Subject: Mathematics (1A1)
Unit: First
Successive Differentiation
n th Derivative:
1)
If y
1
' Then find yn
x 4x 3
2
2) find the nth derivative at y
1
x a2
2
x
( x 1)( x 2)( x 3)
2x
Find nth differential coefficient of tan 1
1 x 2
x
Find nth differential coefficient of 2
x a2
Find nth differential coefficient of
y
y
y
If y=(x-1)n ,Prove that y y1 2 3 n x n
2! 3!
n!
x
If y e (sin x cos x) ,then prove that
3) Find nth differential coefficient of
4)
5)
6)
7)
8)
yn 2
n 1
2
e x sin{ x
(n 1)
}
4
9) If y e x cos x cos 2 x , find yn.
10) If y sin px cos px , then Prove that
y n p n {1 (1) n sin 2 px}1 / 2
# Leibnitzs theorem:
1
m
1
m
1) If y y 2 x ,then prove that
( x 2 1) y n 2 (2n 1) xyn 1 (n 2 m 2 ) y n 0.
1
12) If y e tan x ,show that
i) (1 x 2 ) y 2 (2 x 1) y1 0.
ii) (1 x 2 ) y n 2 [2(n 1) x 1] y n 1 n(n 1) y n 0
13) If y sin 1 x , then prove that
i) (1 x 2 ) y 2 xy1 0.
ii) (1 x 2 ) y n 2 (2n 1) xyn 1 n 2 y n 0
14) If y ( x x 2 a 2 ) m ,prove that ( x 2 a 2 ) y 2 xy1 m 2 y 0.
1
Differentiate this equation n-time and obtain
( x 2 a 2 ) y n 2 (2n 1) xyn 1 (n 2 m 2 ) y n 0.
15) If (sin 1 x) 2 a 0 a1 x a 2 x 2 a 3 x 3
Show that (n 1)( n 2)a n 2 n 2 a n .
16) If f ( x) tan x then prove that
f n (0) c c 2 f
n2
(0) n c 4 f
n4
(0) sin
n
.
2
17) If x sin and y sin 2 show that
2
dy
2 d y
(1 x ) 2 x 4 y 0.
dx
dx
Hence deduce that (1 x 2 ) y n 2 (2n 1) xyn 1 (n 2 4) y n 0.
dn n
x log x , Prove that
dx n
I n nI n 1 n 1! and hence show that
18) If I n
1 1
1
I n n! log x 1
2 3
n
1
19) If y sin m sin x ,Prove that
(1 x 2 ) y n 2 (2n 1) xyn 1 (n 2 m 2 ) y n 0.
20) If y a coslog x b sin log x , Show that
x 2 y n 2 (2n 1) xyn 1 (n 2 1) y n 0 .
n
y
x
21) If cos log , Prove that
n
b
x 2 y n 2 (2n 1) xyn 1 2n 2 y n 0
1
Taylor’s And Maclaurin’s Seris
22) prove that
h2
h 3 cos x
cos ec 2 x
2
3 sin 3 x
x in power of (x-1)
log sin( x h) log sin x h cot x
23) obtain the expansion of tan 1
(m 1) 2 2 ''
x f ( x)
2!
25) If x 3 2 xy 2 y 3 x 1.expand y in ascending power of x.
24) prove that f (mx) f ( x) (m 1) xf ' ( x)
2
26) If
y
sin
1
sin 1 x
1 x
x
1 x2
2
x
,show that (1 x 2 ) y ' xy 1 ,hence show that
2 3 8 5
x x
3
15
x2 x3 x4
,prove that
2! 3! 4!
1
1
1
x y y2 y3 y4
2
3
4
3
2
28) Given f ( x) x 8x 15x 24, find f(11/10) by using Taylor’s
Theorem.
29) Arrange 7 ( x 2) 3( x 2) 3 ( x 2) 4 ( x 2) 5 . in power of x by
Using Taylor’s Theorem.
30) Given y 3 y x 0, show that x being small
27) If
y x
y x x 3 3x 5
1
2
5
31) show that e 1 sin sin 2 sin 3 sin 4
2!
3!
4!
x 3 '''
x5 v
32) prove that [ f (a 2 x) f (a)] 2{xf ' (a x)
f (a x)
f (a x) }
3!
5!
1
26 3
x
33) If x 3 y 3 xy 1 0 ,prove that y 1 x
3
81
34) If x (1 y )(1 2 y ), expand y in ascending power of x.
35) prove that
du x 2
d 2u
x4
d 3u
x6
x
f
u
dx 1 x dx 2 2! (1 x) 2 dx 3 3! (1 x) 3
1 x
Where u=f(x)
36) obtain the expansion of
1) sinx
2)cosx
In power of x
4
#
Indeterminate Form
3
log( 1 x 2 )
37) Find lim
log cos x
x 0
38) Find
lim
x y
xy yx
xx yy
xx x
39) Find lim
x 1 x 1 log x
cosh x cos x
40 Find lim
x sin x
x 0
x
41) Find lim log 2 cot( x a )
a
x 0
tan 2 x
42) Find lim log tan x
x 0
log
2
43) Find lim
tan
2
44) Find
a
x
lim x cot a
x 0
45) Find
lim (1 x
2
)
1
log(1 x )
x 1
1
1
1x
a1 a 2x a nx
46) Find lim
n
x
nx
47) Determine a,b,c so that as 0, the expression
(a b cos ) c sin
.shall tends to unity.
5
1
x
48) Find
lim
x 0
(1 x) e
x
1
x
sin x
49) Find lim
x
x 0
sin 2 x p sin x
50) If lim
is finite .find the value of p and the limit.
x3
x 0
x(1 a cos x) b sin x
51) Find the value of a and b such that lim
1
x3
x 0
4
52) Find
lim 1 sin x
x 0
lim
53) Find
x 0
cot x
1 x cot x
x
n2
54) Find lim cos
n
n
55) Find the value of a,b such that
3
2
lim x sin 3x ax b 0
x 0
56) Prove that
lim (cos ecx)
x
57) Prove that
lim
tan x sin x 1
2
sin 3 x
x
58) Prove that lim 2
a
n 0
lim a
n
e
1
2
2
x 0
59) Evaluate
tan 2 x
x
x
tan
x
2a
2
e
1
x
5
Answer
1) y n
2) y
n
1n n!
1
1
n 1
n 1
x 1
x 3
2
n
1 n! n 1
sin sin n 1
n2
where tan 1 a
x
a
n
1 n!
1
2
3
n
3) y
1 2x 1n 1 x 2n 1 2x 3n 1
4) y n 2 1
5) y n 1
6)
yn
n 1
n 1
(n 1)! sin n sin n where r x 2 a 2 & tan 1 a
(n 1)! sin n sin( n 1) where cot 1 x
1
n
n! cos( n 1)
where r x 2 1 & tan 1 1
n 1
x
r
7)
8)
9)
n
n
1 x 2
n
1
)
9) e 10 cos(3x n tan 3) 2 2 cos( x
2
4
23) tan
1
x
4
x 1 ( x 1) 2
2
4
x 2
25) y 1 x
2! 3
28)3.511
29) 17 11x 38 x 2 29 x 3 9 x 4 x 5
34) y 1 x 2 x 2
2
Indeterminate Forms
37) 2
1 log y
38)
1 log y
39) 2
40) 1
1
41)
a
43) 1
6
x
44) 0
45) 0
46) 0
47) a1 a 2 a n
48) a =120,b=60,c=180
e
49)
2
51)1
5
3
53) a =
,b=
2
2
54)0
55)0
56) e
1
2
57) a=-3,b=
9
2
61) ae
7
\Subject: Mathematics (1A1)
Unit: VI
Differential Equation Of The First Order And Of a Degree Higher Than The First.
# Solvable For p:
Solve the Following.
1) xyp 2 p(3x 2 2 y 2 ) 6 xy 0.
Ans: [( y cx 2 )( y 2 3x 2 c].
2) p 1p
x y
y x
Ans: xy c x 2 y 2 c 0
3) p 3 ( x 2 xy y 2 ) p 2 ( x 3 y x 2 y 2 xy3 ) p x 3 y 3 0
Ans: y ce x 2 3 y x 3 c cy xy 1 0.
