Seshu cheera

Here are some important points about differential equations: Definition of a Differential Equation : A differential equation is an equation that involves the derivative (or derivatives) of a dependent variable with respect to an independent variable (or variables). These equations are fundamental in various fields, including Physics, Chemistry, Biology, Anthropology, Geology, and Economics, highlighting their prime importance in modern scientific investigations . For example, the equation dy/dx = g(x) , where y = f(x) , is a differential equation. Similarly, dy/dx + x/y = 0 is a differential equation because it involves variables and the derivative of y with respect to x . Types of Differential Equations : Ordinary Differential Equation : This type involves derivatives of the dependent variable with respect to only one independent variable . The sources primarily focus on ordinary differential equations. An example is d²y/dx² + (dy/dx)² = 0 . Partial Differential Equati...

Here’s the complete guide for Chapter 8: Application of Integrals — Class 12 CBSE Maths: 🧠 A. Key Concepts Area Under a Curve Area = ∫ a b f ( x )   d x \text{Area} = \int_a^b f(x)\,dx Valid when f ( x ) ≥ 0 f(x) \ge 0 on [ a , b ] [a,b] . Area Between Two Curves If f ( x ) ≥ g ( x ) f(x) \ge g(x) , then Area = ∫ a b [ f ( x ) − g ( x ) ]   d x \text{Area} = \int_a^b [f(x) - g(x)]\,dx Area with Respect to Y-axis Area = ∫ c d [ x right ( y ) − x left ( y ) ]   d y \text{Area} = \int_c^d [x_{\text{right}}(y) - x_{\text{left}}(y)]\,dy Combined Regions Split the integration region when functions intersect; sum individual definite integrals. 📘 B. NCERT Exercises & Solutions ✅ Ex 8.1 – Area under curve Identity: ∫ x 2 , ∫ x \int x^2, \int \sqrt{x} , etc. Check correct application of limits. ✅ Ex 8.2 – Area between curves Determine intersection points Use ∫ ( f − g )   d x \int (f - g) \,dx with proper limits ✅ Ex 8.3 – Area w.r...

 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 ...

 Here’s your comprehensive guide for Chapter 4: Determinants (Class 12 CBSE Maths), covering: 🧠 A. Key Concepts & Properties 1. What Is a Determinant? Assigns a scalar value to a square matrix. For 1 × 1 1 \times 1 : ∣ [ a ] ∣ = a |[a]| = a ; For 2 × 2 2 \times 2 , ∣ a b c d ∣ = a d − b c \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc ; For 3 × 3 3 \times 3 , use expansion by minors and cofactors.  2. Properties of Determinants (Useful for simplifying evaluations) : Transpose Invariance : ∣ A ∣ = ∣ A T ∣ |A| = |A^T| Sign Swap : Swapping two rows flips the sign Zero/Equal Row : Any row (or column) of zeros or two identical rows ⇒ ∣ A ∣ = 0 |A| = 0 Scalar Factor : Multiplying a row by k k scales ∣ A ∣ |A| by k k Linearity : Split row entries into sums to split determinant Triangular Matrix : Determinant equals product of diagonal entries 3. Minors, Cofactors & Expansion Minor M i j M_{ij} : Determinant obt...

 Here’s the comprehensive guide for Chapter 3: Matrices (Class 12 CBSE Maths): 🧠 A. Key Concepts & Properties Matrix Definition & Order A matrix is a rectangular array [ a i j ] [a_{ij}] of order m × n m \times n containing real numbers/functions. Types of Matrices : Row , Column , Square , Rectangular Zero , Identity , Diagonal , Scalar Symmetric ( A T = A A^T = A ), Skew-symmetric ( A T = − A A^T = -A ), Transpose ( A T A^T ) Matrix Operations : Sum/Subtraction : Element-wise, requires same order  Scalar Multiplication Matrix Multiplication : Product defined if columns of A = rows of B; not commutative . Elementary Row/Column Operations & Inverse : Inverse ( A − 1 A^{-1} ) exists only for square matrices; findable via elementary operations  Properties of Transpose : ( A T ) T = A (A^T)^T = A , ( A B ) T = B T A T (AB)^T = B^T A^T   Symmetric/Skew Decomposition : Any square A A can be writ...

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...