2018 CSAT (Korean SAT) EBS mock test (Natural Sciences, Grade 12)

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2018 academic year College Scholastic Ability Test EBS mock test (version 1)\large \text{2018 academic year College Scholastic Ability Test EBS mock test (version 1)}

Math Section (Natural Sciences)\Huge \text{Math Section} ~ \Large \text{(Natural Sciences)}

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Rules:\huge \text{Rules:}

1. This post is for people who really want to experience how Korean SAT works.

2. This test has a time limit of 100 minutes.

3. Any use of calculators is considered cheating.

4. This test consists of 30 questions.

5. The first 21 are multiple choice questions, and the rest, 9, are short-answer questions.

6. The total score is 100 points.

7. The amount of points alloted to each question is marked at the end of the question, so keep that in mind.

8. Answers are in the comments section. If you want full experience, check your answers after you've finished answering all of them, rather than check your answers immediately after solving each question.

9. If you don't have enough time to finish those all, then try #20, #21, #28, #29, #30. Those are the hardest ones among all.

10. Have fun!

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Multiple choice\large \boxed{\text{Multiple choice}}

1. \huge 1.~ There are two vectors a=(1, 2),\vec{a}=(1,~2), and b=(2, 3).\vec{b}=(2,~-3). Find the sum of the components of the vector a+2b.\vec{a}+2\vec{b}.  ~ [2 points]

A.1B.2C.3D.4E.5\begin{aligned}&\boxed{\text{A}}. && 1 \\\\ &\boxed{\text{B}}. && 2 \\\\ &\boxed{\text{C}}. && 3 \\\\ &\boxed{\text{D}}. && 4 \\\\ &\boxed{\text{E}}. && 5 \end{aligned}

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2. \huge 2.~ What's the value of limx0e3x12x2+3x?\displaystyle \lim_{x\to0}\frac{e^{3x}-1}{2x^2+3x}?  ~ [2 points]

A.13B.12C.1D.2E.3\begin{aligned}&\boxed{\text{A}}. && \frac{1}{3} \\\\ &\boxed{\text{B}}. && \frac{1}{2} \\\\ &\boxed{\text{C}}. && 1 \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && 3 \end{aligned}

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3. \huge 3.~ Find the value of 015xxdx.\displaystyle \int_{0}^{1} 5x\sqrt{x}dx.  ~ [2 points]

A.2B.4C.6D.8E.10\begin{aligned}&\boxed{\text{A}}. && 2 \\\\ &\boxed{\text{B}}. && 4 \\\\ &\boxed{\text{C}}. && 6 \\\\ &\boxed{\text{D}}. && 8 \\\\ &\boxed{\text{E}}. && 10 \end{aligned}

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4. \huge 4.~ XCX^C denotes the complementary event of an event X.X. Two events AA and BB are independent, and P(AC)=13, P(AB)=16.\mathrm{P}(A^C)=\dfrac{1}{3},~\mathrm{P}(A\cap B)=\dfrac{1}{6}. Find the value of P(AB).\mathrm{P}(A\cup B).  ~ [3 points]

A.12B.23C.34D.45E.56\begin{aligned}&\boxed{\text{A}}. && \frac{1}{2} \\\\ &\boxed{\text{B}}. && \frac{2}{3} \\\\ &\boxed{\text{C}}. && \frac{3}{4} \\\\ &\boxed{\text{D}}. && \frac{4}{5} \\\\ &\boxed{\text{E}}. && \frac{5}{6} \end{aligned}

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5. \huge 5.~ How many odd, 3-digit natural numbers are there whose digits only consist of 1, 2, 3, 4, 5?  ~ [3 points]

A.55B.60C.65D.70E.75\begin{aligned}&\boxed{\text{A}}. && 55 \\\\ &\boxed{\text{B}}. && 60 \\\\ &\boxed{\text{C}}. && 65 \\\\ &\boxed{\text{D}}. && 70 \\\\ &\boxed{\text{E}}. && 75 \end{aligned}