2
4) 3 p y 2 xyp 4 y x 2 0.
Ans: 3 x 2 y 2 4cx c 2 0.
2
2
2
5) p 2 py cot x y
Ans: y sin 2 x c y cos 2 c 0.
2
6) p p y xx y
2
2
x2
Ans: y
c y x ce x 1 0.
2
# Solvable for y:
7) 16 x 2 2 p 2 y p 3 x 0.
Ans: 16 x 2 2c 2 x 2 y c 3 x 4 0.
8) y 1 p x e p .
1
Ans: y c1 p e p 1 p e p .
2
2
9) xp 2 yp ax 0.
Ans: 2cy c 2 x 2 c.
10) p 3 mp 2 a( y mx).
3
Ans: ax c p 2 mp m 2 log( p m). with the given relation.
2
2
2
11) 4 y x p .
x
c .with the given relation.
Ans: log( p x)
px
# Solvable for x:
8
12) x
p
1 p 2
a.
Ans: x a y c 1.
13) p 3 2 xyp 4 y 2 0.
Ans: 16 y c 2 (2 x c 2 ) 2 .
14) 4 x py( p 2 3).
2
Ans: y c( p 2 4)
2
9
10
3
( p 2 1) 5 .
p
.
15) p tan x
2
1 p
1 c y
Ans:
tan x (c y) (c y) 2 .
c y
16) ayp 2 (2 x b) p y 0.
Ans: ac 2 c(2 x b) y 2 0.
# Clairauts Form Of The Equation:
17) y px 4 p 2
Ans: y cx 4 c 2 .
18) p 2 ( x 2 1) 2 pxy ( y 2 1) 0.
Ans: ( y cx) 2 1 c 2 .
m
19) y px.
p
m
Ans: y cx.
c
dy
dy
e dx .
20) y x
dx
Ans: y cx e c .
a
21) y p ( x b) .
p
a
Ans: y c( x b) .
c
22) sin( px y ) p.
Ans: y cx sin 1 c.
# Method Of Substitution: (Reducible To Clairauts)
23) Solve y 3 px 6 y 2 p 2 .
2
Ans y 3 cx c 2 .
3
9
24) Solve y xp x 4 p 2 .
c
Ans: y c 2 .
x
4x
25) Solve e ( p 1) e 2 y p 2 0.
Ans: e 2 y ce 2 x c 2 .
26) Solve e 3 x ( p 1) p 3 e 2 y 0.
Ans: e y ce x c 3 .
27) Solve y 2 px sin( xp 2 ).
Ans: y 2c x sin c 3 .
28) Solve y 2 px tan 1 ( xp 2 ).
Ans: y 2c x tan 1 c 2 .
# Application :
Find Orthogonal Trajectories Of Curves.
29) r n a n sin n .
Ans: r n b n cos n .
30) r a(1 cos ) .
Ans: r b(1 cos ).
l
31) 1 cos . Where l is the parameter.
r
b
Ans: 1 cos .
r
2
32) y 4ax.
Ans: y 2 2 x 2 b.
33) x 2 4 y 2 c 2 .
Ans: y bx 4 .
34) 2 x 2 y 2 kx.
Ans: x 2 y 2 log cy.
35)Show that the family of confocal parabolas y 2 4a( x a). Is Self Orthogonal.
36) Find Orthogonal Trajectories Of
y2
x2
1. being parameter.
a2 b2
Ans: (Self Orthogonal)
37) The current in a circuit containing an inductance L Resistance R and Voltage E Sinwt is given
di
by L Ri E sin wt , if i=0 at t=0 show that
dt
10
Lw
} .Where tan 1
.
R
R L w
38) A Condensor of Capacity C is Charged through a Resistance R by steady
dQ Q
V .
Voltage V.prove that the charge Q on the plate is given R
dt C
i
E
2
2
2
{sin( wt ) sin e
Rt
L
t
Hence show that if Q=0 at P=0 Q CV (1 e RC ) and find the current flow into the plate.If
C= 5 10 5 farads,V=2000 volts and R=200 ohm.calculate the
Current at the instant’s of closing the switch and after 0.05 sec.
t
V
Ans:- i e RC
R
39)the equation of electromotive force in terms of the current i for an electrical circuit having
resistance R and an condensor of capacity C in
i
Series is E Ri dt .find the current i when E E m sin wt
c
Ans:-
i Ke
wCE m
t
RC 1 R 2 C 2 w 2
sin( wt )
40) When a current is clised in a circuit containing a battery E, a resistance R and an inductance L,
di
the current I builds up nat a given by L Ri E
dt
Find I as a function of t. How long will it be ,before the current has reached one-half of it’s final
value if E=6 volts R= 100 Ohm and L=0.1 henry
Ans(0.0006931sec)
41) when a resistance R Ohm is connected in series with an inductance L henries with e.m.f. of E
di
volts , the current I amperes at time t is given by L Ri E . If E = 10sint Volts and I = 0 when
dt
t=0 ,Find I as a function of t.
Rt
10
L
R
sin
t
L
cos
t
Le
Ans: 2
2
L R
11
Subject: Mathematics (1B1)
Unit: VI
Triple Integration and it’s Application
# Evaluate:
log 2 x x y
e
1)
0
0
2a
2)
x y z
dxdydz .
Ans: 5/8.
0
x
x
dx dy ( xyz)dz .
0
0
Ans:
y
4 6
a
3
1 1 x x y
3)
e
0 0
4)
dxdydz .
Ans:
0
a 2 er 2
2 a sin
0
0
2 4 x
5)
z
Ans:
5a 3
64
Ans:
43
3
a
rdrddz .
0
3
e
x y z
dxdydz .
x 3x
y
2
1 z x z
0
6)
( x y z)dxdydz .
1
2
Ans:0
1 0 x z
1 1 1 x
7)
xdzdxdy.
Ans:
0 y2 0
8)
dx dy
0
0
0
dz
.
2
(1 x y 2 z 2 )
9) Evaluate the integral
Ans:
4
35
2
8
x 2 y 2 dxdydz, where V is the volume bounded
v
by the surface x 2 y 2 z 2 , z 0 and the plane z = 1.
Ans :(
10) Change to polar coordinate and evaluate:
0
1 x 2
1 x1 y 2
1
0
0
2
8
dx.dy.dz
Ans:-
1 x 2 y 2 z 2
11)Evaluate
a
a2 x2
0
0
aq x2 y2
a 5
10
Ans:-
( x 2 y 2 z 2 )dxdydz
0
12
)
6
12) Evaluate :-
z
2
dxdydz taken through out the volume common to the sphere
2a 5
15
Ans:-
x 2 y 2 z 2 = a 2 and the cylinder x 2 y 2 ax
13)show that
dxdydz
( x y z 1)
3
=
1
5
log 2 ,integration being taken through
2
8
Out the volume of tetraheadron bounded by the co-ordinate planes and the
plane x+y+z=1.
# Application of triple integration:14)Find the volume bounded by cylinder y 2 x and x 2 y and the planes
11
30
Z=0 and x+y+z=2
Ans:-
15)Find the volume in the first octant bounded by
x 2 y 2 2, z x y, y x, z 0 and x=0.
1
8
Ans:- ( n 1)
16)Find the volume common to right circular cylinders
16 3
a
3
x2 y2 a2 , x2 z 2 a2
Ans:-
17)Find the volume cut off from the paraboloid
x2
Ans:-
1 2
y z 1 by the plane z=0.
4
#Mean And Root Mean Square Value
18) Find the mean value of xy over the area
y
x2
Of positive quadrant of the ellipse 2 2
a
b
2
1
ab
2
Ans:-
19) Find the mean height of portion of the parabola y ( x 2)(3 x)
Which lies above the x-axis
20) Find the mean value of
e ( x
Ans:2
y2 )
1
6
over the area within the circle
1
e
Ans:- 1
x2 y2 1
13
21) Find the mean value of xyz over the positive octant of the ellipsoid
abc
8
x2 y2 z2
1
a2 b2 c2
Ans:-
22) Find the mena value and RMS value of ordinate y of the cycloid
x a( sin ), y a(1 cos ) over the range to .
