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JEE Maths 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