TS Inter Second Year Maths 2A concept TS Inter Second Year Maths : This note is designed by ‘Basics in Maths’ team. These notes to do help the TS intermediate second year Maths students fall in love with mathematics and overcome the fear. These notes cover all the topics covered in the TS I.P.E second year maths 2A syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations. 1. COMPLEX NUMBERS • The equation x2 + 1 = 0 has no roots in real number system. ∴ scientists imagined a number ‘i’ such that i2 = − 1. Complex number: if x, y are any two real numbers then the general form of the complex number is z = x + i y; where x real part and y is imaginary part. ∗ z = x + iy can be written as (x, y) ∗If z1 = x1 + i y1, z2 = x2 + i y2, then ∗ z1 + z2 = (x1 + x2, y1 + y2) = (x1 + x2) + i (y1 + y2) ∗ z1 − z2 = (x1 − x2, y1 − y2) = (x1 − x2) + i (y1 − y2) ∗ z1∙ z2 = (x1 x2 −y1 y2, x1y2 + x2y1) = (x1x2 −y1 y2) + i (x1y2 +x2 y1) ∗ z1/ z2 = (x1x2 + y1 y2/x22 +y22, x2 y1 – x1y2/ x22 +y22) = (x1x2 + y1 y2/x22 +y22) + i (x2 y1 – x1y2/ x22 +y22) Multiplicative inverse of complex number: Multiplicative inverse of complex number z is 1/z. z = x + i y then 1/z = x – i y/ x2 + y2 Conjugate complex number: The complex numbers x + iy, x – iy are called conjugate complex numbers. The sum and product of two conjugate complex numbers are real. If z1, z2 are two complex numbers then Modulus and amplitude of complex number: Modulus: – If z = x + iy, then the non-negative real number is called the modulus of z and it is denoted by or ‘r’. Amplitude: – The complex number z = x + i y is represented by the point P (x, y) on the XOY plane. ∠XOP = θ is called amplitude of z or argument of z. ∗ x = r cosθ, y = r sinθ ⇒ x2 + y2 = r2 cos2θ + r2 sin2θ = r2 (cos2θ + sin2θ) = r2(1) ⇒ x2 + y2 = r2 ⇒ r = and = r. ∗ Arg (z) = tan−1(y/x) ∗ Arg (z1.z2) = Arg (z1) + Arg (z2) + nπ for some n ∈ { −1, 0, 1} ∗ Arg(z1/z2) = Arg (z1) − Arg (z2) + nπ for some n ∈ { −1, 0, 1} Argand plane: The plane containing all complex numbers is called the Argand plane. This was introduced by the mathematician Gauss (1777-1855), who first thought that complex numbers can be represented as a two-dimensional plane. The square root of a complex number: 2.DE- MOIVER’S THEOREM De- Moiver’s theorem: For any integer n and real number θ, (cosθ + i sinθ) n = cos nθ + i sin nθ. → cos α + i sin α can be written as cis α → cis α.cis β= cis (α + β) → 1/cisα = cis(-α) → cisα/cisβ = cis (α – β) ⟹ (cosθ + i sinθ) -n = cos nθ – i sin nθ ⟹ (cosθ + i sin θ) (cosθ – i sin θ) = cos2θ – i2 sin2θ = cos2θ + sin2θ = 1. → cosθ + i sin θ = 1/ cosθ – i sin θ and cosθ – i sin θ = 1/ cosθ + i sin θ ⟹ (cosθ – i sin θ) n = (1/ (cosθ –+i sin θ)) n = (cosθ + i sin θ)-n = cos nθ – i sin nθ nth root of a complex number: let n be a positive integer and z0 ≠ 0 be a given complex number. Any complex number z satisfying z n = z0 is called an nth root of z0. It is denoted by z01/n or ⟹ let z = r (cosθ + i sin θ) ≠ 0 and n be a positive integer. For k∈ {0, 1, 2, 3…, (n – 1)} let . Then a0, a1, a2, …, an-1 are all n distinct nth roots of z and any nth root of z is coincide with one of them. nth root of unity: Let n be a positive integer greater than 1 and Note: The sum of the nth roots of unity is zero. The product of nth roots of unity is (– 1) n – 1. The nth roots of unity 1, ω, ω2, …, ωn-1 are in geometric progression with common ratio ω. Cube root of unity: x3 – 1 = 0 ⇒ x3 = 1 x =11/3 3.QUADRATIC EXPRESSIONS Quadratic Expression: If a, b, c are real or complex numbers and a ≠ 0, then the expression ax2 + bx + c is called a quadratic expression in variable ‘x’. ∎ A complex number α is said to be a zero of the quadratic expression ax2 + bx + c if aα2 + bα + c = 0. Quadratic Equation: If a, b, c are real or complex numbers and a ≠ 0, then ax2 + bx + c = 0 is called a quadratic equation in variable ‘x’. ∎ A complex number α is said to be root or solution of the quadratic equation ax2 + bx + c = o if aα2 + bα + c = 0. The roots of a quadratic equation: ∎ The zeroes of the quadratic expression ax2 + bx + c are same as the roots of quadratic equation ax2 + bx + c = o.