Hence show that value of later is
2 times that of the former.
a a
Ans:- ,
2 2
23) Find the mean value r 2 over the area of cardiode r a (1 cos )
35 2
a
24
Ans:-
24) A rod of length a is divided into three parts at random.
Find mean value of sum of squares of three parts.
1
Ans:- a 2
2
25) Find the r.m.s values of the expression asinpt+bcosqt over the interval
a2 b2
0 to 2 .
Ans:-
2
26) prove that mean distance of points within a cicular area of radius
32a
a from fixed point on cicumference is
.
9
27) If voltage in an electrical circuit is V volts the current I ampere ,self
dI
Inductance L henry and the resistance R ohm , then L +RI=V,
dt
Where t is time in second.show that if I I 0 sin pt , where I 0 , ,p are
value of power P is
1 2
I0 R .
2
14
constants.the mean
Subject: Mathematics (1B1)
Unit: I
Matrices
Q.1) Find the inverse by method of partitioning.
1 3 3
7 3 3
i) 1 4 3
Ans:- 1 1
0
1 3 4
1 0
1
2 1
1
2
ii) 1 1
2 1 1
3 1 5
Ans:- 5 3 1
1 5
3
1
1
iii)
1
1
4
1
4 6
6 14 11 5
Ans:-
4 11 10 3
3 1
1 .3
1
2 3 4
3 6 10
4 10 20
1
1
1
3
1
iv) 1
3 3
2 4 4
2 1 1
v) 1 3 2
1 2 1
6
12 4
1
Ans:- 5 1 3
4
1 1 1
3 5
1
1
Ans:3 1 5
10
5 5 5
Q.2) Find the rank of the following Matrix.
1
2
i)
3
2
2
1
ii)
3
6
2
5 1
2
8 5
2
11 1 6
3 1 1
1 1 4
1 3 2
3 0 7
2
3
Ans:- 2
Ans:-3
15
1 2 1
iii) 1 0 2
2 3 3
4
3
2
4
iv)
1 2
1 1
1
3 6
6 4
2 3
Ans:-3
1
2 4 6
v) 1 2 3
3 6 9
Ans:-4
Ans:-1
0
1
vi)
3
1
1 3 1
0 1
1
1 0
2
1 2 0
Ans:-2
0
0
vii)
3
1
1 3 1
0 1
1
1 0
2
1 2 0
Ans:-3
1
1
viii)
2
1
2 2 3 1
3 2 3 0
4 3 6 4
1 1 4 6
Ans:-4
1 1 2
ix) 1 2 3
Ans:-3
1 1 1
Q.3) Find the eigen values and eigen vecters of the following matrix.
1 2 1
i) 1 2 1
1 1 0
2,2,1
3
1
3
Ans:-
1 , 2 , 1
4 1 1 2 1 2
16
2 2 3
ii) 2
1 6
1 2 0
1 2 3
iii) 2 4 6
3 6 9
6 2 2
iv) 2 3 1
2 1 3
2 1 1
v) 2 3 2
3 3 4
6
6
4
vi) 1
3
2
1 4 3
2 1 1
vii) 1 2 1
1 1 2
2 2 3
viii) 1 1
1
1 3 1
3,3,5
0
3
1
Ans:-
, 0
, 2
3
2 3 1 3 1 5
0,0,14
2 3 1
Ans:-
1 , 6 , 2
0 5 3
8,8,2
2 1 1
Ans:-
1, 0 , 2
1 2 0
1,1,7
0 1 1
Ans:-
1 , 0 , 2
1 1 3
1,1,4
6 0 3
Ans:-
2 , 1 , 1
7 1 1
1,4,1
1 1 1
Ans:-
1, 1, 1
0 1 2
2,1,3
11 1 1
Ans:-
1 , 1 , 1
14 1 1
17
9 4 4
ix) 8 3 4
16 8 7
8 8 2
x) 4 3 2
3 4 1
1 6 4
2
xi) 0 7
0 4 3
8 6 2
xii) 6 7 4
2 4 3
4 2 2
xiii) 5 3 2
2 4 1
1,1,3
0 1 1
Ans:-
1 , 1, 1
1 1 2
1,2,3
4
3
2
Ans:-
3 , 2 , 1
2 1 1 2 1 3
1,1,3
1
2
2
Ans:-
2 , 2 , 1
3 1 3 1 2 3
8,1,1
2
0
1
Ans:-
, 0
1 , 2
3 8 1 1 1 1
1,2,5
2
1
0
Ans:-
1 , 1 , 1
4 1 2 2 1 5
Q.4) Solve the following system of equation if consistent.
i)
x + 2y + 2z = 1
2x +2y + 3z = 3
x – y + 3z = 5
ii) x + y +z = 3
x + 2y +3z = 4
x + 4y + 9z = 6
Ans:- x = 1, y = -1, z = 1
Ans:- x = 2, y = 1, z = 0
x1 2 x 2 3x 3 4 x 4 0
iii) x1 x 2 3x 3 x 4 0
3x1 x 2 2 x 3 3x 4 0
Ans:- x1
1
2
2
c, x 2 c, x 3
c, x 4 c
3
3
3
18
2 x1 3x 2 4 x3 11
iv) x1 5 x 2 7 x3 15
Ans:- x1 2, x 2 3, x3 4,
3x1 11x 2 13x3 25
Q.5) Determine for what value of & the following system of equation
Having
i)
Unique solution
ii)
No Solution
iii)
Infinite no. of solution
2 x1 3x 2 5 x 3 9
a)
7 x1 2 x 2 2 x 3 8
2 x1 3x 2 x 3
b) x +y +z = 6
x + 2y +3z =10
x + 2y + z =
Ans:- i) 5 & may have any value
ii) 5 & 9
iii) 5 & 9
Ans:- i) 3 & may have any value
ii) 3 & 10
iii) 3 & 10
Q.6) for what value of the equation x y z 1, x 2 y 4 z ,
x 4 y 10 z 2 , has no. solution and solve them.
Q.7) Verify Caley-Hemilton’s theorem for following matrices also find
A 1 .
2 1 1
3 1 1
1
i) 1 2 1
Ans:1 3 1
4
1 1 2
1 1 3
1
ii) 2
0
0
1 0
0 1
2
1 2
Ans:- 2
1
0
0
0
0
3
Q.8) Use Caley-Hemilton’s theorem express the polynomials A 5 4 A 4 7 A 3 11A 2 A 10 I in1 4
terms of matrix A. where A
2 3
19
Ans:- A + 5I.
Q.9) Use Caley-Hemilton’s theorem to find the matrix represented by
2 1 1
8
7
6
5
4
3
2
A 5 A 7 A 3 A A 5 A 8 A 2 A I where A 0 1 0
1 1 2
20
Subject: Mathematics (1B1)
Unit: II
Fourier Series
1)Express f ( x)
1
( x) in a fourier series with period 2 to be valid in the interval (0,2 ) .
2
1
Ans:- f ( x) sin nx
n 1 n
x
2)If f ( x)
in the range (0, 2 ),show that in this range
2
2 cos nx
.
f ( x)
12 n 1 n 2
And hence obtain the follwing relation .
1
1
1
2
i) 2 2 2
6
1
2
3
2
1 1
1
1
2
12
12 2 2 3 2 4 2
3) Obtain the fourier expansion of 1 cos x in the interval 0 x 2
1
1
.
And hence deduce that 2
2
n 1 4n 1
4)Expand f(x)= x sin x in the fourier series in the interval 0 x 2
ii)
5)Expand f(x) in a fourier series where f(x)=sinx
=0
0 x
x 2
1
1
1
1
1.3 3.5 5.7
2
1 sin x 2
1
2 cos 2nx.
Ans:- f ( x)
2
n 1 4n 1
6) If f ( x) mx for 0 x and f ( x) mx 2m in the range x 2
Prove that ,
And deduce that
f ( x)
i)
m 4m
1
1
cos x 2 cos 3x 2
2
3
5
7) Determine the fourier expansion for the following function.
i)
21
f ( x) 1,
- x 0
x
0 x
ii) f ( x) 2 x 2 ,
- x .
8)The function f (x) is defined as fallows.
x
f ( x) x ,
1
2
x
x
2
2
2
2
x .
