Division Festival of Talents - Math Quiz (Team-Orals) for Grade 11

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2016 DIVISION FESTIVAL OF TALENTS - MATH QUIZ (TEAM-ORAL)

GRADE 11

2016 DIVISION FESTIVAL OF TALENTS

MATH QUIZ (Team-Orals)

15-second Questions [2 points each]

15.1. Evaluate 𝑓(−1) in the piecewise function, 𝑓(𝑥) = {𝑥²− 1, 𝑥 ≤ −1

3𝑥 𝑥 > −1 . 0 15.2. For which values of 𝑥 will the function ℎ(𝑥) = ¹²

√8𝑥−7 be defined? 𝒙 >𝟕 𝟖 15.3. 𝑓(𝑥) = 2𝑥 − 1 and 𝑔(𝑥) = 𝑥²− 2. Find (𝑓 ∘ 𝑔)(𝑥). 𝟐𝒙^𝟐− 𝟓 15.4. Given ℎ(𝑥) = √𝑥 − 3, find the value of ℎ(𝑥²+ 4𝑥 + 7). 𝒙 + 𝟐 15.5. What is the horizontal asymptote of the graph of 𝑔(𝑥) =^𝑥²^+2𝑥³^+3𝑥⁴

𝑥⁴+2𝑥−5 ? 𝒚 = 𝟑 15.6. Two functions, 𝑓 and 𝑔 are inverses of each other. Another function, ℎ is

related to 𝑓 in such a way that (ℎ ∘ 𝑓)(𝑥) = 𝑥. What is the value of

(𝑔 − ℎ)(𝑥)? 0

15.7. The function for the height of an object dropped from a 100-meter tall platform at a time of t seconds is approximated by 𝑠(𝑡) = −5𝑡²+ 100.

What is the height of the object after 4 seconds? 20 meters

15.8. A triangle, a circle, a square, a circle, a pentagon, a circle... If this pattern

continues, which nth term would correspond to a dodecagon? 23^rd 15.9. Write the biconditional of the statement, “If 𝒎 = 𝟑, then 𝒏 is

irrational.” 𝒎 = 𝟑 iff n is

irrational

15.10. The graphs of any two inverse functions, 𝑓(𝑥) and 𝑓⁻¹(𝑥), are reflections

of each other across which line? 𝒚 = 𝒙 or 𝒇(𝒙) = 𝒙

identity function

30-second Questions, [3 points each]

30.1. If 𝑓(𝑥) = 𝑥²− 𝑥 + 1, what is 𝑓(𝑥 + 1) − 𝑓(𝑥 − 1)? 𝟒𝒙 − 𝟐 30.2. What is the remainder when 2²⁰¹ is divided by 3? 2

30.3. If 2²^𝑥 = 4¹⁶, what is 𝑥? 5

30.4. A fence is to enclose a rectangular vegetable farm with an area of 400 square meters. If 𝑥 is the length of one side of the fence, find the function 𝑃(𝑥) representing the perimeter of the fencing material required.

𝑷(𝒙) =𝟐𝒙^𝟐+ 𝟖𝟎𝟎 𝒙

30.5. In the Division level DAMATH competition, Matthew has won 6 out of 8 games. How many games must he win consecutively to raise his winning percentage to 90 percent?

12

60.1. Let 𝑎, 𝑏, and 𝑐 be three consecutive even numbers such that 𝑎 > 𝑏 > 𝑐.

What is the value of 𝑎²+ 𝑏²+ 𝑐²− 𝑎𝑏 − 𝑏𝑐 − 𝑎𝑐? 12

60.2. There are 100 people in a room. Sixty of them claim to be good at math but only 50 are actually good at math. If 30 of them truthfully deny that they are good at math, how many people are good at math but refuse to admit it?

10

60.3. If 𝑓(𝑥 + 𝑦) = 𝑓(𝑥) ⋅ 𝑓(𝑦) for all positive integers 𝑥, 𝑦 and 𝑓(1) = 2, 𝟐^{𝟐𝟎𝟏𝟔} Category C:

GRADE 11

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GRADE 11

find 𝑓(2016). Hint: The answer is in exponential form.

60.4. A box with a square base is to have a volume of 8 cubic meters. Let 𝑥 be the length of the side of the square base and ℎ be the height of the box.

What are the possible measurement of the side of the square base if the height should not be longer than the side of the square base?

𝒙 ≥ 𝟐 𝒎

60.5. If 𝑚(𝑥) = 𝑥²+ 4𝑥 − 2, for what values of 𝑥 will is 𝑚⁻¹(𝑥) be defined? ^{𝒚 ≥ −𝟔}

Clincher Questions

C1. Give the inverse of the function 𝑓(𝑥) = 12𝑥 + 3. ^𝒇^−𝟏^{(𝒙) =}^{𝒙 − 𝟑}_𝟏𝟐

C2. Suppose that the half-life of a certain radioactive substance is 10 days and there are 100 grams initially. Determine the amount of substance after 30 days.

