Gujarat Board Class 12 Mathematics — Flashcards for Quick Revision

A: Step 1: For reflexive relation, check if (a,a) ∈ R for all a ∈ A. Step 2: Check each element: - (1,1) ∈ R ✓ - (2,2) ∈ R ✓ - (3,3) ∈ R ✓ Step 3: Since all pairs (a,a) are present, R is reflexive. Ans

A: Step 1: For symmetric relation, if (a,b) ∈ R, then (b,a) ∈ R. Step 2: Check each pair: - (1,2) ∈ R and (2,1) ∈ R ✓ - (2,3) ∈ R and (3,2) ∈ R ✓ Step 3: For every (a,b), the corresponding (b,a) exists.

A: Step 1: For transitive relation, if (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R. Step 2: Check all combinations: - (1,2) and (2,3) → need (1,3) → (1,3) ∈ R ✓ - (1,2) and (2,4) → need (1,4) → (1,4) ∈ R ✓ -

A: Integration is the inverse process of differentiation. While differentiation finds the rate of change of a function, integration finds the original function when its derivative is given. If f'(x) = g(

A: An anti-derivative (or primitive) of a function f(x) is a function F(x) such that F'(x) = f(x). Example: Since d/dx(sin x) = cos x, therefore sin x is an anti-derivative of cos x. Generally, sin x + C

A: The symbol is ∫f(x)dx. Here: ∫ is the integral sign, f(x) is the integrand (function to be integrated), x is the variable of integration, and dx indicates integration with respect to x. The result is

A: Trigonometric functions are not one-one (injective) over their natural domains. For a function to have an inverse, it must be both one-one and onto (bijective). Since sin x, cos x, etc. repeat their v

A: Domain: [-1, 1] Range: [-π/2, π/2] This range is called the principal value branch of sin⁻¹x. The function sin⁻¹: [-1, 1] → [-π/2, π/2]

A: Domain: [-1, 1] Range: [0, π] This range is the principal value branch of cos⁻¹x. The function cos⁻¹: [-1, 1] → [0, π]