Applied Mathematics

other 17 Pages

Maharashtra State Board of Technical Education

Diploma Engineering

Contributed by

Pavan xerox

MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 1 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Important Instructions to Examiners:
1) The answers should be examined by key words and not as word-to-word as given in the model answer scheme.
2) The model answer and the answer written by candidate may vary but the examiner may try to assess the
understanding level of the candidate.
3) The language errors such as grammatical, spelling errors should not be given more Importance (Not applicable for
subject English and Communication Skills.
4) While assessing figures, examiner may give credit for principal components indicated in the figure. The figures
drawn by candidate and model answer may vary. The examiner may give credit for any equivalent figure drawn.
5) Credits may be given step wise for numerical problems. In some cases, the assumed constant values may vary and
there may be some difference in the candidate’s answers and model answer.
6) In case of some questions credit may be given by judgement on part of examiner of relevant answer based on
candidate’s understanding.
7) For programming language papers, credit may be given to any other program based on equivalent concept.

Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
1.

a)

Ans

b)

Attempt any FIVE of the following:
   
 
 
 
 
 
2
2
2
2
If tan , show that 2
1
2
tan2
2tan
1 tan
2
1
fx
f x x f x
fx
fx
x
x
x
fx
fx












OR

 
 
 
2
2
2
1
2tan
1 tan
tan 2
2
fx
fx
x
x
x
fx








-------------------------------------------------------------------------------------------------------------------
State whether the function ( ) is even or odd.
2
xx
ee
fx




10
02

½

1

½

½
1
½

02
22210
Page 1
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 2 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
1.

b)
Ans

c)
Ans

d)
Ans

e)
 
()
2
()
2
()
2
( ) ( )
function is even.
xx
x
x
xx
ee
fx
ee
fx
ee
fx
f x f x







  

  
  


---------------------------------------------------------------------------------------------------------------------------------------
 
 
Find if .
.
.1
x
x
xx
xx
xx
dy
y x e
dx
y x e
dy d d
x e e x
dx dx dx
dy
xe e
dx
dy
xe e
dx


  
  
  

----------------------------------------------------------------------------------------------------------------------------
 
 
 
1
11
11
1
2
1
2
1
2
12
Evaluate tan
tan tan .1
tan 1 1 tan
1
tan
1
tan
1
12
tan
21
1
tan log 1
2
x dx
x dx x dx
d
x dx dx x dx
dx
x x x dx
x
x
x x dx
x
x
x x dx
x
x x x c















   


  




---------------------------------------------------------------------------------------------------------------------------------------
Evaluate 1 sin2 x dx


½

½
½
½

02

2

02

½

½

½
½

02
22210
Page 2
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 3 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
1.

e)
Ans

f)
Ans

g)

Ans

 
 
22
2
1 sin 2
sin cos 2sin cos
sin cos
sin cos
cos sin
x dx
x x x x dx
x x dx
x x dx
x x c

  


   





-------------------------------------------------------------------------------------------------------------------------
 
   
   
0
0
Find the area bounded by the curve sin and the -axis from 0 to
Area
sin
cos
cos cos0
11
2
b
a
y x x x x
A y dx
x dx
x




  



   
    




-----------------------------------------------------------------------------------------------------------------------------------
2
22
1
Express in the form , , where , , IR . 1
2
1

2
12

22
22

2
2 3 1

41
13

5
13

55

i
a ib a b i
i
i
i
ii
ii
i i i
i
i
i
i

     







  









-------------------------------------------------------------------------------------------------------------------------------------

½
½
½
½

02

½
½
½

½

02

½

½

½

½

22210
Page 3
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 4 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q. N.
Answers
Marking
Scheme
2.

a)
Ans

b)
Ans

c)

Ans

Attempt any THREE of the following:
   
   
 
 
 
 
 
If sin , 1 cos find
sin , 1 cos
sin
1 cos
1 cos
0 sin sin
sin
1 cos
sin
1 cos
dy
x a y a
dx
x a y a
xa
dx
a
d
ya
dy
aa
d
dy
dy a
d
dx
dx a
d
dy
dx
  
  












   
   

  

   





-------------------------------------------------------------------------------------------------------------------------------------
 
22
22
If find
2 2 .1
22
22
2
2
dy
x y xy
dx
x y xy
dy dy
x y x y
dx dx
dy dy
x y x y
dx dx
dy
y x y x
dx
dy y x
dx y x


   
   
   




--------------------------------------------------------------------------------------------------------------------------
A metal wire 36 cm long is bent to form a rectangle. Find its dimensions when its area is
maximum.
Let length of rectangle , breadth =
Area

2 2 36
18
xy
xy
A
y
y
x
x

  






12

04

1

1

1

1

04

1

1

1

1

04

1
22210
Page 4
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 5 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
2.

c)

d)

Ans

 
2
2
2
2
2
18
18
18 2
2
Let 0
18 2 0
9
a
Area is maximum at 9
Length 9 ; brea
t 9
20
dth 9
A x x
A x x
dA
x
dx
dA
dx
dA
dx
x
x
x
dA
dx
x

  
  
  

  









-------------------------------------------------------------------------------------------------------------------------------------
2
2
A beam is bent in the form of the curve 2sin sin 2 . Find the radius of curvature of
the beam at this point at
2
2sin sin 2
2cos 2cos2
2sin 4sin 2
at
2
2cos 2cos2
2
y x x
x
y x x
dy
xx
dx
dy
xx
dx
x
dy
dx






  
   


  


 
2
2
3
2
2
3
2
2
2
2
2
2
2sin 4sin 2 2
22
1
12
Radius of curvature
2
dy
dx
dy
dx
dy
dx






   
     
   
   









   
  


1
½
[
½

½

½

04

½

½

½

½

1
22210
Page 5
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 6 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
2.

