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Integral Calculus and Differential Equations Test - 5

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Integral Calculus and Differential Equations Test - 5
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  • Question 1
    2 / -0.83

     f(x, y) = x2  + xyz + z Find fx  at (1,1,1)

    Solution

    fx  = 2x + yz

    Put (x,y,z) = (1,1,1)

    fx  = 2 + 1 = 3.

  • Question 2
    2 / -0.83

    Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons increased by:

    Solution

  • Question 3
    2 / -0.83

    The local maximum point of the function f(x) = (x2/3 ) –x is at  

  • Question 4
    2 / -0.83

    If x=a(θ+ sin θ) and y=a(1-cos θ), then dy/dx will be equal  

    Solution

  • Question 5
    2 / -0.83

    The minimum value of function y = x2 in the interval [1, 5] is  

    Solution

    y =x 2 is strictly increasing function on [1,5]

    ∴y= x 2 has minimum value at x = 1 is 1.

  • Question 6
    2 / -0.83

    The function f(x) = 2x3 –3x2 –36x + 2 has its maxima at  

    Solution

  • Question 7
    2 / -0.83

    What should be the value of λsuch that the function defined below is continuous at x = π/22? 

    Solution

    By the given condition  

  • Question 8
    2 / -0.83

    Consider function f(x) =(x2 -4)2 where x is a real number. Then the function has  

    Solution

  • Question 9
    2 / -0.83

    If f      where ai (i = 0 to n) are constants, then   

    Solution

     - Euler ’s theorem for homogeneous function  

  • Question 10
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    Solution

  • Question 11
    2 / -0.83

    A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct exterma for the curve 3x4 –16x3 –24x2 + 37 is  

    Solution

  • Question 12
    2 / -0.83

    ∇×∇×P, where P is a vector, is equal to  

  • Question 13
    2 / -0.83

    The value of the integral of the function g(x, y) = 4x3 + 10y4 along the straight line segment from the point (0, 0) to the point (1, 2) in the x-y plane is  

    Solution

    The equation of the line passing through (0,0) and (1,2)  is y = 2x  

    Given y x, y ) = 4x3 + 10y4 = 4x3 + 10(2x )4 = 4x3 + 160xy

  • Question 14
    2 / -0.83

    If     is a differentiable vector function and f is a sufficient differentiable scalar function, then curl   

    Solution

  • Question 15
    2 / -0.83

    The temperature field in a body varies according to the equation T(x,y) = x3 +4xy. The direction of fastest variation in temperature at the point (1,0) is given by  

    Solution

  • Question 16
    2 / -0.83

    The divergence of vector   

    Solution

  • Question 17
    2 / -0.83

    The divergence of the vector  

  • Question 18
    2 / -0.83

     Among the following, the pair of vectors orthogonal to each other is  

    Solution

    Then we say that they are orthogonal.  Choice (c) is correct. 

  • Question 19
    2 / -0.83

    The directional derivative of the scalar function f(x, y, z) = x2 + 2y2 + z at the point P = (1,1, 2) in the direction of the vector  

    Solution

    Required directional derivatives at P(1,1,-1) 

    =2

  • Question 20
    2 / -0.83

    The Gauss divergence theorem relates certain  

  • Question 21
    2 / -0.83

    If P, Q and R are three points having coordinates (3, –2, –1), (1, 3, 4), (2, 1, –2) in XYZ space, then the distance from point P to plane OQR (O being the origin of the coordinate system) is given by  

    Solution

    The equation of the plane OQR is (O being origin). 

  • Question 22
    2 / -0.83

    Let x and y be two vectors in a 3 dimensional space and denote their dot product. 

    Then the determinant det  

    Solution

  • Question 23
    2 / -0.83

    If  a  - b  = 3 and  a2  + b2  = 29, find the value of  ab.

    Solution

    2ab  = (a2  + b2 ) - (a  - b )2

      = 29 - 9 = 20

       ab  = 10.

  • Question 24
    2 / -0.83

    If a vector R(t) →  has a constant magnitude, then  

    Solution

    On analysing the given (a) option, we find that     will give constant magnitude, so first  
    differentiation of the integration will be zero. 

  • Question 25
    2 / -0.83

    For the scalar field    magnitude of the gradient at the point(1,3) is   

    Solution

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