Mat1511 Quiz 4

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Precalculus B
4
QUIZ

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Precalculus B
Due date N/A
Time taken 1 hour
Grade out of 50.00


QUESTION 1
Find the ratio of the coefficients of the fourth term and the fifth term in
5
the expansion of (2𝑥 + 1) :

Choose the correct answer:
o a. 3∶1
o b. 1∶2
o c. 4∶3
o d. None of the options
o e. 4∶1
𝑛
𝑛
(𝑎 + 𝑏)𝑛 = ∑ ( ) 𝑎𝑛−𝑟 𝑏 𝑟
𝑟
𝑟=0

(𝑎 + 𝑏)5 = (5) 𝑎5 + (5) 𝑎4 𝑏 + (5) 𝑎3 𝑏 2 + (5) 𝑎2 𝑏 3 + (5) 𝑎𝑏 4 +
0 1 2 3 4
5 5
( )𝑏 4th term 5th term
5

➢ Since 𝑎 = 2𝑥 ; 𝑏 = 1 ; 𝑛 = 5

5 5
𝑇4 = ( ) (2𝑥)2 (1)3 𝑇5 = ( ) (2𝑥)(1)4
3 4
5! 5!
=[ ] 4𝑥 2 × 1 =[ ] × 2𝑥 × 1
3! (5 − 3)! 4! (5 − 4)!
5 × 4 × 3! 5 × 4!
=( ) × 4𝑥 2 =( ) × 2𝑥
3! × 2 × 1 4! × 1!
= 5 × 2 × 4𝑥 2 5
= × 2𝑥
= 40𝑥 2 1
= 10𝑥
• The coefficients for the fourth term and the fifth term are 40 and 10
respectively.

∴ 𝑅𝑎𝑡𝑖𝑜 = 40 ∶ 10
=4∶1

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QUESTION 2
A company wants to advertise a new product using Facebook ads
and Instagram ads. Each Facebook ad reaches 2000 people and
costs R50. Each Instagram ad reaches 1500 people and costs R40.
The advertising budget is R1 000. The company can run at most
22 ads in total.

Question: Determine the number of Facebook and Instagram ads
that maximizes the audience reached.

Choose the correct answer:
o a. 20 Facebook ads and 0 Instagram ads
o b. 22 Facebook ads and 25 Instagram ads
o c. 12 Facebook ads and 10 Instagram ads
o d. 34 Facebook ads and 16 Instagram ads
o e. None of the options


Let: • 𝑥 = Facebook ads 50𝑥 + 40𝑦 ≤ 1 000
• 𝑦 = Instagram ads 𝑥 + 𝑦 ≤ 22
𝑥≥0
𝑦≥0

Objective function: 𝑧 = 2 000𝑥 + 1 500𝑦

5
22 − 𝑥 = 25 − 𝑥
4
5
𝑥 − 𝑥 = 25 − 22
4
1
𝑥=3
4
𝑥 = 12
𝑦 = 22 − (12) = 10




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The vertices of the feasible region are @ (0; 0) , (0; 22) , (12; 10) ,and (20; 0) :

➢ For (12; 10) 𝑧 = 2000(12) + 1500(10) = R39 000
➢ For (20; 0) 𝑧 = 2000(20) + 1500(0) = R40 000
➢ For (0; 22) 𝑧 = 2000(0) + 1500(22) = R33 000
➢ For (0; 0) 𝑧 = 2000(0) + 1500(0) = R0

12 Facebook ads and 10 Instagram
ads are the objective plan that
maximizes the audience reached.
QUESTION 3
If 𝒙 = 𝟏 is a zero of 𝑭(𝒙) = 𝒙𝟑 − 𝟐𝒙𝟐 − 𝟏𝟏𝒙 + 𝟏𝟐. Find the
product of the remaining two zeros?

Choose the correct answer:
o a. −3
o b. None of the options
o c. 4
o d. −18
o e. −12


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1 1 -2 -11 12
1 -1 -12
1 -1 -12 0
𝑃(𝑥) = (𝑥 − 1)(𝑥 2 − 𝑥 − 12) = 0
(𝑥 − 2)(𝑥 + 3)(𝑥 − 4) = 0
𝑥−2=0 OR 𝑥 + 3 = 0 OR 𝑥 − 4 = 0
𝑥=2 𝑥 = −3 𝑥=4

−3 and 4 are the remaining zeros of F
∴ 𝑇ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 − 3 𝑎𝑛𝑑 4 = −3 × 4
= −12

QUESTION 4
Without solving the unknowns, write the following system of equations
in augmented matrix form
𝑥+𝑧 =6−𝑦
−𝑦 + 3𝑧 = 9 − 2𝑥
3𝑥 + 2𝑦 = 5 + 𝑧
Choose the correct answer:
1 1 6 −1
o a. (−1 3 9|−2)
3 2 5 1
1 1 1 6
o b. (2 −1 3 |9)
3 2 −1 5
o c. None of the options




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1 −1 1 −6
o d. (−2 1 −3| 5 )
0 3 1 −9
3 2 1 6
o e. (0 1 −2|9)
1 −4 1 5

𝑥+𝑦+𝑧 =6
2𝑥 − 𝑦 + 3𝑧 = 9
3𝑥 + 2𝑦 − 𝑧 = 5
∴ Augmented matrix form:

1 1 1 6
(2 −1 3 |9)
3 2 −1 5

QUESTION 5
Simplify the following complex number and leave your answer in polar
form.
𝜋 2 5𝜋
(3 ∠ 4 ) (2 ∠ 6 )
𝜋
6∠
3
Choose the correct answer:
5𝜋
o a. 2∠
4

o b. None of the options

𝜋
o c. √2 ∠ 4

𝜋
o d. 4∠
6

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o e. 3∠𝜋

𝜋 5𝜋
(32 ∠ 2 × 4 ) × (2∠ 6 )
= 𝜋
6∠
3
𝜋 5𝜋
(9∠ 2 ) × (2∠ 6 )
= 𝜋
6∠
3
𝜋 5𝜋
(9 × 2)∠ ( + )
= 2 6
𝜋
6∠
3
4𝜋
18∠
= 3
𝜋
6∠
3
4𝜋 𝜋
= (18 ÷ 6)∠ ( − )
3 3
= 3∠𝜋

QUESTION 6
The synthetic division of 𝑷(𝒙) = 𝟐𝒙𝟑 − 𝟗𝒙𝟐 + 𝟕𝒙 + 𝟔 ,by 5 is
shown below.


5 2 -9 7 6
10 𝑎 𝑐
2 1 𝑏 66

Complete the missing entries and determine whether 5 is an upper
or lower bound for the zeros of P?

Choose the correct option:
o a. 𝑎 = 4 , 𝑏 = 9 , 𝑐 = 2 , and 5 is an upper bound
o b. 𝑎 = 3 , 𝑏 = −4 , 𝑐 = 60 , and 5 is a lower bound
o c. 𝑎 = 5 , 𝑏 = 12 , 𝑐 = 60 , and 5 is an upper bound
o d. None of the options

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