3.

d)

a)
Ans

Radius of curvature 5.59
5.59
  


-----------------------------------------------------------------------------------------------------------------
Attempt any THREE of the following:
 
 
 
 
 
22
22
Find the equation of the tangent and normal to the curve 4 9 40 at 1,2
4 9 40
8 18 0
8
18
4
9
at 1,2
41
92
2
9
2
slope of tangent ,
9
Equation of tangent at 1,2 is
xy
xy
dy
xy
dx
dy x
dx y
dy x
dx y
dy
dx
dy
dx
m


  










 
 
 
2
21
9
9 18 2 2
2 9 20 0
19
slope of normal , '
2
Equation of normal at 1,2 is
9
2 1
2
2 4 9 9
9 2 5 0
yx
yx
xy
m
m
yx
yx
xy

  
    
   

  
  
   
   

---------------------------------------------------------------------------------------------------------------------------------------

1

12
04

½

½

½

½

½
½

½

½

22210
Page 6
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 7 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
3.

b)

Ans

c)

Ans

 
 
 
sin
sin
sin
sin
Find if tan
Let
log log
log sin log
11
sin log .cos
1 sin
cos log
sin
cos log
sin
cos log
Let tan
log log tan
log
x
x
x
x
x
x
x
dy
y x x
dx
ux
ux
u x x
du
x x x
u dx x
du x
xx
u dx x
du x
u x x
dx x
du x
x x x
dx x
vx
vx




   
  

  



  





 
 
 
 
   
   
2
2
2
2
2
sin
log tan
11
.sec log tan 1
tan
1 sec
log tan
tan
sec
log tan
tan
sec
tan log tan
tan
sin sec
cos log tan log tan
tan
x
x
x
v x x
dv
x x x
v dx x
dv x x
x
v dx x
dv x x
vx
dx x
dv x x
xx
dx x
dy x x x
x x x x x
dx x x

    
  

  



  




    





----------------------------------------------------------------------------------------------------------------
 
22
22
2 2 2 2
Find if log
log
11
1 2 0
2
dy
y x x a
dx
y x x a
dy
x
dx
x x a x a

  


  


   

  


04

½
1

½

½
1

½

04

2
22210
Page 7
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 8 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
3.

c)

d)

Ans

2 2 2 2
22
2 2 2 2
22
1
1
1
1
dy x
dx
x x a x a
dy x x a
dx
x x a x a
dy
dx
xa

  

  






  




--------------------------------------------------------------------------------------------------------------------------------------
   
2
2
2
2
2
2
22
22
2
22
Evaluate
4 5cos
4 5cos
2
Put tan ,
21
1
cos
1
2
1

1
45
1
2
4 1 5 1
2
4 4 5 5
2
9
2
3
13
2 log
2 3 3
3 tan
1
2
log
3
3 tan
2
dx
x
dx
x
x dt
t dx
t
t
x
t
dt
t
t
t
dt
tt
dt
tt
dt
t
dt
t
t
c
t
x
c
x















  

  


















------------------------------------------------------------------------------------------------------------------------------------

2

04

1

½

1

1

½

22210
Page 8
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 9 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
4.

a)

Ans

b)
Ans

Attempt any THREE of the following:
 
 
 
 
22
22
2
2
2
2
1
Evaluate
sin
1

sin
Put
1
1
1

sin
cos
cot
cot
x
x
x
x
x
xx
x
x
xe
dx
e
x
x
xe
dx
e
x
x
e
t
x
xe e
dx dt
x
xe
dx dt
x
dt
t
ec t dt
tc
e
c
x















  

  







-------------------------------------------------------------------------------------------------------------------------------------
 
 
3
3
33
Evaluate sin
sin
1
since sin3 3sin 4sin sin 3sin sin3
4
1
3sin sin3
4
1 cos3
3cos
43
x dx
x dx
x x x x x x
x x dx
x
xc
    


   






 
3
2
2
sin
sin sin
1 cos n

si
x dxOR
x x dx
x x dx






12

04

1

½
1
1

½

04

2
2

½

1
22210
Page 9
MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(Autonomous)
(ISO/IEC - 27001 - 2013 Certified)
__________________________________________________________________________________________________
Page 10 of 17

WINTER– 19 EXAMINATION

Subject Name: Applied Mathematics Model Answer Subject Code:
Q.
No.
Sub
Q.N.
Answers
Marking
Scheme
4.

b)

c)

Ans

 
 
 
2
2
3
3
Put cos
sin
sin
1
1
3
cos
cos
3
xt
xdx dt
xdx dt
t dt
t dt
t
tc
x
xc

 
  
  
  

   



   





-------------------------------------------------------------------------------------------------------------------------------------
   
   
   
        
2
2
2
2
2
25
Evaluate
1 2 3
25
1 2 3
25
Consider
1 2 3 1 2 3
2 5 2 3 1 3 1 2
Put 1
7 12
7
12
Put 2 13 3
13
3
Put 3 23 4
23
4
25
x
dx
x x x
x
dx
x x x
x A B C
x x x x x x
x A x x B x x C x x
x
A
A
xB
B
xC
C
x
x

  

  

  
     
          



    


   





   
     
13
7 23
3
12 4
1 2 3 1 2 3
7 13 23
log 1 log 2 log 3
12 3 4
dx dx
x x x x x
x x x c



  

     


      


1

1

½

04

½

½

½

½

2

22210
Page 10

Applied Mathematics

Applied Mathematics

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