3 2 1
1
1
2 cos x 2 cos 2 x 2 cos 3x
8 1
2
3
2
9) Find a fourier series with period 3 to represent f ( x) 2 x x in the
range(0,3).
10) find a fourier series to represent x 2 from x l to x l and deduce that
Prove that f ( x)
1 1
1
1
2
.
12
12 2 2 3 2 4 2
11) Find a fourier series with period 2 to represent the function 2 x x
the range (0,2).
12) Determine the fourier expansion for the function
f ( x) 0 ,
,
2 x 1
1 x , 1 x 0
1 x ,
0 x 1
,
0
1 x 2
13) Find the fourier series associated with the function,
f ( x) sin x ,
x .
14) Show that the half- range sin series for x( x) in the interval
0 x is given by
8
1
x( x)
sin( 2n 1) x.
n 0 (2n 1) 3
15) If 0 x ,Then show that
2 2 4
2
4
2
x
_ 2 sin x
sin 2 x 2 sin 3x
1 1
2
3 3
22
in
16) Find the fourier series for the sinx when 0<x< hence deduce that
1 1 1
1 .
3 5 7
4
2x
0 x
17) If f ( x )
, when
3
3
x
x
when
3
3
Prove that if 0 x ,
3 sin x sin 2 x sin 4 x sin 5 x
2 2 .
2
1
2
4
5
18) Obtain half a period sine series for the function.
2d
l
f ( x)
x,
0 x
l
2
2d
l
(l x )
x l.
,
l
2
19) Obtain a cosin series of period 2p for the function ( mx + c) in the
Interval 0<x<p.
20) Analysis the current I into it’s constituent harmonics as far as the fifth
harmonic ,the value of I and being given as fallows.
f ( x)
0
30
60
90
120
150
180
210
240
270
300
360
I
0
24.0
33.5
27.5
18.2
13.0
0
-24
-33.5
-27.5
-18.2
-13.0
Ans : I 5.7 cos 30.33 sin 5.1cos 3 3.2 sin 3 0.6 cos 5 0.4 sin 5
21) The following table gives the variation of a periodic current over a period
t (sec)
A
0
1.98
T/6
1.30
T/3
1.05
T/2
1.30
2T/3
-0.88
5T6
-0.25
T
1.98
Show by numerical analysis that ther is a direct current of 0.75 amp in the variable current and
obtain the amplitude of the first harmonic.
22) Obtain the constant term and the coefficients of first sine and cosine terms in the ssss
Fourier expansion of y as given in the following table .
X
y
0
9
1
18
a 0 41.67 , a1 8.33 ,
2
24
3
28
b1 1.15
23
4
26
5
20
23) The turning moment T on the crank-shaft of a system engine for the crank angle
Degree is given as fallows.
0
15
30
45
60
75
90
105
120
135
150
165
180
T
0
2.7
5.2
7.0
8.1
8.3
7.9
6.8
5.5
4.1
2.6
1.2
0
Expand T in a series of sine upto the fourth harmonic.
24) Develop f(x)in fourier series in the interval (-2,2) if
f ( x) 0 ,
1 ,
25)
If
-2< x< 0
0 < x <2
f ( x) x ,
(2 x) ,
4 cos x cos 3x cos 5x
2
2
2
2 1
3
5
26) Expand the function f ( x) x sin x as a Fourier series in the interval x
1
1
1
1
1
( 2) .
And deduce that
1.3 3.5 5.7 7.9
4
f ( x)
0 x 1
1 x 2 . Show that in the interval (0,2)
27) For a function f(x) define by f ( x) x , < x < .Obtain a Fourier series and
Deduce that
1 1
1
1
2
.
8
12 3 2 5 2 7 2
28) Given
f ( x) x 1 ,
x 0
x 1 ,
0 x
Is the function even or odd ? find the Fourier series for f(x) and hence deduce the
1
1
1
Value of 2 2 2
1
3
5
29) A function is define as fallows.
x 0
0 x
4 cos x cos 3x cos 5x
Show that f ( x) 2
2
2
2 1
3
5
f ( x) x ,
x ,
24
and deduce that
1
2n 1
n 1
30) If f ( x) sin x ,
cos x ,
2
0 x
2
8
.
4
x
4
2
Expand f(x) in a series of sine’s.
25
UNIT III
VECTOR ALGEBRA
rdt
1) Show that the four points whose position vectors 3i-2j+4k, 6i+3j+k,5i+7j+3k
And 2i+2j+6k are coplanar.
2) Find such that the vectors 2i-j+k, i+2j-3k and 3i j 5k are coplanar.
3) Prove that a (b c) (a c)b (a b)c
4) Prove that a (b c) b (c a) c (a b) 0
5) Find wether the vector
6) Prove that
a (b c), b (c a), c (a b) are coplanar. Ans: coplanar.
i (a j ) j (a j ) k (a k ) 2a.
7) If the four point whose p.v are a, b, c, d are coplanar then show that
a, b,c a, b, d a, d , c a, b, c
8) find the volume of the tetrahedron whose edges are represented by a 2i 3 j 4k
b i 2 j k and c 3i j 2k .
Ans:
7
cub.unit
6
9) Find the volume of the tetrahedron whose vertices are the points
A(2,1,3),B(4,1,3) ,C(3,2,-1)and D(1,4,2)
22
Ans: cub.unit
3
10) Prove that a b,b c,c a 2 a, b,c
11)
l a
l b
l c
Prove that l , m,n a,b,c m a m b m c
na
nb
nc
12) Show that the volume of tetrahedron having A B , B C , C A
As concurrent edges is twice the volume of tetrahedron having A, B,C
As concurrent edges.
13) Find the volume of the parallelepiped whose coterminous edges are represented
By the vectors a i j k , b i j k , c i 2 j k
26
Rate of differentiation under integral sign
14) Evaluate by applying the rule of D.U.I.S
tan 1 (ax)
0 x(1 x 2 ) dx
1
15) Prove that
0
Ans:-
2
log( a 1)
x a 1
dx =log(x+1), a>1
log x
log( 1 a sin 3 x)
0 sin 2 x dx = ( a 1 1)
2
16) Prove that
e x (1 e ax )
dx, a>0
0
x
18) Verify the rule of D.U.I.S
17) Evaluate
Ans:-log(a+1)
a2
log( ax)dx
Ans:- RHS=LHS=(6a-2)loga+(a-1)
a
e ax sin x
19) Evaluate
dx and hence show that
x
0
sin x
dx
x
2
xa xb
a 1
0 log x log b 1 , a>0,b>0
21) Verify the rule of D.U.I.S
1
20) Show that
a3
tan
0
1
x
dx
a
27
Curve Tracing
22) Trace the curve xy 2 a 2 (a x)
23) Trace the curve y 2 (2a x) x 3
24) Trace the curve y 2 (a 2 x 2 ) x 2 (a 2 x 2 )
25) Trace the curve x 3 y 3 3axy.
26) Trace the curve r a(1 cos )
27) Trace the curve r 2 a 2 cos 2 .
28
28) Trace the curve y( x 2 4a 2 ) 8a 3 .
29) Trace the curve y 2 x a 2 (a y).
30) Trace the curve y 2 x 5 (2a x).
29
UNIT IV- REDUCTION FORMULAE
GAMMA AND BETA FUNCTION
RECTIFICATION
1) Prove that yB( x 1, y ) xB( x, y 1).
2) If
f (m, n) x m (1 x) n dx , then show that
x m 1 (1 x) n
n
f (m, n)
f (m, n 1)
m n 1
m n 1
1
Hence show that
x
m
(1 x) n dx
0
2
cot
1) If I n
n
m! n!
.
(m n 1)!
d , (n>2), then prove that
4
1
I n2 .
n 1
In
2) Prove that
2
sin
2
5
.
256
cos 6 d
0
3) Prove that sin 2 (1 cos ) 4 d
0
4) Prove that
x sin
5
x cos 4 xdx
0
21
.
16
8
.
315
5 2
Prove that x. cos xdx
.
32
0
6
1
5) Prove that
x
6
1 x 2 dx
0
5
.
256
2a
6) Prove that
x
2ax x dx
0
2
a 3
2
.
