Total No. of Questions : 9] SEAT No.

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23
P3924 [Total No. of Pages : 4

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[6001]-4009

tat
F.E.

7s
ENGINEERING MATHEMATICS-II

3:3
(2019 Pattern) (Semester - I/II) (107008)

02 91
0:3
0
Time : 2½ Hours] [Max. Marks : 70

31
3/0 13
Instructions to the candidates:
1) Q. No.1 is compulsory.
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8/2
2) Solve Q.2 or Q.3, Q.4 or Q.5, Q.6 or Q.7, Q.8 or Q.9.
.23 GP

3) Neat diagrams must be drawn wherver necessary.

4) Figures to the right indicate full marks.
E
80

5) Use of electronic pocket calculator is allowed.

8
C

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6) Assume suitable data, if necessary.

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Q1) Write the correct option for the following multiple choice questions

tat
8.2

7s

2

 sin  dt 
4
.24

3:3
a) [2]
91
0
49

0:3
30

3 3
31

i) ii)
16 8
01
02
8/2
GP

3 3
iii) iv)
3/0

16 8
CE
80

8
23
.23

b) The equation of the tangent to the curve y (1  x 2 )  x at origin, if exist is ic-

16

tat
8.2

[2]
7s

i) X=0 ii) Y=0

.24

3:3
91
49

x  1, x  1 yx
0:3

iii) iv)
30
31
01
02

1 1
1 1
  2 1  y 2 dxdy 
8/2


GP

c) The value of double integration [2]

0 0 1 x
3/0
CE
80

 2
i) ii)
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2 2
16

 2
8.2

iii) iv)
4 8
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49

P.T.O.

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 d) Centre (C) of sphere x 2  y 2  z 2  2 z  4 is [2]

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i) C  (0,0,0) ii) C  (0,0,1)

ic-
tat
iii) C  (0,1,0) iv) C  (1,0,0)

7s
3:3
02 91
e) The curve r  2a sin  is symmetrical about

0:3
[1]

0
31
i) Pole
3/0 13 ii)  0
 
0
8/2
iii)  iv) 
.23 GP

2 4
E
80

8
C

23


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f) dxdydz represents [1]
16

tat
V
8.2

7s
i) Area ii) Mass
.24

3:3
91
iii) Mean Value iv) Volume
49

0:3
30
31
01

 2
02

 n 2
4
n2
8/2

Q2) a) If In =  sec  d , then prove that In =  In-2

n
GP

[5]
n 1 n 1
3/0

0
CE
80

8
  x  2   5  x  dx
3 2

23
b) Evaluate [5]
.23

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2
16

tat

e  x  e ax 1  a2  1 
Using DUIS, prove that  dx  log  ,a  0
8.2

7s

c) [5]
x sec x 2  2 
.24

0
3:3
91
49

0:3

OR
30
31

Q3) a) Evaluate
01
02
8/2

2
 
GP

 sin d
2
i) cos10 [3]
3/0

0 2 2
CE
80


2

 cos t
.23

4
ii) dt [2]
16

 2
8.2
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49

[6001]-4009 2

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 8
1

23
  x log x 
4
b) Evaluate : dx [5]

ic-
0

tat
1 d 1 d
erfc (ax)  

7s
c) Prove that: erf (ax) [5]
x da a dx

3:3
02 91
0:3
Q4) a) Trace the curve x 2 y 2  a 2 ( y 2  x 2 ).

0
[5]

31
b) 3/0 13
Trace the curve r  a (1  cos ). [5]
0
c) Find the are length of Astroid x 2 3  y 2 3  a 2 3
8/2
[5]
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OR
E
80

8
C

23
Q5) a) Trace the curve x 3  y 3  3axy. [5]

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b) Trace the curve r  a cos 2
16

[5]

tat
Trace the curve x  a (t  sin t ), y  a (1  cos t ).
8.2

c) [5]

7s
.24

3:3
91
Show that the plane x  2 y  2 z  7 touches the sphere x 2  y 2  z 2 
49

Q6) a)
0:3
30

10 y  10 z  31  0. Also find the point of contact.

31

[5]
b) Find the equation of right circular cone whose vertex is at origin, whose
01
02

x y 8
8/2
GP

axis is the line   and which has a semi-vertical angle of 60°. [5]
1 2 3
3/0
CE
80

8
c) Find the equation of right circular cylinder of radius 3 and axis is the line

23
.23

x 1 y  2 z  3
  . ic-
16

[5]
tat
2 1 2
8.2

7s

OR
.24

3:3

Q7) a) Show that the two spheres: x 2  y 2  z 2  25 and x 2  y 2  z 2

91
49

0:3

18 x  24 y 40 z  225  0 touches externally. Also find the point of

30
31

contact. [5]
01

b) Find the equation of right circular cone whose vertex is at (0,0,10), axis
02
8/2

 2 
GP

is the Z-axis and the semi-vertical angle is cos 1   [5]

3/0

 5
CE
80

c) Find the equation of right circular cylinder of radius 6, whose axis

.23

1 1 1
, , .
16

passes through the origin and has direction cosines [5]

3 3 3
8.2
.24
49

[6001]-4009 3

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 8
23
Q8) a) Evaluate  xy dx dy, where R is x 2
 y, y 2   x . [5]

ic-
R

tat
b) Find area of cardioide r  a (1  cos ) using double integration. [5]

7s
c) Find the moment of inertia of one loop of the lemniscate

3:3
r 2  a 2 cos 2 about initial line. Givenl that density   2m

02 91
, m is a mass

0:3
a2

0
31
of the area.
3/0 13
0 [5]

OR
8/2
.23 GP

5 2 x

Q9) a) Change order of integration   f ( x, y ) dxdy. [5]

E
80

8
0 2 x
C

23
b) Find the volume bounded by the cone x 2  y 2  z 2 and paraboloid

ic-
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tat
x2  y 2  z . [5]
8.2

7s
c) Find the x - co-ordinate of centre of gravity of one loop of r  a cos 2 ,
.24

3:3  a2
91
which is in the first quadrant, given that area of loop is A  .
49

[5]
0:3
8
30
31
01
02
8/2
GP


3/0
CE
80

8
23
.23

ic-
16

tat
8.2

7s
.24

3:3
91
49

0:3
30
31
01
02
8/2
GP
3/0
CE
80
.23
16
8.2
.24
49

[6001]-4009 4

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