Show that

$$\left|\begin{array}{ccc} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{array}\right|=a b c+a b+a c+b c$$

Find the lexicographic ordering of these n-tuples: a) (1, 1, 2), (1, 2, 1) b) (0, 1, 2, 3), (0, 1, 3, 2) c) (1, 0, 1, 0, 1), (0, 1, 1, 1, 0)

Find a solution of the given boundary-value problem. $y^{\prime \prime}+y=1, y(0)=0, y(1)=0$

Simplify the complex fraction.

$$\frac{\frac{1}{x-1}+1}{\frac{1}{x+1}-1}$$

Find A + B.

$$A=\begin{bmatrix} 0 & 1&2\\ 1&-1& 0\\ 2&1& 1 \end{bmatrix}, \quad B=\begin{bmatrix} 0 & 1&-1\\ 2&1& 1\\ 0&1& 1 \end{bmatrix}$$

Underline the verb in each sentence. Then, above the verb, write AV if the verb is an action verb or LV if it is a linking verb.

Example 1. The tree $\overset{\textit{\color{#c34632}{LV}}}{{\underline{\text{grew}}}}$ tall and sturdy.

I smell the aroma of a freshly mowed lawn.

Graph each of the following functions: $y=\frac{x^{2}-1}{x-1}$. $y=\frac{x^{3}-1}{x-1}$. $y=\frac{x^{4}-1}{x-1}$. $y=\frac{x^{5}-1}{x-1}$. Is x = 1 a vertical asymptote? Why not? What is happening for x = 1? What do you conjecture about $y=\frac{x^{n}-1}{x-1}$, $n \geq 1$ an integer, for x = 1?

What is an alternative-A loan?

Describe the error in solving the equation. $\frac{1}{x+y}+\frac{1}{x-1}=\frac{2}{(x+1)(x-1)},$ $(x − 1) + (x + 1) = 2,$ $2x = 2,$ $x = 1$ The solution is x = 1.

Give the number of vertices in the figure mentioned.

Define the linear transformation T by T(v) = Av. Find (a) ker(T) , (b) nullity(T) , (c) range(T) , and (d) rank(T).

$$A=\left[\begin{array}{rr} {1} & {2} \\ {-1} & {0} \\ {1} & {1} \end{array}\right]$$

Use a Karnaugh map to find the minimal sum-of-products form for the truth functions shown.

$$\begin{array}{|c|c|c|c|c|} \hline x_{1} & x_{2} & x_{3} & x_{4} & f\left(x_{1}, x_{2}, x_{3}, x_{4} )\right. \\ \hline 1 & 1 & 1 & 1 & 1 \\ \hline 1 & 1 & 1 & 0 & 1 \\ \hline 1 & 1 & 0 & 1 & 1 \\ \hline 1 & 1 & 0 & 0 & 1 \\ \hline 1 & 0 & 1 & 1 & 0 \\ \hline 1 & 0 & 1 & 0 & 1 \\ \hline 1 & 0 & 0 & 1 & 0 \\ \hline 1 & 0 & 0 & 0 & 1 \\ \hline 0 & 1 & 1 & 1 & 1 \\ \hline 0 & 1 & 1 & 0 & 1 \\ \hline 0 & 1 & 0 & 1 & 1 \\ \hline 0 & 1 & 0 & 0 & 1 \\ \hline 0 & 0 & 1 & 1 & 0 \\ \hline 0 & 0 & 1 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 \\ \hline \end{array}$$

How do you decide to shade above or below a boundary line? What does this shading represent?

Solve the boundary-value problem, if possible.

$$y’’-8y’+17y=0, y(0)=3, y(π)=2$$