DEPARTMENT OF MATHEMATICS
MNNIT ALLAHABAD
Course Coordinator: ”
B. Tech. II Semester- 2012-2013
MATHEMATICS-II (MAM201)-Tutorial Sheet-Unit-1
Q.l Form the partial differential equation by eliminating the arbitrary constants from the
following
(a) 4 + 4- + 4 = I • (b) (* - A)2 + O’ - *)2 + ?2 = c2 . (C) z = (X2 + a)(y 2 + b).
a2
b2
c2
Q.2 Form the partial differential equation by eliminating the arbitrary functions from the
following
(a) z = en-y/(x-y),(b) z = f(x + ay)+g(x - ay), (c) z = /(x + iy) + F(x -iy),
(d) f(x + y + z,x2 +y2 -z2) = 0.
Q.3 Solve the following PDE’s by Lagrange’s method
(a) p cos(x + y) + q sin(x + y) = z ,
(b) yzp + xzq = xy ,
(c) x(y 2 + z)p - y(x2 + z)$ = z(x2 - y2),
(d) (z2-2yz-y2)p+(xy + xz)q = xy-xzt (e) y2p-xyq = x(z-2y)
■.
Z-xZ^VZS^
9u
du
(f) p+3<y = 5z +tan(y-3x), (g)x—+ y—+ z—- = xyz .
dx
dy
dz
Q.4 Obtain the complete solution of the following PDE’s by using standard form 1,11, III,
IV
(a) p + q = pq , (b) x2p2 + y2q2 = z2 , (c) p2 + q2 = z,(d) (p3 +?3) = 27z,
(e) p2 + q2 =x + y , (f) z = xp + yq + logp<? .
Q.5 Solve the following PDE’s by Charpit’s method
(a)z = p7,
(b) xp + yq = pq, (c) (p2 + q2)y = qz
(e) z = xp + yq + p2 +q2,
(£) z2 = pqxy
(f) (p + q)(xp + yq) = 1.
Q.6 Obtain the general solution of (2y
+ z)p + (y + 2x)q = 4xy - z. Also, find the
particular solution which passes through the straight line z = l,y = x.
3
Q.7 Find the equation of the surface satisfying t = 6x y and containing the two lines
y = 0,z = 0,
y-1, z = l.
(d) (y + z)p - (x + z)q = x - y.
Q3. (a)
cot{y(x + j) + ^},log{cos(x + y)+sin(x+j*)}-x->’] = 0,
(b) ^(x2-j2,x2-z2) = 0,
(c) 0(x2 + y2-2z,xyz) = 0,
(d) 0(x2 + y2 + z2,y2-2yz-z2) = 0,
(e) ^(x2 + y2,yz-y2) = 0,
(f) ^[^-Sx, e-5x{5z+ tan(y-3x)}] = 0, (g) <!>( —,—,xyz-3w) = 0.
y z
Q4. (a) z = ax +
■ + c,
(a-I)
(c) 4(1 +a2 )z = (x + ay + b)2 ,
(b) z = cxay^ , where b = -Jl-a* ,
(d) (l + a3)z2 =8(x + aj> + 6)3,
(e) z + 6 = |-(x + a)3/2 + -|(.y - a)3Z2, (f) z = ax + 6y + logah.
Q5. (a) l4z = ax + — y + b, (b) az =-°X+-^-- + b ,{c) z2 = a2 v2 + (ax + 6)2
a
2
(d) z =bxay^1 a, (e) z = ax + by + a2 + b2 , (f) -Ji + a z - 2y]x + ay +b .
Q6. </>{x-y2 + z,x2 -+z) = 0, z(\-y) +x-y2 + x2 =1
Q7. z = x3y3 + .y(l--v3).
DEPARTMENT OF MATHEMATICS
MNNIT ALLAHABAD
Course Coordinator: '
B. Tech. II Semester- 2012-2013
MATHEM A1 ICS-li (MA-1201)-Tutorial Sheet-Unit-2
Q.l Solve the following PDE’s
. . d2z
d2z
d4z
d^z
d^z
d4z
d4z
d^z
dx2
dy2
dx4
dx^dy
dxdy5
dy4
dx4
dy4
(d) (D2 +DD' +D' -l)z=0,(e) (D2-2D')z = 0.
