Solutions of Final Exam 2016-2017

Solutions of Final Exam 2016-2017


University of Babylon
College of Engineering
Mechanical Engineering Dept.
Final Course Exam/ 1st attempt
Subject : Mathematics III
Class: 2nd
Date: 19 / 01 / 2017 (( A))
Max.Time : 3 hrs.
2016 – 2017 Max. mark : 60 M
(Q1) : Evaluate ?_c??( x+ ?y? ) ds where C is given in the Figure below . ( 10 M)

******************************************************************************(Q2) : ((Answer two branches only)) ( 10 M )
a)) Convert the point ( 6, 5?/3 , 7 ) from cylindrical to rectangular coordinates
b)) Show that the curvature of a circle of radius ( a ) is ( 1/a ) .
c)) Finding limits of vector function if r(t)=(cos t)i+(sin t)j+tk .
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(Q3) : Evaluate the integral ?_0^3??_0^4??_(x=y/2)^(x=(y/2)+1 )??((2x-y)/2+z/3)dx dy dz? By applying the transformation:
u=(2x-y)/2 , v=y/2 , w= z/3
and integrating over an appropriate region in uvw plane . ( 10 M )
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(Q4) : ( 10 M )
a)) A hemispherical bowl of radius 5 cm is filled with water to within 3 cm of the top .
Find the volume of water in the bowl .

b)) Prove or disprove the formula

u X ( u X ( u X v )) .w= - |u|^2 u .v X w


(Q5) : ( 10 M )
Find the work done by the force field F = xi + y j + z k in moving an object
Along the curve C parametrized by r(t)=cos??(?t)i+t^2 j+sin?(?t)k ?
0?t?1 .


b)) Prove that J(? ,?,?)=?^2 sin ? .
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(Q6) : ((Answer two branches only))

At what times in the interval 0?t?? are the velocity and acceleration vectors of the motion r(t)=i+(5cos t)j+(3sint)k orthogonal ?

Show that the planes 2x – 3y –z=5 and -6x+9y+3z= -2 , are parallel and find the distance between the two planes .

Find equations for the osculating , normal , and rectifying planes of the curve
r(t) = ti +t^2 j +t^3 k at the point (1,1,1).

( 10 M )
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Good Luck