I. PART 1
QUESTION 01
The number of digits of \(7^{2677}\) is \(\bbox[10px, border: 2px solid red]{(1-1)}\) and the last digit of it is \(\bbox[10px, border: 2px solid red]{(1-2)}\), where \(\log_{10} 3 = 0.4771, \log_{10} 7 = 0.8451\)
QUESTION 02
Simplify \(\dfrac {1}{1 + \sqrt{2}} + \dfrac {1}{\sqrt{2} + \sqrt{3}} + \dfrac {1}{\sqrt{3} + \sqrt{4}} + \dfrac {1}{\sqrt{4} + \sqrt{5}}\) as briefly as possible.
The result is \(\bbox[10px, border: 2px solid red]{(1-3)}\).
QUESTION 03
Assume that \(0 < \theta < \dfrac {\pi}{4}\). If \(\sin 2 \theta = \dfrac 14\), then \(\dfrac {\sin \theta + \cos \theta}{- \sin \theta + \cos \theta} = \bbox[10px, border: 2px solid red]{(1-4)}\)
QUESTION 04
Let \(P_1, P_2, P_3, P_4, P_5\), and \(P_6\) be the vertices of a regular hexagon in anticlockwise order. We throw a fair dice three times and denote the scores shown on the dice as the ordered triple \((i, j, k)\). In this case, the probability that the three points \(P_i, P_j, P_k\) make a triangle is \(\bbox[10px, border: 2px solid red]{(1-5)}\).
QUESTION 05
For the equation \(4^x - 2^x - 12 = 0\), the real solution is \(x = \bbox[10px, border: 2px solid red]{(1-6)}\).
QUESTION 06
For a pyramid OABC, the centroids of the triangles OAB, OBC, and OCA are F, G, and H respectively. For the centroid P of the triangle FGH, the vector \(\overrightarrow {OP}\) is given by \(\overrightarrow {OP} = \dfrac {2}{\bbox[10px, border: 2px solid red]{(1-7)}} \left(\overrightarrow {OA} + \overrightarrow {OB} + \overrightarrow {OC} \right)\)
QUESTION 07
Let a point O be the origin of xy-coordinate plane. We define four points \(A(1, 0), B(1, 1), C(2,1), D(3,1)\) on the plane. Let us start from C, go through a point on line OA, go through a point on line OB, and reach D with the minimum length of the path. In this path, the point on the line OA is \(\left( \: \bbox[10px, border: 2px solid red]{(1-8)} , \bbox[10px, border: 2px solid red]{(1-9)} \: \right)\), that on the line OB is \(\left( \: \bbox[10px, border: 2px solid red]{(1-10)} , \bbox[10px, border: 2px solid red]{(1-11)} \: \right)\), the length of the path is \(\bbox[10px, border: 2px solid red]{(1-12)}\).
QUESTION 08
Assume that integers \(m\) and \(n\) satisfy \(2 | m | + 3 | n - 1 | \leq 7\). \(m + n\) is maximum when \((m , n) = \left( 3, \bbox[10px, border: 2px solid red]{(1-13)} \: \right)\), \(\left( \: \bbox[10px, border: 2px solid red]{(1-14)} , \bbox[10px, border: 2px solid red]{(1-15)} \: \right)\) and its maximum value is \(\bbox[10px, border: 2px solid red]{(1-16)}\).
QUESTION 09
If a quadratic function \(f(x)\) is maximum at \(x = 1\) with the maximum value \(5\), and satisfies \(f(-2) = -22\), it is given by \(f(x) = \bbox[10px, border: 2px solid red]{(1-17)} \: x^2 + \bbox[10px, border: 2px solid red]{(1-18)} \: x + \bbox[10px, border: 2px solid red]{(1-19)}\).
QUESTION 10
When integers \(k\) and \(n\) satisfy \(1 \leq k \leq n\), we have \(\displaystyle \sum_{l = k}^n 2^l = 2 \: ^{\bbox[10px, border: 2px solid red]{(1-20)}} - 2 \:^{\bbox[10px, border: 2px solid red]{(1-21)}}\).
Therefore, it follows that \(\displaystyle \sum_{k = 1}^n k \: 2^k = \sum_{k = 1}^n \sum_{l = k}^n 2^l = \left( \bbox[10px, border: 2px solid red]{(1-22)} \right) 2 \: ^{\bbox[10px, border: 2px solid red]{(1-23)}} + 2\).
QUESTION 11
decimal number 123456 is shown by a ternary (base 3) number \(\bbox[10px, border: 2px solid red]{(1-24)}
(Describe only the value of the ternary number without describing a notation that indicates a ternary numeral system.)
II. PART 2
For a cubic function \(f(x) = x^3 - 3ax^2 + 3bx - 2\), answer the following questions and fill in your responses in the corresponding boxes on the answer sheet.
QUESTION 01
If \(x = 1,3\) are the extreme points of \(f(x)\), then \(a = \bbox[10px, border: 2px solid red]{(2-1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2-2)}\). In this case, the solutions of \(f(x) = 0\) can be arranged as \( \bbox[10px, border: 2px solid red]{(2-3)} < \bbox[10px, border: 2px solid red]{(2-4)} < \bbox[10px, border: 2px solid red]{(2-5)}\) in increasing order.
QUESTION 02
Assume that \(a = b\). If the function \(f(x)\) is monotonously increasing, then \( \bbox[10px, border: 2px solid red]{(2-6)} < a < \bbox[10px, border: 2px solid red]{(2-7)}\).
III. PART 3
In xyz-coordinate system, we define a solid A by \(\dfrac 19 x^2 + \dfrac 14 y^2 \leq z^4\), \(0 \leq z \leq 1\).
Fill in your responses in the corresponding boxes on the answer sheet.
QUESTION 01
We define a solid B by \(x^2 + y^2 \leq z^4\)
The volume of the solid B is \(\bbox[10px, border: 2px solid red]{(3-1)}\).
QUESTION 02
The solid A is given by elongating the solid B \(\bbox[10px, border: 2px solid red]{(3-2)}\) times in the x-axis direction and \(\bbox[10px, border: 2px solid red]{(3-3)}\) times in the y-axis direction.
QUESTION 03
The volume of the solid A is \(\bbox[10px, border: 2px solid red]{(3-4)}\) times as large as that of the solid B.
QUESTION 04
The volume of the solid A is \(\bbox[10px, border: 2px solid red]{(3-5)}\).