🧮 algebra

1. Problem: Find the unknown matrix $X$ if $$X + \begin{bmatrix}4 & 8 \\ 1 & -2

1. Find the vertical and horizontal asymptotes of the curve defined by $$xy - 5x - 2y + 3 = 0.$$ Step 1: Rearrange the equation to express $y$ in terms of $x$.

1. The problem is to solve the equation involving the absolute value: $|6-3X|=9$. 2. Recall that $|A|=B$ means $A=B$ or $A=-B$. So here, $6-3X=9$ or $6-3X=-9$.

1. Problem 10: Given ratios $x:y=2:3$ and $y:z=2:3$, find ratio $x:z$. Since $x:y=2:3$, we write $\frac{x}{y}=\frac{2}{3}$.

1. Problem statement: Calculate areas A1, A2, A3, total area, and perimeter of the composite shape formed by three rectangles with given dimensions. 2. Calculate area A1:

1. **সমস্যার বিবৃতি:** Cramer's Rule ব্যবহার করে নিম্নলিখিত সমীকরণগুলোর জন্য $x_1,x_2,x_3$ এর মান নির্ণয় করো: $$

1. Let's understand the problem: find the value of $0 \div \infty$ where $\infty$ denotes infinity. 2. Division by infinity is a limit concept; it isn't a standard arithmetic opera

1. The problem gives us two equations involving $x_1$ and $x_2$: $\begin{cases} x_1 + x_2 - x_1 x_2 = 7 \\ x_1 + x_2 + x_1 x_2 = -5 \end{cases}$

1. **Problem 1:** Given $y=12$ when $x=4$, find $y$ when $x=8$ assuming inverse variation. 2. In inverse variation, $y$ varies inversely with $x$ means $y=\frac{k}{x}$ for some con

1. Problem statement: Find the constant of variation $k$ if $y$ varies inversely as $x$, and $y = 12$ when $x = 9$. Since $y$ varies inversely as $x$, the relation is:

1. **Simplify the surds:** (a)(i) Simplify $\sqrt{2} + 3\sqrt{3} - 5\sqrt{2} + \sqrt{3}$.

1. **Problem statement:** Prove by mathematical induction that $$\sum_{i=1}^{n} \frac{1}{i(i+1)} = \frac{n}{n+1}$$

1. **State the problem:** The volume $V$ of a gas is inversely proportional to its pressure $P$. If the volume is $360$ cm³ when the pressure is $20$ g/cm³, find the pressure when

1. **Problem statement:** Illustrate a real-life example of an inverse variation and solve multiple problems applying the concept. **Example:** The time needed to complete a job va

1. The problem states that $f(x) = ax^2 + 2x - 11$ and the point $(1, 0)$ lies on the graph of $f$. This means when $x = 1$, $f(x) = 0$. 2. Substitute $x = 1$ and $f(1) = 0$ into t

1. The problem involves a pattern of hexagonal numbers with each top-right number related to the numbers in the left columns and the middle numbers. 2. To find the rule, observe ea

1. The problem is to solve the equation $$|2x + 5| = 9$$ for $$x$$. 2. Absolute value equations mean the expression inside can be either positive or negative but the result is posi

1. The problem asks us to find the domain of the rational function $$f(x) = \frac{5}{x - 6}$$ and understand its graph. 2. The domain of a rational function includes all real numbe

1. **State the problem:** Simplify the expression $$3^{n+2} + \left(3^{n+3} - 3^{n+1}\right)$$ and determine which choice among (A), (B), (C), or (D) it equals. 2. **Rewrite terms

1. **State the problem:** We need to solve the inequality $4x + 2 > 16$ and determine which of the multiple-choice answers is equivalent. 2. **Solve the inequality:**

1. **State the problem:** Find the domain of the function $$h(x) = \sqrt{\frac{1 - x}{x + 1}}$$ which means we need all values of $$x$$ for which the expression under the square ro