Mathematics for Gate Objective Questions Part 12
Transcript of Mathematics for Gate Objective Questions Part 12
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I
.S.E
Pt,.21l(\4
M THEM TICS
t
L
2.
3.
tf 5[(1C) : ( ( 1
~ ~ l z .
xj'(x).
b ~ n ( ) is eq
u
al to
r- ..r 1
3.
14
b 7
8
c. I
d 11
7
Lei
g(x) be lhe inl'erse of
an
inve.rfible
functJon
f (x) = I t l.r
111en g'(xl
is equal
to
a
I
I l I T]r
b.
I
I+
(/l
JJ
c. I+ Jg(
7\
)j
1
d
+ 1 / ( x l l
A
function from
R
to
R(R
being
th
e
set
of
real
nu
mb
ers) is defined bY
the
following
tonnula;
/(X) = 15 JX I0[
The nwnhcr of poinls 111 \\'bJcb the
functi(lu (")
f l
(x)} is not
diOerentiable ts
a.
v
b. I
c. 2
d. 3
: l
f
y
=
I . - . then
:
is equal1o
II 2
J' m
a
tl
X
c. y
d.
c
' For the funcl10n (x) delined by
/
(X)
{
{
anwmf
1 f l \ f.f INHJII N t/
Which one of 1he following statements i$
corre R ns
fo llows
..(
(x)
=-2x'+
3x+..l
iJ'xe(
x.. J) and.
(x)=Lx
+
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12..
13.
14
.
15.
16.
17
.
31
d. 8
f
(t
.lg"lI
rt>) :(.t))d. s
I:C(Uol O
Ju
' (11
h. / C ~ ) g ( ~ l
c.
.fClilg'(xr/'lx)g(
x}
d.
f'(ll)g(x)-f(x)g'(x)
[he
m a g - ~ of
tit open
interval
( ), I) under
t h ~
tontlnuoos lllllpping
y
x - x
1
i>
' The
open intcrvnl(O.
b 'nte semi-closed interva l (0. ' ]
b.
Thuemi-dosed intcrvnl
LO
, ''fe.
th
e g -.:tl:.
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1.
T. : 4
IL ::i.
7
L
~ :1
,6
26.
nte
c r i ~ t i e
of
the ring
z.
L
n. ()
b.
6
c.
12
d.
24
7.7.
Which one vf llle tollltiVing stntemcnl\
io
no
orr
t?
n, n,e et of rationals
is
a field
b
~ , , .
lhe
rins
of
integefll
modnlo 31
is
field
c.
R[
x).
lite
$C l
of
polynomials QVc-r the
sci
of
N>l numbers is an ntogrnl
do
ma
in but not a field
d. R[xj is not ao iutegra l dom.ain
28. Let R
be
the set of the rea l nu
mbers and
"
r ~ l ) ;
Xt
c R.
1,l>l is
cqunllo
- (a, Zb)
b. (2 . ..,
(- " 2)
d. (a b . b)
32. ifT: v V
is
4 lm.,.lrllnlonnaiion of
vector Spnco V and Uoe C
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3. R
d.
h
1
- h
2
=u' 4g (oos'o. - cosll)
l'wch Qlher. k t W O ~
point
vf in Drsoclion of
be two
n1ds.
Their velochies nt p:>rllculor
time
are v
1
I u(9
v
1
'e"peCCI.ively
,
then they s
hoU
UV\Iid
C
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46 Consider I l l ~ following
sl4
temeniJ
regarding olgorithm vi'a pr
(Jblem:
I. l l begtns wi lh in&lmction 10 accept
iuputs
2. 'fl1c pro
nnd R Is tho
torreut R
'"
1101 A
correct
...
,;-pl
bna
tion
Il
l'A.
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~ ~
b, 25
C
IS i:u
d.
3()"
0
56. I f
co
;
l sin
; 1, r= t. 2.. : .
tbon Xt
. ~ .
.. . . . fupto in:finily) oqu: h
11
.
I
I
i
ll. I
c.. i
.[j
d _ _. _
1
z
S7
.
[he
equation whnse mots are
the
n"'
power> o f the
I Ool.> o f ~ u a l i o n x:2x
cosO I t 0 is
11.
2 . . x cos uO 1 0
bo x
2
f b. n01 I
l
c. xl - '2.'< oos u
l:b
1
0.
u.
x
2
,.
2l . A circle with
C>Jt
p
100
1
is "'lualtu
a. -2
b.
l
"' 0
d. 2
If'
ft... :tnd
y
are the ri>Ol'l
of
the
eC(Ualion
x- ~ 1
._x
~ n=O_thai _. .__ s
equ l
to
.....
f l-1
1l1 n
b. 111/JI
c
3
d -3
T I 1 ~
root ufthc Ct
iU>tiou
.
:oC
- lilt$
"
18x
1
-
3th +25
=I) ore
4 .
- I
2 ~
i
b. 1
:-
2i, -2
i
e. I - 2i, .2 "' i
d, - t 2i..-Hi
In
the group or m ~ e r u
rutitll\nl numhers
under the binary op"ftiou given b,v a
b"''lb/5, the idenlitv element 110\1 lbo
inverse
of I ore l l p e ~ t l v c y
"
5 and
15
1. S and 2518
c.