10)If I n e x sin n xdx, (n>2) then prove that , (1 n 2 ) I n n(n 1) I n 2 and
0
hence evaluate I 4 .
11) If I m, n cos m x sin nxdx, prove that
30
(m+n) I m, n cos x cos nx mI m 1, n 1 and if
I m, n cos m x cos nxdx ,then I m, n
cos m x sin nx
m
I m 1, n 1
mn
mn
2
hence prove that
cos
0
n
x sin nxdx
2 n 1
.
12) If U n x e dx ,prove that U n x n e x nU n 1 hence evaluate U 4 .
n
x
13) If I m, n x m (log x) n dx ,then prove that
x m 1
n
I m, n
(log x) n
I m, n 1
m 1
n 1
14) Prove that B(m,n)=B(m,n+1)+B(m+1,n)
x
B ( x, y ) .
15) Prove that B(x+1,y)=
x y
1
16) Prove that B(m, m) 21 2 m B(m, )
2
1
m 1
n 1
x x
17) Prove that
dx B(m, n)
m n
0 (1 x)
18) Prove that
x m 1
1
0 (a bx) mn dx a n b m B(n, m)
(m) (n)
sin 2 m 1 x cos 2 n 1 x
.
dx
0 (a sin 2 x b cos 2 x) m n
2a m b n ( m n )
2
19) Prove that
20)Evaluate
4
xe x dx
Ans:-
0
31
3
2
RECTIFICATION
t2
21) Find the length of the loop of the curve x t 2 , y t 1
3
Ans:- 4 3
x
22) Show that in the catenary y c cosh , the length of the arc
c
x
From the vertex to any point is given by s c sinh
c
23) Find the length of the cycloid x a ( sin ) , y a(1 cos )
Between two consecutive cusps.
24) Show that the length of the arc of the curve ay 2 x 3 , from the
Vertex to the point whose abscissa is b is
3
1
9b 4a 2 8 a .
27
27 a
25) Find the whole length of the loops of the curves
4a
Ans:3ay 2 x( x a) 2 .
3
y
25) Find the length of the curve 4ax= y 2 2a 2 log a 2 from
a
(0,a) to any point.
Ans: y2 a
x
2a 2
2
2
2
27) Show that in the asteroid x 3 y 3 a 3 , s 3x 2
S being measured from the cusp which lies on the Y-axis.
28) Show that the whole perimeter of the cardioid r a(1 cos )
Is 8a and prove that the arc of the upper half of the cardioide
Is bisected at .
3
29) Find the length of the cadioide r a(1 cos ) lying outside the
Circle r= a cos .
Ans:- 4 3a
30) prove that the length of the archimedian spiral r a in the
a
range 0 is 1 2 log( 1 2 )
2
31) Prove that the length of the equiangular spiral r ae cot
Described as v increases from r1 to r2 is r2 r1 sec
32
UNIT V- DOUBLE INTEGRATION
rdt
1) Evaluate the sintegral ydxdy . over
i)
ii)
the area bounded by y x 2 and x y 2
the area bounded by x = 0, y x 2 and x y 2
36
5
16
ii)
15
ax by
dxdy over the area of a triangle
2) Evaluate the integral e
in the first quadrant.
Ans:i)
1
ab
3) Evaluate the integral sin (ax by )dxdy over the area of triangle
Bounded by x = 0, y = 0 and ax by 1.
Ans:
1
ab
2
2
4) Evaluate the integral ( x y )dxdy over the area in the positive
Bounded by x = 0, y = 0 and ax by 1.
Ans:
ab(a 2 b 2 )
x2 y2
1
.
Ans:
16
a2 b2
3
5) Evaluate the integral x ydxdy over the area in the positive
Quadrant of the ellipse
x2 y2
a 4b 2
1
.
Ans:
24
a2 b2
6) Evaluate the integral xy( x y )dxdy over the area between y x 2
Quadrant of the ellipse
And y = x.
7) Evaluate the integral
Ans:
x
dxdy
where x> 1 and y x 2
4
2
y
Ans:
1
8) Evaluate the integral
a
9) Evaluate the integral
1 x 2
0
0
3
56
0
dydx
1 x 2 y 2
4
Ans:
a2 x2
2
x ydydx
Ans:
0
33
4
a5
15
log( 1 2 )
1 x2
10) Evaluate the integral
e
y
x
Ans:
dxdy.
0 0
a 3
11) Evaluate the integral
0
x2 a2
0
1
2
xdy
dx
y 2 (x 2 a 2 )
Ans:
a
4
a 2a x
xydxdy.
12) Change the order of integration for the integral
0 x2
a
And evaluate the same with reversed order of integration.
3a 4
8
y
1
13) Change the order of integration and evaluate
Ans:
xydxdy.
Ans:
0 y
1 2x
14) Change the order of integration and evaluate
dxdy.
Ans:
0 x
a
15) Change the order of integration and evaluate
x
a
(x
0 x
2
1
24
1
2
y 2 )dxdy.
a
a a
1
4 7 5
2
Ans:
a 2 xa
16) Change the order of integration and evaluate
x
0
Ans:
a
2
dxdy.
0
4
7
a a
17) Change the order of integration and evaluate
x2
x2 y2
0 y
dxdy.
3
Ans:
a
log( 1 2 )
3
a 2a x
18) Change the order of integration and evaluate
xydxdy.
0 x2
Ans:
3a
8
a
4
19) Change the order of integration and evaluate
1
Ans: 1
2
34
1
2 x 2
0
x
dx
x
x2 y2
dy.
20) Change the order of integration and evaluate
ey
0 x y dydx. Ans: 1
2 2 x
21) Change the order of integration and evaluate
xdxdy.
0 2 x
16
3
22) Show the region of integration and change the order of
Ans:
2 1 2 x x 2
f ( x, y)dxdy.
integration.
1 1 2 x x 2
23) Show the region of integration and change the order of
1 2 x
integration.
f ( x, y)dxdy.
2 x2
24) Show the region of integration and change the order of
integration.
2a
2 ax
0
2 ax x 2
f ( x, y)dxdy.
25) Express the following integrals as single integrals and evaluate.
2 2 y
1 y
(x
2
y )dxdy
2
0 0
(x
1
2
y 2 )dxdy.
4
3
Ans:
0
26) Express the following integrals as single integrals and evaluate
1
y
2 1
dxdy dxdy.
Ans:
1 1
0 y
10
3
27) Express the following integrals as single integrals and evaluate
a
0
2 x
xdxdy
0
a2 x2
a
a
xdxdy.
3 2
0
2
rdrd . over the cardioids r 1 cos .
29) Evaluate r cos drd . over the interior of the circle
28) Evaluate
4
a3
Ans:
Ans:
3
2
3
r 2a cos .
Ans:
7 5
a
4
30) Express the integral in polar coordinate showing the region of
a
Integration and evaluate.
0
a2 x2
x2 y2
e
dxdy.
0
35
Ans:
4
(1 e a )
2
31) Express the integral in polar coordinate, showing the region of
a2 x2
a
Integration and evaluate.
0
y 2 x 2 y 2 dxdy.
0
Ans:
a 5
20
32) Express the integral in polar coordinate, showing the region of
a
Integration and evaluate
a2 x2
x 2 y 2 dydx.
Ans:
a 3
3
33) Express the integral in polar coordinate, showing the region of
a a
x 2 dxdy
a3
Integration and evaluate
Ans:
log( 1 2 )
2
2
3
x
y
0 y
a
0
34) Express the integral in polar coordinate, showing the region of
a a
xdxdy
a
Integration and evaluate 2
Ans:
.
2
4
0 y x y
35) Express the integral in polar coordinate, showing the region of
1 x
Integration and evaluate. ( x y)dydx.
Ans:1
0 0
36) Express the integral in polar coordinate, showing the region of
2 a 2 ax x 2
Integration and evaluate
0
2
2
( x y )dxdy.
Ans:
0
3a 4
4
37) Express as a single integral and evaluate.
a
cosk ( x
2 x
0
0
2
a2 x2
cosk ( x
a
y ) dxdy
2
a
2
2
y 2 ) dxdy.
