RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
TUTORIAL-01
[SUCCESSIVE DIFFERENTIATION]
1) If y  ( x  1) n , then P.T.
2) Find nth derivative of y
3) Find yn if y=
4) If
y
y1 y 2 y3


 ....  n  x n
1! 2! 3!
n!
4x
=
( x  1) 2 ( x  1)
y
xn 1
x 1
 x  1
y  x. log 
 then
 x  1
show that yn = (-1)n-2 (n-2)![
5) Find nth order derivative of
( x  n) ( x  n)
]
( x  1) n ( x  1) n
 2x 
Y  sin 1 
2 
1  x 
6) y= tan -1 [ 1  x ] P.T yn= (-1)n-1(n-1)! Sin( nθ ) sin(n) θ, where θ = tan -1 ( 1 x )
1 x
7) U = eax cos(bx+c) then P.T. Un= (a2+b2)n/2 eax cos[bx + c + n tan-1(b/a)]
8) Find yn if y = e5x cosx cos3x
9) If y= xn logx then P.T yn+1 = n!
x
1/k
-1/k
10) If U + U
= 2x then show that (x2-1)Un+2 + (2n+1)xUn+1+ (n2-k2) Un=0
11) If y = a cos (logx) + b sin (logx) then show that x2 yn+2 + (2n+1)x yn+1+
(n2+1)yn =0
12) If y =
1 x
P.T (1-x2) yn -[2(n-1)x+1]yn-1- (n-1)(n-2)yn-2=0
1 x
1
13) If y= eacos x then show that (1-x2) yn+2 + (2n+1)x yn+1- (n2+a2)yn =0
14) If x = sinθ , y=sin(2θ) prove that (1-x2) yn+2 - (2n+1)x yn+1- (n2-4)yn =0
15) If x = sinh(y) prove that (1+x2) yn+2 + (2n+1)x yn+1+ n2yn =0
16) State Leibnitz’s Theorem and use it to find yn for y = e2x (x3+x+1)
17) If y = log x then prove that yn = (1n)1 n! [ log x- (1+ 1 + 1 +---+ 1 )]
n
x
2
x
18) U = (x2-1)n then prove that
d
[(x2-1) d Un]
dx
dx
3
= n(n+1) Un
n
where Un=
dn
dx n
U
19) y = tan-1[ a  x ] then prove that (x2+a2) yn+2 + 2 (n+1)x yn+1+ n(n+1)yn =0
ax
20) If u= sin [ log(1+2x+x2)] then prove that
(x+1)2Un+2 + (2n+1) (x+1)Un+1+ (n2+4) Un=0
Applied Mathematics-I
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RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
TUTORIAL-02
[PARTIAL DIFFERENTIATION]
1) If v  (1  2 xy  y 2 )
1
2
then prove that x
v
v
y
 y 2 v 3 and also find the value of
x
y
 
v    v 
(1  x 2 )    y 2 

x 
x  y  y 
2
2
 2u
=y  u =z u
xz
yz
xy
2z
= -[xlog(ex)] 1 and also
xy
2) If u = log (x2+ y2 + z2) then prove that x
3) If xx yy zz = c, show that at x = y = z,
2z
x 2
2xy
2z
2z
+ 2
y
xy
=
2( x 2  2)
x[log x  1]
at x = y= z
4) x = rcosθ , y = rsinθ then prove that
5) If u= (8x2+ y2)[logx-logy] then find
x y
6) If u= sin 1 [
x y
P.T
] then show that
 2r
 2r
+ 2 = 1 [( r ) 2 +( r ) 2 ]
2
r x
y
y
x
x u +y u
x
y
u
x
u
x y
=-y
7) Verify the Euler’s theorem for u = x2 tan -1 ( y x ) - y2 tan -1 ( x y ) where xy  0
2
2
 2u
= x2  y2
x y
xy
1
+ 1 + log x 2 log y then
2
x
x
xy
and also show that
8) If F(x,y) =
9) If u= tan -1 [ x
10) If
show that x F +y F +2F(x,y) =0
x
 u
y
] Then prove that x2 2 +
x
x y
3
3
2
y
2
 u
+ y2  u2
y
xy
2
2xy
= 2cos3u sinu
1 
 12
x y 2