Q.2 Find the general solution of the following PDE’s
a2
^2
2
(a) ^--a2^- = x ,(b) (D2 -6DD' +9D' )z = 12x2 + 36xy,
dx2
dy2
d2z d2z
(c) —— + —— = cos mx sin nv , (d) 4r - 4s +1 = 16log(x + 2y),
dx2
dy2
(e) (D2-DD,-2£>’2)z = (y-l)ex .(f) (D2-2DD' + D'2)z = ex + 2y + x3 ,
(g) (D2 - DD‘ +D' - l)z = cos (x + 2y) + ey.
Q.3 Classify the equation:
d2u
d2u
dx2
dy2
dt
dx1
dxdy
.
. du
2 d2u
.
dx2
d2u
dt2
dy2
2 d2u
dx2
dx
dy
Q.4 Using the methods of separation of the variables, solve
- 0 ,(b)3^ + 2^ = 0 .where u(x,0) = 4e~x
(a)
, .
(C)
d2u
2 d2u ... du
2 d2u
~T = C T’(d) 17 = C
2 ’
dt2
dx2
dt
dx2
du
d2u
Q.5 Solve the heat equation — = c2 ——, where u(0, /) = 0 , u(l, r) = 0 t > 0,
dt
dx2
when 0 < x<±.
2 •
0, when — < x<l
A,
«(*,0) = {
Q.6 Find the temperature distribution w(x, t) in a thin rod of length /, if the initial
temperature through the rod is f (x) the ends x = 0 and x = /, of the rod are
insulated.
9
i/fi/quis u
w
xwsoo(/f-2/)uquis w(l“)
I= u
**u u»s . „
Z Z
<»
3
t
>+
gO
00
0 /
= (/‘X)n •£&* *7’^7SOO(X)/J’1= uy
£=
I c
0 I
1
‘ *P(*)/ J- = °K 0J9qM ‘ Tin7 503
‘i-<—•,0
i
y
“Is
J=w *
3 —=(^Ksd
V-KM)
z
z
=
'x)n (q)‘
‘x)n
ag +
iJ
3’tf‘iia U> /+ y ‘oqoqBJBd
!l = / + tx ‘an°qj9d^H ‘I < / + zx JI (P) ojIoqjadXH (□) •oiJoqBJBd (q) ‘opdflia (B) -fO
• A-ax - (<Q + x)uis Y - (x + K) c(px a + (O ^ra = z (3)
‘ ^+Zz+xa + (x + <^x + (x + 'f^ = z G)
‘ra<f + (x-X')^ + (xz + /C)^ = z (a)‘X'{.X9+ t7X0i + (x{; + a-)^x + (x£ + 4-)V = z (p)
U+
IU
* K£x9+ t,x0l + (x’£ + '<)^y + (JfE + <Ol^ = z (0)
‘-^r + (x»-A‘)^ + (xp + Z)^ = z (b)’zO
E
•jueisuoo XiBfliqJB 9JB q puB v a-iaqAt ‘
+ ^ay = z (a)
‘(x-K)Z<f>xa + (/Ql0r a = z (p) ‘(x?-X’)^ + (x7 + X’)^ + (x-X’)^ + (x4-X’)10 = z(o)
‘(x + 4)tV_.x + (x + A’)l^x + (x + X’)^ + (x-X’)I0 = z (q)
* (x - A")
+ (x +/)'^ = z(b) •[£) uaMSuy
0 = (X' ‘^)r» = (/C ‘o)A«
‘0 = (^‘x)n ‘
Z
x = (o‘x)m‘jz>X->o 'x>x>q
joj‘
zXQ
0 = -— + -—SAPS 8-0
nzQ »ZQ
•QO ‘I>x>o 3JaqM0 = (:‘l)n PUB o = (/‘O)«
zxe
fQ
‘xzz uis£ = (o €x)n suonipuoo Xiupunoq aqj qjiM ‘------ = —uopraba iroq aqj oajos L'b
n^Q
nQ
DEPARTMENT OF MATHEMATICS
MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY ALLAHABAD
ASSIGNMENT-3, MATHEMATICS-II (MA-201), SEMESTER-n, 2012-2013
Q. 1 Find the Laplace transform of the following functions:
(i) sin/z33z (ii) sin/i3rcos7r (iii) /2e"3,sin5t (iv) e~*' sin 3/sin 5/
Q. 2 Find the Laplace transform of the functions
sm(r-^/3), t >n/3 z..„ .
