Maharashtra Board Class 12 Mathematics & Statistics -Commerce — Flashcards

A: A statement is a declarative sentence which is either true or false, but not both simultaneously. Example: '2 is a prime number' (True), 'The Sun rises in the West' (False). Note: Questions, exclamati

A: A statement has a definite truth value (T or F), while an open sentence's truth depends on variables or conditions. Example: Statement: '5 > 3' (always true). Open sentence: 'x + 4 = 8' (true only whe

A: Conjunction connects two statements with 'and'. p ∧ q is true only when both p and q are true. Truth Table: p | q | p ∧ q T | T | T T | F | F F | T | F F | F | F Example: '2 is even AND 3 is odd' (T

A: Chain Rule Formula: If y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx) Alternatively: If y = f[g(x)], then dy/dx = f'[g(x)] × g'(x) Use when: Differentiating functions within functions, like s

A: Step 1: Identify functions - Outer function: √u where u = x² + 5 - Inner function: u = x² + 5 Step 2: Find derivatives - dy/du = 1/(2√u) = 1/(2√(x² + 5)) - du/dx = 2x Step 3: Apply chain rule dy/dx

A: Inverse Function Theorem: If y = f(x) is differentiable and dy/dx ≠ 0, and x = f⁻¹(y) exists, then: dx/dy = 1/(dy/dx) Geometric Meaning: - The slope of f⁻¹ at point (a,b) is the reciprocal of the sl

A: A matrix is a rectangular arrangement of mn numbers in m rows and n columns, enclosed in [ ] or ( ). Definition: A matrix of order m × n has m rows and n columns. Notation: A = [aij]m×n where aij is

A: This is a Scalar Matrix. Step-by-step identification: 1. Check if it's square: Yes (3×3) 2. Check diagonal elements: 5, 5, 5 (all same) 3. Check non-diagonal elements: All are 0 Definition: A diagon

A: Step 1: Find A^T by interchanging rows and columns A = [1 2 3; 4 5 6]2×3 A^T = [1 4; 2 5; 3 6]3×2 Step 2: Find (A^T)^T (A^T)^T = [1 2 3; 4 5 6]2×3 Step 3: Verify (A^T)^T = A (A^T)^T = [1 2 3; 4 5 6]