Seshu cheera

 Here’s the comprehensive guide for Chapter 5: Continuity & Differentiability (CBSE Class 12 Maths, 2025–26 syllabus): A. 🔑 Key Concepts and Theorems 1. Continuity A function f f is continuous at x = a x = a if lim ⁡ x → a f ( x ) = f ( a ) \lim_{x \to a}f(x) = f(a) , meaning left and right limits equal the function value . Discontinuities : Removable (hole), jump , and infinite/oscillatory . Algebra of continuous functions : Sums, products, quotients (with non-zero denominator), and compositions of continuous functions are continuous . Intermediate Value Theorem / Rolle’s & Mean Value Theorem : continuous on [a,b] ⇒ there exist points satisfying f(b)=f(a) or f′(c) = (f(b)–f(a))/(b–a) . 2. Differentiability f f is differentiable at a a if the limit lim ⁡ h → 0 f ( a + h ) − f ( a ) h \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} exists . Differentiable ⇒ Continuous , but not vice versa (e.g. f ( x ) = ∣ x ∣ f(x)=|x| at 0) . Left ...

Chapter 2: Inverse Trigonometric Functions (Class 12 CBSE Math) ✅ A. Key Concepts & Formulas Definition & Principal Values Inverse trig functions are the restricted inverses of trig functions: sin ⁡ − 1 x \sin^{-1} x : domain x ∈ [ − 1 , 1 ] x \in [-1,1] ; range [ − π / 2 , π / 2 ] [-π/2, π/2] cos ⁡ − 1 x \cos^{-1} x : domain [ − 1 , 1 ] [-1,1] ; range [ 0 , π ] [0, π] tan ⁡ − 1 x \tan^{-1} x : domain all real; range ( − π / 2 , π / 2 ) (-π/2, π/2) Similar definitions apply for cot ⁡ − 1 , sec ⁡ − 1 , csc ⁡ − 1 \cot^{-1}, \sec^{-1}, \csc^{-1} . Properties & Identities Odd–even behavior: sin ⁡ − 1 ( − x ) = − sin ⁡ − 1 x \sin^{-1}(-x)=-\sin^{-1}x , cos ⁡ − 1 ( − x ) = π − cos ⁡ − 1 x \cos^{-1}(-x)=π-\cos^{-1}x , etc. Complementary relations: sin ⁡ − 1 x + cos ⁡ − 1 x = π / 2 \sin^{-1}x + \cos^{-1}x = π/2 ; tan ⁡ − 1 x + cot ⁡ − 1 x = π / 2 \tan^{-1}x + \cot^{-1}x = π/2 . Reciprocal links: sin ⁡ − 1 ( 1 / a ) = csc ⁡ − 1 ( a ) \sin^{-1}(1/a)...

   Unit I – Relations & Functions from Class 12 CBSE Maths , including deeper insights , conceptual clarity , exercise solutions , and top-scoring CBSE answers : 🧠 1. In-Depth Concept Review 📌 Relations A relation on a set A A is any subset of A × A A \times A . Key types include : Empty relation : no pair included. Universal relation : all possible pairs included. ✅ Reflexive : ∀ x x , ( x , x ) ∈ R (x,x) \in R . ✅ Symmetric : if ( x , y ) ∈ R (x,y) \in R , then ( y , x ) ∈ R (y,x) \in R . ✅ Transitive : if ( x , y ) , ( y , z ) ∈ R (x,y), (y,z) \in R , then ( x , z ) ∈ R (x,z)\in R . ✅ Equivalence relation : satisfies all three; partitions A A into equivalence classes . 📌 Functions A relation is a function if each element in the domain corresponds to exactly one value in the codomain. Types of functions: Injective (one‑to‑one) : distinct inputs yield distinct outputs. Surjective (onto) : every element in the codomai...

📘 CBSE Class 10 Maths (Chapter: Statistics & Probability) ✨ 1. Key Concepts of the Chapter ✅ Statistics Statistics is the study of collecting, organizing, analyzing, and interpreting numerical data. In this chapter, we focus on statistical measures: Mean (Average) Median (Middle value) Mode (Most frequent value) Data dealt with here is grouped data (continuous classes). ✅ Probability Probability measures the likelihood of an event. Classical definition: P ( E ) = Number of favorable outcomes Total number of outcomes P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} P ( E ) = Total number of outcomes Number of favorable outcomes Deals with simple experiments : coin toss, dice roll, card pick, etc. 📌 2. Important Formulae 🔷 For Statistics (Grouped Data) 👉 Mean: Mean ,   x ‾ = ∑ f i x i ∑ f i \text{Mean},\ \overline{x} = \frac{\sum f_ix_i}...