When two constants $a , b$ satisfy $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - ( a + 2 ) x + 2 a } { x ^ { 2 } - b } = 3$, what is the value of $a + b$? [2 points] (1) $- 6$ (2) $- 4$ (3) $- 2$ (4) 0 (5) 2
On the coordinate plane, there are two arbitrary distinct vectors $\overrightarrow { \mathrm { OP } } , \overrightarrow { \mathrm { OQ } }$ with initial point at the origin O. When the endpoints $\mathrm { P } , \mathrm { Q }$ of the two vectors are translated 3 units in the $x$-direction and 1 unit in the $y$-direction to points $\mathrm { P } ^ { \prime } , \mathrm { Q } ^ { \prime }$ respectively, which of the following statements in are always true? [3 points] ㄱ. $\left| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OP } ^ { \prime } } \right| = \sqrt { 10 }$ ㄴ. $| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OQ } } | = \left| \overrightarrow { \mathrm { OP } ^ { \prime } } - \overrightarrow { \mathrm { OQ } ^ { \prime } } \right|$ ㄷ. $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OQ } } = \overrightarrow { \mathrm { OP } ^ { \prime } } \cdot \overrightarrow { \mathrm { OQ } ^ { \prime } }$ (1) ㄱ (2) ㄷ (3) ㄱ, ㄴ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two foci of the hyperbola $\frac { x ^ { 2 } } { 5 } - \frac { y ^ { 2 } } { 4 } = 1$, and let Q be the point symmetric to a point P on the hyperbola (not a vertex) with respect to the origin. When the area of quadrilateral $\mathrm { F } ^ { \prime } \mathrm { QFP }$ is 24, and the coordinates of point P are $( a , b )$, what is the value of $| a | + | b |$? [3 points] (1) 9 (2) 10 (3) 11 (4) 12 (5) 13
The figure on the right shows 6 ellipses, each with a side of a regular hexagon ABCDEF with side length 10 as the major axis, and with equal minor axis lengths. As shown in the figure, the sum of the areas of 6 triangles formed by a vertex of the regular hexagon and the foci of the two adjacent ellipses is $6 \sqrt { 3 }$. What is the length of the minor axis of the ellipse? [3 points] (1) $4 \sqrt { 2 }$ (2) 6 (3) $4 \sqrt { 3 }$ (4) 8 (5) $6 \sqrt { 2 }$
For two natural numbers $a , b ( a < b )$, the fractional inequality $$\frac { x } { x - a } + \frac { x } { x - b } \leqq 0$$ is satisfied by exactly 2 integers $x$. What is the number of ordered pairs $( a , b )$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is $$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c }
x ^ { 2 } & ( | x | < 1 ) \\
- 1 & ( | x | > 1 )
\end{array} \right.$$ Which of the following statements in are true? [3 points] ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$. ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$. ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$. (1) ㄱ (2) ㄴ (3) ㄷ (4) ㄱ, ㄷ (5) ㄱ, ㄴ, ㄷ
In coordinate space, the $xy$-plane, $yz$-plane, and $zx$-plane divide the space into 8 regions. How many of these 8 regions does the sphere $$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 24$$ pass through? [4 points] (1) 8 (2) 7 (3) 6 (4) 5 (5) 4
For a positive number $a$, let the characteristic and mantissa of $\log a$ be $f ( a )$ and $g ( a )$ respectively. Which of the following statements in are true? [3 points] ㄱ. $f ( 2006 ) = 3$ ㄴ. $g ( 2 ) + g ( 6 ) = g ( 12 ) + 1$ ㄷ. If $f ( a b ) = f ( a ) + f ( b )$, then $g ( a b ) = g ( a ) + g ( b )$. (1) ㄱ (2) ㄱ, ㄴ (3) ㄱ, ㄷ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
On the coordinate plane, for two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$, what is the total length of the figure represented by points $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points] (가) $x ^ { 2 } + y ^ { 2 } = 4$ (나) For any point $( 1 , a )$ on the line segment AB, the matrix $\left( \begin{array} { l l } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix. (1) $\frac { 1 } { 3 } \pi$ (2) $\frac { 1 } { 2 } \pi$ (3) $\pi$ (4) $\frac { 4 } { 3 } \pi$ (5) $\frac { 3 } { 2 } \pi$
The weight of products manufactured at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people A and B each independently randomly extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means of both A and B are between 10 and 14 inclusive? [3 points]
Q15
4 marksRadians, Arc Length and Sector AreaView