0
Ans:
sin ka 2
8k
x2 y2
dxdy. over the
38) Change to polar coordinates and evaluate 2
3
(x y 2 ) 2
2a
region of circle x 2 y 2 2ax . In the first quadrants.
Ans;
3
2 2
x y
dxdy. over the region
39) Change to polar coordinates and evaluate 2
x y2
Included between the circle x 2 y 2 = a 2 and x 2 y 2 = b 2 (a>b) in the 1st
36
Quadrant.
Ans:-
(a 4 b 4 )
64
40) Change to polar coordinates and evaluate y dxdy. over the area which
2
Lies outside the circle x 2 y 2 ax 0 but inside the circle
x 2 y 2 2ax 0
41) Evaluate
1
(1 x 2 y 2 )
2
Ans:-
dxdy. over one loop of the lemniscates
(x2 y 2 )2 x2 y 2 .
42) Evaluate
Ans:-
43) Evaluate
a 2 x3
0
2
4
(x y )
dxdy. over the area common to circles x 2 y 2 ax ,
2 2
x y
2
2 2
x 2 y 2 by (a>b>0)
a
15 4
a
64
0
Ans:-ab
sin 2 (a 2 x 2 y 2 )dxdy
a
Ans:-
a2
2
a
2
44) Sketch the area of the double integration and evaluate
a2 y2
log
0
e
( x 2 y 2 )dxdy
y
4a
1
log a
2
2
2
(a>0).
Ans:-
45) Find the area of the loop of the curve ay 2 x( x a) 2
Ans:-
8 2
a
15
a2 x2
Ans:- a 2 ( 2)
2
2
a x
47) Find the area between the curve a x y 2 a 2 x and is asymptote .
Ans:- a 2
48) Find the whole area included between the curve x 2 y 2 a 2 ( y 2 x 2 ) and it’s
asymptote
Ans:- 4a 2
49) Find by double integration, the area between the curve y x 2 6 x 3 and
20
y 2x 3
Ans:3
2
50) Show by double integration,that the area between the parabola y 4ax and x 2 4ay
16 2
a .
is
3
4a 2 ( 2 a x )
2
51) Find by double integration ,the area between the curve y
and it’s
x
asymptote
Ans:- 4a 2
52) Find the area between the curve y 2 4 x and 2x-3y+4=0.
46) Find the total area of the curve y 2 x 2
37
Ans:-
1
3
53) Show that the area enclosed by the curve x y 2 a 2 (a x ) and
(a x) y 2 a 2 x is ( 2)a 2 .
3a sin cos
54) Find the area of the closed portion of the folium r
.
sin 3 cos 3
3a 2
Ans:2
55) Find the area common to the cardioides r a(1 cos ) and r a(1 cos ) .
3
Ans:- 2a 2
2
4
56) Find the area inside the cardioide r 2a(1 cos ) and outside the parabola
16
2a
r=
Ans: a 2 3 .
3
(1 cos )
38
UNIT-V ORDINARY DIFFERENTIAL EQUATION OF FIRST
ORDER AND FIRST DEGREE.
dy
=0
dx
2) Solve x(1 y)dx (1 y 2 )( x 1)dy 0
dy
x sin x
y
3) Solve
dx 2e sinh y
1) Solve x cos x cos y sin y
4) Solve ( x 2 y 2 )dx 2 xydy 0
dy
5) Solve ( x y ) ( x y ) 0
dx
6) Solve y x 2 y 2 dx x x x 2 y 2 dy 0
dy 2 x 3 y
7) Solve
0
dx
yz
8) Solve (2x-y+1)dx+(2y-x-1)dy=0
dy
2x 3y 1
9) Solve (6x+9y+6)
dx
10) Solve ( a 2 2 xy y 2 )dx ( x y) 2 dy 0
11) Solve
(2x 2 y + 4x 3 - 12xy 2 + 3y 2 - xe 7 + e 2x )dy + (12x 2 y + 2xy 2 + 4x 3 - 4y 3 + 2ye 2x - e y )dx = 0
12) Solve ( x 4 xy 2 y 2 )dx ( y 2 4 xy 2 x 2 )dy 0
13) Solve ( x 2 y 2 a 2 ) xdx ( x 2 y 2 b 2 ) ydy 0
14) Solve ydx-xdy+logxdx=0
15) Solve y 2 ( xy 2 x 2 y 2 )dx x( xy x 2 y 2 )dy 0
16) Solve ( x 2 y 2 )dx 2 xydy 0 .
17) Solve ( y 3 2 yx 2 )dx (2 xy2 x 3 )dy 0 .
18) Solve ( x 2 y 2 1)dx 2 xydy 0
19) Solve y( xy 1)dx x(1 xy x 2 y 2 )dy 0
20) Solve ye y dx ( y 3 2 xe y )dy
1
21) Solve x
22) Solve
dy
3y x 4e x y 3
dx
dy
x2 y x5
dx
39
23) Solve 3 y 2
2
dy
2 xy3 4 xe x .
dx
24) Solve (1 x 2 )dy (tan 1 x y)dx
dy
x tan( y x) 1
25) Solve
dx
dy
1 x( y x) x 3 ( y x) 2 .
26) Solve
dx
dy
y
y
1 cos
27) Solve
dx
x
x
y
y
y
28) Solve 2 x sinh 3 y cosh dx 3x cosh dx 0
x
x
x
2
29) Solve x ( xdx ydy) y( xdy ydx) 0
dy
y 1
30) Solve
dx ( y 2)e y x
31) Solve xdy ydx ( x 2 y 2 )( xdx ydy)
32) Solve ( x 2 y y 4 )dx (2 x 3 4 y 3 x)dy 0.
dy
y sin x.
33) Solve sec x
dx
dr
cos r 2 .
34) Solve r sin
d
35) Solve y log ydx ( x log y )dy 0.
36) Find the constant n such that ( x y ) n is an integrating factor of
(4 x 2 2 xy 6 y)dx (2 x 2 9 y 3x)dy 0. and hence solve the equation.
37) If y n is an integrating factor of the equation
y(2 x 2 y e x )dx (e x y 3 )dy 0. Find n and hence solve the
Equation.
]
Answer
1)
x sin x cos x log(cos y ) c.
y2
y log( 1 y ) 2 c 0.
2
2y
y
3) e 2e sin x x cos x.
2
2
4) x y cx.
2
2
1 y
c.
5) log x y tan
x
2)
x log( x 1)
6) cx
x 2 y 2 x log( x 2 y 2 x).
7)
(2 x y 4) 2 c( x y 1).
8)
x 2 xy y 2 x y c.
40
9)
(2 x 3 y 1) ce ( x3 y ) .
y3
xy 2 x 2 y c.
3
2 2
3
3
3
y
2x
4
11) x y 4 x y 4 xy y xe e y x c.
10)
a2 x
12)
3x 2 12 x 2 y 12 xy2 y 3 c
x 4 y 4 2 x 2 y 2 2a 2 x 2 2b 2 y 2 c
14) cx y log x 1 0
1
15) 2 log x log y
c
xy
2
2
16) x y cx .
13)
17)
x 2 y 2 ( y 2 x 2 ) c.
18) x
y 2 1 cx .
2
20)
y 2 log y 2 xy 1 cx 2 y 2 .
2
19) 2 x
x
e y c.
2
y
1
21) y
2
6
x (e
x2
c) 1 .
3
22) e
23) y
24) y
25) e
x
3
( y x 3 3) c .
3 x2
e
2 x 2 c.
1
tan 1 x 1 ce tan x .
x2
2
sin( y x) c .
26) c( y x)e
x2
2
( y x)( x 2 2) 1 .
y
log x c.
2x
2
3 y
.
28) x c sinh
x
2
2
2
2
29) ( x y )( x 1) cx .
27) cot
( y 1)(e y x) c.
1 2
1 y
2
31) tan ( x y ) c .
x 2
30)
7
2
y 11 (7 x 2 11y 3 ) c.
sin x
33) y ce
sin x 1.
1
34) c cos sin .
r
2
35) 2 x log y (log y ) c.
32) x
41
36)
n 1, x 4 2 x 3 y x 2 y 2 3x 2 y 6 xy2 3 y 3 c.
37)
n 2,
2x3 e x y 2
c.
