$
(4) $112\sqrt{2}$
(5) $116\sqrt{2}$

Q28 4 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View

There is one bag and two boxes A and B. The bag contains 4 cards with the numbers $1, 2, 3, 4$ written on them, one number per card. Box A contains more than 8 white balls and more than 8 black balls, and box B is empty. Using this bag and the two boxes A and B, the following trial is performed.
A card is randomly drawn from the bag, the number on the card is confirmed, and the card is returned to the bag.
If the confirmed number is 1, 1 white ball from box A is placed into box B. If the confirmed number is 2 or 3, 1 white ball and 1 black ball from box A are placed into box B. If the confirmed number is 4, 2 white balls and 1 black ball from box A are placed into box B.
After repeating this trial 4 times, given that the total number of balls in box B is 8, find the probability that the number of black balls in box B is 2. [4 points]
(1) $\frac{3}{70}$
(2) $\frac{2}{35}$
(3) $\frac{1}{14}$
(4) $\frac{3}{35}$
(5) $\frac{1}{10}$

Q28_calculus 4 marks Differential equations Qualitative Analysis of DE Solutions View

A function $f(x)$ is continuous on the set of all real numbers, $f(x) \geq 0$ for all real numbers $x$, and $f(x) = -4xe^{4x^2}$ for $x < 0$. For all positive numbers $t$, the equation $f(x) = t$ has exactly 2 distinct real roots. Let $g(t)$ denote the smaller root and $h(t)$ denote the larger root of this equation. The two functions $g(t)$ and $h(t)$ satisfy $$2g(t) + h(t) = k \quad (k \text{ is a constant})$$ for all positive numbers $t$. If $\int_0^7 f(x)\,dx = e^4 - 1$, find the value of $\frac{f(9)}{f(8)}$. [4 points]
(1) $\frac{3}{2}e^5$
(2) $\frac{4}{3}e^7$
(3) $\frac{5}{4}e^9$
(4) $\frac{6}{5}e^{11}$
(5) $\frac{7}{6}e^{13}$

Q28_geometry 4 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View

As shown in the figure, there are two distinct planes $\alpha$ and $\beta$ with intersection line containing two points $\mathrm{A}$ and $\mathrm{B}$ where $\overline{\mathrm{AB}} = 18$. A circle $C_1$ with diameter AB lies on plane $\alpha$, and an ellipse $C_2$ with major axis AB and foci $\mathrm{F}$ and $\mathrm{F'}$ lies on plane $\beta$. Let H be the foot of the perpendicular from a point P on circle $C_1$ to plane $\beta$. Given that $\overline{\mathrm{HF'}} < \overline{\mathrm{HF}}$ and $\angle\mathrm{HFF'} = \frac{\pi}{6}$. Let Q be the point on ellipse $C_2$ where line HF intersects it, closer to H, with $\overline{\mathrm{FH}} < \overline{\mathrm{FQ}}$. The circle on plane $\beta$ centered at H passing through Q has radius 4 and is tangent to line AB. If the angle between the two planes $\alpha$ and $\beta$ is $\theta$, what is the value of $\cos\theta$? (where point P is not on plane $\beta$) [4 points]
(1) $\frac{2\sqrt{66}}{33}$
(2) $\frac{4\sqrt{69}}{69}$
(3) $\frac{\sqrt{2}}{3}$
(4) $\frac{4\sqrt{3}}{15}$
(5) $\frac{2\sqrt{78}}{39}$

Q29 4 marks Combinations & Selection Counting Integer Solutions to Equations View

Find the total number of ordered quadruples $(a, b, c, d)$ of natural numbers not exceeding 6 that satisfy the following condition. [4 points] $$a \leq c \leq d \text{ and } b \leq c \leq d$$

Q29_geometry 4 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View

For a positive number $c$, there is a hyperbola with foci $\mathrm{F}(c, 0)$ and $\mathrm{F'}(-c, 0)$ and major axis length 6. Two distinct points $\mathrm{P}$ and $\mathrm{Q}$ on this hyperbola satisfy the following conditions. Find the sum of all values of $c$. [4 points] (가) Point P is in the first quadrant, and point Q is on line $\mathrm{PF'}$. (나) Triangle $\mathrm{PF'F}$ is isosceles. (다) The perimeter of triangle PQF is 28.

Q30 4 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View

For a positive number $t$, the random variable $X$ follows a normal distribution $\mathrm{N}(1, t^2)$. $$\mathrm{P}(X \leq 5t) \geq \frac{1}{2}$$ For all positive numbers $t$ satisfying this condition, find the maximum value of $\mathrm{P}(t^2 - t + 1 \leq X \leq t^2 + t + 1)$ using the standard normal distribution table below, and let this value be $k$. Find the value of $1000 \times k$. [4 points]

$z$	$\mathrm{P}(0 \leq Z \leq z)$
0.6	0.226
0.8	0.288
1.0	0.341
1.2	0.385
1.4	0.419

Q30_geometry 4 marks Vectors Introduction & 2D Optimization of a Vector Expression View

In the coordinate plane, there is an equilateral triangle ABC with side length 4. Let D be the point that divides segment AB in the ratio $1:3$, E be the point that divides segment BC in the ratio $1:3$, and F be the point that divides segment CA in the ratio $1:3$. Four points $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$, and $\mathrm{X}$ satisfy the following conditions. (가) $|\overrightarrow{\mathrm{DP}}| = |\overrightarrow{\mathrm{EQ}}| = |\overrightarrow{\mathrm{FR}}| = 1$ (나) $\overrightarrow{\mathrm{AX}} = \overrightarrow{\mathrm{PB}} + \overrightarrow{\mathrm{QC}} + \overrightarrow{\mathrm{RA}}$ When $|\overrightarrow{\mathrm{AX}}|$ is maximized, let the area of triangle PQR be $S$. Find the value of $16S^2$. [4 points]

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