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Engineering Mathematics I - 2009 - 1st Semester - EduRev Class 12 Notes | EduRev

Created by: Karan Sharma

Class 12 : Engineering Mathematics I - 2009 - 1st Semester - EduRev Class 12 Notes | EduRev

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B.E. 1
st
Semester
Engineering Maths-I
PART A
Q1. (a) Discuss the continuity of the following
(i) ???? ( ???? , ???? ) = ?
???? 2
- ???????? + ???? - ???? ???? - ???? , (???? , ???? ) ? (2,2)
1            ,     ( ???? , ???? ) = (2,2)

(ii) ???? ( ???? , ???? ) = ?
sin
-1
( ???? +2 ???? )
tan
-1
(2 ???? +4 ???? )
, (???? , ???? ) ? (0,0)
1
2
, ( ???? , ???? ) = (0,0)

(b) Find the linearization ???? (???? , ???? , ???? ) of the function ???? ( ???? , ???? , ???? ) = v2 ???????????????? sin ? (???? + ???? ) at ????°(0 ,0,
???? 4
).
Find an upper bound for the magnitude of error in E in the approximation ???? (???? , ???? , ???? ) ˜ ???? (???? , ???? , ???? )
over the region ???? : | ???? | = 0.01, | ???? | = 0.01, | ???? - ???? /4| = 0.01.
Q2. (a) If ???? (???? , ???? , ???? ) is differentiable and ???? = ???? - ???? , ???? = ???? - ???? , ???? = ???? - ???? , prove that

???????? ???????? +
???????? ???????? +
???????? ???????? = 0.
(b) Expand ???? ( ???? , ???? ) = tan
-1
???????? in powers of ???? - 1 ???????????? ???? - 1 upto second degree terms.
Q3. (a) Find the minimum value of ???? 2
+ ???? 2
+ ???? 2
subject to the condition ???????????? = ???? 3
.
(b) Find the volume of the prism whose base is the triangle bound by the ???? - ???????????????? and the lines
???? = ???? ???????????? ???? = 1 and whose top ;ies in the plane ???? = ???? ( ???? , ???? ) = 2 - ???? - ???? .
Q4. (a) Evaluate ? ? ???? ???? - ???? 2
/ ???? ???????????????? ???? 0
8
0
by change of order of integration.
(b) Find the volume of the ellipsoid
???? 2
???? 2
+
???? 2
???? 2
+
???? 2
???? 2
= 1
PART-B
Q5. (a) Define curvature of a smooth curve. Prove that the curvature of a smooth curve
???? ¯( ???? ) = ???? ( ???? ) ???? ^ + ???? (???? )???? ^ defined by twice differentiable function ???? = ???? ( ???? ) ???????????? ???? = ???? ( ???? ) is given by
the formula:
???? =
| ???? ? ???? ¨ - ???? ? ???? ¨|
? ???? 2 ? + ???? 2 ? ?
3/2

Compiled by Akshit Chauhan (IT-3rd Sem) of 4.
Compiled by Akshit Chauhan (IT-3rd Sem) of 4.
Page 2

B.E. 1
st
Semester
Engineering Maths-I
PART A
Q1. (a) Discuss the continuity of the following
(i) ???? ( ???? , ???? ) = ?
???? 2
- ???????? + ???? - ???? ???? - ???? , (???? , ???? ) ? (2,2)
1            ,     ( ???? , ???? ) = (2,2)

(ii) ???? ( ???? , ???? ) = ?
sin
-1
( ???? +2 ???? )
tan
-1
(2 ???? +4 ???? )
, (???? , ???? ) ? (0,0)
1
2
, ( ???? , ???? ) = (0,0)

(b) Find the linearization ???? (???? , ???? , ???? ) of the function ???? ( ???? , ???? , ???? ) = v2 ???????????????? sin ? (???? + ???? ) at ????°(0 ,0,
???? 4
).
Find an upper bound for the magnitude of error in E in the approximation ???? (???? , ???? , ???? ) ˜ ???? (???? , ???? , ???? )
over the region ???? : | ???? | = 0.01, | ???? | = 0.01, | ???? - ???? /4| = 0.01.
Q2. (a) If ???? (???? , ???? , ???? ) is differentiable and ???? = ???? - ???? , ???? = ???? - ???? , ???? = ???? - ???? , prove that

