Solving Polynomial Equations: Descartes and Cardan Methods
1. Nature of Roots: Descartes’s Rule of Signs
Find the nature of the roots for the equation: x⁴ + 2x² + 3x – 2 = 0
Applying Descartes’ Rule of Signs
Given the function: f(x) = x⁴ + 2x² + 3x – 2
Signs of coefficients: +, +, +, –
Count sign changes:
- + to + (No change)
- + to + (No change)
- + to – (1 change)
Therefore, there is exactly one positive real root.
Replace x with -x:
f(-x) = (-x)⁴ + 2(-x)² + 3(-x) – 2 = x⁴ + 2x² – 3x – 2
Signs: +, +, -, –
Count sign changes:
- + to + (No change)
- + to – (1 change)
Refractive Index Determination: Glass Slab and Prism Experiments
Experiment 6: Refractive Index Determination
Part 1: Refractive Index of a Glass Slab (Pin Method)
Aim
To determine the refractive index of a glass slab material using the pin method.
Apparatus and Materials
Glass slab, drawing board, drawing pins, hairpins, paper, scale, and protractor.
Theory
The refractive index, n, of glass with respect to air is defined by Snell’s Law:
n = sin i / sin r
where i is the angle of incidence and r is the angle of refraction.
A graph plotted with sin r along the X-axis and
Read MoreEssential Excel Formulas and Functions Reference
Quick Reference: Excel Formulas
- Adding many cells: =SUM
- Adding 2 cells: + or –
- Finding typical value: =AVERAGE
- Finding true middle: =MEDIAN
- Finding smallest: =MIN
- Finding largest: =MAX
- Counting numbers: =COUNT
- Counting text or anything: =COUNTA
- Applying rule/condition: =IF
- Calculating loan payment: =PMT
- Summarizing multiple sheets: 3D SUM
- Testing different scenarios: Goal Seek / Data Table
- Multiplying by fixed %: Absolute reference ($)
1. Mathematical & Statistical Functions
These are the workhorses of Excel
Read MoreCore Epidemiology Concepts: Definitions, History, and Measures
Lectures 1–2 – Intro to Epidemiology
Key Definitions:
Epidemiology: study of frequency, distribution, determinants of health/disease in populations + application to control.
Frequency: how often events occur (counts, proportions, rates).
Distribution: patterns by person/place/time.
Determinants: factors influencing occurrence (host/agent/environment).
Historical Figures (name + contribution):
Edward Jenner: cowpox → smallpox vaccine (1st vaccination).
James Lind: scurvy + citrus trial.
Percival Pott:
Discrete Mathematics: Sets, Graphs, and Matrix Algebra
Unit 1: Set Theory and Functions
1. Set
A set is a well-defined collection of distinct objects, called elements or members. These objects can be numbers, symbols, people, or even other sets. Sets are usually denoted by capital letters such as A, B, or C, and their elements are written within curly braces. For example, A = {1, 2, 3}. A set must be well-defined, meaning it should be clear whether a given object belongs to the set or not. Sets play a foundational role in mathematics and computer science
Read MoreBusiness Statistics: Central Tendency Measures and Data Insights
Business Statistics: Data-Driven Insights
Business statistics are vital for informed decision-making, performance evaluation, and risk management. They utilize data analysis to provide insights into market trends, customer behavior, and operational efficiency. The scope is broad, covering areas like forecasting, production planning, marketing analysis, and financial modeling, helping businesses make strategic plans based on data rather than intuition.
Importance of Business Statistics
- Informed Decision-