I. PART 1
QUESTION 01
If \(x > 0, y > 0\), and \(x + 3y = 2\), then \(\dfrac {1}{xy} \geq \bbox[10px, border: 2px solid red]{(1)}\) with equality if and only if \(x = \bbox[10px, border: 2px solid red]{(2)}\) and \(y = \bbox[10px, border: 2px solid red]{(3)}\).
QUESTION 02
The real-number solution to the equation \(2^{x + 2} - 2^{-x} + 3 = 0\) is \(x = \bbox[10px, border: 2px solid red]{(1)}\).
QUESTION 03
Let \(a, b\) be constants. If the polynomial \(x^4 + ax^3 + ax^2 + bx - 6\) is divisible by \((x - 1)^2\), then \(a = \bbox[10px, border: 2px solid red]{(1)}, b = \bbox[10px, border: 2px solid red]{(2)}\)
QUESTION 04
The maximum value of the function \(\sin 3x\) for \(\dfrac {5}{18} \pi \leq x \leq \dfrac 23 \pi \) is \(\bbox[10px, border: 2px solid red]{(1)}\), and the minimum value of that is \(\bbox[10px, border: 2px solid red]{(2)}\).
QUESTION 05
For the complex number \(z = \cos \dfrac 13 \pi + \sqrt{-1} \sin \dfrac 13 \pi\), the following equality holds: \(z + z^2 + z^3 + z^4 + z^5 = \bbox[10px, border: 2px solid red]{(1)}\).
QUESTION 06
Let S be a square with side length 1, T a triangle with side lengths \(1, 1, \sqrt{2}\) and R a triangle with side lengths \(1, \sqrt{2}, \sqrt{3}\). Let C be a pyramid with a base that is S and lateral faces consisting of two T's and two R's. The volume of C then is \(\bbox[10px, border: 2px solid red]{(1)}\).
II. PART 2
On the xy-plane, a circle with center \((a, b)\) and radius \(r\) is tangent to the parabola \(y = x^2\) at two distinct points. Fill in the blanks with the answers to the following questions.
QUESTION 01
When one of the two points of tangency is \((t,t^2)\), express \(a, b\), and \(r\) in terms of \(t\).
\(a = \bbox[10px, border: 2px solid red]{(1)}\)
\(b = \bbox[10px, border: 2px solid red]{(2)}\)
\(r = \bbox[10px, border: 2px solid red]{(3)}\)
QUESTION 02
When one of the two points of tangency is \(\left( \dfrac 12, \dfrac 14 \right)\), express the area S of the finite region bounded by the circle and the parabola in terms of \(\pi\).
\(S = \bbox[10px, border: 2px solid red]{(1)}\)
III. PART 3
Let \(f(x) = \int_0^x \dfrac {1}{1 + t^2} \: dt \quad x > 0\) and \(g(x) = f \left( \dfrac 1x \right)\).
QUESTION 01
Compute the derivatives \(f'(x) = \dfrac {d}{dx} f(x)\) and \(g'(x) = \dfrac {d}{dx} g(x)\).
\(f'(x) = \bbox[10px, border: 2px solid red]{(1)}\)
\(g'(x) = \bbox[10px, border: 2px solid red]{(2)}\)
QUESTION 02
Compute the value of \(f(1)\).
\(f(1) = \bbox[10px, border: 2px solid red]{(1)}\)
QUESTION 03
The function \(f(x) + g(x)\) is a constant \(C\). Find the value of \(C\).
\(C = \bbox[10px, border: 2px solid red]{(1)}\)
QUESTION 04
Calculate the limit \(\displaystyle \lim_{x \rightarrow \sim} \:f(x)\)
\(\displaystyle \lim_{x \rightarrow \sim} \:f(x) = \bbox[10px, border: 2px solid red]{(1)}\)