3
y
2
UNIT II
PARTIAL DIFFERENTIATION
1) If u log x 2 y 2 z 2 , then P.T
2u 2u 2u 2
2
2
2 2 2 x y z 1 .
y
z
x
u
u
2) If u f y x 2 y 2 ,Find the value of x
y .
x
x
y
Ans: x 2 y 2
3) If u tan
1
u
u
x3 y 3
, then evaluate x
y .
x
y
x y
4) If u x sin 1
Ans: sin2u
y
u
2u
2u
, then find the value of x 2 2 2 xy
y2 2 .
x
xy
x
y
Ans:-u
2
(x2 y 2 )
5)If log sin u
x y
3
2
, then find x
u
u
y .
x
y
Ans:
6) If t n e
r2
4t
and if
2 log u sin u
cot u
1 2
, then find n.
r
r 2 r r t
Ans:-
3
2
7) If u log( x 3 y 3 z 3 3xyz), prove that
2
9
. u
.
( x y z)3
x y z
8) If u 3(lx my nz ) 2 ( x 2 y 2 z 2 ) where l 2 m 2 n 2 1 ,
Then show that
2u 2u 2u
2 2 2 0.
y
z
x
42
2
z z
z z
9)If z ( x y) x y , then prove that 41
x y
x y
10) If u log(tan x tan y tan z ) then prove that
2
2
u
u
u
sin 2 x sin 2 y .sin 2 z 2 .
x
y
z
1
y
11)If tan 2 , then show that
.
x
y( x y) 2
x y y x
12) If f ( x, y ) 0, and ( x, z ) 0 then show that
f y f
.
x y z x z
13)If x u v w , y= u 2 v 2 w 2 , z= u 3 v 3 w3 then prove that
u
vw
.
x (u v)(u w)
14) If u lx my, v= mx ly and z= f (u , v) then show that
i)
l2
u x
.
2
2
x y u v (l m )
ii)
l 2 m2
u v
.
l
v x y u
iii)
2z 2z
2z
2z
.
lm 2 2 (m 2 l 2 )
xy
uv
v
u
15) If x y 2e cos , and x y 2ie sin then prove that
2v 2v
2v
.
4
xy
2
xy
2
16) If u x 2 y 2 ,v=2xy, f ( x, y ) (u, v) ,
Show that
2
2 f 2 f
2
2
2
4
(
x
y
)
u 2 v 2
x 2 y 2
17) If x u v w then y uv vw uw , z uvw and f is a function of x,y,z
f
f
f
f
f
f
v .w x 2 y
.3 z
Then show that u
.
v
w
x
y
z
u
18) If x vw ,y= uw , z= uv
v
.w x
y
.z
P.T u
.
v
w
x
y
z
u
19) Find the values of the constant a,b such that the substitution
u x ay , v x by transform the
43
2 f
2 f
2 f
2 f
Equation 9 2 9
0.
2 2 0 into
uv
xy
y
x
3
3u
20) If u x y then show that 2
.
x y xyx
21) If x x y y z z c , then show that at x y z ,
2 z
( x. log ex) 1
xy
dy
x
y
x y
22) Find out
if, y x ( x y) .
dx
23) If u log( x 3 y 3 x 2 y xy2 )
2
4
Then prove that u
.
( x y) 2
x y
y
x
24) If u x 2 tan 1 y 2 tan 1 , then show that
x
y
2u x 2 y 2
xy x 2 y 2
x y
, Prove that
25)If u sin 1
x y
2
2
u
u
2u
sin u. cos 2u
.
x 2 2 2 xy
y2 2
xy
x
y
4 cos 3 u
2y 2
1 x
u
cos
ec
26) If
x 13 y 13
1
x2
1
1
2
, then prove that
2
2u
2u
tan u
2 u
2
xy
y
(13 tan 2 u ).
2
2
xy
144
x
y
x4 y4
u
u
, show that x y
27)If u log
3.
x
y
x y
28) If u f (v) , v = being homogenious function of degree in x, y prove that
u
u
u
u
x y
nvf ' (v) . Hence if u log v , then show that x y
n.
x
y
x
y
14 y 14
1 x
, Prove that x z y z 1 tan z.
29) If z sin
1
1
5
x
y 20
5
x y
30) If z f ( x, y ) , Where x e u cos v and y e u sin v , Show that
44
2
2 z 2 z
2 z
2u z
e
u 2 v 2 .
x 2 y 2
Unit-III JACOBIAN OF EXPICT AND IMPLICT FUNCTION
1) Evaluate jacobian of transformation if
x u (1 v) , y uv(1 v) , z uvw .
Ans:- u 2 v(1 v) 2
2) Evaluate jacobian of transformation if
x z sec cos , y z sec sin , z z
Ans:- z 2 sec 2 tan
3) Evaluate jacobian of transformation if
x ar cos sin , y br sin sin , z cr cos
Ans:- abcr 2 sin
4) Evaluate jacobian of transformation if
x u v w , y u 2 v 2 w2 , z u 3 v
Ans:- 2(u w) 6u 2 (v w)
5) Evaluate jacobian of transformation if
Ans:- 2(x+1)
u x2 2y , v x y
6) Evaluate jacobian of transformation if
ux yz ,
vy xz , wz xy.
Ans:-4
2
2
2
2
7) If x y u v 0 , and uv xy 0.
Show that
(u, v) x y 2
.
( x, y) u 2 v 2
8) If x y z u , y z uv , z uvw then show that
( x, y, z )
u 2 v.
(u, v, w)
9) If u 3 v 3 x y , u 2 v 2 x 3 y 3 . then show that
(u, v) 1 y 2 x 2
.
( x, y) 2 uv(u v)
10) If x e v sec u , y e v tan u . Then show that J.J’=1.
11) If x v 2 w 2 , y w 2 u 2 , z u 2 v 2 . Then show that J.J’=1
u
12) If x uv , y . Then show that J.J’=1.
v
13) If x y 2e cos , x y 2ie sin . Then show that J.J’=1
14) If x vw , y uw z uv and u r sin cos ,
45
v r sin sin
( x, y , z )
.
( r , , )
15) If u e x ( x cos y y sin y) ,
v e x ( x sin y y cos y)
(u , v)
And x l m , y l m then find
.
( , )
Ans:- e 2 x ( x 1) 2 y 2 (l 2 m 2 )
16) The root of equation in
( x) 3 ( y) 3 ( z ) 3 0. are u, v, w then prove that
(u, v, w)
( y z )( z x)( x y )
2
.
( x, y, z )
(v w)( w u )(u v)
x
y
z
17) If u
, v
, w
1
1
1
(1 r 2 ) 2
(1 r 2 ) 2
(1 r 2 ) 2
(u , v, w)
1
.
Where r 2 x 2 y 2 z 2 .then show that
( x, y, z ) (1 r 2 ) 5 2
w r cos ,then find
Are the following function , functionally dependent if so find the relation between
Them.
x y
18) u
, v tan 1 x tan 1 y.
Ans:-yes
1 xy
u tan v
x y
yz
y( x y z)
w
19) u
, v
,
Ans:- yes
z
x
xz
(uv 1 w)
x y
x y
20) u
, v
Ans:-yes
x
x y
2
(u 1)
v
2
2
2
21) u x y z , v x y z 2 xy 2 yz 2 zx
Ans:- yes
4w u(u 2 3v)
w x 3 y 3 z 3 3xyz
xy
x y
22) u
, v
( x y) 2
x y
23) u y z
v x 2z 2
Ans:-yes
w x 4 yz 2 y 2
u 2 1 4v
Ans:-yes
2u 2 v w
24)
u
x y
xz
v
xz
yz
Ans:-yes
u 1
25) u sin 1 x sin 1 y
, v x 1 y2 y 1 x2
46
1
v
Ans:-yes
v sin u
# Maxima and Minima.
Maximum and minimum values of function of two independent variables.
1) Find maximum and minimum values of x 3 y 3 3axy.
Ans:- Min: a 3 at (a,a).
2) Find maximum and minimum values of
x 2 y 2 5 x 2 8 xy 5 y 2 .
3) Find maximum and minimum values of
( x y)( x 2 y 2 )( x y 1).
4) Find maximum and minimum values of
Ans:- Max: 0 at (0,0)
Ans:-NO Max and No Min.
