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

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

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\[\huge \text{Rules:}\]

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

2.This test has a time limit of100 minutes.

3.Any use of calculators is consideredcheating.

4.This test consists of30questions.

5.The first21are multiple choice questions, and the rest,9, are short-answer questions.

6.The total score is100 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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\(\large \boxed{\text{Multiple choice}}\)

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

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

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{3} \\\\ &\boxed{\text{B}}. && \frac{1}{2} \\\\ &\boxed{\text{C}}. && 1 \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && 3 \end{align}\)

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

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

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{2} \\\\ &\boxed{\text{B}}. && \frac{2}{3} \\\\ &\boxed{\text{C}}. && \frac{3}{4} \\\\ &\boxed{\text{D}}. && \frac{4}{5} \\\\ &\boxed{\text{E}}. && \frac{5}{6} \end{align}\)

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

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

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{3} \\\\ &\boxed{\text{B}}. && \frac{1}{2} \\\\ &\boxed{\text{C}}. && 1 \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && 3 \end{align}\)

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

\(\begin{align}&\boxed{\text{A}}. && \frac{2}{27} \\\\ &\boxed{\text{B}}. && \frac{1}{9} \\\\ &\boxed{\text{C}}. && \frac{4}{27} \\\\ &\boxed{\text{D}}. && \frac{2}{9} \\\\ &\boxed{\text{E}}. && \frac{7}{27} \end{align}\)

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

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

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

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

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

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

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

\(\begin{align}&\boxed{\text{A}}. && \frac{2\pi - 1}{12} \\\\ &\boxed{\text{B}}. && \frac{2\pi - 3}{12} \\\\ &\boxed{\text{C}}. && \frac{2\pi - 5}{12} \\\\ &\boxed{\text{D}}. && \frac{4\pi - 7}{12} \\\\ &\boxed{\text{E}}. && \frac{4\pi - 9}{12} \end{align}\)

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{6} \\\\ &\boxed{\text{B}}. && \frac{1}{3} \\\\ &\boxed{\text{C}}. && \frac{1}{2} \\\\ &\boxed{\text{D}}. && \frac{2}{3} \\\\ &\boxed{\text{E}}. && \frac{5}{6} \end{align}\)

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\(\huge 13.~\) A sample of size \(16\) and average \(\overline{X}\) is arbitrarily chosen from a population that follows the normal distribution \(\mathrm{N}(m,~1).\) The random variable of the population is \(X.\) Given that \(\mathrm{P}(X\ge5)=\mathrm{P}(\overline{X}\le 1)\) and \(m<5,\) find the value of \(m.\) \(~\) **[3 points]**

\(\begin{align}&\boxed{\text{A}}. && \frac{3}{5} \\\\ &\boxed{\text{B}}. && 1 \\\\ &\boxed{\text{C}}. && \frac{7}{5} \\\\ &\boxed{\text{D}}. && \frac{9}{5} \\\\ &\boxed{\text{E}}. && \frac{11}{5} \end{align}\)

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{4} \\\\ &\boxed{\text{B}}. && \frac{1}{2} \\\\ &\boxed{\text{C}}. && 1 \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && 4 \end{align}\)

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{e^2} \\\\ &\boxed{\text{B}}. && \frac{2}{e^2} \\\\ &\boxed{\text{C}}. && \frac{3}{e^2} \\\\ &\boxed{\text{D}}. && \frac{4}{e^2} \\\\ &\boxed{\text{E}}. && \frac{5}{e^2} \end{align}\)

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

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

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

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

\(\begin{align}&\boxed{\text{A}}. && \frac{\pi}{3} \\\\ &\boxed{\text{B}}. && \frac{\pi}{2} \\\\ &\boxed{\text{C}}. && \frac{2\pi}{3} \\\\ &\boxed{\text{D}}. && \frac{5\pi}{6} \\\\ &\boxed{\text{E}}. && \pi \end{align}\)

