Complex Numbers 02 Class Notes PDF

# AGNIVEER NAVY SSR Varun 2.0, 2024 ## Complex Number - Mathematics - Lecture - 02 - Puneet Dogra Sir ## Topics to be covered 1. Complex No, Conjugate, multiplicative inverse, polar form 2. Questions 3. Sq. Root of Complex No. / संमिश्र एक का वर्गमूल 4. Concept of Omega or Cube Root of unity...

# AGNIVEER NAVY SSR Varun 2.0, 2024 ## Complex Number - Mathematics - Lecture - 02 - Puneet Dogra Sir ## Topics to be covered 1. Complex No, Conjugate, multiplicative inverse, polar form 2. Questions 3. Sq. Root of Complex No. / संमिश्र एक का वर्गमूल 4. Concept of Omega or Cube Root of unity ## Question #Q. - i1 + i² + i3 + i4 = ? - i² + i4 + i6 + i8 = ? - i8 + i9 + i10 + i11 = ? - i10 + i15 + i19 + i23 = ? - i7 + i11 + i15 + i19 = ? - i20 + i30 + i40 + i50 = ? ## Question #Q. - 1/i = ? - (1 + i)² = ? - (1 - i)2 = ? - 1+i/ 1-i = ? - 1-i/ 1+i = ? ## Question #Q. The value of i2n + i2n+1 + i2n+2 + i2n+3, where i = √-1, is: - 0 - 1 - i - -i ## Question #Q. The smallest positive integer n for which (1+1/1-i)n = 1, is: - 1 - 4 - 8 - 16 ## Question #Q. i + i² + i³ +…. + i100 = ? ## Question #Q. i² + i⁴ + i⁶ + ….+ i100 = ? ## Question #Q. (1+i/1-i)³ + (1-i/1+i)³ = ? ## Question #Q. ∑¹³ (iⁿ + iⁿ+²) ## Question #Q. ∑¹⁸ (iⁿ + iⁿ⁺¹) ## Question #Q. ∑²⁰¹⁷⁶³ (iⁿ + iⁿ⁺²) ## Question #Q. ∑¹⁴ (iⁿ + iⁿ⁺¹) ## Question #Q. ∑¹⁵¹ (iⁿ + iⁿ⁻²) ## Question #Q. ∑¹⁵² (iⁿ + iⁿ⁺²) ## Topic: Conjugate सयुग्मी - Z=x+iy - Z = x-iy ## Topic: Multiplicative Inverse गुणात्मक प्रतिलोम - Z⁻¹= Z/ |Z|² = x-iy/ x²+y² ## Topic: Polar Form ध्रुवीय रूप / modulus - argument form मापाक - कोणाक रूप - Argument ## Topic: Modulus - |Z| = √x²+y² ## Topic: Properties of Modulus मापाक - |Z| = |-Z| - Z = |Z| - Z1・Z2 = |Z1| |Z2| m.Imp. - Z1 /Z2 = |Z1|/ |Z2| M.M.Imp. - |Zn| = |Z|ⁿ ## Question #Q. z=1+i |Z+ Z/Z| = ? ## Question #Q. If z = 1+isine/1-isine |Z| = ? ## Question #Q. If z = 1+isine /1- isine '0' = ? If z is purely real. ## Question #Q. z = (1+2i/2-1) * (2- 1/1+2i) Find Z²+ Z.Ż ? ## Topic: Argument कोणांक - I: tanθ = |y|/ |x| - II: '0' → Angle made with the x-Axis. ## Topic: Properties of Argument कोणांक के गुण - Arg(z₁ • z2) = Arg z₁ + Arg z2 - Arg (Z₁ /Z₂) = Arg Z1 - Arg Z2 - Arg zm = m Arg z - Arg (-z) = π + Argz - Arg z + Arg z = 2n - Arg z⁻¹ = Arg z ## Question #Q. The modulus and principal argument of the complex number 1+2i/1-(1-i)² are respectively: - 1, 0 - 1, 1 - 2, 0 - 2, 1 ## Question #Q. If z = 2(1+2i)/(3+i) , where i = √-1, then the argument 0 (-π < 0 ≤ π) of z is: - 3π/4 - π/4 - 5π/6 - 3π/4 ## Question #Q. If z = 1 +i tan α where π < α < 3π/2 then what is |z| equal to? - sec α - -sec α - sec² α - -sec² α ## Question #Q. What is the argument of (1 – sin θ) + i cos θ? (i = √-1) - π/2 - θ/2 - π/2 + θ/2 - π/4 - θ/2 - π/4 + θ/2 ## Question #Q. If z = 1 + cos π/5 + i sin π/5, then what is |z| equal to? - 2 cos π/5 - 2 sin π/5 - 2 cos π/10 - 2 sin π/10 ## Question #Q. What is the modulus of 1/1+3i - 1/1-3i ? - 3/5 - 9/25 - 3/√25 - 5/3 ## Question #Q. Find multiplicative inverse of √5 + 