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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Advanced Math Formulas for Competitive Exams</title>
<style>
body {
font-family: Arial, sans-serif;
background-color: #f4f4f9;
margin: 0;
padding: 20px;
color: #333;
}
h1, h2 {
text-align: center;
color: #007bff;
}
.container {
max-width: 900px;
margin: 0 auto;
padding: 20px;
background: #fff;
border-radius: 10px;
box-shadow: 0 4px 8px rgba(0, 0, 0, 0.1);
}
section {
margin-bottom: 20px;
}
.formula {
padding: 10px 15px;
background: #f9f9f9;
border-left: 4px solid #007bff;
margin: 10px 0;
font-family: "Courier New", monospace;
}
.example {
padding: 10px;
background: #e9ecef;
border-left: 4px solid #28a745;
margin: 10px 0;
}
footer {
text-align: center;
margin-top: 30px;
color: #777;
font-size: 0.9em;
}
</style>
</head>
<body>
<h1>Advanced Math Formulas for Competitive Exams</h1>
<div class="container">
<!-- Algebra -->
<section>
<h2>Algebra</h2>
<div class="formula">Quadratic Formula: \\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\)</div>
<div class="example">
<strong>Example:</strong> Solve the quadratic equation \\( 2x^2 + 3x - 2 = 0 \\).
<br>Here, \\( a = 2 \\), \\( b = 3 \\), and \\( c = -2 \\).
<br>Using the quadratic formula: \\( x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)} \\).
<br>Solution: \\( x = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4} \\).
<br>So, \\( x = \frac{-3 + 5}{4} = \frac{2}{4} = 0.5 \\) or \\( x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2 \\).
</div>
<div class="formula">Sum of Roots of a Quadratic Equation: \\( \alpha + \beta = -\frac{b}{a} \\)</div>
<div class="example">
<strong>Example:</strong> For the equation \\( 2x^2 + 3x - 2 = 0 \\), find the sum of the roots.
<br>Using the formula \\( \alpha + \beta = -\frac{b}{a} \\), where \\( b = 3 \\) and \\( a = 2 \\):
<br>Sum of roots: \\( \alpha + \beta = -\frac{3}{2} = -1.5 \\).
</div>
</section>
<!-- Geometry -->
<section>
<h2>Geometry</h2>
<div class="formula">Area of Circle: \\( A = \pi r^2 \\)</div>
<div class="example">
<strong>Example:</strong> Find the area of a circle with radius \\( r = 7 \\).
<br>Using the formula: \\( A = \pi r^2 \\).
<br>Solution: \\( A = \pi \times 7^2 = 49\pi \\approx 153.94 \\) square units.
</div>
<div class="formula">Volume of Cylinder: \\( V = \pi r^2 h \\)</div>
<div class="example">
<strong>Example:</strong> Find the volume of a cylinder with radius \\( r = 3 \\) and height \\( h = 5 \\).
<br>Using the formula: \\( V = \pi r^2 h \\).
<br>Solution: \\( V = \pi \times 3^2 \times 5 = 45\pi \\approx 141.37 \\) cubic units.
</div>
</section>
<!-- Time and Work -->
<section>
<h2>Time and Work</h2>
<div class="formula">Work Done: \\( W = \text{Rate} \times \text{Time} \\)</div>
<div class="example">
<strong>Example:</strong> If a worker completes 20 units of work in 5 hours, find the rate of work.
<br>Using the formula: \\( \text{Rate} = \frac{\text{Work}}{\text{Time}} \\).
<br>Solution: \\( \text{Rate} = \frac{20}{5} = 4 \\) units per hour.
</div>
<div class="formula">Combined Work Rate: \\( \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} + ... \\)</div>
<div class="example">
<strong>Example:</strong> If two workers complete a task in 6 and 8 hours, find the time taken when they work together.
<br>Using the formula: \\( \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2} \\).
<br>Solution: \\( \frac{1}{T} = \frac{1}{6} + \frac{1}{8} = \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \\).
<br>So, \\( T = \frac{24}{7} \\approx 3.43 \\) hours.
</div>
</section>
<!-- Speed and Distance -->
<section>
<h2>Speed and Distance</h2>
<div class="formula">Speed: \\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \\)</div>
<div class="example">
<strong>Example:</strong> A car travels 120 km in 2 hours. Find its speed.
<br>Using the formula: \\( \text{Speed} = \frac{120}{2} = 60 \\) km/h.
</div>
<div class="formula">Relative Speed (When Two Objects are Moving in Opposite Directions): \\( S_{rel} = S_1 + S_2 \\)</div>
<div class="example">
<strong>Example:</strong> Two trains are moving towards each other with speeds 60 km/h and 40 km/h. Find their relative speed.
<br>Using the formula: \\( S_{rel} = 60 + 40 = 100 \\) km/h.
</div>
</section>
<!-- Profit and Loss -->
<section>
<h2>Profit and Loss</h2>
<div class="formula">Profit Percentage: \\( \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 \\)</div>
<div class="example">
<strong>Example:</strong> A man buys an article for \\( 500 \\) and sells it for \\( 600 \\). Find the profit percentage.
<br>Profit = \\( 600 - 500 = 100 \\).
<br>Profit percentage = \\( \frac{100}{500} \times 100 = 20\% \\).
</div>
<div class="formula">Loss Percentage: \\( \text{Loss \%} = \frac{\text{Loss}}{\text{Cost Price}} \times 100 \\)</div>
<div class="example">
<strong>Example:</strong> A man buys an article for \\( 800 \\) and sells it for \\( 600 \\). Find the loss percentage.
<br>Loss = \\( 800 - 600 = 200 \\).
<br>Loss percentage = \\( \frac{200}{800} \times 100 = 25\% \\).
</div>
</section>
</div>
<footer>
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</footer>
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