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 v 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) = e^x
   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 ƒ_x1 = ƒ_x2 = ƒ_x2 = 0
   Therefore, ƒ_x1 = 2x₁ − 2 = 0 ⇒ x₁ = 1
   ƒ_x2 = 2x₂ − 4 = 0 ⇒ x₂ = 2
   ƒ_x3 = 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 = ƒ_x1x1(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
   1. 
   2. 
   3. 
   4. 

   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
   1. 
   2. 
   3. 
   4. 

   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 k₂ is equal to
    1. k₁
    2. 2k₁
    3. k₁/4
    4. 4k₁

    Answer:- 4k₁