GATE 2011 AEROSPACE ENGINEERING (Q 1-10)

  1. Consider x, y, z to be right-handed Cartesian coordinates. A vector function defined in this coordinate system as v = 3xi + 3xyj − yz2k, where i, j and k are unit vectors along x, y and z axes, respectively. The curl of is given by
    1. z2i − 3yk
    2. z2j + 3yk
    3. z2i + 3yj
    4. −z2i + 3yk

    Answer:- −z2i + 3yk
    ∇ × v =
    =
    = −z2i + 3yk

  2. Which of the following functions is periodic?
    1. ƒ(x) = x2
    2. ƒ(x) = log x
    3. ƒ(x) = ex
    4. ƒ(x) = const.

    Answer:- ƒ(x) = const.

  3. The function ƒ(x1, x2, x3) = x12 + x22 + x32 − 2x1 − 4x2 − 6x3+ 14 has its minimum value at
    1. (1, 2, 3)
    2. (0, 0, 0)
    3. (3, 2, 1)
    4. (1, 1, 3)

    Answer:- (1, 2, 3)
    The critical points satisfy ƒx1 = ƒx2 = ƒx2 = 0
    Therefore, ƒx1 = 2x1 − 2 = 0 ⇒ x1 = 1
    ƒx2 = 2x2 − 4 = 0 ⇒ x2 = 2
    ƒx3 = 2x3 − 6 = 0 ⇒ x3 = 3
    So, (1, 2, 3) (denoting by p) is a critical point. Now, check whether it is maximum, minimum or saddle point.
    Δ1 = ƒx1x1(p) = 2 > 0
    Δ2 = = = 4 > 0
    Δ3 = = = 8 > 0
    As Δ1 > 0, Δ2 > 0 and Δ3 > 0, (1, 2, 3) is the local minimum of the given function.

  4. Consider the function ƒ(x1, x2) = x12 + 2x22 + e− x1 − x2. The vector pointing in the direction of maximum increase of the function at the point (1, -1) is

    Answer:- 
    ∇ƒ(1, -1) =
    =
    =

  5. Two simultaneous equations given by y = π + x and y = x − π have
    1. a unique solution
    2. infinitely many solutions
    3. no solution
    4. a finite number of multiple solutions

    Answer:- no solution

  6. In three-dimensional linear elastic solids, the number of non-trivial stress-strain relations, strain-displacement equations and equations of equilibrium are, respectively,
    1. 3, 3 and 3
    2. 6, 3 and 3
    3. 6, 6 and 3
    4. 6, 3 and 6

    Answer:- 6, 6 and 3

  7. An Euler-Bernoulli beam in bending is assumed to satisfy
    1. both plane stress as well as plane strain conditions
    2. plane strain condition but not plane stress condition
    3. plane stress condition but not plane strain condition
    4. neither plane strain condition nor plane stress condition

    Answer:- neither plane strain condition nor plane stress condition

  8. A statically indeterminate frame structure has
    1. same number of joint degrees of freedom as the number of equilibrium equations
    2. number of joint degrees of freedom greater than the number of equilibrium equations
    3. number of joint degrees of freedom less than the number of equilibrium equations
    4. unknown number of joint degrees of freedom, which cannot be solved using laws of mechanics

    Answer:-

  9. Consider a single degree of freedom spring-mass-damper system with mass, damping and stiffness of m, c, and k, respectively. The logarithmic decrement of this system can be calculated using

    Answer:-
    logarithmic decrement =
    ζ = , substituting in above equation we get
    logarithmic decrement =

  10. Consider a single degree of freedom spring-mass system of spring stiffness k1 and mass m which has a natural frequency of 10 rad/s. Consider another single degree of freedom spring-mass system of spring stiffness k2 and mass m which has a natural frequency of 20 rad/s. The spring stiffness kis equal to
    1. k1
    2. 2k1
    3. k1/4
    4. 4k1

    Answer:- 4k1

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