Algebra


  1. If   x =
    5 − √3
    and y =
    5 + √3
    then the value of
    x2 + xy + y2
    = ?
    5 + √35 − √3x2 − xy + y2









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    x =
    5 − √3
    , y =
    5 + √3
    5 + √35 − √3

    ∴  x + y =
    5 − √3
    +
    5 + √3
    5 + √35 − √3

    =
    (√5 − √3)2 + (√5 + √3)2
    (√5 + √3) × (√5 − √3)

    =
    2[(√5)2 + (√3)2]
    5 − 3

    = 5 + 3 = 8
    xy =
    5 − √3
    ×
    5 + √3
    = 1
    5 + √35 − √3

    =
    x2 + xy + y2
    x2 − xy + y2

    ∴ 
    (x + y)2 − xy
    (x + y)2 − 3xy

    =
    (8)2 − 1
    (8)2 − 3

    =
    64 − 1
    64 − 3

    =
    63
    61

    Correct Option: A

    x =
    5 − √3
    , y =
    5 + √3
    5 + √35 − √3

    ∴  x + y =
    5 − √3
    +
    5 + √3
    5 + √35 − √3

    =
    (√5 − √3)2 + (√5 + √3)2
    (√5 + √3) × (√5 − √3)

    =
    2[(√5)2 + (√3)2]
    5 − 3

    = 5 + 3 = 8
    xy =
    5 − √3
    ×
    5 + √3
    = 1
    5 + √35 − √3

    =
    x2 + xy + y2
    x2 − xy + y2

    ∴ 
    (x + y)2 − xy
    (x + y)2 − 3xy

    =
    (8)2 − 1
    (8)2 − 3

    =
    64 − 1
    64 − 3

    =
    63
    61


  1. If p = 99 then, the value of p(p2 + 3p + 3) is :









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    p = 99 (Given)
    Expression = p(p2 + 3p + 3)
    = p3 + 3p2 + 3p
    = p3 + 3p2 + 3p + 1 – 1
    = (p + 1)3 – 1
    = (99 + 1)3 – 1 = (100)3 – 1
    = 1000000 – 1 = 999999

    Correct Option: C

    p = 99 (Given)
    Expression = p(p2 + 3p + 3)
    = p3 + 3p2 + 3p
    = p3 + 3p2 + 3p + 1 – 1
    = (p + 1)3 – 1
    = (99 + 1)3 – 1 = (100)3 – 1
    = 1000000 – 1 = 999999



  1. If   x +
    1
    = 1 then the value of
    x2 + 3x + 1
    is
    xx2 + 7x + 1









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    x +
    1
    = 1   (Given)
    x

    Expression =
    x2 + 3x + 1
    x2 + 7x + 1

    = x +
    1
    + 3
    x
    x +
    1
    + 7
    x

    (Dividing numerator and denominator by x)
    =
    1 + 3
    =
    4
    =
    1
    1 + 782

    Correct Option: C

    x +
    1
    = 1   (Given)
    x

    Expression =
    x2 + 3x + 1
    x2 + 7x + 1

    = x +
    1
    + 3
    x
    x +
    1
    + 7
    x

    (Dividing numerator and denominator by x)
    =
    1 + 3
    =
    4
    =
    1
    1 + 782


  1. If  
    m − a2
    +
    m − b2
    +
    m − c2
    = 3 , then the value of m is
    b2 + c2c2 + a2a2 + b2









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    m − a2
    +
    m − b2
    +
    m − c2
    − 3 = 0
    b2 + c2c2 + a2a2 + b2

    ⇒ 
    m − a2
    − 1 +
    m − b2
    − 1 +
    m − c2
    − 1 = 0
    b2 + c2c2 + a2a2 + b2

    ⇒ 
    m − a2 − b2 − c2
    +
    m − b2 − c2 − a2
    +
    m − c2 − a2 − b2
    = 0
    b2 + c2c2 + a2a2 + b2

    ⇒ 
    m − (a2 + b2 + c2)
    +
    m − (a2 + b2 + c2)
    +
    m −(a2 + b2 + c2)
    = 0
    b2 + c2c2 + a2a2 + b2

    ∴  Each term = 0
    ∴ 
    m − (a2 + b2 + c2)
    = 0
    b2 + c2

    ⇒  m − (a2 + b2 + c2) = 0
    ⇒  m = (a2 + b2 + c2)

    Correct Option: C

    m − a2
    +
    m − b2
    +
    m − c2
    − 3 = 0
    b2 + c2c2 + a2a2 + b2

    ⇒ 
    m − a2
    − 1 +
    m − b2
    − 1 +
    m − c2
    − 1 = 0
    b2 + c2c2 + a2a2 + b2

    ⇒ 
    m − a2 − b2 − c2
    +
    m − b2 − c2 − a2
    +
    m − c2 − a2 − b2
    = 0
    b2 + c2c2 + a2a2 + b2

    ⇒ 
    m − (a2 + b2 + c2)
    +
    m − (a2 + b2 + c2)
    +
    m −(a2 + b2 + c2)
    = 0
    b2 + c2c2 + a2a2 + b2

    ∴  Each term = 0
    ∴ 
    m − (a2 + b2 + c2)
    = 0
    b2 + c2

    ⇒  m − (a2 + b2 + c2) = 0
    ⇒  m = (a2 + b2 + c2)



  1. If x =
    1
    , y =
    1
    , then the value of 8xy (x2 + y2) is
    2 + √32 − √3









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    x =
    1
    2 + √3

    =
    1
    ×
    1
    =
    2 − √3
    2 + √3 2 − √ 3 4 − 3

    = 2 − √ 3
    ∴  y =
    1
    = 2 + √ 3
    2 − √ 3

    ∴  x + y = 2 – √ 3 + 2 + √ 3 = 4
    xy = (2 – √ 3 ) (2 + √ 3 )
    = 4 – 3 = 1
    ∴  8xy (x2 + y2)
    = 8xy [(x + y)2 – 2 xy]
    = 8 × 1 (42 – 2 × 1)
    = 8 (16 – 2) = 8 × 14 = 112

    Correct Option: C

    x =
    1
    2 + √3

    =
    1
    ×
    1
    =
    2 − √3
    2 + √3 2 − √ 3 4 − 3

    = 2 − √ 3
    ∴  y =
    1
    = 2 + √ 3
    2 − √ 3

    ∴  x + y = 2 – √ 3 + 2 + √ 3 = 4
    xy = (2 – √ 3 ) (2 + √ 3 )
    = 4 – 3 = 1
    ∴  8xy (x2 + y2)
    = 8xy [(x + y)2 – 2 xy]
    = 8 × 1 (42 – 2 × 1)
    = 8 (16 – 2) = 8 × 14 = 112