(i) /(O=|r-l|+|r+l|,^O(ii) /(0 = (in) sintcostlogtd(t-x)where,
0
,t<Jtl3
S(t-n) is Unit Impulse function.
Q.3 Find the Laplace transform of the functions
(i) f (0 = cos^ (ii) f(j) = s^nflZ where, a is constant. Does the Laplace transform of /(/) = ^^ exist?
ft
t
t
Q. 4 By using the Laplace transform evaluate the following integrals
co •
co
00 -/ • 2 4
z,. rsinz . .... r 3 . ,
. _z.... re sin 1 .
(1) j------ dt (11) J t3 e sintdt =0 (111) |------------- dt
Q. 5 Find the Laplace transform of the erf ft
Q. 6 Find a function /(/) for each of the given function F(s)given bellow such that £{/(/)} = F(s)
(i»)4+
(0-4=
x/s2+4
Sm
-1
V
52
7
z. x 3s + 7 z v
32
z ,x
e4-” , m
s2
--------- (iv) —----------- (v)
:------ 7 (vi)-------- — (vil)—------- 3s + 2
s2-25-3
(16s2+1)2
(s + 4)5'2
(s2+4)2
Q. 7 Find the values of the following by using the convolution theorem
- -](u) r‘[
jciiiir't-] (ivjr’tcot-'^j (v)£-‘[iog^]
1
(s-2)(s +1)
(s +16)
2
s(s+4)
s+b
Q. 8 Find solution of each of the following initial value problem by using Laplace transform
X0) = l,y'(0) = -l
(а) y"-3y'+2y = 4t + e3'
(б)
ym+2yn-y'-2y = 0
y(0) = l, /(0) = 2, /'(0) = 2
(c)
ty"+(l-2t)y,-2y = 0
XO)=1, y(0)=2
X0) = l, y'(0) = 2
(d) y"-ty'+y = l
t, 0£t<l,
X0) = 2
0,■* t>l
(/)/’+5/+6y = l-u(r-3)-u(r-5) X0) = 0, /(0) = 0
(e) y'+3y = <
Q. 9 Solve the following simultaneous equations by using Laplace transform
(a)
(D2-3)x-4y = 0, x+(D2 + l).v = 0 given thatx = y = — = 0 and — = 2 art = 0;Z) s —
dt
dt
dt
(b)
(D2 -\)x + 5Dy = t, 2Dx-(D2-4)y = 2 given thatx = y = ^- = ^ = 0 at/ = 0; Da-y-
dt
di
dt
Q. 10 Solve the following initial-boundary value problems.
(a) . u„ + xu, = 0,
u(x, 0) = 1, u(0, /) = t
(b) . u„ =ua, x>0, t>0, w(x,0) = e"\ u,(x,0) = 0, u(Q, t) = 0, u(x, t) is bounded as x -> 00
DEPARTMENT OF MATHEMATICS
MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY ALLAHABAD
ASSIGNMENT-4, MATHEMATICS-II (MA-201), SEMESTER-II, 2012-2013
Q. 1 Find the Fourier series of the given function on the given interval.
0, -7r<x<-n/2
-x, -2<x<0
kt ~n/2<x<rr/2 (ii) /(x) =|cosx|, -;r<x<;r (iii)/(x) =
x2, 0 < x < 2
0, x/2<x<ir
/ —._ -A 2
Q. 2 Find the Fourier series to represent f (x) = ■ '
, 0 < x < 2n. Hence obtain the following relations:
K_ 11.11
1
I
1 1
(i)-yl——rH—r +....... =---- (ii) —z---- 5"*—5----- » +
..........
I2 22 32 42
6
l2 2’ 32 42
12
Q. 3 Find the Fourier series for the function/(x) = x + x2, -n<x<n . Hence show that
„ 1
l2
1
22
1
32
1
4
<2
12
<*,1111
— H—— ———-F
6
I2 2!2 32 42
Q. 4 Find the Fourier series of the function f(x) -
0,
-;r£x£0
sin x,
0<x£<
and hence show that
_1___ 1__ J___ 1_
4~2 + 1.3 3.5 + 5.7 7.9 +
Q. 5 Obtain a half-range cosine series for /(x) = -
kx,
0£x£f/2
and hence sum of the series
k(t-x),
1111
I2 32 52 72
Q. 6 Write the Fourier cosine series and Fourier sine series for the following function
-74- —4- —4—7 +...........