3i? ## Question #Q. 1+i/2-i = x +iy find (x,y) ## Question #Q. 1 + i² + i⁴ + i⁶ + ..... i2n ## Question #Q. Find Real '0' 1+icosθ/ 1-icosθ is Real No. ## Question #Q. Find conjugate of 2 - 1/(1 - 2i)2 ## Question #Q. x +iy = 3/ 2+cosθ+isinθ x²+y² = |z|² ## Question #Q. The modulus-amplitude form of √3 +i. ## Remaining - Comparison of Two Complex Numbers - Square Root of Complex Numbers / समिश्र एक का वर्गमूल - Locus of Point / बिंदु का ठिकाना - Cube Root of Unity / इकाई का धन‌मूल - Triangle Related Modulus Properties ## Topic: Polar Form - Modulus - |z| = r = √x2 + y2 - Argument - (4 cases) - z = x + iy ## Topic: Polar Form - r(cos θ + i sin θ) - |z| - Argument ## Topic: Euler’s form - Euler’s form - r(cos θ + i sin θ) = e^iθ = e^iθ - DeMorvi's - [r(cos θ + i sin θ)]^n = r^n [cos(nθ)+ i sin(nθ)] ## Topic: Argument ’0’ - Step1: tanα = |y|/|x| - Step II: '0' → Angle made with the x-Axis. - Case I: z = x + iy, x = +ve, y = +ve, α = θ - Case II: z = -x+iy, α = θ - Case III: z = -x – iy, θ = α - π, θ = π + α - Case IV: z = x-iy, θ = - π & θ = 2π - α ## Topic: Important Result - Z1/Z2 = (a + ib)/(c +id) = (ac + bd) / (c² + d²) + i(bc - ad) / (c² + d²) ## PYQ #Q. The modulus-amplitude form of √3 + li, where i = √–1 is: - 2 (cos π/3 + i sin π/3) - 2 (cos π/6 + i sin π/6) -4 (cos π/3 + i sin π/3) - 4 (cos π/6 + i sin π/6) ## PYQ #Q. If z = -2(1+2i) / (3 + i) , where i = √-1, then the argument 0 (-π < 0 ≤ π) of z is: - 3π/4 - π/4 - 5π/6 - 3π/4 ## PYQ #Q. What is the real part of (sin x + i cos x)³ where i = √–1? - -cos 3x - -sin 3x - sin 3x - cos 3x ## PYQ #Q. If z is a complex number such that |z| = 4 and arg (z) = 5π/6, then what is z equal to? - 2√3 + 2i - 2√3 - 2i - 2√3 - 2i - -√3 + i ## PYQ #Q. What is the argument of the complex number –1 – i, where i = √−1 ? - 5π/4 - 5π/4 - 3π/4 - None of these ## PYQ #Q. What is the modulus of √2+i/√2-i where i = √-1? - 3 - 1/2 - 1 - None of these ## Topic: Square Root of Complex Number समित स० का वर्गमूल - Z=√a+ib → √|Z|+a / 2 + √|Z|-a/ 2 ## PYQ #Q. What is one of the square roots of 3 + 4i, where i = √−1 ? - (2+i)² - (2-i)² - (-2+i)² - (-3-i)² ## PYQ #Q. What is the value of √-i where i = √-1? - (1- i/√2)² - 1+i/√2 - 1 – i/2 - 1 + i/2 ## PYQ #Q. What is the square root of √3 / 2 - i / 2 ? - (√3 / 2 +i/2) - (√3 / 2 + i/2) - (1/2 + √3 / 2 i) - (1/2 - √3 / 2 i) ## Topic: Cube Root of Unity इकाई का धनमूल - w³ = 1, w=1⅓, w³-1=0 - (w-1)(w²+w+1)=0, w²+w+1=0, w² + w + 1 = 0 - w= -1 ± √1-4(1)(1) / 2(1) - w = -1 ± √-3 / 2 - w= -1 ± i√3/ 2, w= 1+i√3/ 2, w= - 1 - i√3/ 2 - ①: w = 1 - ②: w²+w+1 = 0 - ③: w = -1 + i√3 / 2 - ④: w = -1 - i√3 / 2 ## PYQ #Q. What is w100 + w200 + w300 equal to, where w is the cube root of unity? - 1 - 3w - 3ω² - 0 ## PYQ #Q. If α is a complex number such that α² + α + 1 = 0, then what is α³¹ equal to? - α - α² - 0 - 1 ## PYQ #Q. If 1, ω, ω² are the cube roots of unity, then (1 + ω)(1 + ω²)(1 + ω³)(1 + ω + ω²) is equal to: - -2 - -1 - 0 - 2 ## PYQ #Q. If 1, w and w² are the cube roots of unity, then the value of (1 + ω) (1 + ω²) (1 + ω⁴) (1+ ω⁸) is: - -1 - 0 - 1 - 2 ## PYQ #Q. If w is the imaginary cube root of unity, they what is (2 – ω + 2 ω²)27 equal to? - 327ω - -327ω² - 327 - -327 ## PYQ #Q. If w is a complex cube root of unity, then what is w10 + w⁻¹⁰ equal to? - 2 - -1 - -2 - 1 ## PYQ #Q. If ω is complex cube root, then what is the value of 1 - 1 / (1 + ω) -1 / (1 + ω²) ? ## PYQ #Q. Roots of the equation x2017 + x2018 + 1 = 0 are: - -ω - ω - ω² - ω, ω² ## PYQ #Q. What is the value of (i +√3 / 2 )²⁰¹⁹ + ( i - √3 / 2 )²⁰¹⁹ = ? - 1 - -1 - 2i - -2i ## PYQ #Q. The value of (√3 / 2 + i / 2 )¹⁰⁷ + (√3 / 2 - i / 2 )¹⁰⁷ then what is the imaginary part of z equal to? - 0 - 1/2 - √3 - 1 ## PYQ #Q. What is the value of (1 + √3 / -i + √3)²⁰⁰ + (1 - √3 / i + √3)²⁰⁰ + 1? - 1 - 1 - -1 - -1 ## PYQ #Q. If w is the cube root of unity, then what is the conjugate of 2w² + 3i? - 2ω – 3i - 3ω + 2i - 2ω + 3i - 3ω – 2i ## PYQ #Q. If 1, w and w² are the three cube roots of unity, then what is the value of (aw⁶+ bw³+ cw²) / (b⁶+ cw¹⁰ + aw⁸)? - a/b - b/b - ω - ω² ## Doubt Ques #Q. The value of (√3 / 2 + i / 2 )¹⁰⁷ + (√3 / 2 - i / 2 )¹⁰⁷ then what is the imaginary part of z equal to? - 0 - 1/2 - √3 - 1 ## Doubt Ques #Q. If z = 1+i√3 / 1 - i√3 where i = √-1, then what is the argument of z? - π/3 - 2π/3 - 4π/3 - 5π/6 ## Doubt Ques #Q. The smallest positive integer n for which (1 - i / 1 + i)ⁿ = 1 where i = √-1 is: - 2 - 4 - 6 - 8 ## Doubt Ques #Q. If z = 1 + i, where i = √–1, then what is the is the modulus of z + z/z ? - 1 - 2 - 3 - 4 ## Doubt Ques #Q. Consider the following in respect of a complex number z: 1. (z⁻¹ )⁻¹ = (z)-¹ 2. zz⁻¹ = |z|² Which of the above is/are correct? - 1 only - 2 only - Both 1 and 2 - Neither 1 nor 2 ## PYQ #Q. Consider the following statements in respect of an arbitrary complex number z: 1. The difference of z and its conjugate is an imaginary number. 2. The sum of z and its conjugate is a real number. Which of the above statements is/are correct? - 1 only - 2 only - Both 1 and 2 - Neither 1 nor 2 ## PYQ #Q. What is the modulus of the complex number i2n+1(– i)2n−1, where n ∈ N and i = √-1? - -1 - 1 - √2 - 2 ## PYQ #Q. Which one of the following is a square root of 2a + 2√a² + b², where a, b ∈R? - √a + ib + √a - ib - √a + ib - √a - ib - 2a + ib - 2a - ib ## PYQ #Q. What is the principal argument of 1/ 1 + i where i = √–1? - 3π/4 - -π/ 4 - π/4 - 3π/4 ## PYQ #Q. What is the modulus of (√3/2 + i /2)²⁰⁰? - 1/4 - 1/4 - 1 - 2200 ## PYQ #Q. The common roots of the equation z³ + 2z² + 2z + 1 = 0 and z2017 + z2018 + 1 = 0 are: - –1, ω - 1, ω² - -1, ω² - ω, ω² ## PYQ #Q. If x = 1 + i, then what is the value of x⁶ + x⁴ + x² + 1? - 6i – 3 - –6i + 3 - –6i – 3 - 6i + 3 ## Important Question #Q. What is [sin π/6 + i (1 - cos π/6)]³ / [sin π/6 - i( 1 - cos π/6)]³ where i = √-1, equal to? - 1 - -1 - i - -i ## PYQ #Q. If w³ = 1 and w ≠ 1, then (a + bω + cω²)/ (c + aω + bω²) + (a + bω + cω²)/ (b + cω + aω²) = ? - -2 - -1 - 1 - 2 ## JOIN US ON TELEGRAM - X: https://t.me/agniveervayu_x - Y: https://t.me/agniveervayu_Y ## JAI HIND !