Kerala Board Class 12 Mathematics — Flashcards for Quick Revision

A: A mathematical statement (or proposition) is a sentence that is either true or false, but not both. It must be a definite assertion that can be evaluated for its truth value. Example: '5 is an even n

A: Let p(n) be a statement involving a natural number n. If: (i) p(1) is true (Base case), and (ii) For any k ≥ 1, if p(k) is true, then p(k+1) is also true (Inductive step) Then p(n) is true for all n

A: Given: p(n): 2ⁿ > n + 1 Step 1: Replace n with 1 p(1): 2¹ > 1 + 1, which is 2 > 2 (False) Step 2: Replace n with k p(k): 2ᵏ > k + 1 Step 3: Replace n with k+1 p(k+1): 2ᵏ⁺¹ > (k+1) + 1, which is 2ᵏ⁺

A: A complex number is a number of the form z = a + bi, where: • a and b are real numbers • i = √(-1) is the imaginary unit • a is called the real part [Re(z) = a] • b is called the imaginary part [Im(z)

A: Powers of i follow a cyclic pattern: • i¹ = i • i² = -1 • i³ = i² · i = (-1) · i = -i • i⁴ = (i²)² = (-1)² = 1 • i⁵ = i⁴ · i = 1 · i = i • i⁸ = (i⁴)² = 1² = 1 Pattern: i, -1, -i, 1 (repeats every 4 p

A: For z = 3 - 5i, the conjugate z̄ = 3 + 5i General Rule: The conjugate of z = a + bi is z̄ = a - bi • Change the sign of the imaginary part only • Real part remains unchanged Properties of Conjugates