x,
0£x<l
(D/W = 1,
l£x<2
(ii)/(x) = e“\ 0<x£2
3-x, 2^x<3
Q. 7 Find the complex form of Fourier series of /(x) on the given interval
0)/W =
(ii)/(x) = ^, -2<x<2
0,
1
x<2
Q. 8 Find the Fourier integral representations of the following functions
sinx, - 2 < x < 0
cosx, 0<x^2
0)/W =
0,
|x|>2
Q. 9 Find the Fourier sine integrals of the following function
sinhx, 0^x£3
(»V(x) = In
1
0,
x>3
,
,
0,
0<x<l
= ' 1,
l<x<2
0,
x>2
Q. 10 Find the complex form of the Fourier integrals of the following function
ri+x,
|x|^l
; (ii)Z(x) = 4
(i)/W =
w»
|x|>fl
DEPARTMENT OF MATHEMATICS
MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY ALLAHABAD
ASSIGNMENT-5, MATHEMATICS-II (MA-201), SEMESTER-II, 2012-2013
Q. 1 Find the Fourier transform of the following function
0,
(0/(0 =
0<t <a
a^t^b-
(ii)/(0 = e-?/2 (iii)/(t)=/7(t-3)^' (iv) /(f)
t>b
0,
Q. 2 Find the Fourier transform of the function /(x) = •
sin sa cos sx^
1,
W<a
l<>.
|x|>«
and hence evaluate
(H) j sin 5 ,
------- ds
s
Q. 3 Find the Fourier sine transform of the following function
(i)/(x) = - (ii)/(x) = —(iii)/(x) =
-2-x
X
x{a +x )
Q. 4 Find the Fourier cosine transform of the following function
ax + P~ax
1
2
Q. 5 Find the finite Fourier sine transform of the following function
(i) /(x) = cos kx (u) /(x) = X1 (Hi) /(x) = ec x
Q. 6 Find the finite Fourier cosine transform of the following function
- x + y- (ii) /(x) = sin mx (iii) /(x) = -
(i) /(x) =
3
2zr
k sin kn
Q. 7 Find the solution of the following differential equation
(i) y•-4y = H(t)e*, -n<t«n (u)y"+3y'+2y = <5(t-3)
Q. 8 Find the solution of the following initial boundary value problem
... du
2 ^2u
. n
(l) —= C --,-co<x<oo,t >0,
dt
dx
-2|x| ,-oo<x<oo
u(x,0) = e
■ • •<
.... du d2u .
a „
(n)—=—?)0<x<oo,r >0,
dt
dx
w(x,0) = < 1, 0<x^£
;
u(0,z) = 0, t>0
0,
x>
£
< *
.... d2u d2u
n
(m) —7+—7 = 0,—oo < x < oo,0 < y<n
dx
dy
u(x,0) = e’21 H(x)
u(x,tr) = 0, -oo<x<oo
«>0
Answer:
Assignment-3.
, 5
3(52-58)
.u 10(3?+65+ 2)
_______________________
>62
30(5 + 4)
z
» 11*
(?+58)
2
-365
2
’
1
'
(?+6s
+
34)
2
’
‘
V
’
(52+85 +20)(52 + 85 + 80)’
(52 - 81)(52 - 9)’
1» 1»“ z
2
e~*
e~"n
2. i.-(l + —)ii. £—in. 0.3. (i) J-e-|M,(ii)tan-‘(l/5)(iii)no; 4. (i)- (ii) 0 (Hi) (>/<) log 5; 5. —2=
5
5
5+1
V5
2
5V5 + I
6i. (l/2)sin2r; ii. (8 /15) >/7/7;
iii. 6Z + 1-4V77JT-(7/3)e-27'3;
iv.4e5'-e";
v. (/ / 4)sin(r / 4); vi. (4/3V#)(Z-3)lJ/2e-*t'-4,(;(r-3); vii. (l/4)(sin2/ + 2Zcos2r);
y 5)
7. zsin4/.
