Request a call back

Join NOW to get access to exclusive study material for best results

Complex Numbers and Quadratic Equations

Complex Numbers and Quadratic Equations PDF Notes, Important Questions and Synopsis

 

SYNOPSIS

  1. A number of the form x + iy, where x, y Î  and begin mathsize 12px style text i= end text square root of negative 1 end root end style (i is iota), is called a complex number.
    It is denoted by z, and a set of complex numbers is denoted by ℂ.
    x = real part or Re(z), y = imaginary part or Im(z)

  2.  

    Complex conjugate

    Argument

    Magnitude

     

    If z = x + iy, then the conjugate of z is
    = x - iy

    amp(z) = arg(z) = q = begin mathsize 12px style text tan end text to the power of negative 1 end exponent straight y over straight x end style
    General argument: 2nπ + θ, n ϵ ℕ
    Principal argument: -π < θ ≤ π
    Least positive argument: 0 < θ ≤ 2π

    z = x + iy
    |z|=begin mathsize 12px style square root of straight x squared plus straight y squared end root end style

    begin mathsize 12px style open vertical bar straight z close vertical bar minus open vertical bar straight z with bar on top close vertical bar end style

     

  3. Representation of Complex Number

    Polar Representation

    Exponential Form

    Vector Representation

     

    x = r cos  θ, y = r sin  θ

    z = r e

    (where = cos e + I sin θ)

     

     

    z = x + iy is considered a position vector of point p

     


  4. Square roots of a complex number
    Let z = x + iy, then square root of z is

    begin mathsize 12px style square root of straight x plus iy end root equals plus-or-minus open square brackets square root of fraction numerator vertical line straight z vertical line plus straight x over denominator 2 end fraction end root plus straight i square root of fraction numerator vertical line straight z vertical line minus straight x over denominator 2 end fraction end root close square brackets end style,for y>0

      begin mathsize 12px style square root of straight x plus iy end root equals plus-or-minus open square brackets square root of fraction numerator vertical line straight z vertical line plus straight x over denominator 2 end fraction end root minus straight i square root of fraction numerator vertical line straight z vertical line minus straight x over denominator 2 end fraction end root close square brackets end style, for y<0

  5.  Properties of the argument of a Complex Number: 
    • arg(any real positive number) = 0   
    • arg(any real negative number) = π
    •  begin mathsize 12px style text arg end text open parentheses straight z subscript 1. straight straight z subscript 2 close parentheses straight equals straight arg open parentheses straight z subscript 1 close parentheses plus arg open parentheses straight z subscript 2 close parentheses end style
    •  begin mathsize 12px style text arg end text open parentheses straight z subscript 1. straight stack straight z subscript 2 with bar on top close parentheses straight equals straight arg open parentheses straight z subscript 1 close parentheses minus arg open parentheses straight z subscript 2 close parentheses end style
    •  begin mathsize 12px style text arg end text open parentheses straight z minus straight z with bar on top close parentheses straight equals straight plus-or-minus straight pi over 2 end style
    •  begin mathsize 12px style text arg end text open parentheses straight z subscript 1 over straight z subscript 2 close parentheses straight equals space arg left parenthesis straight z subscript 1 right parenthesis minus arg left parenthesis straight z subscript 2 right parenthesis end style
    •  begin mathsize 12px style text arg end text open parentheses straight z with bar on top close parentheses straight equals straight minus arg left parenthesis straight z right parenthesis equals arg open parentheses 1 over straight z close parentheses end style
    •  begin mathsize 12px style text arg end text open parentheses negative straight z close parentheses straight equals space arg left parenthesis straight z right parenthesis plus-or-minus straight pi end style
    •  begin mathsize 12px style text arg end text open parentheses straight z to the power of straight n close parentheses straight equals space straight n space arg left parenthesis straight z right parenthesis end style
    •  begin mathsize 12px style text arg end text open parentheses straight z close parentheses straight plus straight arg open parentheses straight z with bar on top close parentheses straight equals straight 0 end style