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6. \huge 6.~ Let g(x)g(x) be the inverse function of f(x)=ex3+1.f(x)=e^{x^3+1}. What's the value of g(1)?g'(1)?  ~ [3 points]

A.13B.12C.1D.2E.3\begin{aligned}&\boxed{\text{A}}. && \frac{1}{3} \\\\ &\boxed{\text{B}}. && \frac{1}{2} \\\\ &\boxed{\text{C}}. && 1 \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && 3 \end{aligned}

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7. \huge 7.~ When throwing a dice 3 times, find the probability of the product of the three results being a multiple of 9.  ~ [3 points]

A.227B.19C.427D.29E.727\begin{aligned}&\boxed{\text{A}}. && \frac{2}{27} \\\\ &\boxed{\text{B}}. && \frac{1}{9} \\\\ &\boxed{\text{C}}. && \frac{4}{27} \\\\ &\boxed{\text{D}}. && \frac{2}{9} \\\\ &\boxed{\text{E}}. && \frac{7}{27} \end{aligned}

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8. \huge 8.~ There are two points A(1, 2, 3), B(2, 5, 3)\mathrm{A}(1,~2,~3),~\mathrm{B}(-2,~5,~-3) in a coordinate space. A point P(a, b, c)\mathrm{P}(a,~b,~c) is on segment AB,\mathrm{AB}, satisfying AP=2PB.\overline{\mathrm{AP}}=2\overline{\mathrm{PB}}. Find the value of a+b+c.a+b+c.  ~ [3 points]

A.2B.1C.0D.1E.2\begin{aligned}&\boxed{\text{A}}. && -2 \\\\ &\boxed{\text{B}}. && -1 \\\\ &\boxed{\text{C}}. && 0 \\\\ &\boxed{\text{D}}. && 1 \\\\ &\boxed{\text{E}}. && 2 \end{aligned}

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9. \huge 9.~ What's the value of 1eln(x2)dx?\displaystyle \int_{1}^{e} \ln(x^2)dx?  ~ [3 points]

A.1B.2C.3D.4E.5\begin{aligned}&\boxed{\text{A}}. && 1 \\\\ &\boxed{\text{B}}. && 2 \\\\ &\boxed{\text{C}}. && 3 \\\\ &\boxed{\text{D}}. && 4 \\\\ &\boxed{\text{E}}. && 5 \end{aligned}

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10. \huge 10.~ The position of a moving point P(x, y)\mathrm{P}(x,~y) at time t (t>0)t~(t>0) is {x=2ty=4t+1.\cases{x=2\sqrt{t} \\\\ y=\dfrac{4}{t+1}}. What is the speed at which P\mathrm{P} moves at t=1?t=1?  ~ [3 points]

A.1B.2C.3D.2E.5\begin{aligned}&\boxed{\text{A}}. && 1 \\\\ &\boxed{\text{B}}. && \sqrt{2} \\\\ &\boxed{\text{C}}. && \sqrt{3} \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && \sqrt{5} \end{aligned}

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11. \huge 11.~ As shown on the right, there is a solid figure whose base side is the area enclosed by two curves y=sinx, y=cosx,y=\sin x,~y=\cos x, the yy-axis and a line x=π6.x=\dfrac{\pi}{6}. Every longitudinal section perpendicular to the xx-axis is a square. What's the volume of this solid?  ~ [3 points]

A.2π112B.2π312C.2π512D.4π712E.4π912\begin{aligned}&\boxed{\text{A}}. && \frac{2\pi - 1}{12} \\\\ &\boxed{\text{B}}. && \frac{2\pi - 3}{12} \\\\ &\boxed{\text{C}}. && \frac{2\pi - 5}{12} \\\\ &\boxed{\text{D}}. && \frac{4\pi - 7}{12} \\\\ &\boxed{\text{E}}. && \frac{4\pi - 9}{12} \end{aligned}

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12. \huge 12.~ θ\theta is the acute angle that line x=2y+1=z+3x=2y+1=z+3 and the xx-axis form, in a coordinate space. What's the value of cosθ?\cos \theta ?  ~ [3 points]

