Seshu cheera

 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...

  Class 10 CBSE Maths – Chapter 7: Coordinate Geometry 🔹 Key Concepts and Formulas ✅ Coordinate Plane A plane with two perpendicular lines (axes): x-axis (horizontal) and y-axis (vertical) The point of intersection is called the origin (0, 0) Coordinates are written as (x, y) ✅ Quadrants I Quadrant: (+x, +y) II Quadrant: (–x, +y) III Quadrant: (–x, –y) IV Quadrant: (+x, –y) ✅ Distance Formula To find distance between two points A ( x 1 , y 1 ) A(x_1, y_1) and B ( x 2 , y 2 ) B(x_2, y_2) : A B = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ✅ Section Formula To find coordinates of a point P which divides the line segment AB in the ratio m : n m:n : P = ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ✅ Midpoint Formula Special case of section formula when m = n: Midpoint = ( x 1 + x 2 2 , y 1 + y 2 2 ) \text{Midpoint} = \left( \frac{x_1 + ...