???????? ???????? +
???????? ???????? +
???????? ???????? = 0.
(b) Expand ???? ( ???? , ???? ) = tan
-1
???????? in powers of ???? - 1 ???????????? ???? - 1 upto second degree terms.
Q3. (a) Find the minimum value of ???? 2
+ ???? 2
+ ???? 2
subject to the condition ???????????? = ???? 3
.
(b) Find the volume of the prism whose base is the triangle bound by the ???? - ???????????????? and the lines
???? = ???? ???????????? ???? = 1 and whose top ;ies in the plane ???? = ???? ( ???? , ???? ) = 2 - ???? - ???? .
Q4. (a) Evaluate ? ? ???? ???? - ???? 2
/ ???? ???????????????? ???? 0
8
0
by change of order of integration.
(b) Find the volume of the ellipsoid
???? 2
???? 2
+
???? 2
???? 2
+
???? 2
???? 2
= 1
PART-B
Q5. (a) Define curvature of a smooth curve. Prove that the curvature of a smooth curve
???? ¯( ???? ) = ???? ( ???? ) ???? ^ + ???? (???? )???? ^ defined by twice differentiable function ???? = ???? ( ???? ) ???????????? ???? = ???? ( ???? ) is given by
the formula:
???? =
| ???? ? ???? ¨ - ???? ? ???? ¨|
? ???? 2 ? + ???? 2 ? ?
3/2

Compiled by Akshit Chauhan (IT-3rd Sem) of 4.
Compiled by Akshit Chauhan (IT-3rd Sem) of 4.
(b) Without finding ???? ?
???????????? ???? ?
write the acceleration of the motion
???? ?( ???? ) = ( ???? ???? cos ???? ) ???? ^ + ( ???? ???? sin ???? ) ???? ^ + v2 ???? ???? ???? ?

???? = 0 ???????? ????h???? ???????????????? ???? ? = ???? ???? ???? ? ?
+ ???? ???? ???? ? ? ?

Q6. Find the directional derivative of ?
( ???? , ???? , ???? ) = ???? ???? 2
+ ???? ???? 3
at the point (2,-1,1) in the direction
of the vector ???? ^ + 2 ???? ^ + 2 ???? ?
.
Q7. (a) Prove that the differential form in the integral below is exact. Then evaluate the integral.
? 2 cos ???? (1,
???? 2
,2)
(0,2,1)
???????? + ?
1
???? - 2 ???????????????? ???? ? ???????? +
1
???? ????????
(b) State Green’s theorem. Apply it to evaluate the integral ?
( ???? 2
???????? + ???? 2
???????? ), ???? :

???? the triangle
bounded by ???? = 0, ???? + ???? = 1, ???? = 0.
Q8. (a) Evaluate ? ???? ?
. ???????? ? ? ? ? ?

???? by Stoke’s theorem, where ???? ?
= ???? 2
???? ^ + ???? 2
???? ^ - (???? + ???? )???? ?
and ???? is the
boundary of the triangle with vertices at (0,0,0), (1,0,0), and (1,1,0).
(b) State Gauss divergence theorem. Use it to find the outward flux of ???? ?
across the boundary of
the region D:
???? ?
= ( ???? - ???? ) ???? ^ + ( ???? - ???? ) ???? ^ + (???? - ???? )???? ?

D: the cube bounded by the planes ???? = ±1, ???? = ±1, ???? = ±1.
30/Oct/2011
Disclaimer: Well this looks a little stupid but we are not responsible if any portion of this paper is
not relevant with today’s syllabus. Due care has been taken to make this paper error free, but
there always remains a chance of error so please ignore any typographical error if present..
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Compiled by Akshit Chauhan (IT-3rd Sem) of 4.
Compiled by Akshit Chauhan (IT-3rd Sem) of 4.
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