I. PART 1
QUESTION 01
When \(\dfrac {a + b}{4} = \dfrac {b+c}{5} = \dfrac{c+a}{6}\), then ratio \(a : b : c = 1 : \bbox[10px, border: 2px solid red]{(1)} : \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 02
When \(a = 3+2 \sqrt{2}\) and \(b = 3 - 2 \sqrt{2}\), then \(a^2 + b^2 = \bbox[10px, border: 2px solid red]{(1)}\), \(\dfrac {a^2}{b} + \dfrac {b^2}{a} = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 03
\(\cos 30^{\text{o}} \sin 45^{\text{o}} \tan 60^{\text{o}} + \cos 135^{\text{o}} \sin 120^{\text{o}} \tan 150^{\text{o}} = \bbox[10px, border: 2px solid red]{(1)}\)
QUESTION 04
The solutions of an equation \((x + 1)^2 + 9 (x + 1) + 20 = 0\) are \(\bbox[10px, border: 2px solid red]{(1)}\) and \(\bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 05
When the range of \(x\) determined by \(-ax^2 + bx + 4 \geq 0\) is \(- \dfrac 13 \leq x \leq 4\), then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 06
When \(a = 3\) and \(b = 2\), then \(\log_a b^a \times \log_b a^b = \bbox[10px, border: 2px solid red]{(1)}\)
QUESTION 07
Let \(f(x) =-x^2 - 2ax + b, (a \neq 0)\).
When \(f (1) = 3\) and the maximum value of \(f (x)\) is \(4\), then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 08
Let a sequence \(a_n = 2, 5, 8, 11, \dotso , a_n\). If \(a_n > 100\), the minimum value of \(n\) is \(\bbox[10px, border: 2px solid red]{(1)}\).
QUESTION 09
Let \(f(x) = x^3 - 2x + 4\).
(i) \(f(2) = \bbox[10px, border: 2px solid red]{(1)}\).
(ii) Differential coefficient \(f'(2) = \bbox[10px, border: 2px solid red]{(2)}\).
(iii) If \(f(x) = 0\), the real value of \(x = \bbox[10px, border: 2px solid red]{(3)}\).
(iv) The definite integral \(\displaystyle \int_0^2 f(x) \: dx = \bbox[10px, border: 2px solid red]{(4)}\).
II. PART 2
On the plane \(xy\), there are four points: \(A (a, b)\), \(B (-1, 0)\), \(C (2, 1)\) and \(D (0,2)\).
QUESTION 01
If point \(D\) is the center of \(\Delta ABC\), then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 02
If quadrilateral \(ABCD\) is a parallelogram, then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 03
If \(\angle ABC = 90^{\text{o}}\) and point \(D\) is on the side \(AC\), then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 04
If vector \(\overrightarrow {CA} = 2 \: \overrightarrow {CB} - 3 \: \overrightarrow {CD}\), then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 05
If scalar product \(\overrightarrow {BA} \cdot \overrightarrow {BC} = -2\) and scalar product \(\overrightarrow {BA} \cdot \overrightarrow {BD} = 1\), then \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(b = \bbox[10px, border: 2px solid red]{(2)}\).
III. PART 3
Choose two inequalities which represent the hatched area not containing the border, from (1) to (10).
