Engineering Mathematics Miscellaneous


Engineering Mathematics Miscellaneous

Engineering Mathematics

  1. For the matrix A =
    5
    3
    ,
    13

    ONE of the normalized eigen vectors is given as









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    | A - λI | = 0

    ⇒ (5 – λ) (3 – λ) = 3 = 0
    ⇒ λ2 – 8λ + 15 – 3 = 0
    ⇒ λ2 – 8λ + 12 = 0
    ⇒ λ = 2, λ = 6
    Now, at λ = 2, eign vector:

    Hence required vector is

    Correct Option: B

    | A - λI | = 0

    ⇒ (5 – λ) (3 – λ) = 3 = 0
    ⇒ λ2 – 8λ + 15 – 3 = 0
    ⇒ λ2 – 8λ + 12 = 0
    ⇒ λ = 2, λ = 6
    Now, at λ = 2, eign vector:

    Hence required vector is


  1. Laplace transform of the function f(t) is given by F(s) = L{f(t)} =

    Laplace transform of the function shown below is given by










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    f(t) = 2; 0 < t < 1 > 0; otherwise

    ∴ L[f(t)] =

    =
    2 - 2e-s
    s

    Correct Option: C

    f(t) = 2; 0 < t < 1 > 0; otherwise

    ∴ L[f(t)] =

    =
    2 - 2e-s
    s



  1. The Laplace transform of ei5t where i = √-1, is









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    ∴ L[ei5t] =
    1
    s + 5i

    =
    1
    ×
    (s + 5i)
    (s - 5i)(s + 5i)

    =
    s + 5i
    s2 + 25

    Correct Option: B

    ∴ L[ei5t] =
    1
    s + 5i

    =
    1
    ×
    (s + 5i)
    (s - 5i)(s + 5i)

    =
    s + 5i
    s2 + 25


  1. Laplace transform of cos(ωt) is
    s
    s2 + ω2

    The laplace transform of e–2t cos(4t) is









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    We know that if L{f(t)} = F(s)
    Then L{eat f(t)} = F(s – a)

    ∴ L[e-2t cos 4t ] =
    s + 2
    (s + 2)2 + 42

    =
    s + 2
    (s + 2)2 + 16

    Correct Option: D

    We know that if L{f(t)} = F(s)
    Then L{eat f(t)} = F(s – a)

    ∴ L[e-2t cos 4t ] =
    s + 2
    (s + 2)2 + 42

    =
    s + 2
    (s + 2)2 + 16



  1. Laplace transform of cos (ωt) is









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    Laplace transform of cos (ωt) is

    L[ cos (ωt) ] =
    S
    s2 + ω2

    Correct Option: A

    Laplace transform of cos (ωt) is

    L[ cos (ωt) ] =
    S
    s2 + ω2