Punjab Board Class 12 Mathematics — Flashcards

A: Step 1: Check reflexivity - Does (a,a) ∈ R for all a? Test (1,1): 1-1=0≠10. So NOT reflexive. Step 2: Check symmetry - If (a,b) ∈ R, is (b,a) ∈ R? Since no pairs exist in R (no two elements differ by

A: Step 1: Reflexivity - Does 2 divide (a-a) = 0? YES, 0 is divisible by 2. ✓ Step 2: Symmetry - If (a,b) ∈ R, then 2|(a-b). Does 2|(b-a)? Since (b-a) = -(a-b) and 2 divides (a-b), it divides -(a-b). So

A: Step 1: Reflexivity - Can a line be perpendicular to itself? NO. (L₁, L₁) ∉ R. NOT reflexive. Step 2: Symmetry - If L₁ ⊥ L₂, is L₂ ⊥ L₁? YES, perpendicularity is mutual. (L₁, L₂) ∈ R ⟹ (L₂, L₁) ∈ R.

A: Step 1: Let y = sin⁻¹(1/√2). Then sin(y) = 1/√2 Step 2: The principal value branch of sin⁻¹ has range [-π/2, π/2] Step 3: Find angle in [-π/2, π/2] where sin equals 1/√2 Step 4: sin(π/4) = 1/√2 Step 5

A: Step 1: Let y = cos⁻¹(-1/2). Then cos(y) = -1/2 Step 2: Principal value branch of cos⁻¹ has range [0, π] Step 3: Find angle in [0, π] where cosine equals -1/2 Step 4: cos(2π/3) = -1/2 (since cos(π/3)

A: Step 1: Let y = tan⁻¹(-√3). Then tan(y) = -√3 Step 2: Principal value branch of tan⁻¹ has range (-π/2, π/2) Step 3: tan(π/3) = √3, so tan(-π/3) = -√3 Step 4: -π/3 lies in (-π/2, π/2) ✓ Step 5: Therefo

A: Step 1: Count the number of rows = 3 Step 2: Count the number of columns = 4 Step 3: Order of matrix = m × n = 3 × 4 Answer: The matrix A is of order 3 × 4 (read as 3 by 4)

A: Step 1: If matrix has order m × n, then total elements = m × n = 24 Step 2: Find all factor pairs of 24: - 1 × 24 - 24 × 1 - 2 × 12 - 12 × 2 - 3 × 8 - 8 × 3 - 4 × 6 - 6 × 4 Answer: Po

A: Step 1: Set up general form: A = [[a₁₁, a₁₂], [a₂₁, a₂₂]] Step 2: Calculate each element: - a₁₁ = (1 + 1)²/2 = 4/2 = 2 - a₁₂ = (1 + 2)²/2 = 9/2 - a₂₁ = (2 + 1)²/2 = 9/2 - a₂₂ = (2 + 2)²/2 = 1