u= cos ec
then prove that
 x 13  y 1 3 


2
2
2
2
x2  u2 + 2xy  u + y2  u2 = 1 tan u[ 13  tan u ]
12
12
12
y
xy
x
1
11) If x = e r cos cos(rsinθ) and y = e r cos sin(rsinθ) then prove that
x
= 1 y
r
r 
and
y
= - 1 x
r
r 
Hence deduce that
 2 x 1 x 1  2 x
+
+
r 2 r r r 2  2
=0
12) If z = f(u,v) where u = x2-2xy-y2 and v=y then show that
Applied Mathematics-I
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RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
(x+y)
z
+
x
(x-y)
z
=
y
0 is equivalent to
13) If logeθ = r-x where r2= x2+ y2 then show that
z
=0
v
 2
y 2
=
 [ x 2  ry 2 ]
r3
14) If F=F(x,y) and If x+y = 2eθcos(Ф) , x-y= 2ieθsin(Ф) then prove that
F
F
F
i
 2y


y
15) State and Prove Euler’s Theorem on Homogeneous Functions of Three
Independent Variables.
16) If θ= t e
n
r 2
4t
17) If u = f(r) and x = rcoθ , y = rsinθ , then
2 
18) If
1  2 

[r
]
2
r
t
r r
P.T.  2 u  F ' ' (r )  1 F ' (r ) ,
r
then find the value of ‘n’ for which
where
2
2

x 2 y 2
x2 y2 z2


 1, and lx+my+nz=0
a2 b2 c2
dx
dy
dz


ny mz lz nx mx ly



b2 c2
c2 a2
a2 b2
then prove that
TUTORIAL NO -03
[Applications of Partial Derivatives, Jacobian]
1] Show that the minimum value of u =xy+a 3 ( 1 + 1 ) is 3a 2
x
y
2] Find the dimensions of the rectangular box with open top of maximum
capacity whose surface area is 432 cm2
3] Find maximum value of xm yn zp where x + y +z = a
4] Find the minimum and maximum distances from the origin to the
Curve 3x2+4xy+6y2=140
5] Divide 24 into three parts such that the continued product of the first,
square of the second and cube of the third may be maximum.
6] Determine the point on paraboloid z = x2+y2 which is closest to the
Point ( 3, -6, 4)
Applied Mathematics-I
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RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
7] Find the minimum distance of any point from the origin on the plane
x +2y +3z = 14.
8] Find the minimum and maximum distance of the point (3,4,12) from
The Sphere x2 + y2+ z2 = 1
9] Examine the minimum and maximum on the surface u  xy  a3 ( 1 x  1 y)
10] Examine the minimum and maximum on the surface
z  sin x. sin y. sin( x  y )
11] A rectangular box , open at top is to have a volume of 108 m3. Find
the dimensions of the box so that the surface area is minimum.
 (u , v )
12] Find
if
 ( x, y )
1) u = x siny, v = y sinx, 2)
3)
13]
u  e x sin y, v  x  log sin y
x y
, v  tan 1 x  tan 1 y , 4) u = x + y,
1  xy
Find  (u, v) = J and  ( x, y ) = J’ for
 ( x, y )
 (u , v )
u
1)
14] Find
15] Find
16] find
u  x
y2
y2
,v 
x
x
y = uv.
u
u
, 2) x  e cos v, y  e sin v
( x, y, z )
2 yz
3xz
4 xy
for..u 
..v 
..w 
(u, v, w)
x
y
z
(u, v)
(r , )
if u = 2axy, v = a (x2 + y2) and x = r cos ө , y = r sin ө.
 ( x, y , z )
for.. x  vw ,.. y  uw ,.. z  uv ,.
 (u , v, w)
and
u = r sinө cosφ, v = r sinө sinφ , w = sinө cosφ
TUTORIAL NO -04
[Indeterminate Forms,Expansions of functions Fitting of curves]
A] Evaluate the following limits:
1)
lim
x 0
x  sin x cos x
x3
2)
tan x  x
x 0 x  sin x
lim
3)
e x sin x  x  x 2
x 0 x 2  x log( 1  x )
lim
Applied Mathematics-I
4)
lim
x 0
x 2  2 cos x  2
x sin 3 x
4
RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
5)
6)
7)
8)
ae x  b cos x  ce  x
2
x 0
x sin x
 [a  b cos  ]  c sin 
1
Find the constants a, b, c if lim
 0
5
sin 2 x  p sin x
Find value of ‘p’ if lim
is finite.
x 0
x3
1
1
x
1
lim [ x  ][log( x  )  log x]
Evaluate lim
9)
Evaluate
[

