Vectors
Matrices
Sequences and Series
Counting and Probability
100

u=langle-2,5rangle" Find "2u.





2langle-2,5rangle=langle2(-2),2(5)rangle=langle-4,10rangle

100

Which of the following are NOT elementary row operations used to solve systems of linear equations?

a. Interchange two rows
b. Multiply a row by a nonzero constant
c. Add a multiple of a row to another row
d. All are elementary row operations



d. All are elementary row operations.


100

"Write the first 5 terms of the geometric "

"sequence if "aunderset(1)=3 and r=4.

a underset(n) ""=aunderset(1)""r^(n-1)" "implies" "a underset(2)""=3(4)^(2-1)=3(4)=12

a underset(3)""=3(4)^(3-1)=3(16)=48

a underset(4)""=3(4)^(4-1)=3(64)=192

a underset(5)""=3(4)^(5-1)=3(256)=768

100

"Calculate "underset(9)""Cunderset(7)


(9!)/(2!7!)=36

200

u=langle-2,5rangle and v=langle3,1rangle"


"Find "u+v.



langle-2+3, 5+1rangle=langle1,6rangle

200

"Find "A+B if A=[[3,5],[-1,1]] and B=[[-3,-2],[4,2]]




[[3+(-3),5+(-2)],[-1+4,1+2]] =[[0,3],[3,3]]

200

"Find the sum "sum_(i=1)^8(13-2i)




S underset(n) ""=n/2(aunderset(1)""+aunderset(n)"")" "implies" "S underset(8) ""=8/2(aunderset(1)""+aunderset(8)"")

=4(11+(-3))=4(8)=32

200

There are 8 bicycles entered in a race.  In how many different ways could the top three places be decided?


underset(8)""Punderset(3)""=(8!)/(3!)=336

300

v=langle3,1rangle" Find "norm(v)




sqrt(3^2+1^2)=sqrt(9+1)=sqrt(10)

300

"Find the inverse of " A=[[1,2],[4,7]] 


A^(-1)=1/(7-8)[[7,-2],[-4,1]]=-1[[7,-2],[-4,1]]=[[-7,2],[4,-1]]

300

"Find the formula for the nth term"

"of the arithmetic sequence where"

aunderset(1)""=5 and d=-2.




aunderset(n)""=aunderset(1)""+(n-1)d" "implies" "aunderset(n)""=5""+(n-1)(-2)

aunderset(n)""=5+(-2n)+2


aunderset(n)""=7-2n

300

A child returns a five-volume set of books to a bookshelf.  The child is not able to read, and so cannot distinguish one volume from another.  What is the probability that the books are shelved in reverse order?


"Number of ways 5 books can be arranged is 5!=120."

"Number of ways books can be arranged in reverse order is 1."

"Probability "=1/120

400

u=langle-2,5rangle and v=langle3,1rangle


"Find the dot product of "u and v.




(-2)(3)+(5)(1)=-6+5=-1

400

"Find the determinant of "B=[[1,1,3],[-1,2,5],[-2,3,2]].


|[1,1,3],[-1,2,5],[-2,3,2]||[1,1],[-1,2],[-2,3]|

downs-ups

=(4-10-9)-(-12+15-2)=(-15)-(1)=-16

400

"Find the sum of the infinite geometric series "sum_(i=0)^oo (1/3)^i


S=(aunderset(0)"")/(1-r)=1/(1-1/3)=1/(2/3)=3/2


400

A six-sided die is rolled five times. What is the probability that each side appears exactly once? (Leave your answer as a fraction.)



(underset(6)""Punderset(5)"")/6^5=720/7776=5/54

500

u=langle-2,5rangle and v=langle3,1rangle

"Find the angle between the vectors "

u and v" to the nearest degree."




costheta=(ucdotv)/(norm(u)norm(v))=(-1)/(sqrt(29)sqrt(10))=-0.0587 


arccos(-0.0587)=93^o

500

"Use the Binomial Theorem to "

"expand the binomial "(x+3y)^4.



1x^4(3y)^0+4x^3(3y)^1+6x^2(3y)^2+4x^1(3y)^3+1x^0(3y)^4

x^4+12x^3y+54x^2y^2+108xy^3+81y^4

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