🧮 algebra

1. The problem is to solve the first question from the provided document link. Since the link is inaccessible, I will solve a common algebra problem as an example. 2. Let's solve t

1. समस्या: हमें एक सरल समीकरण हल करना है। 2. नियम: समीकरण को हल करने के लिए, हम दोनों पक्षों में समान क्रियाएं करेंगे ताकि समीकरण संतुलित रहे।

1. **State the problem:** Solve the equation $$\frac{2x - 4}{3} - \frac{3x + 2}{4} = \frac{x - 5}{6} - 3$$ given $x = 12$. 2. **Substitute $x = 12$ into the equation:**

1. The problem gives three quadratic functions: $$J = x^2 + 6x + 9$$, $$K = 5x^2 - 2\sqrt{5} x + 1$$, and $$L = 7x^2 - 1$$. We are asked to analyze their shapes and properties. 2.

1. **State the problem:** Find the equation of the ellipse with axes along the coordinate axes, vertices at $(\pm 5, 0)$, and foci at $(\pm 4, 0)$.\n\n2. **Recall the standard form

1. Problem: A bookworm bores horizontally from the first page of Volume 1 to the last page of Volume 3. Each volume is 1 inch thick without covers, and each cover is \frac{1}{16} i

1. **Problem statement:** Find the length of the major and minor axes, coordinates of foci and vertices, and eccentricity for the ellipses: (i) $16x^2 + 25y^2 = 400$

1. **Problem:** A shopkeeper earns a profit of 70 for every 3 bags of wheat sold. How many bags must be sold to make a profit of 1400? 2. **Formula:** Profit per bag = \frac{70}{3}

1. The problem is to solve the equation $2x + 3 = 7$ for $x$. 2. We use the basic algebraic principle of isolating the variable $x$ by performing inverse operations.

1. **Problem statement:** Find the area of the triangle formed by the vertex of the parabola $x^2 = 12y$ and the ends of its latus rectum. 2. **Identify the parabola parameters:**

1. **State the problem:** We have a parabolic arc with a vertical axis, 10 m high and 5 m wide at the base. We want to find the width of the arc 2 m from the vertex. 2. **Set up th

1. **Problem Statement:** Find the coordinates of the foci, the equations of the directrices, and the length of the latus rectum for each parabola. 2. **General formulas for parabo

1. **State the problem:** Find the equation of the circle passing through points $(1,-2)$ and $(4,-3)$ with its center on the line $3x + 4y = 7$. 2. **Formula and rules:** The gene

1. Synthetic division ek tarah ka shortcut method hai jo hum polynomial division ke liye use karte hain, khas taur par jab divisor linear form mein ho, jaise $x - c$. 2. Iska maksa

1. **State the problem:** Divide the polynomial $p(x) = x^3 - x^2 + x - 1$ by $x - 1$ using synthetic division and find the quotient and remainder. 2. **Recall synthetic division:*

1. **State the problem:** Simplify the expression $2\sqrt{5} \cdot 3\sqrt{40}$.\n\n2. **Recall the multiplication rule for radicals:** $a\sqrt{b} \cdot c\sqrt{d} = (a \cdot c) \sqr

1. **State the problem:** Rationalise the denominator of the expression $$\frac{6 + 5\sqrt{2}}{3\sqrt{2} + 4}$$ and express the answer in the form $$a + b\sqrt{2}$$. 2. **Formula a

1. প্রশ্ন: $x^3 + y^3$, $r^3 - s^3$ এবং $x^6 - y^6$ এর সমানু কত? 2. সূত্র:

1. **Problem statement:** Find the equation of the circle that passes through the origin and cuts off intercepts 3 and 4 from the positive parts of the x-axis and y-axis respective

1. **Problem statement:** Given the sequence $u_n = \frac{n^2}{2^n}$ for $n > 0$, define $v_n = \frac{u_{n+1}}{u_n}$. We need to find $\lim_{n \to +\infty} v_n$, prove inequalities

1. **Problem statement:** Given that $\sqrt{5}$ is irrational, prove that $3 - 2\sqrt{5}$ is irrational. 2. **Recall the definition:** A number is irrational if it cannot be expres