Question

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Answer 1

Answer:

LIST OF IMPORTANT MATH FORMULAS AND RESULTS

Algebra:

● Laws of Indices:

(i) aᵐ ∙ aⁿ = aᵐ + ⁿ

(ii) aᵐ/aⁿ = aᵐ - ⁿ

(iii) (aᵐ)ⁿ = aᵐⁿ

(iv) a = 1 (a ≠ 0).

(v) a-ⁿ = 1/aⁿ

(vi) ⁿ√aᵐ = aᵐ/ⁿ

(vii) (ab)ᵐ = aᵐ ∙ bⁿ.

(viii) (a/b)ᵐ = aᵐ/bⁿ

(ix) If aᵐ = bᵐ (m ≠ 0), then a = b.

(x) If aᵐ = aⁿ then m = n.

● Surds:

(i) The surd conjugate of √a + √b (or a + √b) is √a - √b (or a - √b) and conversely.

(ii) If a is rational, √b is a surd and a + √b (or, a - √b) = 0 then a = 0 and b = 0.

(iii) If a and x are rational, √b and √y are surds and a + √b = x + √y then a = x and b = y.

● Complex Numbers:

(i) The symbol z = (x, y) = x + iy where x, y are real and i = √-1, is called a complex (or, imaginary) quantity;x is called the real part and y, the imaginary part of the complex number z = x + iy.

(ii) If z = x + iy then z = x - iy and conversely; here, z is the complex conjugate of z.

(iii) If z = x+ iy then

(a) mod. z (or, | z | or, | x + iy | ) = + √(x² + y²) and

(b) amp. z (or, arg. z) = Ф = tan−1−1 y/x (-π < Ф ≤ π).

(iv) The modulus - amplitude form of a complex quantity z is

z = r (cosф + i sinф); here, r = | z | and ф = arg. z (-π < Ф <= π).

(v) | z | = | -z | = z ∙ z = √ (x² + y²).

(vi) If x + iy= 0 then x = 0 and y = 0(x,y are real).

(vii) If x + iy = p + iq then x = p and y = q(x, y, p and q all are real).

(viii) i = √-1, i² = -1, i³ = -i, and i⁴ = 1.

(ix) | z₁ + z₂| ≤ | z₁ | + | z₂ |.

(x) | z₁ z₂ | = | z₁ | ∙ | z₂ |.

(xi) | z₁/z₂| = | z₁ |/| z₂ |.

(xii) (a) arg. (z₁ z₂) = arg. z₁ + arg. z₂ + m

(b) arg. (z₁/z₂) = arg. z₁ - arg. z₂ + m where m = 0 or, 2π or, (- 2π).

(xiii) If ω be the imaginary cube root of unity then ω = ½ (- 1 + √3i) or, ω = ½ (-1 - √3i)

(xiv) ω³ = 1 and 1 + ω + ω² = 0