A.16B.13C.12D.23E.56\begin{aligned}&\boxed{\text{A}}. && \frac{1}{6} \\\\ &\boxed{\text{B}}. && \frac{1}{3} \\\\ &\boxed{\text{C}}. && \frac{1}{2} \\\\ &\boxed{\text{D}}. && \frac{2}{3} \\\\ &\boxed{\text{E}}. && \frac{5}{6} \end{aligned}

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13. \huge 13.~ A sample of size 1616 and average X\overline{X} is arbitrarily chosen from a population that follows the normal distribution N(m, 1).\mathrm{N}(m,~1). The random variable of the population is X.X. Given that P(X5)=P(X1)\mathrm{P}(X\ge5)=\mathrm{P}(\overline{X}\le 1) and m<5,m<5, find the value of m.m.  ~ [3 points]

A.35B.1C.75D.95E.115\begin{aligned}&\boxed{\text{A}}. && \frac{3}{5} \\\\ &\boxed{\text{B}}. && 1 \\\\ &\boxed{\text{C}}. && \frac{7}{5} \\\\ &\boxed{\text{D}}. && \frac{9}{5} \\\\ &\boxed{\text{E}}. && \frac{11}{5} \end{aligned}

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14. \huge 14.~ As shown on the right, let D, E, F\mathrm{D},~\mathrm{E},~\mathrm{F} be the point of contact between the inscribed circle of triangle ABC\mathrm{ABC} and AB, BC, CA,\overline{\mathrm{AB}},~\overline{\mathrm{BC}},~\overline{\mathrm{CA}}, where ABC=θ, BCA=θ, BC=2.\angle \mathrm{ABC}=\theta,~\angle \mathrm{BCA}=\theta,~ \overline{\mathrm{BC}}=2. Define S(θ)S(\theta) as the area of triangle DEF.\mathrm{DEF}. Find the value of limθ0+S(θ)θ3.\displaystyle \lim_{\theta\to 0^+ }\frac{S(\theta)}{\theta^3}.  ~ [4 points]

A.14B.12C.1D.2E.4\begin{aligned}&\boxed{\text{A}}. && \frac{1}{4} \\\\ &\boxed{\text{B}}. && \frac{1}{2} \\\\ &\boxed{\text{C}}. && 1 \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && 4 \end{aligned}

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15. \huge 15.~ There's a point P(t, lnt) (0<t<1)\mathrm{P}(t,~-\ln t)~(0<t<1) on the curve y=lnx.y=-\ln x. Q\mathrm{Q} is the foot of perpendicular from P\mathrm{P} to the xx-axis, and R\mathrm{R} is the point of intersection of the tangent line from P\mathrm{P} and the xx-axis. What's the maximum area of triangle PQR?\mathrm{PQR}?  ~ [4 points]

A.1e2B.2e2C.3e2D.4e2E.5e2\begin{aligned}&\boxed{\text{A}}. && \frac{1}{e^2} \\\\ &\boxed{\text{B}}. && \frac{2}{e^2} \\\\ &\boxed{\text{C}}. && \frac{3}{e^2} \\\\ &\boxed{\text{D}}. && \frac{4}{e^2} \\\\ &\boxed{\text{E}}. && \frac{5}{e^2} \end{aligned}

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16. \huge 16.~ Two fixed points, A, B,\mathrm{A},~\mathrm{B}, satisfy AB=2.\overline{\mathrm{AB}}=2. A point P\mathrm{P} satisfies the below conditions.

(1) APPB=0.\overrightarrow{\mathrm{AP}}\cdot\overrightarrow{\mathrm{PB}}=0.

