Unit 3 of CSE Btech questions answers

1. ABES ENGINEERING COLLEGE GHAZIABAD
B. TECH FIRST SEMESTER 2021-22
ENGINEERING MATHEMATICS-I (KAS-103T)
UNIT-3 (DIFFERENTIAL CALCULAS-II)
QUESTION BANK
SECTION –A
1. Find the Taylor’s series expansion of 2
3
)
,
( xy
x
y
x
f 
 about point (2 , 1).
2. If
x
u
find
then
bv
au
y
bv
au
x





 ,
, 2
2
.
3. Find the condition for the contour onxy plane where the partial derivative of 2
2
y
x  with
respect to y is equal to the partial derivative of x
y 4
6  with respect to x .
4. If )
1
(
,
)
1
( u
v
y
v
u
x 


 , then find the Jacobian of v
u , with respect to y
x, .
5. Verify the chain rule for Jacobians if w
z
v
u
y
u
x 

 ,
tan
, .
6. If
x
w
w
z
z
y
y
x
find
then
w
z
y
x
f








 ,
0
)
,
,
,
( .
7. If
z
y
x
u
find
then
e
u z
y
x




 

3
,
2
2
2
.
8. If
y
u
y
x
u
x
of
value
the
find
then
y
x
u






  5
1
2
2
1
)
(
sin .
9. What is the degree of homogeneous function
3
2
2
2
3
1
2
2
2
)
(
)
(
)
,
(
y
x
y
x
x
y
x
u


 ?
10. If k
pv 
2
and the relative errors in p and v are respectively 0.05 and 0.025 , show that the
error in k is 10% .
11. Find approximately : (i) 3
)
98
.
2
( (ii) 5
1
3
2
]
)
1
.
2
(
2
)
82
.
3
[(  .
12. At what point , 2
2
2
3
2 2
2
2





 yz
xy
z
y
x
u is maximum ?
13. If
5
)
(
)
,
( y
x
y
x
u 
 , find the value of 2
2
2
2
2
2
2
2
y
u
y
y
x
u
xy
x
u
x









.
14. Determine the points where the function 12
6
2
2



 x
y
x
u has a maximum or minimum.
Find the possible percentage error in computing the parallel resistance r of three
resistances 𝑟1, 𝑟2, 𝑟3 from the formula
1
𝑟
=
1
𝑟1
+
1
𝑟2
+
1
𝑟3
if 𝑟1, 𝑟2, 𝑟3 are each in error by
1.2%.
SECTION –B
1. If )
(
1
)
(
,
,
)
( 2
2
2
2
2
2
2
r
f
r
r
f
y
u
x
u
that
prove
y
x
r
where
r
f
u 











 .

2. 2. If
y
x
y
x
y
x
u 1
2
1
2
tan
tan 


 , then prove that 2
2
2
2
2
y
x
y
x
y
x
u






.
3. If
u
x
v
y v
y
y
v
u
x
x
u
that
prove
bv
au
y
bv
au
x 







































2
1
.
,
, 2
2
.
4. Show that 




















 
cz
by
ax
z
y
x
u
where
u
z
u
z
y
u
y
x
u
x
3
3
3
1
sin
,
tan
2 .
5. If












 
6
1
6
1
4
1
4
1
1
sin
y
x
y
x
u ,then evaluate 2
2
2
2
2
2
2
2
y
u
y
y
x
u
xy
x
u
x









.
6. If )
2
4
,
4
3
,
3
2
( x
z
z
y
y
x
f
u 


 ,prove that 0
4
1
3
1
2
1









z
u
y
u
x
u
.
7. If uv
z
v
u
y
v
x
and
z
y
x
w 




 ,
sin
,
cos
2
2
2
then prove that
2
1 v
u
v
w
v
u
w
u







.
8. Express the function 26
9
9
3
)
,
( 2
2




 y
x
y
x
y
x
f as Taylor’s series expansion about
the point (1,2).
9. Expand y
e
y
x
f x 1
tan
)
,
( 
 in power of )
1
(
)
1
( 
 y
and
x up to two terms of degree 2.
10. Find first six terms of the expansion of 𝑒𝑥
log⁡
(1 + 𝑦) in the neighbor-hood of (0,0)
by taylor’s series.
11. If v
u
w
v
u
z
y
x
that
show
then
uvw
z
uv
z
y
u
z
y
x 2
)
,
,
(
)
,
,
(
,
,
, 







 .
12. If w
v
u ,
, are the roots of the cubic equation (𝑥 − 𝑎)3
+ (𝑥 − 𝑏)3
+ (𝑥 − 𝑐)3
= 0 in x then
find
𝜕(𝑢,𝑣,𝑤)
𝜕(𝑎,𝑏,𝑐)
.
13. If 2
2
4
2
,
3
2
,
2 z
yz
xz
xy
w
z
y
x
v
z
y
x
u 








 , show that they are not
independent . Find the relation between w
and
v
u, .
14. What error in the common logarithm of a number will be produced by an error of 1% in the
number?
15. Find an approximate value of  2
1
2
2
2
)
94
.
1
(
)
01
.
2
(
)
98
.
0
( 
 .
16. Examine for minimum and maximum values : )
sin(
sin
sin y
x
y
x 

 .
17. Find the maximum and minimum distances of the points (3,4,12) from the sphere
1
2
2
2


 z
y
x .
18. Divide a number into three parts such that the product of first , square of the second and
cube of the third is maximum .
19. Find the extreme values of function : axy
y
x 3
3
3

 .

3. ANSWERS: SECTION-A
1. .....
..........
)]
1
)(
2
(
)
1
(
)
2
(
3
[
2
)]
1
(
4
)
2
(
13
[
10
)
,
( 2
2












 y
x
y
x
y
x
y
x
f
2.
a
x
x
u



3. 2

y
4.
v
u 

1
1
6.1
7. xyzu
8
8. u
tan
5
2
9. 4/3
11. (i)26.46 (ii) 2.012
12. (1,1)
13. u
4
15
14. (-3,0) , minimum , min value=3
15. 3%
SECTION – B
5.
)
12
(sec
tan
144
1 2

u
u
8. 2
2
)
2
(
3
)
1
(
)
2
(
3
)
1
(
7
12
)
,
( 







 y
x
y
x
y
x
f
9. ........
)
1
(
)
1
)(
1
(
2
2
)
1
(
4
)
1
(
2
)
1
(
2
4
tan 2
2
1


























y
y
x
x
e
y
x
e
e
y
ex 


12. 















)
)(
)(
(
)
)(
)(
(
2
)
,
,
(
)
,
,
(
u
w
w
v
v
u
x
z
z
y
y
x
z
y
x
w
v
u
13. w
v
u 4
2
2

 14. 0.0043429
15. 2.96 16. maximum value=
2
3
3
17. min distance=12 ,max distance=14
18. a
is
number
if
a
a
a
2
,
3
,
6
19. max. value 3
a
 , min value 3
a