2 sin r - cos /); iii. (1 /16)(1 -1 sin 2t - cos 2/); (iv) —sin2<; (v) e* e‘
8
t
t
8 a. 3 + 2/ + (l/2)(eJ'-e')-2e2'; b. (1/3) (5er+e’2')-<?"; c. e2';
d. 2Z + 1; e.[(3r-l + 19e’J')u(/) + (l-3i + 2e-3O’l))M(f-l)]/9; f. (1 / 6)[1 + 2*?~3'-3e~2,]M(r)
- (1 / 6)[1 + 2e_3(,_3) - 3e”2(,'3) ] u(r - 3) - (1 / 6)[1 + 2e’1(M) - 3e*2(”5) ] u(t - 5);
9. a. x = 2rcosh/,j' = (l-/)sinh/;
b. x = -/+5sinf-2sin2t, y = l-2cos/ + cos2/
10. a. u(x, r) = 1+ [(r—x2/2) —l]w(r —x2/2); b.
w(x, z) = e"Jcoshr-cosh(r-x)u(Z-x)
Assignment-4
, ... k 2k-^rl . ,nn.
,,..x2 4.cos2x cos4x
.
UO “ + —2J~sin(—)cosnx](n)-+-(— --------- 77~+-)
2
tc
n
2
n n
3
15
.... 7
2
..
,nnx. . 2
8(1 —(—1)”). . ,nnx.
n2 -^cosnx
(m) - + 2J^-r(5cosn^-l)cos(—)-{—cosnn~ + --<}sin(—); 2. — + 2}——
(n«-)3
6
nn
2
nn
(nxy
2
12 Zi «
1
1.
i^(-ir'-i
3. — + 42,—— cos nx- 22,-—— sinnx;4. —+ -sinx + —> -—------- cosnx
n 2
n2-1
5.
kf
2ktr„
n2x2
4
,nn.
,
.
2
.
2
65-. 1 .
nn
2/ur
,nnx. 6
I . . nn
. 2nn
. ,nnx.
0. (1) — + —-y —{cos------- COS717T + COS —— 1} cos(—-),—2,—{sin — + sin—— 1} sin (—)
3 n
n
3
3
n
3
3
3
3
(n)^(l-e"2) + 4£
1
,{l-e cos nn} cos------ , 2/r> -------- z—^{l-e cos nx} sin (------ )
4+n n
2
4 + nrC
2
7. (i)4+J-£l(e""-e-""/2)e'1""2 (ii)V —^-y[{l-(-l)”e'2}e",x'/2]
4
^4 + nsr
2m
8. (i)/j(ta) = 2[zasin2®cosh2+cos2tusinh2]/(l + tn2), B(cd) = 2[sin2tycosh2-tacos2o>sinh 2]/(1 + <w2)
(ii) A(cd) = —[—y2—+—— {cos 2(co +1) + sin 2(<y +1)} + —!— {sin 2(cd -1) - cos 2(<a -1)} 1
2 CD -I
l + O)
CD-I
B(co) = -[—7^-------- — {cos 2{cd + 1) + sin 2(fi>+l)} + —{sin2(cy-l)-cos2(<a-l)}]
2 co -1 l + O)
CD-I
9. (i) B(cd) = 2[ sin 3<y cosh 3 - cd cos 3<y sinh 3] I (1 + cd2 ), (ii) B(cd) = 2[ cos cd - cos 2cd] I cd
10. (i)c(cd) = —-—sinh(a(2 + io))) + —-—sinh(a(2 - ita))] (ii)c(ta) = 2i[sin a)-cDe"u]/ cd2
2 + ico
2-icD
Assignment-5
xx .
|2 r-b cos ab + a cos co a sinab-sincoa
2
] (ii) e-’/2
Q.l (i) -[-------------------------------+
<o
Vn
co
- , 2sinfi>a
_ ... [*♦ |x|<
O'2^’’“’*0 (,,lo. |x|>
(ii)^
a
2
_(iv)-4P(4 + jcj)
^2) J^tan-'^/aXiioPPd-e"
”)
Q.3 (i).P(ii)
J
V2
Vn
(iii) -f—e
[2cos™^e^ - (ii) Jj-e"'"
,-«2/4
2cosa
e‘u + e
e'"
2
cos a +
4(ii)(-l)"(4-—
n
n
n+c
(iiO-P-T, k *0,1, 2, 3
n2-k2'
N2
(a +0) )
a<2;r
a >I2n
■a’a Q.5(i)-P-r[l-(-ircos^]
n -k
2m T, If m-n is odd
“T> n>0
-(-1)’e"] Q.6(i)p-n
2
. (»i)- n
•
L
0, n = 0
0,
If m-n is even
Q.7 (i).y(Z) = ._(1/8)e*’ t<0 (H) (0 = l[e-</-3)_e-4«-3)]H(f_3)
—e
[-(1/8)^', r>0
3
2 00
1
Q.8 (i)u(xj) = — f-------- -cos(iyx)e■ cmlda)t (ii)M(x,7) = —[—(l-cos(ty^))sin(a>x)e‘" 'dco
zr —'30 4 + <y
(iii)w(x,,y) = ~ j s|nh[p > )1—[2cos(q?x) + c?sin (a)x)]da)
2 4, (4 + <» ) sinh(<v zr)
i
Motilal Nehru National Institute of Technology, Allahabad
Department of Mathematics
Tutorial Questions for First B.Tech
Sub: Mathematics- II (MA-1201), Units -V & VI
1) Let V be the set of all ordered pairs (x,y), where x, y are real numbers .Let a»(xl>gyl) and b=(x2,<y2)
be two elements in V. Define the addition as a + 6 = (xp>'I) + (x2,j'2) = (2xl -3^,^ -/2)and the
scalar multiplication as a(x|t/() = («Xj !3.ayx /3). ShowthatV is not a vector space. Which of the
properties that are not satisfied?
2) Let V be the vector space of all 2X2 real matrices.
Show that the sets
And
3) Let V be the vector space of all polynomials of degree S3.Determine whether or not the set
S = p3,/2+f,?4-f + l} spansV?
4) Let v, =(1,-1,0),v2 =(0,1,-1) and v3 =(0,0,1) be elements of 1R* .Show that the set of vectors
{vp v2, v3} Is linearly independent.
5) Find the dimension of IR* spanned by the set
{(1
0
0
0),(0
1
0
0),(l
2
0
l),(0
0
0
1)}.
Ans:<4
6) Let T be a linear transformation function defined by
T
'r
2
u
-2
<3,
'r
J
fJ
-2 ,T
I? i
lo 1/JJ
7) Let T be a linear transformation form IR* into IK2 defined by TX = AX. A =
X = (x,^,z)r. Find Ker(T), ran(T) and their dimensions.
2
L\
I
1
0‘
-1
0
I
8)
For the set of vectors
(4.6) are ,njjncj
/'*
\where
transformation T: IR2-+IR\ such that TX^^~2
matrix of linear
~7) and7*2^"2
2
-1
2
-3
9) Write f* + / + l as a linear combination of the elements of the set S:|3ZJ2 -l,t2 + 2/4-2} .Show that
5 is the spanning set for all polynomials of degree 2 and can be taken as its basis.
Ans:
r2+r + l = [-r+(/2-l) + 2(/24-2r + 2)]/3
10) Reduce the following matrices to row echelon form and find their ranks.
■ 1
3
5'
2
-1
4
-2
8
2
’1
2
3
4'
2
I
4
5
1
5
5
7
_8
1
14
17
(ii)
Ans:2,2
11) Solve the following systems of equations using Gauss elimination method
(i)
4x-3<y-9z + 6w = 0
2x + 3y + 3z + 6w = 6
4x~21j'-39z-6w = -24
(ID
x+2y-2z = 1
2x-3j> + z = 0
3x-f-14y-5z = 5
Ans:( i)x = l + z-2w and y = (4-5z-2w)/3 where?, w are arbitrary.
(ii) ^ = 1.7 = 1.2=1
12) Solve the following homogeneous system of equations AX = 0 where A is given by
2
1 '
’1
2
<-3"
'1
1
-1
r
1
-1
1
1
-1
2
3
1
4
3
2
1-1
1
3
2—61
(ii)
(iii)
Find the rank () and nullity ().
Ans:(i)2,0 (ii)3,0 (iii)2,2
13) Determine which of the following systems are consistent and find all the solutions of the consistent
system
i
1
1
1
1 ‘
X
T
I
-1
1
1
0
1
1
-1
1
y
z
1
1
1
-1
vv
0
0
An^l1’3’3)
>'«)
14) Using Gauss-Jordan method find the inverses of the following matrices
1 ■
'l 1 0 l'
-1
1
1
1
-1
1
1
1
1
1
1
1
1
-1
1
4
4
1
1
1
-1
1
0
0
1
1
1
(i)
Ans:( i)
'-1
1
(H)
1
r
'-1
-1/3
1/3
1 '
1
-1
1
i
I
0
0
-1
1
1
-1
i
-1
1
0
0
-1/3
0
‘ 1
2
O'
16) Verify Cayley-Hamilton theorem for the matrix A = -I
1
2
1
2
1
1/4
1
1/3
(ii)
15) State and Prove Cayley-Hamilton theorem
1
1
1
-i
Also
(i) Obtain A ',A} (ii)find Eigen values of A, A2 and verify that Eigen values of /l2 are squares of those
of A ,(iii)find the spectral radius of A .
