Solved question paper for MATHS-1 May-2018 (B-TECH 1st-2nd)

Engineering mathematics-1

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Question paper 1

  1. Section A

    1. a) Find asymptotes, parallel to axes, of the curve: y = 

    Answer:

    (a) yx- y = x+ 1

    || to x axis

              X2=0

              X = 0                    No asymptote || to x axis

    || to y axis

            X2-1 = 0

                X= 1

                X = 1

  2. b) Write a formula to find the volume of the solid generated by the revolution, about
    x-axis, of the area bounded by the curve y = f(x), the x-axis and the ordinates
    x = a and x = b.

    Answer:

    (b) Volume will be given by integrating the terms

                 =  

                 =  

  3. c) Find the value of  , where x = rcosθ & y = rsinθ .

    Answer:

  4. d) If an error of 1% is made in measuring the major and minor axes of an ellipse, what is the
    percentage error in its area?

    Answer:

    (d) let x and y be semi major and semi minor axes of an ellipse.

                                            Area of ellipse = A=

                                            Log A = Logπ + Log x +Log y

     1+1=2

                                             Error in area = 2 %

  5. e) Is the function  ? If yes, what is its degree?

    Answer:

    Yes, Degree = 2

  6. f) What is the value of   over the positive quadrant of the circle x2 + y2 = 1?

    Answer:

    =

  7. g) Give geometrical interpretation of  

    Answer:

     

      =  

      =    

      =    

      =   2

  8. h) Show that for the vector field  (x2 - y2 + x)  — (2xy + y)   = 0.

    Answer:

        =

           =

            = -2y + 2y

               [  ]

  9. i) Show that the vector field  = (-x2 + yz)   + (4y — z2x) + (2xz — 4z)  is solenoidal.

    Answer:

    For solenoidal function

    div 

    div f =   

    = -2x + 4 + 2x -4

    = 0

    So it is solenoidal

  10. j) State Green’s theorem in plane.

    Answer:

    Green’s theorem in plane:-

    Let R bed closed region of x-y plane bounded by a simple closed cense c and let M and N be continuous function of x and y having continuous partial derivative  and  in R then:

  11. Section B

    2. Trace the following curves by giving their salient feature:
    a) y(a — x) = x(a + x).
    b) r = a (1 — cosθ)

    Answer:

    (a) Symmetry:  Symmetrical about x-axis

    Origin: passes through origin and tangent at origin

           y= x and y =-x   therefore origin is a node.

          Asymptotes x = a

     Points:  it crones the Ares at (0,0) and  (-a,0 )

          When x >a  or  <-a    y is imaginary

    Shape of curve is strophoid      

    (b) Given figure is a cardioid and symmetrical about q = 0 i.e. initial line.

    q  varies from (0-π/2 ) or (0-180).

     

  12. 3. a) Find the whole length of the curve  x2/3  + y2/3 = a2/3 .

    b) Use definite integral to find the area of ellipse  

    Answer:

    (a) Given curve is asteroid. Which is symmetrical about x-axis and y-axis therefore entries length of curve is four times the length of one part.

       Diff given curve w.r.t x

    2/3x-1/3+ 2/3y-1/3dy/dx =0

              2/3y-1/3dy/dx = -2/3x-1/3

    dy/dx=-(y/x)1/3

    total length = 4    = 4 

     = 4 

    = 6a

    (b)   

          Therefore area of ellipse = 4x area of ellipse in I quadrant.

            =

            =

    =

     

    Therefore required area = area of region PQRS

     = 4 x area of region OFPB in the first quadrant


      =

      = 

      = 

  13. 4. a) lf u = log (x3 +y3 +z3 —3xyz), show that  =   -9(x+y+z)-2

    b) State Euler’s theorem for homogeneous functions and apply it to show that
      ,   where sin u  =

    Answer:

    (a) 

           =

     

                                            = 

                                            =

                                            =

    (b)  Euler’s theorem  

                  If H= f (x, y, z)is ahomogeneous function of x, y and z of degree n, then

    U= sin-1(x2+y2/x+y)

           Sin u = x2+y2/x+y

          F =   x2+y2/x+y

    Which is homogeneous of degree 1.