12.5 grams

C3. What is the domain of 𝑓(𝑥) = √ ⁵

𝑥²−7𝑥+12? (−∞, 𝟑)

∪ (𝟒, ∞)

Do-or-Die Question

DOD1. Give the range of the function, 𝑘(𝑥) = 𝑥²+ 4𝑥 + 7 in interval notation. _{𝒚 ≥ 𝟑}

2016 DIVISION FESTIVAL OF TALENTS Category C:

GRADE 11

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GRADE 11 MATH QUIZ (Team-Orals)

(Quizmaster’s Copy) 15-second Questions [2 points each]

15.1. Evaluate 𝑓(−1) in the piecewise function, 𝑓(𝑥) = {𝑥

²

− 1, 𝑥 ≤ −1 3𝑥 𝑥 > −1 . Read as: 𝑓 of 𝑥 equals 𝑥

²

− 1 if 𝑥 ≤ −1 and 3x if 𝑥 > −1.

15.2. For which values of 𝑥 will the function ℎ(𝑥) =

¹²

√8𝑥−7

be defined?

15.3. 𝑓(𝑥) = 2𝑥 − 1 and 𝑔(𝑥) = 𝑥

²

− 2 . Find (𝑓 ∘ 𝑔)(𝑥).

15.4. Given ℎ(𝑥) = √𝑥 − 3, find the value of ℎ(𝑥

²

+ 4𝑥 + 7).

15.5. What is the horizontal asymptote of the graph of 𝑔(𝑥) =

^𝑥²^+2𝑥³^+3𝑥⁴

𝑥⁴+2𝑥−5

? 15.6. Two functions, 𝑓 and 𝑔 are inverses of each other. Another function, ℎ is

related to 𝑓 in such a way that (ℎ ∘ 𝑓)(𝑥) = 𝑥. What is the value of (𝑔 − ℎ)(𝑥)?

15.7. The function for the height of an object dropped from a 100-meter tall

platform at a time of t seconds is approximated by 𝑠(𝑡) = −5𝑡

²

+ 100. What is the height of the object after 4 seconds?

15.8. A triangle, a circle, a square, a circle, a pentagon, a circle... If this pattern continues, which nth term would correspond to a dodecagon?

15.9. Write the biconditional of the statement, “If 𝒎 = 𝟑, then 𝒏 is irrational.”

15.10. The graphs of any two inverse functions, 𝑓(𝑥) and 𝑓

⁻¹

(𝑥), are reflections of each other across which line?

30.1. If 𝑓(𝑥) = 𝑥

²

− 𝑥 + 1, what is 𝑓(𝑥 + 1) − 𝑓(𝑥 − 1)?

30.2. What is the remainder when2

²⁰¹

is divided by 3?

30.3. If 2

²^𝑥

= 4

¹⁶

, what is 𝑥?

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GRADE 11 30.4. A fence is to enclose a rectangular vegetable farm with an area of 400 square

meters. If 𝑥 is the length of one side of the fence, find the function 𝑃(𝑥) representing the perimeter of the fencing material required.

30.5. In the Division level DAMATH competition, Matthew has won 6 out of 8 games. How many games must he win consecutively to raise his winning percentage to 90 percent?

60.1. Let 𝑎, 𝑏, and 𝑐 be three consecutive even numbers such that 𝑎 > 𝑏 > 𝑐. What is the value of 𝑎

²

+ 𝑏

²

+ 𝑐

²

− 𝑎𝑏 − 𝑏𝑐 − 𝑎𝑐?

60.2. There are 100 people in a room. Sixty of them claim to be good at math but only 50 are actually good at math. If 30 of them truthfully deny that they are good at math, how many people are good at math but refuse to admit it?

60.3. If 𝑓(𝑥 + 𝑦) = 𝑓(𝑥) ⋅ 𝑓(𝑦) for all positive integers 𝑥, 𝑦 and 𝑓(1) = 2, find 𝑓(2016). Hint: The answer is in exponential form.

60.4. A box with a square base is to have a volume of 8 cubic meters. Let 𝑥 be the length of the side of the square base and ℎ be the height of the box. What are the possible measurement of the side of the square base if the height should not be longer than the side of the square base?

60.5 . If 𝑚(𝑥) = 𝑥

²

+ 4𝑥 − 2, for what values of 𝑥 will is 𝑚

⁻¹

(𝑥) be defined?

Clincher Questions

C1. Give the inverse of the function 𝑓(𝑥) = 12𝑥 + 3.

C2. Suppose that the half-life of a certain radioactive substance is 10 days and there are 100 grams initially. Determine the amount of substance after 30 days.

C3. What is the domain of 𝑓(𝑥) = √

⁵

𝑥²−7𝑥+12

? Do-or-Die Question

DOD1. Give the range of the function, 𝑘(𝑥) = 𝑥

²