(2) ABAP2+3\overrightarrow{\mathrm{AB}}\cdot\overrightarrow{\mathrm{AP}}\ge2+\sqrt{3}

What is the length of the locus of P?\mathrm{P}?  ~ [4 points]

A.π3B.π2C.2π3D.5π6E.π\begin{aligned}&\boxed{\text{A}}. && \frac{\pi}{3} \\\\ &\boxed{\text{B}}. && \frac{\pi}{2} \\\\ &\boxed{\text{C}}. && \frac{2\pi}{3} \\\\ &\boxed{\text{D}}. && \frac{5\pi}{6} \\\\ &\boxed{\text{E}}. && \pi \end{aligned}

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17. \huge 17.~ For a natural number n2,n\ge2, there's a pocket that contains nn cards where each card is written a different natural number from 11 to n.n. Choose 2 cards randomly, at the same time, from the pocket. Let the random variable XX be the product of the two numbers written on them. Below is the process of finding E(X).E(X).

When choosing 2 cards randomly, at the same time, from nn cards, the probability of choosing 22 particular cards is 2 (a) .\dfrac{2}{\boxed{\text{ (a) }}}.

Therefore, if we let SS be the sum of all product of the two chosen cards, E(X)=2 (a) ×S.E(X)=\dfrac{2}{\boxed{\text{ (a) }}}\times S.

Meanwhile, if we let r (2rn)r~(2\ge r \ge n) be the bigger number between the two chosen numbers,

S=r=2n[r×{1+2+3++(r2)+(r1)}]=r=2n{r×r(r1)2}=12r=2n(r3r2)=12r=1n(r3r2)=n(n+1)(n1)( (b) )24\begin{aligned} S & = \sum_{r=2}^{n} \left[r\times\{1+2+3+ \cdots +(r-2)+(r-1)\}\right] \\ & =\sum_{r=2}^{n} \left\{r\times \frac{r(r-1)}{2}\right\} \\ & = \frac{1}{2}\sum_{r=2}^{n} (r^3-r^2) \\ & = \frac{1}{2}\sum_{r=1}^{n} (r^3-r^2) \\ & = \frac{n(n+1)(n-1)(\boxed{\text{ (b) }})}{24}\end{aligned}

To sum up,

E(X)= (c) 12.E(X)=\dfrac{\boxed{\text{ (c) }}}{12}.

The correct expressions that fit in the blanks (a), (b), (c)\text{(a), (b), (c)} are f(n), g(n), h(n),f(n),~g(n),~h(n), respectively. Find the value of f(2)+g(3)+h(4).f(2)+g(3)+h(4).  ~ [4 points]

A.80B.81C.82D.83E.84\begin{aligned}&\boxed{\text{A}}. && 80 \\\\ &\boxed{\text{B}}. && 81 \\\\ &\boxed{\text{C}}. && 82 \\\\ &\boxed{\text{D}}. && 83 \\\\ &\boxed{\text{E}}. && 84 \end{aligned}

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18. \huge 18.~ A random variable XX that follows the normal distribution and a random variable ZZ that follows the standard normal distribution satisfy the below conditions for some constant a.a.

(1) X=a+2ZX = a+2Z

(2) P(X1)=P(X5)\mathrm{P}(X\le 1)=\mathrm{P}(X\ge 5)

Find the value of P(a2X2a+1)\mathrm{P}(a-2\le X \le 2a+1) using the standard normal distribution table shown on the right.  ~ [4 points]

A.0.5328B.0.6687C.0.7745D.0.8185E.0.9104\begin{aligned}&\boxed{\text{A}}. && 0.5328 \\\\ &\boxed{\text{B}}. && 0.6687 \\\\ &\boxed{\text{C}}. && 0.7745 \\\\ &\boxed{\text{D}}. && 0.8185 \\\\ &\boxed{\text{E}}. && 0.9104 \end{aligned}

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19. \huge 19.~ A parabola y2=4xy^2=4x has F\mathrm{F} as its focus, and some point in the first quadrant, P,\mathrm{P}, is on the parabola satisfying PF=4.\overline{\mathrm{PF}}=4. Let Q\mathrm{Q} be where the tangent line from P\mathrm{P} intersects the xx-axis, and let R\mathrm{R} be where that tangent line intersects the circle with QF\overline{\mathrm{QF}} as its radius. Find the area of triangle FRQ.\mathrm{FRQ}. (RQ\rm R\neq Q)  ~ [4 points]

