key to correction g8

MATH QUIZ (Individual-Written) Contestant’s Code Number: KEY TO CORRECTION

GRADE 8

SCORE

Category B:

2016 DIVISION FESTIVAL OF TALENTS

General Instruction: Write your final answer on the space provided before each item. Part I. No solution is required for this part of the quiz. Two points will be given for every correct answer. 𝟓 𝟑 1. Rewrite 5𝑥 − 7𝑦 − 3 = 0 in the slope-intercept form. 𝒚= 𝒙− 𝟕 𝟕 2. Suppose that 3𝑥 + 5 > 0 and 𝑥 is an integer, what is the minimum -1 value of 𝑥? 3 3. When 𝑥−5 is subtracted from a rational algebraic expression, the 𝟐 𝒙 √𝟏𝟑 − 𝟑 𝟐 𝟓𝒙 + 𝟏𝟏𝒚 = 𝟐𝟗 87 𝒂𝒙 𝒙+𝟑 𝟑 𝒙 = 𝟎, − , −𝟓 𝟓

−𝑥−10

result is 𝑥 2 −5𝑥 . Find this other rational algebraic expression.

4. Give all real numbers 𝑥 for which the reciprocal of 𝑥 is three less than 𝑥. 5. Give the standard form of the equation of the line that passes through the points (−3,4) and (8, −1). 6. If 𝑎 and 𝑏 are the 𝑥-intercept and 𝑦-intercept of 5𝑥 − 2𝑦 = 30, respectively, what is the value of 2𝑎 − 5𝑏? 7. If 𝑥 men can do a job in 𝑎 days, how long will it take for 𝑥 + 3 men to do the same job? 8. For which values of 𝑥 will 2𝑥 3 + 13𝑥 2 + 15𝑥 = 0 be true?

4𝑎4 𝑏2 − 12𝑎2 𝑏𝑐 3 + 9𝑐 6

9. Expand: (2𝑎2 𝑏 − 3𝑐 3 )2

𝟐(𝒙 − 𝟐)(𝒙𝟐 + 𝟐𝒙 + 𝟒)

10. Factor completely: 2𝑥 3 − 16.

Part II. Each question is worth a maximum of three points. Solutions are needed to earn partial scores. Write your solutions on the spaces provided after each item. 1. What is the standard form of the equation of a line that is 𝟓𝒙 + 𝒚 = 𝟏 perpendicular to the graph of 𝑥 − 5𝑦 = 8 and passing through the point (1, −4)? 1 point 2 points

𝒚𝟏𝟎 𝒛𝟑 𝒙𝟏𝟎

The slope of the perpendicular line is determined. 𝑚 = −5 Final answer is not in standard form but is correct slope-intercept form, 𝑦 = −5𝑥 + 1

2. Simplify: [𝑦 −2 (2𝑥 3 )]−5 [4𝑧 3 (64𝑥10 )1/2 ] No partial score will be given for this item.

(

𝟑𝟕 𝟐𝟗 ,− ) 𝟏𝟕 𝟏𝟕

3𝑥 + 5𝑦 = −2 3. Solve the system of linear equations: { 𝑥 − 4𝑦 = 9 1 point

1|P ag e o f 2

The student’s solution evidently shows he has an understanding of elimination, comparison, substitution, or Cramer’s rule in solving systems of linear equations.

2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (INDIVIDUAL-WRITTEN) GRADE

8

𝟐𝒃 − 𝟔 𝒃𝟐 − 𝟕𝒃 + 𝟏𝟎

2 𝑏−3 4. What is the simplest form of 2 −1 𝑏−3 1 point

𝟐√𝟐

The contestant simplified the denominator to

−𝑏+5 𝑏−3

5. What is the shortest distance from the point (3,1) to the line 𝑦 = 2𝑥 + 5? 1 point 2 points

The intersection between the line and its perpendicular passing through the given point is determined, (1,3) The student used the distance formula but was unsuccessful.

𝒅 = √(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐

Part III. Each question is worth a maximum of five points. Solutions are needed to earn partial scores. Write your solutions on the spaces provided after each item. 𝑨 = 𝟑, 𝑩 = 𝟔, 𝑪 = 𝟖 𝒐𝒓 1. The 6-digit number 739𝐴𝐵𝐶 is divisible by 7, 8, and 9. Find one set 𝑨 = 𝟖, 𝑩 = 𝟕, 𝑪 = 𝟐 of values that 𝐴, 𝐵, and 𝐶 can take. 2 points

𝟗𝟕

𝟐𝟏 𝟑𝟐

2. Give the sum of all the numerical coefficients and constant in the 1 5

expansion, (2𝑥 + ) . Express answer as a mixed number. 2

1 point

3 points

4 points

46

The contestant determined that the 6-digit number should be divisible by 504.

The student used binomial expansion through the Pascal’s triangle The expanded form is indicated in the solution: 1 1 1 1 32𝑥 5 + 5(16𝑥 4 ) ( ) + 10(8𝑥 3 ) ( ) + 10(4𝑥 2 ) ( ) + 5(2𝑥) ( ) 2 4 8 16 1 + 32 Student showed the process of getting the sum of the correct coefficients as simplified from the above form, however th final answer is incorrect.

3. What is the smallest positive integer that leaves a remainder of 4 when divided by 7 and a remainder of 2 when divided by 11. No partial score will be given for this item.

2|P ag e o f 2

2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (INDIVIDUAL-WRITTEN) GRADE

8