mathematics objective questions part 13
TRANSCRIPT
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8/18/2019 Mathematics Objective Questions Part 13
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C.S.f.. MATHEMATICS (Prelim.) - 2005
C.S.E.
MATHEMATICS 2005
(PREUMINARY)
11me Allowed :
wo
lours
I.
2.
3.
4.
J:or what value
on
. arc the the lines x 3y • 3. z
>
y · 6
an
-
8/18/2019 Mathematics Objective Questions Part 13
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6.
I
...
tf f
x )
--------
. x >
0
l + x
then
what
is the
least
val ue o
f / {
/(x)}
+ f
{ f l x
}
(a
) I
(b) 2
(c)
3
(d)
0
7.
If
/
(K }
is
-defin
ed Over
j(),
l j then
wha
t
is
the
dom
ain of'
the
8.
f
\mct
io
n
f
(2
x .; 3)
?
a)
.1 ·'2,
I l
(h)
f3, 5
]
(c)
R
{.3i2}
d)
(
312. I )
(
i) /( :<
. , y
x) - f y
)
(n) f
( x) ·
I
•
x
g(x)
\.V
hat
is
d
1
······ f
(x } equ
al to'>
J
X
(a
) g(x)
h ~
(
c) g (
x)
V
· x
g (x )
where
p
o
W h
at is t
he val
ue of
- ..
R
x) a t
x 0
dx
a)
0
(h)
p
p
1
(
C)
; p-
(d )
6p
3
I 0.
I ·
an
g.(x
)
a
rc
two
"uncti
ons
co
minuo
m ev
eryw
he
re
and
. /
X ) I X
1
'g_
(:x)
(x lu
n
· ...
;;;· ·--
·
···--
th
ell F x
) is continu
ou s
ever
ywhere
. . ~
t x.
~
'
' '
' '
of
rollo
wi g
'
'
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8/18/2019 Mathematics Objective Questions Part 13
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(a
)
2
(b
) -
2
(
c)
.:: I
{ d )
« l
I I.
L
et f :
R
R
be
a mo
llotc
mic
nlly
d
ec re
as in
g
fun c tio
n
{
dclin
ed
every
wh
ere
()n
the
rea
l
I it le)
an
d le
f
be
di
sco n
tirtu
ous at
.
::e
rta
i n
po
int X
••• a
. W
hm
is
th
e: nat
ure
o r
th i
s
di
scon
tinu
ity•)
( a
)
R
emov
abl
e di
scon
t
in
uity
(b)
Either
removable
d i
iCO
nt·inuity
or
d iscon tin\ li ty
of
2nd
k in d
(c )
Di s
c o
n
l
in
uit
y of 1s
t k in
d
(d
) E
hcr
.
(c ) f
I . ·
l.
·
I )
.(d )
{
l •
I.
+
(a)
1
1)
(b)
{
1 . ·
1
0
13
.
/\
rod
o f le n
gth
o
ne
metr
e
begin s
·nic:al
p lane
p
1-.4
8
-
(a)
·
(b
)
5
(
•
( d)
) irb
>
0
L
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8/18/2019 Mathematics Objective Questions Part 13
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B.
y
1
"'
b (
x•+
x·)
i
fb
0
y
3
b(
xz
•
.xJ_ i
fb
> 0
A
B
c
2
.
Cu
sp
3
Co
nj
ug
at
e
po
int
4.
N
ei
th
er a
no
de
no
r a
cu
sp
no
r
a c
on
ju
g<
tte
po
in
t
A
B
C
(
a)
4
3
(b
)
3
4
i
(c)
2 3 (d) 3 2 I
16
.
Re
su
ha
nt
or
a for
ce
j
) a
lo
ng
x
-a
xis
. a
f
or
ce
Q
-
8/18/2019 Mathematics Objective Questions Part 13
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21.
.2
2.
2
3 .
2 4
.
25
.