A.23B.32C.33D.42E.43\begin{aligned}&\boxed{\text{A}}. && 2\sqrt{3} \\\\ &\boxed{\text{B}}. && 3\sqrt{2} \\\\ &\boxed{\text{C}}. && 3\sqrt{3} \\\\ &\boxed{\text{D}}. && 4\sqrt{2} \\\\ &\boxed{\text{E}}. && 4\sqrt{3} \end{aligned}

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20. \huge 20.~ For a function f(x)=(ex1)cos(π2x)+π2x,f(x)=(e^x-1)\cos\left(\dfrac{\pi}{2}x\right)+\dfrac{\pi}{2}x, which of the followings are true?  ~ [4 points]

(a) f(1)>0f(1)>0

(b) There exists a value for aa in the open interval (0, 1)(0,~1) that satisfies f(a)>0.f'(a)>0.

(c) There exists a value for bb in the open interval (0, 1)(0,~1) that satisfies f(b)=0.f'(b)=0.

A.(a)B.(c)C.(a), (b)D.(b), (c)E.(a), (b), (c)\begin{aligned}&\boxed{\text{A}}. && \text{(a)} \\\\ &\boxed{\text{B}}. && \text{(c)} \\\\ &\boxed{\text{C}}. && \text{(a), (b)} \\\\ &\boxed{\text{D}}. && \text{(b), (c)} \\\\ &\boxed{\text{E}}. && \text{(a), (b), (c)} \end{aligned}

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21. \huge 21.~ For a function f(x)f(x) that's continuous in closed interval [0, 1],\left[0,~1\right], define F(x)=0xf(t)dt (0x1).\displaystyle F(x)=\int_{0}^{x}f(t)dt~(0\le x \le 1). These two functions satisfy the below conditions.

(1) Function f(x)f(x) increases in closed interval [0, 1],\left[0,~1\right], and 01{f(x)+f(x)}dx=6.\displaystyle \int_{0}^{1}\{f(x)+|f(x)|\}dx=6.

(2) Function F(x)F(x) has its local minimum 1-1 at x=a (0<a<1).x=a~(0<a<1).

What's the value of a1F(x)f(x)dx?\displaystyle \int_{a}^{1}F(x)f(x)dx?  ~ [4 points]

A.12B.1C.32D.2E.52\begin{aligned}&\boxed{\text{A}}. && \frac{1}{2} \\\\ &\boxed{\text{B}}. && 1 \\\\ &\boxed{\text{C}}. && \frac{3}{2} \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && \frac{5}{2} \end{aligned}

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Short-answer\large \boxed{\text{Short-answer}}

22. \huge 22.~ In the expansion of (x+1)7(x+1)^7, find the coefficient of the x3x^3 term.  ~ [3 points]

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23. \huge 23.~ Find the sum of all the values for positive integer xx that satisfies the inequality (12)x5log381.\left(\dfrac{1}{2}\right)^{x-5}\ge \log_{3} 81.  ~ [3 points]

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24. \huge 24.~ The circle made by plane x+y+zk=0x+y+z-k=0 and sphere x2+y2+z22x3=0x^2+y^2+z^2-2x-3=0 intersecting has a radius of 1.1. Find the sum of all the values for real number k.k.  ~ [3 points]

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25. \huge 25.~ The sum of the xx-coordinate of all the points that two functions y=3tanxy=3\tan x and y=2cosxy=2\cos x meet at 0<x<2π0<x<2\pi is aπ.a\pi. Find the value of a constant a.a.  ~ [3 points]

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26. \huge 26.~ Five natural numbers a, b, c, d, ea,~b,~c,~d,~e satisfy (a+b+c)(d+e)2=36.(a+b+c)(d+e)^2=36. How many ordered pairs (a, b, c, d, e)(a,~b,~c,~d,~e) are there?  ~ [4 points]

