AJAY PATHAK (MATH)

Best Online Mathematics Tutor – Ajay Pathak Learn Mathematics from School Level to University Level with Conceptual Clarity 👨‍🏫 About Me I am Ajay Pathak , an experienced online Mathematics tutor with a strong academic background in Pure Mathematics, Applied Mathematics, and Physics . I teach students from school level to university level with a focus on conceptual understanding, problem-solving skills, and exam-oriented preparation . Currently, I also explore the use of Python programming in Mathematics and mathematical modeling to help students connect theory with real-world applications. 📘 Subjects I Teach 🔹 School Level Mathematics Basic Mathematics Arithmetic & Number System Algebra (Linear & Quadratic Equations) Polynomials Trigonometry Coordinate Geometry Mensuration Statistics & Probability 🔹 International Curriculum IGCSE Mathematics Cambridge Mathematics Concept-based exam preparation ...

Solution: Triangle Inequality Statement: For all real numbers x and y, |x + y| ≤ |x| + |y| (a) Verification Case (i): x = 2, y = 3 |x + y| = |2 + 3| = |5| = 5 |x| + |y| = |2| + |3| = 2 + 3 = 5 ∴ |x + y| = |x| + |y| Case (ii): x = −2, y = −3 |x + y| = |−2 − 3| = |−5| = 5 |x| + |y| = |−2| + |−3| = 2 + 3 = 5 ∴ |x + y| = |x| + |y| Case (iii): x = −2, y = 3 |x + y| = |−2 + 3| = |1| = 1 |x| + |y| = |−2| + |3| = 2 + 3 = 5 ∴ |x + y| ≤ |x| + |y| (b) Proof for all real numbers We prove the inequality by considering different cases. Case 1: x ≥ 0, y ≥ 0 |x + y| = x + y = |x| + |y| Hence, |x + y| ≤ |x| + |y|. Case 2: x ≤ 0, y ≤ 0 |x + y| = −(x + y) = −x − y = |x| + |y| Hence, |x + y| ≤ |x| + |y|. Case 3: x and y have opposite signs In this case, the sum x + y is reduced in magnitude. Therefore, |x + y| < |x| + |y| Hence, the inequality holds. Therefore, the Triangle Inequality holds for all real numbers x and ...

Max and Min of Two Numbers Using Absolute Value 📘 Need help with Mathematics? Learn Mathematics with Ajay Pathak on Preply. 👉 Click here to book a lesson Maximum and Minimum of Two Numbers Using Absolute Value The maximum and minimum of two real numbers a and b can be expressed using the absolute value function. These formulas are very useful in mathematical analysis and proofs. 1. Formula for Maximum max(a, b) = (a + b + |a − b|) / 2 Proof Case 1: a ≥ b a − b ≥ 0 ⇒ |a − b| = a − b (a + b + |a − b|) / 2 = (a + b + (a − b)) / 2 = a Hence, max(a, b) = a . Case 2: b > a a − b < 0 ⇒ |a − b| = b − a (a + b + |a − b|) / 2 = (a + b + (b − a)) / 2 = b Hence, max(a, b) = b . 2. Formula for Minimum min(a, b) = (a + b − |a − b|) / 2 Proof Case 1: a ≤ b a − b ≤ 0 ⇒ |a − b| = b − a (a + b − |a − b|) / 2 = (a + b − (b − a)) / 2 = a Hence, min(a, b) = a . ...

📘 Learn Mathematics With Ajay Pathak Hello everyone! 👋 My name is Ajay Pathak , and I am a Mathematics tutor with experience in teaching: School-level Mathematics IGCSE & O Level Additional Mathematics BSc & Engineering Mathematics Calculus, ODE, Linear Algebra & More If you are looking for clear explanations, structured lessons, weekly tests, and strong exam preparation, I would be happy to guide you. 👉 Book Your Lesson You can check my Preply tutoring profile and book a lesson using the link below: 📌 Click Here to Visit My Preply Profile Feel free to contact me anytime if you have questions. Looking forward to helping you succeed in Mathematics!

Probability and Statistics — Formula Book (GTU: BE03000251) Probability and Statistics — Formula Book GTU: BE03000251 — Semester 3 Contents Unit 1: Basic Probability Unit 2: Special Probability Distributions Unit 3: Basic Statistics (Grouped and Ungrouped Data) Correlation and Regression Unit 4: Hypothesis Testing (Applied) Unit 5: Curve Fitting by Least Squares Appendices Unit 1: Basic Probability Basic concepts Experiment , Outcome , Sample space \(S\). Event : subset \(A\subseteq S\). Probability measure \(P\) satisfies: \(0\le P(A)\le1\), \(P(S)=1\), countable additivity. Axioms and simple results \(P(\varnothing)=0,\qquad P(S)=1.\) \(P(A^c)=1-P(A).\) Addition rule \(P(A\cup B)=P(A)+P(B)-P(A\cap B).\) For mutually exclusive events (\(A\cap B=\varnothing\)): \(P(A\cup B)=P(A)+P(B)\). Conditional probabili...

: South Korea's Declining Population South Korea is experiencing a significant demographic shift characterized by a declining population. This trend is driven by several socio-economic and cultural factors, posing challenges for the country's future. Current Population Trend As of 2024, South Korea's population has slightly declined by 0.02% from the previous year, bringing the total population to approximately 51.7 million. Projections indicate that this trend will accelerate, with the population potentially dropping to around 34.4 million by 2070. Key Causes of Population Decline Low Fertility Rate: South Korea's fertility rate is one of the lowest globally, at 0.78 children per woman, far below the replacement level of 2.1. High Costs of Living: Child-rearing costs are prohibitively expensive, discouraging families from having children. Cultural Shifts: Many young adults prior...

The Golden Ratio in Art, Architecture, and Nature The Golden Ratio in Art, Architecture, and Nature The Golden Ratio , also known as the divine proportion, has fascinated artists, architects, and mathematicians for centuries. This special number, approximately equal to 1.618 , appears in many forms in art, architecture, and nature. Let's explore how it shapes our world. What is the Golden Ratio? The Golden Ratio is derived from the Fibonacci sequence, where each number is the sum of the two preceding ones. If you divide a number in the sequence by its previous number, the result converges to 1.618 as the numbers grow larger. "Mathematics is the language of the universe, and the Golden Ratio is its poetry." Golden Ratio in Art Many artists have used the Golden Ratio to create aesthetically pleasing compositions. Some notable examples include: Leonardo da Vinci's "Vitruvian Man,...