]
x 
x 1 x  1
2
2
log x
Find values of a, b, c such that
10) Find a, b if
12)
Prove that
lim [
x 0
lim
cos x
11) Evaluate lim [tan x]
a cot x b
1
 2]
x
3
x
lim (1  x 2 )
1
log(1 x )
x 1
x
e
13)
2
tan
Evaluate lim [cos ecx]
x
14)
Evaluate
16)
1 a
x x a
[ (

)]
Evaluate lim
x a 2
x
a
18)
Evaluate
x
2
1
2 cosh x  2 1 x 2
lim [
]
x 0
x2
2
1
1
1
1x  2 x  3 x  4 x 4x
lim [
]
x 
4
15)
Evaluate
17)
x tan( 2 a )
lim
(
2

)
Evaluate x a
a
x
1
1x  2 x  3 x  4 x x
lim [
]
x 0
4
1
B]
1) Prove that e ax cosbx = 1+ ax + (a
2) Prove that e x sin x  1  x 2 
3) Show that log(secx)=
4)
5)
6)
7)
8)
2
 b 2) 2 a(a 2  3b 2 ) 3
x 
x 
2!
3!
x4
x6


3 120
1 2 1 4 1 6
x  x 
x 
2
12
45
x2 5 4
P.T sin[e -1]= x   x    
2 24
P.T. log[1+e x ]= log 2  1 x  1 x 2  1 x 4    
2
8
192
3
P.T. sin 1 x = x+ x  3 x 5  5 x 7    
6 40
112
3
5
S.T. cos 1 [tanh(logx)]=  2[ x  x  x    ]
3
5
Expand x x in powers of x and Hence show
e 1
x ex 1
x2
x4
[ x ]  1


2 e 1
12 720
x
Applied Mathematics-I
that
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RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
9) Expand log[ 1  x  x 2  x 3 ] upto term in x 8
10) If y =
x2 x3
x


2! 3!
then prove that x=
xcosecx = 1  x
11) Show that
2
12

y2 y3 y4
y



2
3
4
7 4
x 
360
12) Express 2x +3x -8x+7 in terms of (x-2) by using Taylor’s Theorem.
13) By Using Taylor’s Theorem arrange the function in Powers of x
[7+(x+2) +3(x+2) 3 +(x+2) 4 -(x+2) 5 ]
14)Calculate the value of 10 to four decimal places using Taylor’s
Theorem
1
15) Use TSE to calculate approximately (63.7) 3
16) Use TSE to calculate approximately (2.98) 3
3
2
C) 1) Fit a straight line for given value.
x
12
15
21
25
y
50
70
100 120
2) Fit a straight line for given value.
Year
Production
in ton
1941
8
1951
10
1961
12
1971
10
1981
16
Find expected production in 2021.
3) Fit a straight line for given value
x 0 1 3 6 8
Y 1 3 2 5 4
Find y(15)
D] Fit a Parabolic equation for the data given below
1) Find y(10)
x
y
1
2
2
6
3
7
4
8
5
10
6
11
Applied Mathematics-I
7
11
8
10
9
9
6
RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
2) Fit x=a+by+cy2
X
Y
-3
4.63
-2
2.11
-1
0.67
0
0.09
1
0.63
2
2.15
3
4.58
Find x(2)
3) Using least squarer method fit y=a+bx+cx2 for (-1,2) (0,0) (0,1) (1,2)
E) 1) Fit x=aby for
Y
X
2
144
3
172.8
4
207.4
5
248.6
6
298.5
2) Fit y=abx for the data. Find y at 25.
X
Y
1
98.2
2
91.7
5
81.3
10
64.0
20
36.4
30
32.6
40
17.1
50
11.3
3) Fit y=aebx for
X
Y
0
0.5
1
3.69
2
27.3
3
201.7
TUTORIAL NO -05
[Matrices]
1) Find the rank of the matrix using Row-echelon form of following
matrix.
i)
2 3 1 4 


