🧮 algebra

1. **Stating the problem:** We have a point $P$ moving in the $Oxy$-plane with parametric equations: $$x(t) = \frac{1}{2} \sin(t), \quad y(t) = \sin\left(t + \frac{1}{3} \pi\right)

1. The problem asks for the common ratio $r$ of the geometric sequence $(42, 21, 10.5, 5.25, \ldots)$.\n\n2. The common ratio $r$ in a geometric sequence is found by dividing any t

1. The problem asks to identify two points on the graph of the function $f(x) = 2 \cdot 3^{x-2} - 2$ and determine if the asymptote is vertical or horizontal. 2. The function is an

1. **State the problem:** We need to classify each given sequence as either an arithmetic sequence or a geometric sequence. 2. **Recall definitions:**

1. The problem asks to find two points with integer coordinates on the graph of the function $f(x) = 3 \left(\frac{1}{2}\right)^x - 5$. 2. The function is an exponential decay func

1. **Stating the problem:** We have two sequences: (0, 6, 12, 18, 24, ...) and (1, 6, 36, 216, 1296, ...). We need to identify which is arithmetic, find its common difference, and

1. The problem is to solve the equation or expression given previously (user did not specify, so assuming continuation of a prior problem). 2. Since the user asked to "solve the re

1. The problem asks to match each function with the correct rule describing the sequence. 2. Let's analyze each function:

1. The problem asks to identify whether the asymptote of the function $f(x) = 3 \left(\frac{1}{2}\right)^x - 5$ is vertical or horizontal. 2. The general form of an exponential fun

1. The problem is to solve the equation or expression given by the user, but since no specific equation or expression was provided, I will demonstrate a general approach to solving

1. **State the problem:** We have a linear equation with points $(0,a)$, $(2,b)$, and $(6,8)$ on its graph. 2. **Recall the formula for the rate of change (slope) of a linear funct

1. **State the problem:** We have two functions, Function A represented by the table of values and Function B represented by the equation $y=\frac{3}{2}x+2$. We need to find the ra

1. Express each of the following as a single integer: (i) $6 - 4$

1. **Stating the problem:** We need to identify the type of variable that a function assigns to exactly one value of the dependent variable.

1. **State the problem:** We want to find how much percent more men earn than women in the Bachelor's degree category. 2. **Given data:**

1. **State the problem:** We are given a graph of a vertical line at $x=3$ and asked to determine which statements about this graph are true. 2. **Recall the definition of a functi

1. **State the problem:** We need to find the domain of the function $$f(x) = 2x^2 + 5\sqrt{x - 2}$$. 2. **Recall the domain rule for square roots:** The expression inside the squa

1. The problem asks which student correctly solved the expression $$x^{\frac{1}{3}} \cdot x^{\frac{1}{4}}$$. 2. The rule for multiplying powers with the same base is to add the exp

1. **Problem:** Identify characteristics of the quadratic function $f(x) = -(x + 6)(x + 2)$. 2. **Step 1: Expand the function**

1. **State the problem:** We are given the area of a rectangle as $$6x^3 - 2x^2 + 4x$$ and the width as $$2x$$. We need to find the length of the rectangle. 2. **Formula used:** Th

1. **State the problem:** Find the acute angle $\theta$ between two lines $a$ and $b$.