Math141 College algebra

Math 141 Exam 4: 8.2-9.5 Name:________________________ SCIENTIFIC CALCULATORS ONLY! YOU MUST SHOW WORK AS DONE IN CLASS VIDEO INSTRUCTION FOR CREDIT!!!

1. [6 pts] Solve for . 2. [6 points] Find the determinant of the following Matrix.

! 𝑥 1 1 2 1 4 −1 4 3

! = 3 ) 1 3 −2 6 1 −5 8 2 3

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3. [10 pts] Solve . 𝑥! − 4𝑥𝑦 + 3𝑦! = 0

𝑥! + 𝑥𝑦 = 12

x

4. [20 pts] Find the inverse of the matrix using Gauss-Jordan elimination BY HAND.

) 1 −1 −1 0 2 1 3 2 0

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Extra Credit: Given the system: 𝑥" − 3𝑥! = −2

2𝑥" − 5𝑥! = 1

a) Write a matrix equation in the form of 𝐴𝑋 = 𝐵.

b) Find 𝐴#".

c) Use 𝐴#" to solve the system.

5. [8 pts each] Refer to the following matrices below:

𝐴 = #2 −1 3 0 4 −2

*;𝐵 = #−3 1 2 5

*;𝐶 = / −1 0 2 4 −3 1 −2 3 5

0;𝐷 = / 3 −2 0 −1 1 2

0

Find the following: a) B+AD b) DB 6. [3 pts] Write down the first 3 terms of the sequence.

{𝑆$} = T (−1)$

(𝑛 + 6)(𝑛 + 3) W

7. [6 pts] Express the sum using summation notation.

1 2ln2

− 1

3ln3 +

1 4ln4

− 1

5ln5 + ⋯+

1 100ln100

8. [12 pts] Write the partial fraction decomposition of !"

#%!$%

9. [8 pts] Find the sum of ∑ 𝑘&'%()* 10. [4 each] Given: 9 + 12 + 16 + !"

# + ⋯

a) Determine whether the following series is arithmetic b) Find the sum of the series. or geometric. If arithmetic, find its common difference,

, and if geometric, find its common ratio, .

d r

11. [8 pts] The first term of the geometric series is 3 and the third term is " # . Find the fifth term.

12. [8 pts] Find the nth term of the arithmetic sequence given that the first term 𝑎" = 6, and common difference 𝑑 =

$ %

Find the 53rd term: 𝑎&'.

13. [7 pts] Expand Z1 − " !( [ &

using the Binomial Theorem. Simplify completely. (Make sure you give the set-up)

14. [12 pts] Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. (Must show work as done in class video instruction.) 2' + 4' + 6' + ⋯(2𝑛)' = 2𝑛!(𝑛 + 1)!