  6. Inequalities


    I.
    Triangle inequalities
          1. |z1 ± z2| £ | z1| ± | z2|
          2. |z1
    ± z2| ³ | z1| - | z2|

    II. Parallelogram inequalities
          | z1
    + z2|2+ | z1 - z2|2 = 2 [|z1|2+| z2|2]

     
  7. If ABC is an equilateral triangle having vertices z1, z2, z3, then begin mathsize 12px style text z end text subscript 1 squared plus straight z subscript 2 squared plus straight z subscript 3 squared equals straight z subscript 1 straight z subscript 2 plus straight z subscript 2 straight z subscript 3 plus straight z subscript 3 straight z subscript 1 end style or begin mathsize 12px style fraction numerator 1 over denominator straight z subscript 1 minus straight z subscript 2 end fraction plus fraction numerator 1 over denominator straight z subscript 2 minus straight z subscript 3 end fraction plus fraction numerator 1 over denominator straight z subscript 3 minus straight z subscript 1 end fraction equals 0 end style

  8. If z1, z2, z3, z4 are vertices of a parallelogram, then z1 + z3 = z2 + z4.

  9. If z1, z2, z3 are affixes of the points A, B and C in the Argand plane, then 

    i. ÐBAC = begin mathsize 12px style arg open parentheses fraction numerator straight z subscript 3 minus straight z subscript 1 over denominator straight z subscript 2 minus straight z subscript 1 end fraction close parentheses end style
    ii.  begin mathsize 12px style fraction numerator straight z subscript 3 minus straight z subscript 1 over denominator straight z subscript 2 minus straight z subscript 1 end fraction equals fraction numerator open vertical bar straight z subscript 3 minus straight z subscript 1 close vertical bar over denominator open vertical bar straight z subscript 2 minus straight z subscript 1 close vertical bar end fraction open parentheses cosα plus straight i straight sinα close parentheses end style , where α = ÐBAC
  10. The equation of a circle whose centre is at a point having affix z0 and radius R = |z - z0|.
  11. If a, b are positive real numbers, then. begin mathsize 12px style square root of negative straight a end root cross times square root of negative straight b end root equals negative square root of ab end style
  12. Integral powers of iota
    Error converting from MathML to accessible text.
    Hence,begin mathsize 12px style text i end text to the power of 4 straight n plus 1 end exponent equals straight i comma straight straight i to the power of 4 straight n plus 2 end exponent equals negative 1 end style

Quadratic Equations

  1. An equation of the form begin mathsize 12px style text ax end text squared plus bx plus straight c equals 0 end style is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.
  2. Values of the variable which satisfies the quadratic equation are called its roots.
  3. Nature of Roots
    Let f(x) = begin mathsize 12px style text ax end text squared plus bx plus straight c equals 0 end style be the quadratic equation, the discriminant D = begin mathsize 12px style text b end text squared minus 4 ac end style.

    If a > 0

    If a < 0

    1.

    1.

    2.

    2.

    3.

    3.







  4. Let α, β be the roots of the quadratic equation begin mathsize 12px style text ax end text squared plus bx plus straight c equals 0 comma end stylethen

      i. Roots are given by the quadratic formula:
          formula:
         a, b = begin mathsize 12px style fraction numerator negative straight b plus-or-minus square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction end style
     ii.  Relation between roots and coefficients:
    1. Sum of the roots =a+= -begin mathsize 12px style straight b over straight a end style
    2. 
     Product of the roots = a×bbegin mathsize 12px style straight c over straight a end style

    Note: Quadratic equation can be rewritten as begin mathsize 12px style text x end text squared minus left parenthesis straight alpha plus straight beta right parenthesis straight x plus straight alpha times straight beta equals 0 end style.

  5. Quadratic inequalities
    Let y = begin mathsize 12px style text ax end text squared plus bx plus straight c end style be the quadratic polynomial. There are two inequalities:

     begin mathsize 12px style text ax end text squared plus bx plus straight c greater than 0 end style  begin mathsize 12px style text ax end text squared plus bx plus straight c less than 0 end style
       
       
       
Download complete content for FREE PDF
JEE Main - Maths
Asked by kadamnarayan867 | 29 Jul, 2024 09:54: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE Main - Maths
Asked by padmapadmavati952 | 19 Jun, 2024 08:52: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE Main - Maths
Asked by mohanvaidya6501 | 15 Jun, 2024 01:09: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE Main - Maths
Asked by bhavanilankepss18 | 12 Jun, 2024 07:09: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE Main - Maths
Asked by priyadharshinigopinath27 | 03 May, 2024 08:44: PM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE Main - Maths
Asked by ashwinskrishna2006 | 14 Apr, 2024 11:58: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT
JEE Main - Maths
Asked by spurthipuli | 27 Mar, 2024 07:58: AM
ANSWERED BY EXPERT ANSWERED BY EXPERT