T
w
ha
t
doc
s J
u
1
d t
eq
ua
l
t
o (
u
be in
g
vcit
y o
f
the
m o
tio
n) ?
0
na
'
a·
(c )
T
n
a •
T'
b)
(d
)
l t i
2T
n
1
-
-
··-
T
A particula r is m ov ing und
er
the
i
nl1ucncc
or
l
centra l
fo
rce
alo
ng
th
e
wrv
c r
·
ae
"
0
,
th en
fim:<
? d
irec
ted
tow
ar
d
s
n
i
s p
ro p
ort
ion
al
to
:
(a
}
r
3
(
b
)
r
2
c )
l f
r
(d)
1/ rl
0
If
a
p a r
tic l
e mo
ve
s alo
ng
a s
trai
gh t
iinc
ac
co r
d
i
n
'
s
2
•• at
1
•
. 2
bt ·
· c
wh
ere
s
is th
e
dis
plac
em
en
t a
t a
· i
ns
n
t t.
(a)
A
s
Its
(b)
A
s lis
+
(c) As IN (
d}
As
FQ
r w
ha
t v
a lue
of
(,
m ay
th
e f
orce
s
o
f
ag n
t
d e f
··
1.
f-
I
am
i
f
• 2
.
act
i ng
a t
a p
o in
t b
e i
n
c
qt
t
1
.
(a )
·2
5
·.
(c )
1·7
5
·5
T
hre
e
for
ces
3
.
4
a
nd
1its
ac
ting
a
t
a
po
·in
t
o
f a
b()d
y,
ma
kin
g e
qua
l
an
gle
s ·lh
·h
ot
he1
.
Wh
at
is
t
he r
na
gn itu
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8/18/2019 Mathematics Objective Questions Part 13
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2
6.
Fo
r<
:c
s o
f
m
ag
.n
d
I , 2
.
3,
4
,
5
N
ac
t
a t
an
:an
gu
la
r p
oi
n
t
o
r
a r
eg
ul
ar
h
e
xa
go
n
t
o
w
ard
s th
e
Qt
h
er
an
g
ul
ar
p
o
in
ts.
W
h
a
t
is
t
he
ap
p
n>x
i
ma
tc
va
lu
e
o
f m
a
gni
t
ude
o
f
t
hei
r
r
es
ul
ta
n
t
,,
(a
)
12
N
(
c )
7
·5
N
(
b)
/
SS
N
d)
IS
N
27.
Let
rnl
rn
1 , . • . ,
m"
he
it.
mass partic
le
s
at
p
os
it
io
ns
w
it
h
'to
• )
jl()S
it
io
n
ve
c t
o r
s
r,
,
r
2
, .
.. ,
•·
.,
re
s p
ec
t
ive
)
·
re
fe
rre
d
to
a
p
o
im
;s
o
r
igi
n.
t
f: ea
cl
\
m
o
ve
s
wi
.t
h
ve
lo
c
ity
i\
·.
ti
r (
w
h
er
·
;
.n
d
B
ar
c
con
s
t
an
t
vc
ct
or
s)
at
t
nn
c 1
th
en
wh
tc
h
o
li
l l1o
w
11
1g
is
c-o
r
re
c t
?
,\
J
[a
)
cc
n t
rl
o
r
o
f t
he
p
ar
tic
le
s m
o
v
es
~
o
c
a
y
A
(b
)
ce
n t
re
o
f m
n
's
o
r t
he
p
a1
·ti
c
e
:s
it
c
nc
it
)
1
1.?
r ~
(<
.:) T
h
e
e
'
k
i
(b
)
IO
Ox
"
0>
+
1
0
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8/18/2019 Mathematics Objective Questions Part 13
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c) n x ~ · l )
(d) 7x ' 'nx+3
under matrix
multiplication
and
N
·
r
J
y
( )
•\
.
\
'
~ }
where
a,
b.
c.
d
arc
I'c:1l mtmhct·.
(\msidcr
the f(>ll
-
8/18/2019 Mathematics Objective Questions Part 13
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35
.
