2012 澳洲 AMC E-Senior 真题 答案 详解

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中英双语真题

Advanced Exam (11-12th Grade)

1-10 questions, 3 points each

1. Calculate (2012 - 2.012): (A) 2010.012 (B) 2010.998 (C) 2009.998 (D) 2012.012 (E) 2009.988

2. What number divided by 6 equals (\frac{1}{3})? (A) 18 (B) (\frac{1}{2}) (C) (\frac{1}{18}) (D) 2 (E) 9

3. In the right triangle, find the value of (x) given angles: (A) 90 (B) 95 (C) 100 (D) 110 (E) 120

4. Find which point lies on the line connecting (0, 2) and (2, 8): (A) (4, 14) (B) (3, 14) (C) (3, 10) (D) (4, 10) (E) (2, 0)

5. Determine the number between (\frac{a}{b}) and (\frac{c}{b}): (A) 1 (B) (\frac{a-c}{b}) (C) (\frac{a+c}{b}) (D) (\frac{a-c}{2b}) (E) (\frac{a+c}{2b})

6. Solve for (x) if (3^{16} \times 27^{10} = 9^x): (A) 18 (B) 21 (C) 23 (D) 26 (E) 27

7. Given (\frac{p}{p-2q} = 3), determine (\frac{p}{q}): (A) 3 (B) (\frac{1}{3}) (C) (\frac{2}{3}) (D) 2 (E) -3

英文真题

Senior Division
Questions 1 to 10, 3 marks each

1. The expression (2012 - 2.012) equals
   (A) (2010.012)
   (B) (2010.998)
   (C) (2009.998)
   (D) (2012.012)
   (E) (2009.988)

2. By what number must 6 be divided to obtain (\frac{1}{3}) as a result?
   (A) (18)
   (B) (\frac{1}{2})
   (C) (\frac{1}{18})
   (D) (2)
   (E) (9)

3. In the diagram, the sizes in degrees of three angles are given. Find the value of (x).
   (A) (90)
   (B) (95)
   (C) (100)
   (D) (110)
   (E) (120)

4. The straight line joining the points ((0,2)) and ((2,8)) also passes through
   (A) ((4,14))
   (B) ((3,14))
   (C) ((3,10))
   (D) ((4,10))
   (E) ((2,0))

5. What is the number that is halfway between (\frac{a}{b}) and (\frac{c}{b})?
   (A) (1)
   (B) (\frac{a-c}{b})
   (C) (\frac{a+c}{b})
   (D) (\frac{a-c}{2b})
   (E) (\frac{a+c}{2b})

6. If (3^{16} \times 27^{10} = 9^x), then (x) equals
   (A) (18)
   (B) (21)
   (C) (23)
   (D) (26)
   (E) (27)

7. If (\frac{p}{p-2q} = 3) then (\frac{p}{q}) equals
   (A) (3)
   (B) (\frac{1}{3})
   (C) (\frac{2}{3})
   (D) (2)
   (E) (-3)

真题文字详解(英文)

SOLUTIONS - SENIOR DIVISION 1. The value of 2012 - 2.012 is 2009.988, hence (E). 2. (Also J10 & I5) From ( \frac{6}{x} = \frac{1}{3} ) we get ( x = 3 \times 6 = 18 ), hence (A). 3. Alternative 1 Join the dotted line as shown. Then, using the exterior angle of a triangle twice, we get ( x = \alpha + 30 + \beta + 40 = 70 + \alpha + \beta = 120 ), hence (E). Alternative 2 The sum of the interior angles is 360°. So ( 30 + 40 + 50 + (360 - x) = 360 ), ( x = 120 ), hence (E). 4. The line joining (0,2) and (2,8) has gradient ( \frac{8-2}{2} = 3 ). So, any point on this line when joined to (0,2) will form a line with a gradient of 3. Checking, with (A) gradient is ( \frac{14-2}{3} = 4 ), with (B) gradient is ( \frac{14-2}{4} = 3 ), with (C) gradient is ( \frac{10-2}{3} = 2\frac{2}{3} ), with (D) gradient is ( \frac{10-2}{4} = 2 ) and with (E) gradient is ( \frac{2-0}{0-2} = -1 ), hence (A). 56