key to correction g9

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

GRADE 9

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. 0 1. How many real roots does 2𝑥 2 − 5𝑥 + 4 = 0? (zero, or none) 𝒚 ≥ −𝟏 or 2. Give the range of the graph of the quadratic function 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 8. [−𝟏, ∞] 3. A camera’s aperture determines the size of the circular opening in the lens that allows light in. if we want to allow twice as much light in, what should be the ratio of the new radius to the current radius? √𝟐 4. When the function 𝑓(𝑥) = 𝑥 2 − 𝑥 + 1 is evaluated at 𝑥 = 𝑎, 𝑓(𝑎) = 7. What is the value of 𝑎 if it is a positive integer?

3 𝒙𝟐 + 𝟑𝒙𝟐

𝟏𝟔 𝒙 𝟑

− 𝟒 = 𝟎 or

2 3

5. The roots of a quadratic equation are -6 and . Find the equation.

+ 𝟏𝟔𝒙 − 𝟏𝟐 = 𝟎

3

6. For which value of x will 𝑓(𝑥) = 4(𝑥 − 3)2 + 1 have its minimum point?

12

7. If 𝑟 is the greater of two roots of the quadratic equation, 𝑥 2 + 7𝑥 + 10 = 0, and 𝑠 is the lesser root, then what must be the value of (𝑟 − 1)(𝑠 + 1)?

−

𝟑 𝟓

1 5

8. If 𝑦 varies inversely as 𝑥 and 𝑦 = when 𝑥 = 9, find 𝑦 when 𝑥 = −3. 9. TRUE or FALSE: If 𝑚 varies inversely as 𝑛 and directly as 𝑝, then 𝑚 = is the constant of variation.

FALSE

𝑘𝑛 𝑝

where 𝑘

10. Simplify: (3√2 + 2√3)(2√3 − 3√2).

-6

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. An object is fired straight up with a velocity of 64 ft/s. Its altitude (height) h after t seconds is given by ℎ(𝑡) = −16𝑡 2 + 64𝑡. How long will it take the object to hit the ground? 4 seconds 1 point

𝟏𝟓𝟗°

Contestant looked for the zero of the function thus, the solution contains

0 = −16𝑡 2 + 64𝑡 or 16𝑡 2 − 64𝑡 = 0

2. Given parallelogram ABCD, find 𝑚∠𝐶. A B 𝒙 − 𝟐𝟔°

1 point 2 points

𝟑𝒙 + 𝟏𝟖°

D 4

C

3. Find the value of 2√2√2√2 … . 1 point 2 points

28 m

Contestant adds the measures of the two angles B and C and equates it to 180. (𝑥 − 26) + (3𝑥 + 18) = 180 The final answer is just the value of x, 𝑥 = 47°

The contestant expressed the expression using rational exponents: 1

1

1

2 ∙ 22 ∙ 24 ∙ 28 ∙ … The contestant expressed this progression by employing powers of a product: 1 1 1

21+2+4+8+⋯

4. The length of a rectangular plot is 5 m longer than its width and the area is 14 𝑚2 . What is its perimeter? 1 point 2 points

1|P age of 2

The correct working equation was written, 𝑤 2 + 5𝑤 − 14 = 0 or a variation of that, such as 𝑥 2 + 5𝑥 − 14 = 0 or 𝑙 2 − 5𝑙 − 14 = 0 Either the correct value of the length or the width is obtained. 𝑤 = 2; 𝑙 = 7

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

9

𝟐 ± √𝟕𝒊 or 𝟐 ± √−𝟕

5. What are the solutions to the quadratic equation, 𝑥 2 − 4𝑥 + 11 = 0? No partial scored is given for this item.

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. 1. Sixty men working on a construction job have done 1/3 of the work in 18 days. The project is behind schedule and must be accomplished in the next twelve days. How many more workers need to be hired? 120 2 points 3 points

The contestant determined that it would take 54 days for the 60 workers to finish 1 the job: 54 The contestant determined that to finish the job in 12 days, the rate must be 1 increased to 18 days: 18

2. The figures below consisting of unit squares are the first four in a sequence. How many unit squares are there in the 2016th figure? 2017 × 1008 or 2 033 136 1 point 3 points

3

The correct sequence was determined: 1, 3, 6, 10, … The contestant determined that the 2016th figure would be composed of 1 + 2 + 3 + ⋯ + 2015 + 2016 blocks 1 1 3 2

1 1 4 3

1 1 5 4

1 1 6 5

3. Evaluate (27 ) (27 ) (27 ) (27 ) . 1 point

2|P age of 2

1

1

1

1

Expressed as 276 ∙ 2712 ∙ 2720 ∙ 2730 1

1

1

1

2 points

Expressed as 26+12+20+30

3 points

Expressed as 260

20

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

9