H
=
{a +
ib
(a
1
b
)=
l}
Co
ns
i
d
er
the
fo
llo
wi
n
g
s
tate
m
ent
s :
.
.
H
is
a
subg
ro
up
o
f G
.
l
H
is
A
be l
ian.
W
hi
ch
o
th
e sta
tem
e
nts
g
ive
n
a
b
ove
is
ar
e c
orre
c
t
?
ta
}
l on iy
(
b)
2 ·o
nly
(c)
>h I
ana
2
d)
N
ei
ther
I
nor
2
r:}
Ab
e
lian g
ro
ups
m
ay
ha
ve
n
on
-Abel
i n
su
bg
rou
ps
\b)
No
n
-
A
;
.cli
an
g
rou
p
s
ma
y
have
Ab
elia
n su
bg
ro
ups
c)
Cy
cli
c
grou
ps
m
ay
hav
e
no
n-c
yc
lic
su
bg
ro
ups
i
;i)
No
n
-
}
l ic
gro
up
s
ca
n
n
ot
hav
e
c
yc
lic
su
bg
rou
p
s
1
n
t
he
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8/18/2019 Mathematics Objective Questions Part 13
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38.
39.
40.
41
(a) 2
(c)
4
(b)
3
(d) 5
Each
root
of the
equation
v
· 2 2 v ~ ' 1 3 0 v ~ - 2 4 3 y + 6 1 · 0
. '
'
i:; greater by p .tltan each root o f
the
equation
X • 6 x ~ · 38
··
) X 17 ' ()
What
is
the
value
of
p
?
(a) 16 (h)
8
(c)
4
d) 4
If the
mots
of
1hc equation (x a)
(x
b) k ·
c and d, then
x . c) x d) ·
k,
0 ?
(a) a and
·b
(b) a and l>
(c) a and b (d) a and +
What
is
the im:aginar;
pan
of
·
·-·(
:·
.
. .
n\ ;
"
·r
s i n ~ · -
s
)<
Sl
O$c1
(a) sin I 10 h ~
(c) s in i O
.tT(:J IIO
\Vhat rs the angle
a
whicl ~ v r · c. 2cosec 20 intersects
the line 0 '
r 16
?
(a)
0
(c) 2nl: i
42.
What is the n .1 o cal roots lying in the interval ( I, )
forthefun
·
· · · · x ~ ;
15x
.8?
(a) I (b) 2
(c) 3 (d) 4
•
•
·
J
(t
..
I) (t
2)dt
where x > 0 '
, I
0
b)
I /2
(c) :.V2 (d) 5 2
length s - 0 when 0 0) ?
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8/18/2019 Mathematics Objective Questions Part 13
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(
a)
(r
+
a)
s
in
o
.
c)
(
r
•
a)
c
os
ec
(b)
(r
..
. a
)
co
s o
.
(d
)
(r
--
a)
se
c
o
.
4
5.
W
ha
t i
s
th
e
ar
ea
bo
w1d
e
d
b
y
tJ
1e
c
ur
v e
y
...
x
s
in x
an
d
th
e
x-
a
xi
s
be
tw
ee
n
x
=
0
a
nd
x
=
2
x
?
u
n
(
c)
3
rt
4 6. W hat
is
(b
)
2
7t
(
d)
7
C
4
7.
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et
be
a
p
o
sit
iv
e
n u
m
be
r.
D
el
in
e
a f
un
c
ti
on
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····
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1
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ch
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a
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)
f {
a)
.
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y
Ro
lle
's
t
he
·•
h
er
e
ex
is
ts
a
nu
m
be
r
/c ly
in
g
in
th
e
o
pe
n
in
te
rv
e
·
u
h
th
a t
.f
(k
a)
·· 0
.
W
ha
t
is
t h
e
va
lu
e
o
f k
if
a
·)
>
l/
4
(
b)
11 r
(c
}
J
/ 4
4
8.
