ESAT Math 2: Definite Integrals & Probability

Consider the integral $\int_0^1 \frac{3x^2 + 2x - 1}{\sqrt{x}} dx$. Which of the following represents the correct value of this definite integral?

A bag contains g green beans, r red beans, and b blue beans. If three beans are chosen at random without replacement, what is the probability that exactly two of the three beans are blue?

A function is defined as $f(x) = \frac{x^2 - 4}{x - 2}$ for $x \neq 2$. What value should be assigned to $f(2)$ to make $f(x)$ continuous at $x = 2$?

Given the function $g(x) = x^3 - 3x^2 + 2x$, determine the interval(s) where $g(x)$ is increasing.

A geometric sequence has a first term of 3 and a common ratio of 2. What is the sum of the first 5 terms of this sequence?

Determine the derivative of the function $f(x) = \ln(\sin(x))$, where $0 < x < \pi$.

If $\tan(\theta) = \frac{3}{4}$ and $\theta$ is in the third quadrant, find the value of $\cos(\theta)$.

Determine the area of the region enclosed by the curves $y = x^2$ and $y = 2x$.

Solve the differential equation $\frac{dy}{dx} = xy$ with the initial condition $y(0) = 2$. What is the solution $y(x)$?

Given the vectors $\vec{a} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\vec{b} = -\mathbf{i} + 5\mathbf{j} - 2\mathbf{k}$, find the dot product $\vec{a} \cdot \vec{b}$.