As shown in the figure, a sector $\mathrm { OA } _ { 1 } \mathrm { B } _ { 1 }$ with radius equal to the line segment $\mathrm { OA } _ { 1 }$ connecting the origin O and the point $\mathrm { A } _ { 1 } ( 0,8 )$, and with central angle $\theta$, is drawn. The foot of the perpendicular from point $\mathrm { B } _ { 1 }$ to the $x$-axis is $\mathrm { A } _ { 2 }$, and a sector $\mathrm { OA } _ { 2 } \mathrm { B } _ { 2 }$ with radius equal to the line segment $\mathrm { OA } _ { 2 }$ and central angle $\theta$ is drawn. The foot of the perpendicular from point $\mathrm { B } _ { 2 }$ to the $y$-axis is $\mathrm { A } _ { 3 }$, and a sector $\mathrm { OA } _ { 3 } \mathrm { B } _ { 3 }$ with radius equal to the line segment $\mathrm { OA } _ { 3 }$ and central angle $\theta$ is drawn. Continuing this process of alternately dropping perpendiculars to the $x$-axis and $y$-axis in the clockwise direction, let $l _ { n }$ be the length of the arc $\mathrm { A } _ { n } \mathrm { B } _ { n }$ of the sector $\mathrm { OA } _ { n } \mathrm { B } _ { n }$. When $\sum _ { n = 1 } ^ { \infty } l _ { n } = 12 \theta$, what is the value of $\sin \theta$? (Here, $0 < \theta < \frac { \pi } { 2 }$.) [4 points] (1) $\frac { 1 } { 7 }$ (2) $\frac { 1 } { 6 }$ (3) $\frac { 1 } { 5 }$ (4) $\frac { 1 } { 4 }$ (5) $\frac { 1 } { 3 }$
As shown in the figure, a rectangular solid is made from 12 transparent glass boxes in the shape of cubes of equal size. When 4 of these glass boxes are replaced with black glass boxes of the same size, and the view from above looks like (가) and the side view looks like (나), how many ways can this be done? [4 points] (1) 54 (2) 48 (3) 42 (4) 36 (5) 30
For the function $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$, find the value of $$\lim _ { h \rightarrow 0 } \frac { f ( 1 + 2 h ) - f ( 1 ) } { h }$$. [3 points]
When a solid of revolution is created by rotating the figure enclosed by the curve $y = a \left( 1 - x ^ { 2 } \right)$ and the $x$-axis around the $y$-axis, and the volume of the solid of revolution is $16 \pi$, find the positive value of $a$. [3 points]
The graph of the function $f ( x ) = x ^ { 3 }$ is translated $a$ units in the $x$-direction and $b$ units in the $y$-direction to obtain the graph of the function $y = g ( x )$. $$g ( 0 ) = 0 \text { and } \int _ { a } ^ { 3 a } g ( x ) dx - \int _ { 0 } ^ { 2 a } f ( x ) dx = 32$$ Find the value of $a ^ { 4 }$. [3 points]
Two spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81 , x ^ { 2 } + ( y - 5 ) ^ { 2 } + z ^ { 2 } = 56$ are denoted by $S _ { 1 } , S _ { 2 }$ respectively. Let P be a point on the circle formed by the intersection of the two spheres $S _ { 1 } , S _ { 2 }$, and let $\mathrm { P } ^ { \prime }$ be the orthogonal projection of point P onto the $xy$-plane. Let Q and R be the points where the sphere $S _ { 1 }$ intersects the $y$-axis. Find the maximum volume of the tetrahedron $\mathrm { PQP } ^ { \prime } \mathrm { R }$. [4 points]
There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, color region A in the figure on the right; if the number is 2, color region B. Continue throwing this box until both regions are colored. Find the probability that the process is completed on the 3rd throw, expressed as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]
Let C be the circle formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the plane $z = - 1$. When a plane $\alpha$ containing the $x$-axis intersects the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ to form a circle that meets C at exactly one point, and a normal vector to plane $\alpha$ is $\vec { n } = ( a , 3 , b )$, find the value of $a ^ { 2 } + b ^ { 2 }$. [4 points]
There is a line $l$ passing through the origin O with slope $\tan \theta$. Let $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ be the feet of the perpendiculars from two points $\mathrm { A } ( 0,2 ) , \mathrm { B } ( 2 \sqrt { 3 } , 0 )$ to line $l$, respectively. What is the value of $\theta$ that maximizes the sum of the distances from the origin O to point $\mathrm { A } ^ { \prime }$ and to point $\mathrm { B } ^ { \prime }$, $\overline { \mathrm { OA } ^ { \prime } } + \overline { \mathrm { OB } ^ { \prime } }$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [3 points] (1) $\frac { \pi } { 12 }$ (2) $\frac { \pi } { 6 }$ (3) $\frac { \pi } { 4 }$ (4) $\frac { \pi } { 3 }$ (5) $\frac { 5 } { 12 } \pi$
The distance between point O and point E is 40 m. As shown in the figure on the right, person A departs from point O and runs along the half-line OS perpendicular to segment OE at a constant speed of 3 m/s, and person B departs from point E 10 seconds after person A starts and runs along the half-line EN perpendicular to segment OE at a constant speed of 4 m/s. The angle formed by the intersection of the segment connecting the positions of persons A and B with segment OE is $\theta$ (in radians). What is the rate of change of $\theta$ at the moment 20 seconds after person A departs? [4 points] (1) $\frac { 21 } { 290 }$ radians/second (2) $\frac { 13 } { 290 }$ radians/second (3) $\frac { 7 } { 290 }$ radians/second (4) $\frac { 3 } { 290 }$ radians/second (5) $\frac { 1 } { 290 }$ radians/second
For a positive number $a$, on the closed interval $[ - a , a ]$, the function $$f ( x ) = \frac { x - 5 } { ( x - 5 ) ^ { 2 } + 36 }$$ has maximum value $M$ and minimum value $m$. Find the minimum value of $a$ such that $M + m = 0$. [4 points]