key to correction g10

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

Category B:

SCORE

GRADE 10

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. 1. What is the tenth term in the Fibonacci sequence if its first term is 1?

55 2016

2. The first term of an arithmetic sequence with a common difference of 250 is 16. What is its 9th term?

4n + 6

3. What is the nth term of the sequence 10, 14, 18, 22, 26, …? 4 8 3 9

4. A geometric sequence has the following consecutive terms − , , − 4 3

16 … 27

. What term

precedes − ?

2 √𝟐

5. What is the distance between the points A(−3,2) and B(−2,3)?

-16

6. There are two geometric means between -1 and 64. Find the lesser of these means.

60

7. In how many distinguishable ways can the letters of the word SEVEN be arranged?

-4

8. What is the remainder when 𝑥 5 + 2𝑥 3 + 3𝑥 2 − 4 is divided by 𝑥 + 1? 2 2 2 2 2 , … is 3 5 7 9 11

9. TRUE or FALSE: The sequence , , , ,

TRUE

a harmonic sequence.

10. What is the sum of all of the terms of the infinite geometric sequence whose first five 1 1 1 1 1 terms are , , , , , …?

1

2 4 8 16 32

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. 𝒙𝟐 + 𝒚𝟐 + 𝟒𝒙 − 𝟏𝟐 = 𝟎

1. A circle with a 4-unit long radius has its center at (−2,0). Give its general equation. 1 point 2 points

Contestant correctly determined the standard form of the equation of the given circle. (𝒙 + 𝟐)𝟐 + (𝒚 − 𝟎)𝟐 = 𝟒𝟐 All terms of the equation are written but not in the proper order (e.g. 𝒙𝟐 + 𝟒𝒙 + 𝒚𝟐 − 𝟏𝟐 = 𝟎). Constants must have been combined.

2. The second term of an arithmetic sequence is 𝑥 − 2𝑦. What is the sum of its first four terms if its common difference is 𝑦 − 2𝑥?

−𝟔𝒚

1 point 2 points

24 (𝑤𝑎𝑦𝑠) 𝑜𝑟 16 (𝑤𝑎𝑦𝑠)

3. In how many ways can three couples be seated around a table if the all of them insist on sitting beside their partner? 1 point

𝟖 (𝒅𝒂𝒚𝒔)

The contestant correctly identified two or three of the terms in the sequence. 𝟑𝒙 − 𝟑𝒚, 𝒙 − 𝟐𝒚, −𝒙 − 𝒚, 𝒂𝒏𝒅 − 𝟑𝒙 The contestant identified all four terms but the final answer indicated is incorrect.

The contestant recognized circular permutations and evidently used the formula for circular permutation, (3 − 1)!

4. A student on vacation for x days observed the following: (a) it rained five times, either in the afternoon or in the morning; (b) there were five clear afternoons; and (c) there were six clear mornings. Determine x. No partial score is given for this item.

𝟕 𝒂𝒏𝒅 −

𝟏 𝟐

5. One of the roots of the polynomial equation 2𝑥 3 + 9𝑥 2 − 33𝑥 + 14 = 0 is 2. Find the other roots. 1 point 2 points

1|P age of 2

The quotient when the polynomial equation is divided by 𝑥 − 2 which is 𝟐𝒙𝟐 + 𝟏𝟑𝒙 − 𝟕 = 𝟎 is found in the solution. Only one of the two correct answers is correct or given.

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

10

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. Find all values of k so that the numbers 2𝑘 + 1, 3𝑘 + 4, 𝑎𝑛𝑑 7𝑘 + 6 form a geometric 2 and-1 sequence. 3𝑘 + 4 7𝑘 + 6 1 point = 2𝑘 + 1 3𝑘 + 4 2 points (3𝑘 + 4)2 = (2𝑘 + 1)(7𝑘 + 6) 3 points 5𝑘 2 − 5𝑘 − 10 = 0 𝑜𝑟 𝑘 2 − 𝑘 − 2 = 0 4 points

𝟏𝟐 𝟒𝒏 − 𝟏

2. What is the nth term of the harmonic sequence 4, 1 point

𝟑𝟓 𝟐𝟔𝟒

Only one of the answers is correct

Contestant got all reciprocals correct:

1

,

7

,

12 12 12 , , , …? 7 11 15

11 15

,

4 12 12 12 1

2 points

The correct common difference was identified and shown in the solution, 𝑑 = 3

3 points

The answer given is

4𝑛−1 12

𝑛

1

3

12

or −

3. A box contains 3 red marbles, 5 blue marbles, and 4 yellow marbles. What is the probability of randomly picking out either a red marble or a yellow marble, returning it then picking a blue marble, not replacing it and picking one last marble which happens to be not blue? 1 point each

2|P age of 2

The contestant got the following probabilities. Give one point each. 7 5 4 , , 12 12 11

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

10