     By Euler’s theorem  

              =

  14. 5. a) The temperature T at any point (x, y, z) in space is T = 400xyz2 . Find the highest
    temperature on the surface of the unit sphere x2 + y2 + z2= 1.

    b) If f(x,y) = tan-1 xy, compute f(0.9, –1.2) approximately.

    Answer:

    (a) Temperature at any point on sphere is given by

                  T= 400 xy (1-x2-y2)

      = 400y -400y3-1200x2y

    400x – 1200xy2-400x3

     For critical points

    400y-400y3-1200x2y

      400x(1-3y2-x2)=0

    critical points after solution above equation are

    (0,0),(±1,0),(0, ±1),( ±1/2, ±1/2) out of which first three given the value of T to be zero. For other four point (±1/2, ±1/2). Now A = =  2400 xy, B =   = 400-1200y2-1200x2, C= = -2400xy

    At (1/2,1/2,) A= -600 <  0, B= -200, C = -600 so that AC-B2= 320000> 0

     T is maximum at(1/2,1/2)

                                          Whose maximum value will be given by

                                   = 400 ¼(1 -1/4 - 1/4) = 100 x ½ =50

     

    (b)  f(x,y) = tan-1xy

       Let x=0.9  , y=-1

    X+δx= 0.9      1+δx=0.9                =δx=-1            =   

         -y+ δy=-1.2       δy=-1.2+1            = δy = -0.2

    df=                        = y/1+x2y2  dx + x/1+x2y2dy = ydx+xdy/1+x2y2

        When x =1

                   Y = -1

    df =

                           =   =   = 

  15. Section C

    6. a) Evaluate the following integral by changing the order of integration:

    b) Evaluate the triple integral   

    Answer:

    (a) Given region of integration

             0

    it is a circle of radius 1

    we will introduce a vertical strip which will vary from

    order to change the order into integration

    = 2

    (b  =

                     = 

                     = 

                     = 

                    = 

                    = 

                    = 

  16. 7. a) Find a unit vector normal to the surface x2+y2+z2= 9 at the point (2, -1, 2) .

    b) If u = x2+y2+z2 & Show that Ñ.  = 5u

    Answer:

    (a) f(x,y,z) = x2+y2+z2 = 9

    =2x

     = 2y

     = 2z

                      Vector =

         A point (2,-1,2)

    =

     Unit vector normal to it =      

    (b) u= x2+y2+z2

    (by identity)

    )

             =

                    [ ]

  17. 8. a) If   evaluate,  where C is the curve in the xy-plane y = 2x2
    from (0,0) to (1,2).

    (b) Compute  , where   and S is the triangular surface with vertices (2, 0,0), (0, 2, 0) and (0, 0, 4).

    Answer:

    (a)    where

                  y =

       and  

     

    (b)   

    Equation of given plane x/2+y/z+z/4=1

    Let R be orthogonal projection of x+y +z/2=1in XoY plane i.e. z=0 given by x+y=1

    Therefore R=0{(x.y):0 y 1-x;0 x 1}

    =x +yx

    =(x )

    ds =

                                     [Put x=2,y=2,z=neglected ]

  18. 9. State Gauss Divergence theorem and verify it for    taken over
    the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

    Answer:

    Surface contains six faces

     (i) OABC faces Z = 0,

     .

    (ii) On face DEFG Z = 1,

    (iii) On face OAFG y = 0,

    (iv) On face DEBC y = 0,

    (v) On face ABEF x = 1,

    (vi) On face OCDG x = 0,

    Gauss Divergence theorem

    If   be a vector point function having continuous first partial derivative in reason v bounded by surface (s).

    Then