Vectors
u=langle-2,5rangle" Find "2u.
2langle-2,5rangle=langle2(-2),2(5)rangle=langle-4,10rangle
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.
"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
"Calculate "underset(9)""Cunderset(7)
(9!)/(2!7!)=36
u=langle-2,5rangle and v=langle3,1rangle"
"Find "u+v.
langle-2+3, 5+1rangle=langle1,6rangle
"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]]
"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
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
v=langle3,1rangle" Find "norm(v)
sqrt(3^2+1^2)=sqrt(9+1)=sqrt(10)
"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]]
"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
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
u=langle-2,5rangle and v=langle3,1rangle
"Find the dot product of "u and v.
(-2)(3)+(5)(1)=-6+5=-1
"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
"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
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
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
"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