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Elementary Algebra

  1. The degree f polynomial p(x) = x3 + 1 + 2x = 6x + 1/x is ?
    1. 2
    2. 4
    3. 3
    4. 5
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    x3 + 1 + 2x =6x + 1/x
    x3 + 1 + 2x = (6x2 + 1)/x
    (x3 + 1 + 2x)x = 6x2 + 1
    x4 - 4x2 - 6x2 -1 =0
    x4 - 4x2 + x - 1 =0

    Correct Option: B

    x3 + 1 + 2x =6x + 1/x
    x3 + 1 + 2x = (6x2 + 1)/x
    (x3 + 1 + 2x)x = 6x2 + 1
    x4 - 4x2 - 6x2 -1 =0
    x4 - 4x2 + x - 1 =0
    Degree of polynomial is highest exponent degree term i.e.,4.

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  1. If x = 1 + √2, then what is the value of x4 - 4x3 + 4x2 ?
    1. -1
    2. 0
    3. 1
    4. 2
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    x = 1 + √2
    ∴ x4 - 4x3 + 4x2 = x2(x2 - 4x + 4)
    = x2(x - 2)2
    = (1 + √2)2(1 + √2 - 2)2

    Correct Option: C

    x = 1 + √2
    ∴ x4 - 4x3 + 4x2 = x2(x2 - 4x + 4)
    = x2(x - 2)2
    = (1 + √2)2(1 + √2 - 2)2
    =(√2 + 1)2 (√2 - 1)2
    =[(√2)2 - (1)2]2
    =(2 - 1)2 =1

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  1. What is the solution of the equations x - y = 0.9 and 11(x + y)-1 = 2 ?
    1. x = 3.2, y = 2.3
    2. x = 1, y = 0.1
    3. x = 2, y = 1.1
    4. x = 1.2, y = 0.3
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    x - y = 0.9 ...(i)
    and 11(x + y)-1=2
    ⇒ 11/ (x + y) = 2
    ⇒ 2(x + y) =11

    Correct Option: A

    x - y = 0.9 ...(i)
    and 11(x + y)-1=2
    ⇒ 11/ (x + y) = 2
    ⇒ 2(x + y) =11
    ⇒ x + y = 11/2 ...(ii)
    On solving Eqs.(i) and (ii),we get
    x = 3.2
    and y = 2.3

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  1. The product of two alternate odd integers exceeds three times the smaller by 12. What is the larger integer ?
    1. 9
    2. 7
    3. 3
    4. 5
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    Let two alternate odd integers odd integers be (2x+1) and (2x+5).
    Then according to the question,
    (2x + 1) (2x + 5) = 3(2x + 1) + 12
    ⇒ (2x + 1)(2x + 5 - 3) = 12
    ⇒ 2x2 + 3x - 5 = 0

    Correct Option: B

    Let two alternate odd integers odd integers be (2x+1) and (2x+5).
    Then according to the question,
    (2x + 1) (2x + 5) = 3(2x + 1) + 12
    ⇒ (2x + 1) (2x + 5 - 3) = 12
    ⇒ 2x2 + 3x - 5 = 0
    On solving this quadratic equation,we get
    x = 1 and x = -5/2
    x = -5/2 is not a integer ∴ x = 1
    Then, larger integer = 2x + 5 = 2 x 1 + 5 = 7

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