115
nd 1140
d. 1 and 118
U:t x
1
>
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al11ays
e l l ~
b, an inl'egt111domain
c.. an Interval ring
d. licld
68. '
Ote
rln
g of integers (n1od 6lis
a. finite intcgT3l donlllin
b.
tt11
inlioitc i
nl
cgro llfomo in
e.
alie
ld
d.
1101 311
integral doma in
(i\1,
II'
n
donate,; the number
vF
dcmentll in
n ~ J d . litrintc
of
the
form
~ k 1
c. a d u ~ t ol dislinct ~
d. power ofa prime
70. If
th
e
pair
of Rtruight
l i n ~
x
0
2pxy-
rU
ond X-;; 2qxy- . Y ~ he
s:
uch tho I
eaclt pail' bisects ~ 1 e ougl
b
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n. CA ) - I ,
g'(
p)=4 then lim &I J/ (
JI
) glpl/(O
-r I
equal to
lL 5
b. 10
e. -10
d. 5
8.4. An int"' rating
foe
tor of the dill"erenl
irol
e q o ; ~ t i o n
sinh v 1>: + cosh v d1= II is
4
"
"'
. X
)
d 'Y
85
. l'he degree ol' the differential equahOn
...1.. ......1.
-
Ui
(
J
) ' (d'.
)n
.
rr
u
3
b.
5
'
4
d. 'J
Rur9
.
Ti
tc c u r v ~ In
wldcb the
llluJ ent of the
angle between
the
tllngent and
the
radius
at an y point
i'l
equ;tl
to the tangent
nJthe vccll)rio l angle
ate
b. Sytem o f ~ t r . l i g
l l t l i n . , ;
h. S)'lltem
ofcircles
c.. Systerru of
pnrabl)ltL'
d. System
of
ellipses
87
, rr
e be an
r u r y c()nstunt, the general
wlutlon
of tho dlffo:tcn
tinl
cquolion
.x(x
1
3l):d.>t
3 ~ ) d y
o i5
. ... ..., .
\
..
n. tx
-y)'
b
~ . .
.. 2
. y ~ ~
e; (
.-?+y
1
).
x
1
yl:e
d. (x
2
-
y )
1
+x'l
=
1
88. tf c be au nrbitr:lry ""ustauL
Uu: ~ ; , : n . : r n l
l u t i Qfthe dlffet-enliol
equnlion
l t o r ~ l l x i s
a, xy
(c - log I81J
= I
b. xy(c l
og
lxl)=
I
C;
X}' (C 2 Jog X) 1
d.
X ) ( t .
0 8 X ~
I
89.
Tit
gcrtc'l'lll solution of the di.fftn:ntil
90.
91
.
. .i . \ ,, J
eqmUton ...
,m
~ . I S
,;..
,
a.
l
=.'\cos (3;-:+
B)-
,
\sin
X ~ n 3x
b. Auk.t B -
+ 2
si
n 1< ~ sin
d A
. ( ' BJ 3 . + .t
. " sm ~ X - t - stn x - cos
:n ~ - 1 .
3 1
The
l l n i - v e r t i c a l g l ~ of ~ i g l l l circulor
cone v i n g sebl of' three mutually
Jlerpendlcu
.largeneratol"
Is
given
by
a.
fDn
(
./2)
b. tnnl ..fi)
C.
T
d.
' 2
By mQans ol'
il
~ u h a b l c
tr.ln.limn
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b. i l ' ; t l i e ~ I
Jt oft
21
c. _ ;
=
6c log I
/(11
cl.
111
J -
6 isr
d.
ls
none ~ m
93
A
uniform straight beol
m
A.B < f
eight \\
and
length 2
l.s
111nd.s w
th.:
cn
ll A fi.
(o
d
lo
ground.
AB h
in
e
Inoo at
nn
30'
1
10
the
Tho c
ml
A i
ub
jcctcd
IO
H
a. ~ up
wa
nl f4rct W
b. vertical upw>td force \\' nnd n
horizontn l force
\V"2
"
vertica
l up
ward
fo
rce \V
and 4 co
upl
e
of mo
ment \V
U2
lL cnuple 11f 11C)t11 ~ ~
hod
zo
r$lly on lwo 5upports at c.n whW"d
AB= 2 , AC=B0=/12
'
11J
e
weigh of the
plank is \V and a man
of
w stam
wnlkin
g
from
end
W
th
o:.
oth
dr.
Tb
o plane
d
oll l
no t overl
W11
if
n. w
~
h. w< \V
c w > W
t
l. w
W/2
95,
rwu
Jor
ot.os P
and Q
I J I ~
- < ~ > U J t R.
lf P be
~ < l l , U1e oc\v ro
iiUllllnt i s ~ c b th
o
angle be tw
n
R
nnd
P
Then in
t> is given
by
u. 3R
b. R/2
c.
2.R
d. R
% .
AO)Aeni uf lorc
ply
p.1
n
b
3lan
ces wcighiS W
1
ond
\\1
in the atbcr
pan
. \V is given by
(U a J}IY-l
a.
Ill
b.
(If
.-
w )sr.
w
c.
.l..::.. . .
r z
d.
Wi
ITO )I
:
11 2
tOO. < : o K
clovercd
h) in
lime
t
is given b
yx
= 2 c
os
cit
(2a-
l )t . (a
"' I), .If motion is required to be simple
hannonic. llteo n.sbould be
a. I
.fj
b. I - fi
c.
L
;;: .(;
d - 1 t .f'.
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