QUESTIONS AND SOLUTIONS

 

I. PART 1

 

QUESTION 01

\(\log_5 0.008 = \bbox[10px, border: 2px solid red]{(1)}\), \(\left( \sqrt[6] {16} \right)^3 = \bbox[10px, border: 2px solid red]{(2)}\)

 

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QUESTION 02

\(\sin 75^{\text{o}} + \sin 120^{\text{o}}  - \cos 150^{\text{o}} + \cos 165^{\text{o}} = \bbox[10px, border: 2px solid red]{(1)}\).

 

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QUESTION 03

\(\dfrac {1}{3 \:.\: 6} + \dfrac {1}{6 \:.\: 9} + \dfrac {1}{9 \:.\: 12} + \dfrac {1}{12 \:.\: 15} = \bbox[10px, border: 2px solid red]{(1)}\)

 

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QUESTION 04

The number of integers that satisfy the following inequalities \(-x < x^2 < 6\) is \(\bbox[10px, border: 2px solid red]{(1)}\)

 

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QUESTION 05

Among four-digit integers where digits are all different numerals, the total possible number of integers that are greater than or equal to 5000 is \(\bbox[10px, border: 2px solid red]{(1)}\).

 

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QUESTION 06

When \(\vec a + \vec b + \vec c = 0\) and \(\| \vec a \| = \| \vec b \| = \| \vec c \| = 1\), ，then the degree measure of the angle between \(\vec a\) and \(\vec b\) is \(\bbox[10px, border: 2px solid red]{(1)}\) and \(\| \vec a - \vec b \| = \bbox[10px, border: 2px solid red]{(2)}\).

 

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QUESTION 07

In the progression \(3, 4, 6, 10, 18, \dotso\), the numeral of the 8th term is \(\bbox[10px, border: 2px solid red]{(1)}\), and the number of term that is 1026 is \(\bbox[10px, border: 2px solid red]{(2)}\).

 

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QUESTION 08

Let \(f(x) = x^2 - 4x + 1\).

(i)   \(f(-2) = \bbox[10px, border: 2px solid red]{(1)}\)

(ii)   If \(f(x) = 0, x = \bbox[10px, border: 2px solid red]{(2)} \text{ or } x = \bbox[10px, border: 2px solid red]{(3)}\) , where \((2) < (3)\)

(iii)   The area bounded by the parabola \(y = f(x)\) and the x-axis is \(\bbox[10px, border: 2px solid red]{(4)}\)

 

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QUESTION 09

In a space with a coordinate system, there are three points \(A(0,1,1), B(-1,-1,2)\) and \(C(2,3,1)\). The area of \(\Delta ABC\) is \( \bbox[10px, border: 2px solid red]{(1)}\).

 

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II. PART 2

 

A quadrangle ABCD which is inscribed in a circle on a plane satisfies 2AB＝AC, BC＝3, BD = DC and ∠BAC＝60º.

 

QUESTION 01

(i)   The radius of the circumscribed circle of the quadrangle ABCD is \(\bbox[10px, border: 2px solid red]{(1)}\)

(ii)   \(AC = \bbox[10px, border: 2px solid red]{(2)}\)

(iii)   \(\angle BDC = \bbox[10px, border: 2px solid red]{(3)}\)

(iv)   The area of \(\Delta BDC = \bbox[10px, border: 2px solid red]{(4)}\)

(v)   The scalar product of two vectors \(\overrightarrow {DC} \cdot \overrightarrow {CA} = \bbox[10px, border: 2px solid red]{(5)}\)

 

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III PART 3

 

QUESTION 01

On the plane \(xy\), the graph of the parabola \(y = ax^2 + bx + c\) is shown in the figure below. Judge whether the following expressions are larger than or smaller than zero.

Fill in the blanks with the correct marks: > or < .

 

 

 

 

 

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