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t
is
tl
·
v
al
ue
o f
r•
r
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r
X
w
he
re
n
E
?
a
-
.
n
.
a
)
n
n
2
.
y
a
1
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8/18/2019 Mathematics Objective Questions Part 13
11/22
C
.
x
·
a
(
0
·
s
in
6
)
,
y
'
a
(1
·t ·
co
s
0
)
.
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.
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),
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I
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o
s
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4
.
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C
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C
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}
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I
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ts
?
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-
8/18/2019 Mathematics Objective Questions Part 13
12/22
(a)
1
6
(c)
12
8
(
b)
64
(d
) 2
56
Dir
eetill
IIS : T
he
fo
llo
w
ing
5 (
FIV
E)
item
s. co
nsist
of tw
st
a
m n
t ~
on
e la
bell
ed s
th
e 'A
sse r
t ion
(A )
' and
th
e oth
er
as ' .R
eas
on (
R}'.
You
arc
t.o
ex
am in
e th
ese
tw o
sta
tem e
nts
care
fu ll
y an
d St
:lect
the
an5
wcr
s t
o
i tem
s us
ing tl
tc
cod
es g
iven
bel
ow :
(a) Both A
and
R arc' in divid ually tru e
an
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8/18/2019 Mathematics Objective Questions Part 13
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58
.
5
9.
6
0
.
.(
) 1
.
.
>
to
re
d
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ll
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-
8/18/2019 Mathematics Objective Questions Part 13
14/22
(c
} T
he
kern
el
o f
con
sis ts
o
f
on
l
v tw
o e
le
me
nt
s
o
f R
d)
Nul
li ty o
f
i s z
ero
6
3.
L e t
T
: R
1
;
R
1
be
a li
near tr
ansf
om
l a
t ion
s uc
h th
at
64
.
·{
(
= (
)a
nQ ·
r
=
n
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s
r
:)) e
qua
l to
?
a
)
G
b)
3
2
)
c
;
c)
3
-
8/18/2019 Mathematics Objective Questions Part 13
15/22
Lo
nsw
cr tli
f"
l
ollo
wmg
stat
em
en
ts :
l.
y is
a lin
e
a
r
co
mb i
n
a
tion o
f
a
an
d
2.
i is
a lin
ear c
omb
inati
on
or
I
a
nd
Whi
ch
o
f
the
follo
win
g siate
mcn
t
s g
iv
en ab
l)ve
i
s/arc
corr
ccl?
(a)
1
ly
(b)
2
()n
ly
(c
) Bo
th 1
and
2
(d)
Neith
er I nor
2
6
6. W
hat
is
t
he
inve
rse of
the pe
rmu
tation
G
?
(a) (
;
; :)
(b
)
;
:
: )
«) ( ;
; ;
;)
(d
) ;
;
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; )
<
.
67
. Le
t II
and
B
he an
y \W
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n
matric
es a
nd
)
tr
A ~
an
d tr
b
~
l j
,,
.
~ '
'
i
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I
i ' I
Conside
r t
he fh
low
ing
s
t a
~ c
s
:
.
tr (
liB)
tr
( BA
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2
tr
(A
U)
tr
W
hioh
o
f th
<
' :
, 1 t
~ ( j
j F ab
o '
'
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-
8/18/2019 Mathematics Objective Questions Part 13
16/22
I ( 0
·-3
(c
) 0"
'
2l
3
0
l
2 3)
(d)
Q
2 0
s
69.
t\. hat i
s
th e
d eterm
inant
of th
e follo
wing
matrix
?
1
0
0
0
2
0
0
;
l
2 I
3
0
I
I
2
3
I
I 4
0
I
· ·
... .
l
I
2 l
i3
I
4
...
I n
(t )
70.