A  5 2 3 0
9 8 0 8


ii)
6

4
A
10

16
6

1
3 9 7

4 12 13
1
2
3
6
2) Reduce the matrix to normal form and find it’s rank
i)
1 1 0


A  2 2 0
0 1 0


ii)
1 1

4 1
A
0 3

0 1
2 3

0 2
1 4

0 2
3) Find the non-singular matrix P and Q such that PAQ is in normal form,
Applied Mathematics-I
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RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
i)
1 2 3 4 


A  2 1
4 3 
3 0 5  10


ii)
1 1 1 


A  1  1  1
3 1 1 


4) Solve the following equations
i) x+2y+3z=0; 2x+3y+z=0; 4x+5y+4z=0; x+2y-2z=0
ii) x+y+2z=0; x+3y+4z=0; x+2y+3z=0; 3x+4y+7z=0
iii) 2x-y+3z=0; 3x+2y+z=0; x-4y+5z=0
iv) 3x+y-5z=0; 5x+3y-6z=0; x+y-2z=0; x-5y+z=0
5) Examine whether the following vectors are linearly dependent or
independent
i) [2,1,1] , [1,3,1], [1,2,-1]
ii) [3,1,-4], [2,2,-3], [0,-4,1]
6) Test the following equations for consistency
i) x+2y-z=1; x+y+2z=9; 2x+y-z=2
ii) x-3y-3z=-10; 3x+y-4z=0; 2x+5y+6z=13
iii) 6x+y+z=-4; 2x-3y-z=0; -x-7y-2z=7
7) Find the values of λ for which the system of equations x+y+4z=1;
X+2y-2z=1; λx+y+z=1.
8) Solve the following equations by Gauss elimination method;
i) x+y+z=2; 2x+2y-z=1; 3x+4y+z=9
ii) 2x+y+z=10; 3x+2y+3z=18; x+4y+9z=16
9) Solve the following equations by Gauss-Jorden method,
i) 3x+2y-2z=4; x-2y+3z=6; 2x+3y+4z=15
ii) X+2y+z=8; 2x+3y+4z=20; 4x+3y+2x=16
10) Solve the equations by using Gauss-Seidel method,
i) 15x+y+z=17; 2x+15y+z=18; x+2y+15z=18
ii) 10x+2y+z=9; 2x+20x-2z=-44; -2x+3y+10z=22
TUTORIAL NO -06
[Complex Numbers]
Part I
1) Express the complex numbers in the form x+iy
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RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
i)
( 3  i)(2  i 3)
ii)
1 i
1 i
iii)
(1  i) 2 (1  i )
(1  i) 2 (2  i )
2) Find modulus and amplitude of
i)
i9  i6  2
3  i 20  i 5
ii)
(1  i )(1  i 3)
(1  i 3 ) 3 (1  i) 2
iii)
3 i
( 3  i)
3) Prove that the statements ‘ Re(z)>0’ and z  1  z  1 are equivalent
1  2i
(3  4i)( 2  11i )
ii)
4) Express in polar form i)1-cosα+isinα
5) If α & β are roots of x2-2x+2=0 then show that
6) If z=cosθ + isinθ , show that
n
1
 n   n  2 2 . cos(
z