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27. \huge 27.~ Pocket A\mathrm{A} and B\mathrm{B} both contain 44 balls numbered from 11 to 4.4. Amy randomly chooses 22 balls from pocket A\mathrm{A} and Bob randomly chooses 22 balls from pocket B\mathrm{B} at the same time. The probability of the biggest number among the chosen numbers coming only from among the balls Bob chose is pq.\dfrac{p}{q}. Find the value of p+q.p+q. (pp and qq are coprime natural numbers.)  ~ [4 points]

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28. \huge 28.~ Let F1, F1\mathrm{F}_1,~\mathrm{F}_1 ' be the two foci of an ellipse x2a2+y2b2=1 (a>b>0)\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1~(a>b>0) and let F2, F2\mathrm{F}_2,~ \mathrm{F}_2 ' be the two foci of a hyperbola x2a2y2b2=1.\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1. Then, define P1\mathrm{P}_1 as the point where the circle that is centered at F1\mathrm{F}_1 and passes through the origin intersects with the ellipse, and define P2\mathrm{P}_2 as the point where the circle that is centered at F2\mathrm{F}_2 and passes through the origin intersects with the hyperbola. The ellipse and the hyperbola satisfy the below conditions.

(1) The asymptotes of the hyperbola are y=±155x.y=\pm\dfrac{\sqrt{15}}{5}x.

(2) P1F1+P2F2=9.\overline{\mathrm{P}_1\mathrm{F}_1}+\overline{\mathrm{P}_2\mathrm{F}_2}=9.

Given that two points P1\mathrm{P}_1 and P2\mathrm{P}_2 are in the first quadrant, the xx-coordinates of F1\mathrm{F}_1 and F2\mathrm{F}_2 are positive, O\mathrm{O} is the origin, and OF1>a2,\overline{\mathrm{OF}_1}>\dfrac{a}{2}, find the value of P2F2P1F1.\overline{\mathrm{P}_2\mathrm{F}_2 '}-\overline{\mathrm{P}_1\mathrm{F}_1 '}.  ~ [4 points]

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29. \huge 29.~ There's a regular tetrahedron ABCD\mathrm{ABCD} with side length 2.2. Let P\mathrm{P} be the point that divides AB\overline{\mathrm{AB}} internally with a ratio of 2:1,2:1, and let Q\mathrm{Q} be a point on the triangle BCD.\mathrm{BCD}. Given that two vectors AP\overrightarrow{\mathrm{AP}} and PQ\overrightarrow{\mathrm{PQ}} are perpendicular, the maximum of PQ|\overrightarrow{\mathrm{PQ}}| is M,M, and M2=pq,M^2=\dfrac{p}{q}, for some coprime positive integer pp and q.q. Find the value of pq.pq.  ~ [4 points]

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30. \huge 30.~ For a 4th degree polynomial f(x)f(x) whose leading coefficient is 1,1, define a function g(x)g(x) whose domain is the set of all positive reals as g(x)=lnf(x)x.g(x)=\ln \dfrac{f(x)}{x}. Then, g(x)g(x) satisfies the below condition.

Function g(x)g(x) has its minima of 00 at both x=1x=1 and x=2.x=2.

Given that all the function values of f(x)f(x) are positive, find the value of f(3).f(3).  ~ [4 points]

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Well done! This is the end of the test.

Hope you had a lot of fun solving these!

Note by Boi (보이)
1 year, 7 months ago

No vote yet
1 vote

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This is the answer sheet. Ask for solutions if you're curious, or you can submit your solution as well, if you want to.

To prevent any possible spoiler, I will hide it into the reply. Click on the blue button below to show the answer sheet.

Boi (보이) - 1 year, 7 months ago

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(circled numbers from 1~5 are equivalent to A~E respectively.)

Boi (보이) - 1 year, 7 months ago

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These are nice questions.What would be a respectable score?What do top rankers get?

Anandh Rajan - 1 year, 5 months ago

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@Anandh Rajan I believe the top 4% score would be around 94~100.

Boi (보이) - 1 year, 5 months ago

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