🧮 algebra

1. **State the problem:** Find the angle between the two lines given by their parametric equations: Line 1: $x = -2t - 9$, $y = 10t - 1$, $z = -2t - 9$

1. समस्या: एक व्यक्ति ने दो किस्म के सेब खरीदे, पहली किस्म 3 सेब 2 रु. में और दूसरी किस्म 1 रु. प्रति सेब की दर से। दोनों किस्म के सेब समान संख्या में खरीदे। सभी सेबों को 1 रु. प्र

1. **State the problem:** Express the given series in summation notation and find the general term. 2. **Identify the sequences:**

1. **State the problem:** Simplify the expression $$\frac{x^2 - xy + y^2}{x - y} - \frac{x^2 + xy + y^2}{x + y}$$. 2. **Recall the formula and rules:** We want to simplify each rat

1. The problem asks to analyze and sketch the graph of the function $$y = 21 (x - 3)^2 + 2$$ by translating, reflecting, compressing, and stretching the graph of the base function

1. **State the problem:** Simplify the expression $$\left(\frac{48}{2}\right) - \frac{17}{1} + 14^{-2} \times \frac{2}{1}$$. 2. **Recall the order of operations:** Perform division

1. **State the problem:** Simplify the expression $$\frac{3}{x^2 - 4} + \frac{1}{(x-2)^2}$$. 2. **Recall the formula and rules:** Recognize that $$x^2 - 4$$ is a difference of squa

1. **Determine whether the following statements are functions of $x$ or not.** (i). $y = x^2 + 1$ for each $x \in \mathbb{R}$.

1. **State the problem:** Find the value of $?$ in the equation $$45.24 \times 89.92 + 28.39 \times 90.23 = ? - 3.44.$$ 2. **Rewrite the equation:** Add $3.44$ to both sides to iso

1. **State the problem:** Find the value of $\sqrt{24}$.\n\n2. **Recall the formula and rules:** The square root of a number $x$ is a value that, when multiplied by itself, gives $

1. The problem is to simplify the expression \(\sqrt{x}24\). 2. We interpret this as \(24\sqrt{x}\), which means 24 times the square root of \(x\).

1. The problem is to sketch the graph of the function $$y = -2(x + 1)^2 - 3$$ by transforming the graph of $$y = x^2$$. 2. The base function is $$y = x^2$$, which is a parabola ope

1. **State the problem:** We are given the sum of the first 10 terms ($S_{10}$) of an arithmetic progression (AP) as 460 and the sum of the next 10 terms ($S_{20} - S_{10}$) as 126

1. The problem is to analyze the system of 10 linear equations in variables $x$, $y$, and $z$: $$\begin{cases}

1. **Stating the problem:** We are given the sequence of numbers 4, 6, 10 and asked to find the pattern. 2. **Observing the sequence:** The numbers are 4, 6, 10.

1. The problem asks to find the pattern of a sequence of numbers, but no numbers were provided. 2. To find a pattern, we typically look for relationships such as arithmetic progres

1. **Stating the problem:** We are given the sequence 2, 8, 18, 32, 50 and need to find the formula for the nth term. 2. **Look for a pattern:** Let's examine the differences betwe

1. **Stating the problem:** We are given the sequence 2, 8, 18, 32, 50 and asked to find the pattern of the terms. 2. **Look for a pattern:** Let's denote the $n$th term as $a_n$.

1. **Problem statement:** Solve the equation $$\sqrt{3} - x = x\sqrt{3} + x.$$\n\n2. **Rewrite the equation:** Move all terms to one side to isolate $x$:\n$$\sqrt{3} - x - x\sqrt{3

1. **State the problem:** Given that $\left(x+\frac{1}{x}\right)^2=40$, find the value of $x^2+\frac{1}{x^2}$.\n\n2. **Recall the formula:** We know that $\left(a+b\right)^2 = a^2

1. **State the problem:** We are given the system: