PART 1
QUESTION 01
\(\log_5 0.008 = \bbox[10px, border: 2px solid red]{(1)}\), \(\left( \sqrt[6] {16} \right)^3 = \bbox[10px, border: 2px solid red]{(2)}\)
Solution
\begin{equation*}
\begin{split}
& \log_5 0.008 \\\\
& \log_5 \frac{8}{1000} \\\\
& \log_5 \frac{1}{125} \\\\
& \log_5 5^{-3} \\\\
& -3 \:.\: \log_5 5 \\\\
& \bbox[5px, border: 2px solid magenta] {-3}
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& \left( \sqrt[6] {16} \right)^3 \\\\
& \left( \sqrt[6] {2^4} \right)^3 \\\\
& \left( 2^{\frac 46} \right)^3 \\\\
& 2^{\frac 42} \\\\
& 2^2 \\\\
& \bbox[5px, border: 2px solid magenta] {4}
\end{split}
\end{equation*}
QUESTION 02
\(\sin 75^{\text{o}} + \sin 120^{\text{o}} – \cos 150^{\text{o}} + \cos 165^{\text{o}} = \bbox[10px, border: 2px solid red]{(1)}\).
Solution
[s2If current_user_can(access_s2member_level3)]
\begin{equation*}
\begin{split}
& \sin 75^{\text{o}} + \sin 120^{\text{o}} – \cos 150^{\text{o}} + \cos 165^{\text{o}} \\\\
& \sin 75^{\text{o}} + \sin (180^{\text{o}} – 60^{\text{o}}) – \cos (180^{\text{o}} – 30^{\text{o}}) + \cos (180^{\text{o}} – 15^{\text{o}}) \\\\
& \sin 75^{\text{o}} + \sin 60^{\text{o}} – (- \cos 30^{\text{o}}) + (- \cos 15^{\text{o}}) \\\\
& \sin (90^{\text{o}} – 15^{\text{o}}) + \sin 60^{\text{o}} + \cos 30^{\text{o}} – \cos 15^{\text{o}} \\\\
& \cancel{\cos 15^{\text{o}}} + \sin 60^{\text{o}} + \cos 30^{\text{o}} – \cancel{\cos 15^{\text{o}}} \\\\
& \sin 60^{\text{o}} + \cos 30^{\text{o}} \\\\
& \frac 12 \sqrt{3} + \frac 12 \sqrt{3} \\\\
& \bbox[5px, border: 2px solid magenta] {\sqrt{3}}
\end{split}
\end{equation*}
[else]
Require Platinum Membership
[/s2If]
QUESTION 03
\(\dfrac {1}{3 \:.\: 6} + \dfrac {1}{6 \:.\: 9} + \dfrac {1}{9 \:.\: 12} + \dfrac {1}{12 \:.\: 15} = \bbox[10px, border: 2px solid red]{(1)}\)
Solution
[s2If current_user_can(access_s2member_level3)]
\begin{equation*}
\begin{split}
& \frac {1}{3 \:.\: 6} + \frac {1}{6 \:.\: 9} + \frac {1}{9 \:.\: 12} + \frac {1}{12 \:.\: 15} \\\\
& \frac {1}{3 \:.\: 3 \:.\: 2} + \frac {1}{3 \:.\: 2 \:.\: 3 \:.\: 3} + \frac {1}{3 \:.\: 3 \:.\: 3 \:.\: 4} + \frac {1}{3 \:.\: 4 \:.\: 3 \:.\: 5} \\\\
& \frac {1}{3^2} \left( \frac {1}{2} + \frac {1}{6} + \frac {1}{12} + \frac {1}{20} \right) \\\\
& \frac {1}{9} \left( \frac {30 + 10 + 5 + 3}{60} \right) \\\\
& \frac {1}{9} \left( \frac {48}{60} \right) \\\\
& \frac {1}{9} \left( \frac {4}{5} \right) \\\\
& \bbox[5px, border: 2px solid magenta] {\frac {4}{45}}
\end{split}
\end{equation*}
[else]
Require Platinum Membership
[/s2If]
QUESTION 04
The number of integers that satisfy the following inequalities \(-x < x^2 < 6\) is \(\bbox[10px, border: 2px solid red]{(1)}\)
Solution
[s2If current_user_can(access_s2member_level3)]
\begin{equation*}
\begin{split}
& x^2 < 6 \\\\
& x^2 – 6 < 0 \\\\
& (x + \sqrt{6})(x – \sqrt{6}) < 0
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
& -x < x^2 \\\\
& 0 < x^2 + x \\\\
& 0 < x(x + 1)
\end{split}
\end{equation*}
Solution:
\(\{- \sqrt{6} < x < 0\}\), the integers are \(-2, -1\)
\(\{1 < x < \sqrt{6}\}\), the integers are \(2\)
The number of integers are \(\bbox[5px, border: 2px solid magenta] {3}\)
[else]
Require Platinum Membership
[/s2If]
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 \(\bbox[10px, border: 2px solid red]{(1)}\).
Solution
[s2If current_user_can(access_s2member_level3)]
\begin{equation*}
\begin{split}
& 5 \:.\: 9 \:.\: 8 \:.\: 7 \\\\
& \bbox[5px, border: 2px solid magenta] {2520}
\end{split}
\end{equation*}
[else]
Require Platinum Membership
[/s2If]
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 \(\| \vec a – \vec b \| = \bbox[10px, border: 2px solid red]{(2)}\).
Solution
[s2If current_user_can(access_s2member_level3)]
[else]
Require Platinum Membership
[/s2If]
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 \(\bbox[10px, border: 2px solid red]{(2)}\).
Solution
[s2If current_user_can(access_s2member_level3)]
[else]
Require Platinum Membership
[/s2If]
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 \(\bbox[10px, border: 2px solid red]{(4)}\)
Solution
[s2If current_user_can(access_s2member_level3)]
[else]
Require Platinum Membership
[/s2If]
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 \( \bbox[10px, border: 2px solid red]{(1)}\).
Solution
[s2If current_user_can(access_s2member_level3)]
[else]
Require Platinum Membership
[/s2If]
