QUESTIONS AND SOLUTIONS

 

I. PART 1

 

QUESTION 01

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

 

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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 \(\bbox[10px, border: 2px solid red]{(1)}\).

 

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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 \(\left( x - \bbox[10px, border: 2px solid red]{(1)} \right)^2 + \left( y - \bbox[10px, border: 2px solid red]{(2)} \right)^2 = \bbox[10px, border: 2px solid red]{(3)}\)

 

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

 

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QUESTION 05

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

 

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

 

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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 \(x = \bbox[10px, border: 2px solid red]{(1)}, y = \bbox[10px, border: 2px solid red]{(2)}, z = \bbox[10px, border: 2px solid red]{(3)}\)

 

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QUESTION 08

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

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

 

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

 

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II. PART 2

 

A triangle ABC on a plane satisfies \(AC = BC\) and \(\angle ACB = 90^{\text{o}}, DC = 1, \angle AHD = 90^{\text{o}}\) and \(\angle ADC = 60^{\text{o}}\).

 

QUESTION 01

(i)   The radius of the circumscribed circle of \(\Delta ADC = \bbox[10px, border: 2px solid red]{(1)}\).

(ii)   The radius of the circumscribed circle of \(\Delta ABC = \bbox[10px, border: 2px solid red]{(2)}\).

(iii)   The radius of the inscribed circle of \(\Delta ABC = \dfrac {\bbox[10px, border: 2px solid red]{(3)} \sqrt{\bbox[10px, border: 2px solid red]{(4)}} - \sqrt{\bbox[10px, border: 2px solid red]{(5)}}}{\bbox[10px, border: 2px solid red]{(6)}} \).

(iv)   \(DH = \dfrac {\sqrt{\bbox[10px, border: 2px solid red]{(7)}} - \sqrt{\bbox[10px, border: 2px solid red]{(8)}}}{\bbox[10px, border: 2px solid red]{(9)}} \)

(v)   \(\cos DAH = \dfrac {\sqrt{\bbox[10px, border: 2px solid red]{(10)}} + \sqrt{\bbox[10px, border: 2px solid red]{(11)}}}{\bbox[10px, border: 2px solid red]{(12)}}\), \((10) > (11)\)

 

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III. PART 3

 

QUESTION 01

On the plane \(xy\), there are two straight lines (1) and (2), two parabolas (3) and (4) and a circle (5) as shown in a lower figure. Choose the correct equation from (1) to (10) to satisfy each graph and fill in the blank with the number.

 

 

 

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