 1  i tan
1 z
2
n
)
4
7) If z1, z2 are two complex numbers then show that
2
z1  z 2  z1  z 2
2

2
 2 z1  z 2
2

8) If x and y are real, solve the equation
iy
3 y  4i

0
ix  1 3x  y
9) If (1+cosθ+isinθ)(1+cos2θ=isin2θ)=u+iv prove that
i) u 2  v 2  16 cos 2  cos 2  ii) v  tan 3 
2
u
2
10) If sinθ+sinФ=0 & cosθ+cosФ=0 then prove
cos2θ+cos2Ф= 2cos(Π+θ+Ф) & sin2θ+sin2Ф = 2 sin(Π+θ+Ф)
11) Use De`moivre`s Theorem to solve equation x 4 -x 2 +1=0
12) solve x10  11x 5  10  0
13) Express cos 5 θ in term of multiple of θ
14) Express sin 7 θ in term of multiple of θ
15) Express sin 7 in powers of sinθ only
16) Show that
17) Show that
sin 
sin 5
 16 cos 4   12 cos 2   1
sin 
sin 
sin 2
sin 3


     c. sin(tan  )
2
cos  2!cos  3!cos 3 
4
18) Simplify
(cos 3  i sin 3 ) 4 (cos 5  i sin 5 ) 5
3
3 2
4
4 10
(cos( )  i sin( )) a (cos
 i sin
)
2
2
5
5
Part II
Applied Mathematics-I
9
RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
1] Find tanhx if 5sinhx-coshx=5
2] S.T [1  tanh  ]3 = cosh6θ+sinh6θ
1  tanh 
3] If y=log(tanx) P.T i) sinhny =
4]
5]
1
[tan n x  cot n x ]
2
ii) 2coshnycosec2x = cosh(n+1)y+cosh(n-1)y
tan  ii) coshu = secθ
If u= log tan[    ] P.T i) tanh y =
4 2
2
n


i

S.T tan 1 (e i )  [  ]  log[tan(   )]
2
4
2
4 2
2
6] If tan(α+iβ)=x+iy show that i) x2+y2+2xcot(2α)=1
ii) x2+y2-2ycoth(2β)+1=0
7] P.T i) cos 1 z  i log[ z  z 2  1] ii) sinh 1 (tanx) = log tan[   x ]
8] P.T
4
sin 1 (cosecθ)=   i log(cot  )
2
2
2
9] If log[ cos(x-iy)]= α+iβ , P.T α= 1 log[ cosh 2 y  cos 2 x and find β.
2
2
10] Separate into real & imaginary part of ii
11] S.T log[ sin( x  iy ) ]  2i tan 1 (cot x. tanh y)
i
sin( x  iy )
12] P.T e 2 ai cot 1 (b)[
bi  1 n
] 1
bi  1
13] S.T i) sin(log i i )=1 ii) cos(log i i )=0
14] If i
i i ...
   i Then S.T (α +β )= e
2
15] P.T
x  iy
2 xy
tan[ i log{
)]  2
x  iy
x  y2
16] P.T
1 1 z2
cos ech z  log[
],
z
1
2
 ( 4 m 1) 
and tan(   )= 
is it defined for all values of z
Tutorial 7
Topic: Scilab
//Programme for Crout's Method ( Decomposition Method )
Applied Mathematics-I
10
2

RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
//Name:
//Roll No.:
//Batch: And Div:
clc
A=input('Enter 3*3 coefficient matrix A:')
l(1,1)=A(1,1);
l(1,2)=0;
l(1,3)=0;
u(1,1)=1;
u(1,2)=A(1,2)/l(1,1);
u(1,3)=A(1,3)/l(1,1);
u(2,1)=0;
u(2,2)=1;
u(3,1)=0;
u(3,2)=0;
u(3,3)=1;
l(2,1)=A(2,1);
l(2,2)=A(2,2)-(l(2,1)*u(1,2));
l(2,3)=0;
l(3,1)=A(3,1);
u(2,3)=(A(2,3)-l(2,1)*u(1,3))/l(2,2);
l(3,2)=A(3,2)-l(3,1)*u(1,2);
l(3,3)=A(3,3)-(l(3,1)*u(1,3)+l(3,2)*u(2,3));
L=l;
U=u;
B=input('Enter 3*1 constants matrix B:')
v=L\B;
x=U\v;
disp('A')
disp(A)
disp('B')
disp(B)
disp('Lower triangular Matrix L is')
disp(L)
disp('Upper triangular Matrix U is')
disp(U)
disp('By Crout Method,')
disp('Ans')
disp(x)
Applied Mathematics-I
11
RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
Tutorial 8
Topic: Scilab
//Programme for Cuvrve fitting 1 (line)
//Name:
//Roll No.:
//Batch: And Div:
//Fitting a straight line y= a + bx
clc
x=input('Enter the values of x:')
y=input('Enter the values of y:')
n=length(x)
sx=sum(x)
sx2=sum(x^2)
sy=sum(y)
sxy=sum(x.*y)
A=[n sx;sx sx2]
B=[sy;sxy]
c=linsolve(A,-B)
disp(c)
//Programme for Curve fitting 2( Parabola)
//Name:
//Roll No.:
//Batch: And Div:
//Fitting a parabola y = a + bx +c x^2
clc
x=input('Enter the values of x:')
y=input('Enter the values of y:')
n=length(x)
sx=sum(x)
sx2=sum(x^2)
sy=sum(y)
sxy=sum(x.*y)
sx3=sum(x^3)
sx4=sum(x^4)
sx2y=sum((x^2).*y)
A=[n sx sx2;sx sx2 sx3; sx2 sx3 sx4]
Tutorial 9
Applied Mathematics-I
12
RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
Topic: Scilab
// Programme for Gauss Joran Method
clc
A=input('Enter 3*3 coefficient matrix A:')
B=input('Enter 3*1 constants matrix B:')
A_Aug=[A,B]
a=rref(A_Aug)
disp('A')
disp(A)
disp('B')
disp(B)
disp('Agmented Matrix [A:B] is')
disp(A_Aug)
disp('By Gauss Jordan Ellimination,')
disp(a)
disp('Ans')
disp(a(1:3,4))
Tutorial 10
Topic: Scilab
//Programme for Gauss Jacobi Method
//Name:
//Roll No.:
//Batch: And Div:
clc
A=input('Enter 3*3 coefficient matrix A:')
B=input('Enter matrix B:')
i=input('Enter initial values')
for j=1:7
x1=(B(1)-(A(1,2)*i(2))-(A(1,3)*i(3)))/A(1,1);
x2=(B(2)-(A(2,1)*i(1))-(A(2,3)*i(3)))/A(2,2);
x3=(B(3)-(A(3,1)*i(1))-(A(3,2)*i(2)))/A(3,3);
i(1)=x1;
i(2)=x2;
i(3)=x3;
b=i;
disp(b)
end
//Programme for Gauss Seidal Method
//Name:
Applied Mathematics-I
13
RAMRAO ADIK INDTITUTE OF TECHNOLOGY
NERUL, NAVI MUMBAI
//Roll No.:
//Batch: And Div:
clc
A=input('Enter 3*3 coefficient matrix A:')
B=input('Enter matrix B:')
i=input('Enter initial values')
for j=1:5
x1=(B(1)-(A(1,2)*i(2))-(A(1,3)*i(3)))/A(1,1);
i(1)=x1;
x2=(B(2)-(A(2,1)*i(1))-(A(2,3)*i(3)))/A(2,2);
i(2)=x2;
x3=(B(3)-(A(3,1)*i(1))-(A(3,2)*i(2)))/A(3,3);
i(3)=x3;
b=i;
disp(b)
end
Applied Mathematics-I
14