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\(\huge 17.~\) For a natural number \(n\ge2,\) there's a pocket that contains \(n\) cards where each card is written a different natural number from \(1\) to \(n.\) Choose 2 cards randomly, at the same time, from the pocket. Let the random variable \(X\) be the product of the two numbers written on them. Below is the process of finding \(E(X).\)

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

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

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

\(\begin{align} S & = \sum_{r=2}^{n} \left[r\times\{1+2+3+ \cdots +(r-2)+(r-1)\}\right] \\ & =\sum_{r=2}^{n} \left\{r\times \frac{r(r-1)}{2}\right\} \\ & = \frac{1}{2}\sum_{r=2}^{n} (r^3-r^2) \\ & = \frac{1}{2}\sum_{r=1}^{n} (r^3-r^2) \\ & = \frac{n(n+1)(n-1)(\boxed{\text{ (b) }})}{24}\end{align}\)

To sum up,

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

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

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

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\(\huge 18.~\) A random variable \(X\) that follows the normal distribution and a random variable \(Z\) that follows the standard normal distribution satisfy the below conditions for some constant \(a.\)

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

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

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

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

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

\(\begin{align}&\boxed{\text{A}}. && 2\sqrt{3} \\\\ &\boxed{\text{B}}. && 3\sqrt{2} \\\\ &\boxed{\text{C}}. && 3\sqrt{3} \\\\ &\boxed{\text{D}}. && 4\sqrt{2} \\\\ &\boxed{\text{E}}. && 4\sqrt{3} \end{align}\)

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

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

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

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

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

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

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

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

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

\(\begin{align}&\boxed{\text{A}}. && \frac{1}{2} \\\\ &\boxed{\text{B}}. && 1 \\\\ &\boxed{\text{C}}. && \frac{3}{2} \\\\ &\boxed{\text{D}}. && 2 \\\\ &\boxed{\text{E}}. && \frac{5}{2} \end{align}\)

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

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

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

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\(\huge 24.~\) The circle made by plane \(x+y+z-k=0\) and sphere \(x^2+y^2+z^2-2x-3=0\) intersecting has a radius of \(1.\) Find the sum of all the values for real number \(k.\) \(~\) **[3 points]**

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

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

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\(\huge 27.~\) Pocket \(\mathrm{A}\) and \(\mathrm{B}\) both contain \(4\) balls numbered from \(1\) to \(4.\) Amy randomly chooses \(2\) balls from pocket \(\mathrm{A}\) and Bob randomly chooses \(2\) balls from pocket \(\mathrm{B}\) at the same time. The probability of the biggest number among the chosen numbers coming only from among the balls Bob chose is \(\dfrac{p}{q}.\) Find the value of \(p+q.\) (\(p\) and \(q\) are coprime natural numbers.) \(~\) **[4 points]**

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\(\huge 28.~\) Let \(\mathrm{F}_1,~\mathrm{F}_1 '\) be the two foci of an ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1~(a>b>0)\) and let \(\mathrm{F}_2,~ \mathrm{F}_2 '\) be the two foci of a hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1.\) Then, define \(\mathrm{P}_1\) as the point where the circle that is centered at \(\mathrm{F}_1\) and passes through the origin intersects with the ellipse, and define \(\mathrm{P}_2\) as the point where the circle that is centered at \(\mathrm{F}_2\) 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 \(y=\pm\dfrac{\sqrt{15}}{5}x.\)

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

Given that two points \(\mathrm{P}_1\) and \(\mathrm{P}_2\) are in the first quadrant, the \(x\)-coordinates of \(\mathrm{F}_1\) and \(\mathrm{F}_2\) are positive, \(\mathrm{O}\) is the origin, and \(\overline{\mathrm{OF}_1}>\dfrac{a}{2},\) find the value of \(\overline{\mathrm{P}_2\mathrm{F}_2 '}-\overline{\mathrm{P}_1\mathrm{F}_1 '}.\) \(~\) **[4 points]**

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

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

Function \(g(x)\) has its minima of \(0\) at both \(x=1\) and \(x=2.\)

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

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Well done! This is the end of the test.

Hope you had a lot of fun solving these!

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