If c and d are different positive integers and

$$4c + d = 26,$$

what is the sum of all possible values of c?

$$A \ 6 \\ B \ 10 \\ C \ 15 \\ D \ 21$$

Determine whether W is a subspace of V.

$$V = M _ { 22 } , W = \left\{ \left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] : a + b = c + d= a + c = b + d \right\}$$

Cost, revenue, and profit A company can manufacture $x$ items at a cost of $c(x)$ thousand dollars, a sales revenue of $r(x)$ thousand dollars, and a profit of $p(x)=r(x)-c(x)$ thousand dollars. Find $d c / d t, d r / d t$, and $d p / d t$ for the following values of $x$ and $d x / d t$

a. $r(x)=9 x, c(x)=x^3-6 x^2+15 x,$ and $d x / d t=0.1$ when $x=2$ b. $r(x)=70 x, c(x)=x^3-6 x^2+45 / x, and$d x / d t=0.05$when$x=1.5$

If

$$\frac { a } { b } = \frac { c } { d }$$

will

$$\frac { a } { c } = \frac { b } { d } ?$$

Assume that

$$b \neq 0 , c \neq 0$$

and

$$d \neq 0$$

Explain your reasoning.

The table shows the number of points selected players have at the end of a game.

$$\begin{array}{|c|c|} \hline \text { Player } & \text { Points } \\ \hline A & -10 \\ \hline B & -50 \\ \hline C & -5 \\ \hline D & 0 \\ \hline E & -15 \\ \hline \end{array}$$

Which list shows the order of the players from greatest to least points? A. D, C, A, E, B. B. B, E, A, C, D. C. D, C, A, B, E. D. B, C, A, D, E.

Suppose $\mathrm{P}(C)=\frac{9}{20}, \mathrm{P}\left(C | D^{\prime}\right)=\frac{3}{7}$, and $\mathrm{P}(C | D)=\frac{6}{13}$. Find: (i) P(D) (ii) P $(C^{\prime} \cup D^{\prime})$

Factor the polynomial completely: $6 c^3-27 c^2 d+12 c d^2$

Evaluate the expression $(4 c d)^{-2}$ for c = 2 and d = 1.

Find general solutions of the differential equation. Primes denote derivatives with respect to $x$.

$$4 y ^ { \prime \prime } + 8 y ^ { \prime } + 3 y = 0$$

Find the sum or difference.

$$\left( c ^ { 2 } - 6 d ^ { 2 } \right) + \left( c ^ { 2 } - 2 c d + 2 d ^ { 2 } \right)$$

Solve each expression if $a=3, b=6, c=5$, and $d=9$.

$$6 d-c \cdot c$$

Let A[-3], B[-1], C[6], and D[5] denote four collinear points, with their coordinates. If P[x] denotes any other point on $\overleftrightarrow{A B}$, find the range of x allowable if P lies on

(a) $\overrightarrow{A B}$

(b) $\overrightarrow{B D}$

(c ) $\overline{D C}$

(d) $\overrightarrow{A B} \cup \overrightarrow{C D}$.

(a). In the coordinate plane, graph points A(5,2), B(2,2), C(2,-1), D(-1,-1). (b). Draw $\overleftrightarrow{A B}$ and $\triangle B D C$. ($c$). Explain how you know that $\triangle B D C$ is an isosceles right triangle. (d). What is the measure of $\angle B D C$ ? Justify your answer. (e). What is the measure of $\angle D B A$ ? Justify your answer.

Let $U=\{ a,b,c,d,e,f,g,h\},\quad K=\{c,d,e,f,h\}$ and $R=\{ a,c,d,g \}$. Find the following:

$$K^{\prime}$$