PART 1

QUESTION 01

Find the range of \(x\) that satisfies the following inequality \(| x + 3 | < 4x\). The answer is \(\bbox[10px, border: 2px solid red]{(1)}\)

QUESTION 02

The number of solutions \((x,y,z)\) of the equation \(x + y + z = 4\), where \(x\), \(y\) and \(z\) are zero or positive integers is \(\bbox[10px, border: 2px solid red]{(1)}\).

QUESTION 03

On the plane \(xy\), there are two points: \(0(0,0), A(6,8)\). The equation of the circle with a diameter of the line segment \(OA\) is \(\left( x - \bbox[10px, border: 2px solid red]{(1)} \right)^2 + \left( y - \bbox[10px, border: 2px solid red]{(2)} \right)^2 = \bbox[10px, border: 2px solid red]{(3)}\)

QUESTION 04

\(\log_4 9 = \log_2 \: \bbox[10px, border: 2px solid red]{(1)}\), \(\log_9 4 = \log_3 \: \bbox[10px, border: 2px solid red]{(2)}\), hence \(\left( \log_2 3 + \log_4 9 \right) \left( \log_3 2 + \log_9 4 \right) = \bbox[10px, border: 2px solid red]{(3)}\)

QUESTION 05

\(\sqrt [6] {25} \times \sqrt [3] {25} : \sqrt {5} = \bbox[10px, border: 2px solid red]{(1)}\)

QUESTION 06

Let the sequence \(a_n\) \(n = 1, 2, 3, \dotso\) be a geometric progression satisfying \(a_1 + a_2 + a_3 = 14\), \(a_2 + a_3 + a_4 = -42\). When we denote the first term of \(a_n\) by \(a\), and the common ratio by \(r\), we have \(a = \bbox[10px, border: 2px solid red]{(1)}\) and \(r = \bbox[10px, border: 2px solid red]{(2)}\).

QUESTION 07

Let \(\vec a = (1,0,-1), \vec b = (-2,2,1), \vec c = (x,y,z) \: x> 0\) and \(\| \vec c \| = 3\). When \(\vec c\) is perpendicular to both \(\vec a\) and \(\vec b\), then \(x = \bbox[10px, border: 2px solid red]{(1)}, y = \bbox[10px, border: 2px solid red]{(2)}, z = \bbox[10px, border: 2px solid red]{(3)}\)

QUESTION 08

Let M denote the midpoint of side BC of a triangle ABC.

When \(BC = 8, CA = 4, AB = 6\), then \(\cos ABC = \bbox[10px, border: 2px solid red]{(1)}, AM = \bbox[10px, border: 2px solid red]{(2)}\)

QUESTION 09

The equation of the tangent to the curve \(f(x) = -x^2 + x + 2\) at the point \((0,2)\) is \(y = \bbox[10px, border: 2px solid red]{(1)}\), and the area of the region bounded by the curve \(f(x)\), the tangent and the x axis is \(\bbox[10px, border: 2px solid red]{(2)}\).