3 3
at (m , n )
8
3
3
3 3
at (m , n )
Min:
8
3
3
sin x sin y sin( x y ).
Ans:- Max:
5) Find maximum and minimum values of
2( x 2 y 2 ) x 4 y 4 .
6) Find maximum and minimum values of
x 3 3xy2 15 x 2 15 y 2 72 x.
Ans:- Max: at ( 1,0)
Min: at(6,0)
Ans:-Max: at (4, 0)
Min: at (6, 0)
7) Examine the function for the extreme value
xy(a x y )
Ans:-Stationary
8) Examine the function for the extreme value
x 3 y 2 (1 x y)
9) In a plane triangle, find the maximum value of
a3 a a
at ,
27 3 3
Ans:- Max: at (1,1)
1
8
10) The temperature T at any point (x, y, z) in space is T = 400xyz2. Find the highest
Temp. on the surface x 2 y 2 z 2 1.
5 xyz
11) If xyz 8 , find the values of x,y,z for which u
is a maximum.
x 2 y 4z
Ans:- x = 4, y = 2, z = 1.
3 4 5
12) If 6. Find the values of x, y, z, which makes x y z a minimum.
x y z
Ans: - x = 1.723, y = 1.990, z = 2.223
13) Use Lagrange’s method to determine the minimum distance from the origin to the
12
Plane 3x 2 y z 12.
Ans:14
14) Find the greatest rectangular parallelepiped that can be inscribed in given ellipsoid
Ans:-
cos A cos B cos C
47
x2 y2 z 2
2a 2b 2c
Ans: - Sides
2 2 1.
,
,
2
a
b
c
3 3 3
15) Divide 24 into three parts such that the continued product of the first square of the
Second and the cube of the third may be maximum.
Ans:- 4,8,12
# Lagrange’s Method of undetermined multipliers.
1 1 1
1. show that stationary value of u is given by .
x y z
abc
abc
abc
x
, y
, z
b
a
c
2) If r is the distance of the point on the conic ax 2 by 2 cz 2 1. , lx my nz 0.
From the origin, then the stationary values of r are given by.
l2
m2
n2
0.
1 ar 2 1 br 2 1 cr 2
x2 y2 z 2
4) Prove that the stationary values of u 4 4 4 where lx my nz 0.
a
b
c
2
2
2
x
y
z
2 2 1. are the roots of the equation.
2
a
b
c
2 4
l a
m 2b 4
n2c 4
0.
1 a 2u 1 b 2u 1 c 2u
5) show that the stationary value of u x m y n z p . where x y z a. is
1) If u a 3 x 2 b 3 y 2 c 3 z 2 where
a
m n p
mn p
m
n
p
m n p
.
48
UNIT- COMPLEX NUMBER
1) Express the following in the form a+ib.
2 3i
1 i
(cos 5 i sin 5 ) 2 (cos 7 i sin 7 ) 3
2)Prove that
1.
(cos 4 i sin 4 ) 9 (cos i sin )5
cos i sin
cos 8 i sin 8 .
3) Prove that
sin i cos
4) If x r cis r , Show that lim x1, x2, x3 xn 1.
2
n
m
m
m
b
m
5) Prove that (a ib ) n (a ib ) n 2(a 2 b 2 ) 2 n cos tan 1 .
a
n
4
n
1
6) Prove that (1 i ) n (1 i ) n 2 2 cos
n
.
4
n n n
cos
7) Prove that (1 sin i cos ) n (1 sin i cos ) n 2 n 1 cos n
.
2
4 2 4
1 sin i cos
n
n
cos
n i sin
n .
8) Prove that
1 sin cos
2
2
n
x 1
y 1
and 2 cos
. Show that one of the valuesof
x
y
2 cos( m n ).
9) If 2 cos
xm yn
1
xm yn
x 1
xm yn
y 1
and 2 cos
. Show that one of the values of n m is
x
y
y
x
2 cos( m n ).
10) If 2 cos
11) Find all the value of (1 i )
1
4
1
1
12) Find all the value of (1 i 3 ) 3 (1 i 3 ) 3 .
13) Use De-Moiver’s theorem to solve the equation x 7 x 4 x 3 1 0.
14) Use De-Moiver’s theorem to solve the equation ( x 1) 5 x 5 0.
15) Solve the equation x12 1 0 and find which of it’s roots satisfy
the equation x 4 x 2 1 0.
16) Prove that cos 6 32 cos 6 48 cos 4 18 cos 2 1.
17) Prove that 32 cos 6 cos 6 6 cos 4 15 cos 2 10.
1
18) Prove that cos 7 (cos 7 7 cos 5 21cos 3 35 cos ).
64
1
(sin 7 3 sin 5 sin 3 5 sin ).
19) Prove that sin 5 cos 3
64
49
is
20) If (a1 ib1 )(a2 ib2 ) (an ibn ) A iB. , then prove that
tan 1
b
b1
b
B
tan 1 2 tan 1 n tan 1
a1
a2
an
A
21) If i i i , then prove that 2 2 e ( 4 n1)
i
A B
and A 2 B 2 e .
22) If i i
A iB. then prove that
2
A
23) If (a1 ib1 )(a2 ib2 ) (an ibn ) A iB. Then prove that
(a12 b12 )( a 22 b22 ) (a n2 bn2 ) A 2 B 2 .
24) Simplify in the form ( a ib ).
Ans:-
i
4
(1 i) 6 (1 i 3 ) 4
.
(1 i) 8 (1 i 3 ) 5
25) Simplify in the form ( a ib ).
Ans:- -3-2i
1 i
1 i
3
2
.]
1 i
1 i
26) If x 2 y 2 1. Show that
1 x iy
1 y ix
x iy ,
y ix.
1 x iy
1 y ix
2
2
i sin
. then prove that ( x 2 x 3 )( x 4 x) 5.
27) If x cos
5
5
2
3
n
n
1 i 3 1 i 3
.Has the value -1 if n = 3k 1 and 2 if n = 3k where
28) Prove that
2
2
k is an integer.
29) Prove that
n
x iy n x iy has n real values and find those of 3 1 i 3 3 1 i 3 .
30) Find all the values of (36 i 64)
1
2
2
2
31) Find all the values of cos
i sin
3
3
1
4
n
1
32) If n is a positive integer show that (1 i ) n (1 i ) n 2 2 cos
n
4
1
And show that the continued product of all the values of (1 i ) 5 is (1+i).
33) Prove that
n
n
(1 cos i sin ) n 2 n cos n cos
i sin
2
2
2
34) Prove that cos( i ) i sin( i ) e (cos i sin ).
35) If y log tan x , Prove that
50
1
(tan n x cot n x).
2
ii) 2 cosh ny cos ec2 x cosh( n 1) y cos( n 1) y.
3i
36) ) Find in the form a +ib. the expression cos 1
4
i) sinh ny
Ans:- i log 2
2
2
2
2
2
x
y
x
y
37) If sin( i ) x iy , show that
1
1 and
2
2
2
sin cos 2
cosh sinh
38) If tan( i ) x iy , show that x 2 y 2 2 x cot 2 1. and
x 2 y 2 2 y coth 2 0.
39) If sin( i ) R(cos i sin ), Prove that
1
R 2 cosh 2 cos 2 and tan tanh cot .
3
40) Prove that log( 1 cos 2 i sin 2 ) log( 2 cos ) i .
cos( x iy )
41) Prove that log
2i tan 1 (tan x tan y ).
cos( x iy )
4n 1
.
42) Show that log ii
4m 1
43) Solve for z if e z 1 i 3 Ans:- z log 2 i 2n .
3
44) Find in the form a +ib.
3i
i) log
3i
ii) (1 i)
i
Ans:i) -0.6+34i
4
1
1
cos 2 log 2 i sin 2 log 2
log 2
iii) e cos(log 2) i sin(log 2) .
ii) e
iii) 2 (1i )
45) Show that all the n th roots of unity are given by expression 1, , 2 n1
2
2
i sin
. and those of -1 are given by , 3 , 5 2 n1 .
Where cos
n
n
Where cos i sin .
n
n
2
46) Prove that the continued product of the three values of i 3 is -1.
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51
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