If the
system
of e
q
uat
io
ns
X i 2y ·
z '' I , (
p · 1 2 )
Z
;
•
is
inco
nsiste
nt, the
n wh
(a ) 2
(b)
(c) 0
(d
) 2
7
1. Ir
th e e
quat
on
+ bxy
2
+ y3 '
0
represents
tl ·
two of
wh
i
ch
are perpc
nd
icul r
tO
ea
ch
other
.
is the equ
a
tion
M th e L
hird li
ne ?
a)
y ·
(h )
y -
b
x
•
(
d)
y
·X
n
h
t l ine is
define
d by th
e ime
rscctio
n of a
plan e
paralle
l
l
lane and
a p
l;inc thr
nn
} f dir
ec
ti
on
ratios.
of t
his stra
i ght lir1e
a) (0
,
p,
I)
b)
(p,
0. 1)
1. ;
) l. 0,
p )
(d)
(L
p, 0)
73.
For w
hat v
alues o
r A an
d B,
a re th
e two
p lane
s repre
sented
by
t
he
equ
-
8/18/2019 Mathematics Objective Questions Part 13
17/22
ort
h
o
go
n
a
l '
?
(<
l
A
'
l
,
B
,
I
(c
)
A
··•
2
,
B
·
·
I
(
b
)
A
·••• ·
··
2
,
B
·
I
(d
)
A
"
-
1
,
B
'
2
7
4
.
W
h
a
t
i
s
th
e
n
m
xi
m
um
d
i
st
a
nc
e
fi
·o
m
t
h
e
p
oi
n
t
(1
, 2
,
I
)
to
h
e
s
u
r f
ac
e
o
f
th
e
x
2
+
y
2
+·
z
2
·•
••
• 2
4
?
(
a)
.J
6
(
h) 2.J6
(
c
)
3
i6
(d
)
4
/
6
7
5
.
F
o
r
w
h
a
t
v
a
lu
c
(
s
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o
f
a
,
v
. i
ll
th
e
tw
o
p
o
m
ts
(
I
. a
.
I
) a
n
d
C
3
,
0
, a
)
lie
o
n
o
p
p
o
si
te
s
i
d
e
s
o
f
<
he
p
la
n
e
3
x
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4
y
.
. 1
2
7..
'
1 3
•
0
?
(
a)
a
·
I
or
a >
ii
3
·
(
b
)
a
"
0
o
n
ly
r
:,
(c
)
0
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a
<
i
(
d
)
I
<
a
<
I
-
J
7
6
.
T
h
re
e
ve
c
to
r
s
o
f
m
ag
n
i
tu
d
e
s
I,
2
,
3
u
n
i
ts
m
ee
t
c
Q
rn
er
o
f
a
cu
b
e
a
re
d
ir
e
c
te
d
a
lo
ng
th
e
d
ia
g
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a
ls
·
I
e
lit
ces
me
et ing at the corner. W
hat
is
the
m a
~
e
ir
r
c
su
l
ia
n
t
?
(a
)
4
u
ni
t>
(b)
5
~
(
c j
6
(
d
)
~
h
~
..
4
- -
...
...,
7
7
.
O
A
B
C
i
s
a
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ar
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l
le
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g
r
am
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i
th
0
.
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o
,
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c
.
W
h
a
t
is
th
e
e
x
p
r
es
s
io
n
o
f
a
+
1
2
1
ta
l
to
?
•
>
b
1
2
(
l:
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·
;
2
~
7
8
.
t\B
CD
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is
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.
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A
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A
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F
.
}
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a
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n
d
J
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-
8/18/2019 Mathematics Objective Questions Part 13
18/22
79.
80.
81.
82.
liJ.
84.
• )' \.
l
···l>
..
•
···•
.... • •
If p ··· b x c. q c ' a and r ,, a .,
hare three
v ~ c t o r s then
which
'
is parallel
w
q " r
·a pctvcnd
ic
u lar to
q
··•
(b) <
(C)
·l> .
a is para lcl tc• r
:<
p
{
is J>Crpcndintlar
to
r
•
.
.
