Admas University Bishoftu Campus
College of Business and Management
Mathematics for Social Science
Assignment for BMGT Sections

1. Let [tex]$P(n): \frac{5n-6}{3}$[/tex] be an open sentence over the domain [tex]$Z$[/tex]. Determine, with explanations, whether the following statements are true or false:
a. [tex]$(\forall n \in Z) P(n)$[/tex]
b. [tex][tex]$(\exists n \in Z) P(n)$[/tex][/tex]

2. Give a formal proof to show that the following argument forms are valid:
[tex]$p \Rightarrow q, \neg r \Rightarrow \neg q \vdash \neg r \Rightarrow \neg p$[/tex]

3. A truck carries a load of 50 boxes; some are 20 kg boxes and the rest are 25 kg boxes. If the total weight of all boxes is 1175 kg, how many of each type are there?

4. The product of two numbers is 5. If their sum is 92, find the numbers.

5. Find the quotient and remainder and verify the Remainder Theorem by computing [tex]$p(a)$[/tex]:
a. Divide [tex]$p(x) = x^5 - 2x^2 - 3$[/tex] by [tex][tex]$x + 1$[/tex][/tex]

6. Determine the rational zeros of the polynomial [tex]$P(x) = 2x^3 - 5x^2 - 28x + 15$[/tex]

7. Evaluate the following limits, if they exist:
a. [tex]\lim_{x \rightarrow 2} \frac{x - 2}{\sqrt{x + 2} - 2}[/tex]
b. [tex]\lim_{x \rightarrow 1} \frac{\frac{1}{x} - 1}{x - 1}[/tex]
c. [tex]\lim_{x \rightarrow \infty} \frac{2x^3 + 3x - 5}{5x^3 + 1}[/tex]

8. Evaluate: [tex]\int \frac{x^3 - 4\sqrt{x}}{x^2} \, dx[/tex]

9.
a. Find the integral by the method of integration by parts: [tex]\int x^2 \sin x \, dx[/tex]
b. Find the area of the region under the graph of [tex]f[/tex] and the [tex]x[/tex]-axis on the given interval: [tex]f(x) = x^2 + 1[/tex] on [1, 3]

10. Use Cramer's rule (if possible) to solve the following linear systems:
[tex]\begin{cases}
4x_1^3 - 2x_2 + 3x_3 = -2 \\
2x_1 + 2x_2 + 5x_3 = 16 \\
8x_1 - 5x_2 - 2x_3 = 4
\end{cases}[/tex]