Algebra


  1. The value of
    1
    +
    1
    +
    1
    is
    ( a² + ax + x² )( a² - ax + x² )( a4 + a2x2 + x4 )










  1. View Hint View Answer Discuss in Forum

    Expression =
    1
    +
    1
    +
    2ax
    a² + ax + x²a² - ax + x²a4 + a2x2 + x4

    Expression =
    ( a² - ax + x² - a² - ax - x² )
    +
    2ax
    ( a² + ax + x² )( a² - ax + x² )( a4 + a2x2 + x4 )

    Expression =
    -2ax
    +
    2ax
    = 0
    ( a4 + a2x2 + x4 )( a4 + a2x2 + x4 )

    Correct Option: D

    Expression =
    1
    +
    1
    +
    2ax
    a² + ax + x²a² - ax + x²a4 + a2x2 + x4

    Expression =
    ( a² - ax + x² - a² - ax - x² )
    +
    2ax
    ( a² + ax + x² )( a² - ax + x² )( a4 + a2x2 + x4 )

    Expression =
    -2ax
    +
    2ax
    = 0
    ( a4 + a2x2 + x4 )( a4 + a2x2 + x4 )


  1. The factors of (a2 + 4b2 + 4b – 4ab – 2a – 8) are









  1. View Hint View Answer Discuss in Forum

    a2 + 4b2 + 4b – 4ab – 2a – 8 = a2 + 4b2 – 4ab – 2a + 4b – 8
    = (a – 2b)2 - 2(a – 2b) – 8
    Let (a – 2b) = x
    ∴ Expression = x2 – 2x – 8
    Expression = x2 – 4x + 2x – 8
    Expression = x (x – 4) + 2(x – 4)
    Expression = (x – 4)(x + 2)
    Expression = (a – 2b – 4)(a – 2b + 2)

    Correct Option: A

    a2 + 4b2 + 4b – 4ab – 2a – 8 = a2 + 4b2 – 4ab – 2a + 4b – 8
    = (a – 2b)2 - 2(a – 2b) – 8
    Let (a – 2b) = x
    ∴ Expression = x2 – 2x – 8
    Expression = x2 – 4x + 2x – 8
    Expression = x (x – 4) + 2(x – 4)
    Expression = (x – 4)(x + 2)
    Expression = (a – 2b – 4)(a – 2b + 2)



  1. If 3x +
    3
    = 1, then x3 +
    1
    + 1 is :
    xx3










  1. View Hint View Answer Discuss in Forum

    3x +
    3
    = 1
    x

    ⇒ x +
    1
    =
    1
    x3

    On cubing both sides,
    ⇒ x3 +
    1
    + 3x +
    1
    =
    1
    x3x27

    ⇒ x3 +
    1
    + 3 ×
    1
    =
    1
    x3327

    ⇒ x3 +
    1
    + 1 =
    1
    x327

    Correct Option: B

    3x +
    3
    = 1
    x

    ⇒ x +
    1
    =
    1
    x3

    On cubing both sides,
    ⇒ x3 +
    1
    + 3x +
    1
    =
    1
    x3x27

    ⇒ x3 +
    1
    + 3 ×
    1
    =
    1
    x3327

    ⇒ x3 +
    1
    + 1 =
    1
    x327


  1. If ab + bc + ca = 0 then the value of
    1
    +
    1
    +
    1
    is
    ( a² - bc )( b² - ac )( c² - ab )










  1. View Hint View Answer Discuss in Forum

    ab + bc + ca = 0
    ⇒ ab + ca = – bc
    ∴ a2 – bc = a2 + ab + ac = a(a + b + c)
    Similarly,
    b2 – ca = b(a + b + c)
    c2 – ab = c(a + b + c)

    ∴ Expression =
    1
    +
    1
    +
    1
    ( a² - bc )( b² - ac )( c² - ab )

    Expression =
    1
    +
    1
    +
    1
    a(a + b + c)b(a + b + c)c(a + b + c)

    Expression =
    1
    1
    +
    1
    +
    1
    (a + b + c)abc

    Expression =
    1
    bc + ca + ab
    (a + b + c)abc

    Expression =
    1
    ×
    0
    = 0
    (a + b + c)abc

    Correct Option: A

    ab + bc + ca = 0
    ⇒ ab + ca = – bc
    ∴ a2 – bc = a2 + ab + ac = a(a + b + c)
    Similarly,
    b2 – ca = b(a + b + c)
    c2 – ab = c(a + b + c)

    ∴ Expression =
    1
    +
    1
    +
    1
    ( a² - bc )( b² - ac )( c² - ab )

    Expression =
    1
    +
    1
    +
    1
    a(a + b + c)b(a + b + c)c(a + b + c)

    Expression =
    1
    1
    +
    1
    +
    1
    (a + b + c)abc

    Expression =
    1
    bc + ca + ab
    (a + b + c)abc

    Expression =
    1
    ×
    0
    = 0
    (a + b + c)abc



  1. If a2 + b2 + c2 = ab + bc + ca ,then the value of
    a + c
    is
    b










  1. View Hint View Answer Discuss in Forum

    a2 + b2 + c2 = ab + bc + ca
    ⇒ a2 + b2 + c2 - ab - bc - ca = 0
    On multiplying by 2,
    2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
    ⇒ a2 + b2 - 2ab + b2 + c2 - 2bc + c2 + a2 - 2ca = 0
    ⇒ (a - b)2 + (b - c)2 + (c - a)2 = 0
    ⇒ a – b = 0 ⇒ a = b
    b – c = 0 ⇒ b = c
    c – a = 0 ⇒ c = a

    a + c
    =
    2a
    = 2
    ba

    Correct Option: B

    a2 + b2 + c2 = ab + bc + ca
    ⇒ a2 + b2 + c2 - ab - bc - ca = 0
    On multiplying by 2,
    2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
    ⇒ a2 + b2 - 2ab + b2 + c2 - 2bc + c2 + a2 - 2ca = 0
    ⇒ (a - b)2 + (b - c)2 + (c - a)2 = 0
    ⇒ a – b = 0 ⇒ a = b
    b – c = 0 ⇒ b = c
    c – a = 0 ⇒ c = a

    a + c
    =
    2a
    = 2
    ba