In order for your answers to earn full credit, they must be fully supported by clear, neat, well organized work. Each problem is worth 10 points.

- Find the sum of the vectors [ – 1 , 4] and [6, – 2 ] and illustrate geometrically on the x-y

plane.

Solution:

x=[-1,4]; y=[6, – 2 ]

x-y=[-1,4]-[6, – 2 ]

x-y=[-1-6,4-(-2)]=[-7,6]

**2. Determine whether the vectors [– 1, 2, 5] and [3, 4, – 1] are orthogonal. Your workmust clearly show how you are making this determination.**

Two vectors are said to be orthogonal if their dot product is equal to zero

[-1,2,5]*[3,4,-1]=(-1)(3)+(2)(4)+(5)(-1)=-3+8-5=0

so required vectors are orthogonal.

\begin{equation} \text { 3. Find the determinant of the matrix }\left[\begin{array}{ll} 1 & 2 \\ 4 & 8 \end{array}\right] \text {. Is it invertible? } \end{equation}

\begin{equation} \operatorname{det}\left[\begin{array}{ll} 1 & 2 \\ 4 & 8 \end{array}\right]=(1)(8)-(2)(4)=8-8=0 \end{equation}

4. Find and sketch the domain of f(x, y) = sqrt(2x – y)

- Find any local max/mins for 𝑓(𝑥, 𝑦) = 𝑥^3 -12 𝑥𝑦 + 8𝑦^3

- Given the system of differential equations 𝑑𝑥⃗/𝑑𝑡= 𝐴𝑥⃗ with 𝐴 = [−3 12 −2] , graph the nullclines and label the equilibrium point.

- Solve the system of differential equations 𝑑𝑥⃗/𝑑𝑡= 𝐴𝑥⃗ with 𝐴 = [1 0 4 −1]. (Note: as no initial condition is specified, your solution will contain constants c1 and c2 .)

- 4-digit PIN codes are commonly used. How many 4-digit PIN codes can be made?

Remember to show explanatory work for your answer.

Solution:

There are 10 digits, so using the multiplication principle:

the codes = 10*10*10*10 = 10,000 answer

- In a raffle with 10 entries, in how many ways can three winners be selected? Show

work!

- A ball is drawn randomly from a box containing 5 red balls, 2 white balls, and 1 yellow

ball. Find the probability that the ball drawn is not white.

In order for your answers to earn full credit, they must be fully supported by clear, neat, well organized work. Each problem is worth 10 points.

- Find the sum of the vectors [ – 1 , 4] and [6, – 2 ] and illustrate geometrically on the x-y

plane.

2. Determine whether the vectors [– 1, 2, 5] and [3, 4, – 1] are orthogonal. Your work

must clearly show how you are making this determination.

3. Find the determinant of the matrix