( I)
..
p
b
'
:
b "
a
is C
-
8/18/2019 Mathematics Objective Questions Part 13
19/22
a
)
+
s
(
;
•
·
y
·
e
8
6
.
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-
8/18/2019 Mathematics Objective Questions Part 13
20/22
C •
. •
s h:
x
·
(d )
y c , e
+ c
1
e ·
- -
2
-
+ · · ··
88 .
Curv
es pa
ssing
through
the p
oint
(0, I
) and
sat is
fying &he
(
dy)
equ
at ion s
in
~
•• c
r
e
gi ve
n b
y
(b)
CQS ;;: l)
; : ~ c
89.
W
hat is
th e s
ingula
r sol ut
ion of
the d
i
ff
eren
tial u
p ···
In
( p x
- y)?
.
r
> •
(
a}
y ·
ex
·
e
(b
)
y
~
(C )
y
·.··
X
h1 X ·
X
(d)
)
C
dy
~
l
T
he di11
c-rem
ial e
qu
atio
n
· c
os
y
i s r
ed
u c
e
d
dv
~
to
li neat lo
rrn
·
r
c P an
d Q
arc. fim
ct io n
of
d
x:
90
.
x
alone.
by ch
angt
c
· r ia b c
as
(a )
sin ·
v
(b )
cos y
::: v
(c)
tan
y
(d )
sin
2y
'
v
9 1
.
.
·
{
-- y (l
n
y
In
x .; l) >
x
e'
b)
y ·:
xc••
'· · 0
) •
(;e
•
··
0
d)
x
···· e
-
8/18/2019 Mathematics Objective Questions Part 13
21/22
a
r
b
it
r
a
r
y
c
o
T
h
e
n
i
n
g
e
n
e
r
a
l
1
a
)
y
1
+
y
z
i
s
i
t
s
s
o
l
u
t i
o
n
,
b
u
t
c
y
1
i:
s
t
b
)
i
s
it
s
s
o
l
u
t
io
n
.
b
u
t
y
1
+
y
2
i
s
n
(
)
t
(
c
)
b
o
t
h
y
1
•
·
y
z
a
n
d
c
y
1
a
r
c
i
ts
s
o
l
u
t
i
o
n
s
(
d
)
n
e
i
E
h
c
r
y
1
+
y
1
i
o
r
c
y
1
i
s
it
s
s
o
iu
t
i
o
n
9
3
.
C
o
n
s
i
d
e
r
th
e
s ta tements regardi ng t
he;
di fferenti
al
e
q
u
a
t
i
o
n
'
\
d
y
;
I
l
d
x
j
y
'
0
,
0
l
0
n
~
y
(
O
)
:
I
:
r
"
.
.
I
t
is
-
8/18/2019 Mathematics Objective Questions Part 13
22/22
(
a)
b
·
a
(c
)
c
a
)
1
/4
(t:)
I
f
(b
) a
···
· b
(d
)
1
c
(b
)
l
/
3
{
d)
l
W
h
at i
s t
he
le
ng
th
of
a
n a
rc
o
r t
he
cu
rv
e
y
=
x
-
1
X
· 2
1
l·i
·om
x
I
to
x
3
?
(a )
J
/
3
(
c)
2
(
b)
1
2
(
d)
3
9
9.
W
hat
a
re
th
e a
sy
m
pt
ote
s
o
f th
e
c
urv
e
(X
' 4
)
'
I· I
)
(
X
···
4y ·
I )
"
4
?
(
a)
y
•·· i4
(
x
I
)
(b
) y
·' ·:
:
4 x
C
J
C
)
X
. . ±
4
y
(d)
X
i =
·
i 4
)'
•
1
00.
At
w
ha p
o
int
o
n
he
c
ur
ve
y
((
e '
c
- )1
2F
, i
s
fn
ur
e
maxim um ?
~
a)
tJ
f2
,
I
1
2
)
(b
)
(0
,
I)
(
C)
(2
, 0
)
(d
)
;t q}