- Consider x, y, z to be right-handed Cartesian coordinates. A vector function defined in this coordinate system as
**v**= 3x**i**+ 3xy**j**− yz^{2}**k**, where**i**,**j**and**k**are unit vectors along x, y and z axes, respectively. The curl of**v**is given by- z
^{2}**i**− 3y**k** - z
^{2}**j**+ 3y**k** - z
^{2}**i**+ 3y**j** - −z
^{2}**i**+ 3y**k**

- z
- Which of the following functions is periodic?
*ƒ(x) = x*^{2}*ƒ(x) = log x**ƒ(x) = e*^{x}*ƒ(x) = const.*

Answer:-

*ƒ(x) = const.* - The function ƒ(x
_{1}, x_{2}, x_{3}) = x_{1}^{2}+ x_{2}^{2}+ x_{3}^{2}− 2x_{1}− 4x_{2}− 6x_{3}+ 14 has its minimum value at- (1, 2, 3)
- (0, 0, 0)
- (3, 2, 1)
- (1, 1, 3)

Answer:- (1, 2, 3)

The critical points satisfy ƒ_{x1}= ƒ_{x2}= ƒ_{x2}= 0

Therefore, ƒ_{x1}= 2x_{1}− 2 = 0 ⇒ x_{1}= 1

ƒ_{x2}= 2x_{2}− 4 = 0 ⇒ x_{2}= 2

ƒ_{x3}= 2x_{3}− 6 = 0 ⇒ x_{3}= 3

So, (1, 2, 3) (denoting by p) is a critical point. Now, check whether it is maximum, minimum or saddle point.

Δ_{1}= ƒ_{x1x1}(p) = 2 > 0

Δ_{2}= = = 4 > 0

Δ_{3}= = = 8 > 0

As Δ_{1}> 0, Δ_{2}> 0 and Δ_{3}> 0, (1, 2, 3) is the local minimum of the given function. - Consider the function ƒ(x
_{1}, x_{2}) = x_{1}^{2}+ 2x_{2}^{2}+ e^{− x1 − x2}. The vector pointing in the direction of maximum increase of the function at the point (1, -1) is - Two simultaneous equations given by y = π + x and y = x − π have
- a unique solution
- infinitely many solutions
- no solution
- a finite number of multiple solutions

Answer:- no solution

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

Answer:- 6, 6 and 3

- An Euler-Bernoulli beam in bending is assumed to satisfy
- both plane stress as well as plane strain conditions
- plane strain condition but not plane stress condition
- plane stress condition but not plane strain condition
- neither plane strain condition nor plane stress condition

Answer:- neither plane strain condition nor plane stress condition

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

Answer:-

- 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 usingAnswer:-

logarithmic decrement =

ζ = , substituting in above equation we get

logarithmic decrement = - Consider a single degree of freedom spring-mass system of spring stiffness
*k*and mass_{1}*m*which has a natural frequency of 10 rad/s. Consider another single degree of freedom spring-mass system of spring stiffness*k*and mass_{2}*m*which has a natural frequency of 20 rad/s. The spring stiffness*k*is equal to_{2 }*k*_{1}*2k*_{1}*k*_{1}/4*4k*_{1}

Answer:-

*4k*_{1}

# GATE 2011 AEROSPACE ENGINEERING (Q 1-10)

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may i know why u r complicated 3rd problem.trial&error is enough know

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I have given the method by which answer can be obtained. One should also know the reason of the answer.

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where will i can found this method

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Any engineering mathematics book which has the chapter for function of 2 variables.

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