'-1
10
12'
1
11
10
-1
16
17
Ans:(i)
(ii)9,l,-l
(»i)3
'1
0
0
17) If A = 1
0
1 , then show that A" - A" ~+A2-I for n£3.Hence find A50.
0
1
0
' 1
0
O'
25
1
0
25
0
1
Ans:
18) Find the Eigen values an-! the corresponding Eigen vectors of the following matrix
1
0
1
0
0
1
Ans;^=[o.o,i]r,x,-[i,o,o]r
1
-1'
19) Show that the matrix A = -2
1
2
0
1
2
is diagonalizable. Hence, find P such that P~'AP is a
diagonal matrix. Then, obtain the matrix B = A2 + 5A 4- 3/ .
20) Examine whether the matrix
,where
is given by
'1
2
2'
0
2
1
-1
2
2
(i) A =
(ii) J =
-2
2
-3‘
2
1
-6 is diagonalizable. If so, obtain the matrix P such that
-1
-2
0
P~'AP is a diagonal matrix.
(il) diagonalizable, P~'AP = <//ag(5,-3,-3)
Ans: (i) not diagonalizable
21) The Eigen vectors of a 3X3 matrix A corresponding to the Eigen values 1,1, 3 are
[l,0, — l]r,[0,1,— l]r ,[l,l,0]r respectively. Find the matrix A
22) obtain the symmetric matrix B for the quadratic form
(I) Q = 2x ^3x^2 + x2
B=
Ans: (i)
2
3/2
3/2
1
(ii) B =
1
1
-2'
1
-5
3
-2
3
4
23) Verify Cayley-Hamilton theorem for the matrix A .Find A 1,if exists, where Jis
' 1
0
—4
"1
i
i
0
5
4
i
1
i
-4
4
3
i
i
1
(il)
Ans: (i)
P(2) = A3-92?-9A + 8Z = 0;J-’ =1/81
16
16
-20
13
4
4
-5
(ii)
P(A) = ^’-3A2 + 6A-4 + 2t = 0;J"1 =l + 3i/10
7-1
1
1
r-1
1
1
1
1
i-1
24) Find all the Eigen values and corresponding eigenvectors of the matrices given below, which of the
matrices are diagonalizable.
(i)
’1
1
1
0
i
—i
—i
1
i
3'
1 0
3
-2 0 1
’1
'0
1
(ii)
-1
-1
2
1
1
1
o'
0 -1
-2 1
-2 1
-2
(iii)
2
Ans: (i)
(«)
z.0.0:[l,0,0,l]r, [1,-1,-1.0]r;A-2:[l. 1,0,0)r^ = -2:[1.0.1.ir;diago„ali,aWe
1,1,1: [0,3, 2] ’> not (jiagOna|jZab|e
A
25) Show that the matrices given below are diagonalizable .Find the matrix P such that P 1AP is a
diagonal matrix.
0
2
1
2
0
3
1
-3
0
(H)
5
-6
-6
-1
,1
IT
2
J
-6
-4
Ans: (i)
A = 0: [3,1,-2]r; A = 2i: [3+i,l + 3i,-4f ;A = -2i: [3-i, 1 -3/,-4]r
p=
’3
3+/
3-i'
24
-8
16
1
1 + 3/
1—3/
=1/32 2/-6
2-6/
-8
-2
-4
-4 .
-2i -6
2 + 6/
-8
(ii)
A = l:[3,-l,3]r;A = 2.2:[2,0,l]r,[2,l,0]r
’3
2*
2
'-I
2
2*
-6
-5
3
2
-1
0
1 ;P-'=
rx
J
3
1
0
-1
26) Eigen values: 1,1,1; Eigen vectors:
Z1,
.
Ans:
-ii
P=
r
'0
2
2'
'1
0
0‘
1
-1
1
■,p-' =1/4 2
0
2 A = PDP~' = 0
1
0
1
1-1
2
2
0
0
0
1
27) Let a 3X3 matrix A have Eigen values l;2,-l.Find the value of the determinant of the matrix
B^A-A-'+A1
Using Eigen values :ntd 1? igen vectors to solve linear first order systems