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8/11/2019 JEE Questions Determinants
1/10
IN
T
l1
t I:
I
'-
'
+3
}.
,_
,
.
,
'
L
e
t
p
+
qA
3
+
r
J
2
~
s
A
).
+I
-n
).
-4
_
,
t
. +
4
be
a
n
id
en
ti t
y
in)
w
h
er
ep
q
r
.
an
d
I ar
8/11/2019 JEE Questions Determinants
2/10
/
J
e
t
e
r
m
m
a
m
s
I
0
0
'
a
1
1
"
"
a
:
-
b
c
l
1
1
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I
"
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I
I
"
0
I
b
.
b
-
1
-
1
1
i
'
c
-
a
b
l
1
1
c
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I
1
1
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,
I
.
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l
o
g
,
y
l
o
g
,
:
z
i
I
.
,
\
=
l
l
o
g
,
,
x
'
:
z
l
I
o
g
o
x
l
o
g
,
y
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I
o
g
y
i
I
_
I
l
o
g
.
x
,
y
,
z
a
r
e
i
n
G
.
p
_
T
h
e
r
e
f
o
r
e
(
b
)
i
s
t
h
e
=
v
c
r
.
"
3
.
L
e
t
l
J
.
=
c
o
s
p
-
d
c
o
s
p
x
l
s
i
n
(
p
-
d
)
x
s
i
n
p
x
A
p
r
l
y
i
n
g
C
1
_
.
c
+
C
,
i
1
+
a
'
"
-
o
o
s
(
p
+
B
(a)
c(Cl)
A (a
)
B' (a )
C
(a )
j A (
a )
B (
u } C'( o
:)
oj l'(
a)=r
A(o.)
B
(a) C
(a)
IA
'(a
) B
'(a)
C'
(a)
as
R
1
and
R,
ar
c id en
ticaL
Thus. a a repeated
rl
oro)>
(x)
=
0.
H
ence .
>
x)
is d iv
isib le
by
f(x
).
c
,
'C
>+ l
C
,,,
l.e
L\ =
)c,
rc .
1
c,
.,
C,
'
C,.,
c,
l
Ap
plyin
gc, -
>C
3
+
C,
-
0,
'C
,.
I
c c,.
,
,
y-'c
r -2
C
C
' '
'
1
O S,
,
T
- L(
'"'
' r
I
-
2
'
'C,
' ,
'c,
'
-' 'c ,
,
'
I
'C,
''c
,
,_
,c ,.
,
appl
yiTig
c, >C,
'
Co
I
'C ,
,.;c
'. 'c
'
'
,
y
( ,
J
c
'
,c
,
,
-
+'
,
'
' ' 'c
,.,
'
C,
c ,.
,
'
'
'
C,
'C
,
.
,
'c,
.,
'
C
,
c,
_,
C,.
,
'c ,
, c,+
,
'c,.
,
c,
x.'c
,
'
'r
.,
'C ,
'c
) +
0
,,
-c,
,
c
'c
, '
' .
c
r
'
6. T
he
sy
slem
of
"'1uat
ions ho s
non-tr
ivia l
soltLti
on if,
6.
sm 3
8
_,
co
slfl
'
r
expan
ding
along
C
1
w
e ge t
co
sin
JEI (2
8 -2
1)-c
oslf l
(-1 -7 )
+ 2
(
-3-
4)=
---
'
7sin3
EI
ti4c
os: G
-14
=0
=
>
sin3
l+2 c
os20
-2= 0
=
3sin
l 4s
in '
1
()
9 -
2 ~ 0
=>
h t 9
- 3 ) ~
=>
s
in9(2
sin9
-1)(2
;infl
13
)=0
=
>
sm
l=O
,s in 0
-11 2
{n
eglect
m g
H
= - 3
O-n11;mt+ (-l)'
7
8/11/2019 JEE Questions Determinants
6/10
"
l.:
(a-1)
a=l "
'
"
~ (a-1)
2
l.:
,.,
=
'"'
4fl
-2
" = 1 "
_,
"
.; (a -1)
3
o= I
'"'
3n
1
-3n
"'"
'l
"
'
'
~ ( n - 1 ) ( 2 n - l )
' '
(4n-2)
'
1
(n
l
J'
'"'
'
'
Jn- -3n)
'
I
' '
' (n-1)
Go
"
'
(4n-2)
'
)
n(n -1)
3n On - Jn)
'
j I
'
'
(n- l ) : (2n- l )
"'
12n-6
" I
_ n
- I )
'
C,
- 6C
, I '
,
=.::._j_".-21m - I
'
o o
" I
'
0,
,_,
"
'
\ , = C (C = 0
i "- consran )
"='
8.
We know,
A28=Axl00+2x10< 8
389=h 100 +
B x 10+9
an.i 62C=6x100+2xlO+C
Since ;
A211 3B9
and
62C
are divisible by K th..-efore
I h ~ r e exi>< positive integers m
1
, m
1
and m, il= IO lA+2dO
Hl{lx3-rJOxB+9
' .. (1)
' I
'
0 0 x 6 + ~ 0 x 2 + C I
A
' '
A28 lEO
62C
(usin
'
'
'
'
)
'
I'
)
=
m
1
K
m
1
K m,K ~ K m
m, m,
'
B
'
I '
B
,
'
. l = mK, Hence dctcrrnimml is divisible byK.
lp h ,.]
9.
Let,
1'1 =
a
q
c
'
a
h "i
Applying R
1
-->
R
1
- R
1
and R
3
-->
R, - R
1
we ge
p
h :
I
t'l.=
a - p q - b u
a - p
0 r - 1
~ c ~ p
q-h:l+(r-c)' l
' o
I
a - p 0 a - p q-b
,__,.(a- p) (q - b
-.-(r-c) {p q
- b ) -
b(a-
~ - c a -
p)
(q -
h)+
p(r-
c)
(q -
h
-b r -c a - r -
d - 0
=> -< a -p) q -b)+p r -c i q -b)
- h ( a - p } ( r c ) ~ O
c p b
_ _ _
r - c
p - a
q-b
{Ou Jividing both sides by a-
p}(q-
b) r- c
p b
,.
- - - - - . - 1 - - - 1 ~ 2
p a
q - h
r - c
- ' - - - - ' - ~ 2 .
p - a
q -b r -c
I
n (rl+l) (n+2J j
10_ D=
(11+1)
(n+2)'
(n-.-3)
I (n+2)
{n-t-3) (n+4) 1
( g i
J'akiTig n , {n + 1) and n +
2)
common from R
1
, R
2
R
3
=pectively.
(n+l)
( n + l ) ( n + 2 ~
lhn {n+l) '(n+2) (n+l)
n+ l ) nd )
il {n+3)
{n+3)(n+4)
ApplyingR
2
-;.R
2
-R
1
andR
3
-;.R
3
-R,,weget
l j (u+l)
(n+l)(n+2J
D=n {n+l)l(n+2) 0 l 2n+4
0
2n+6
Expanding
alongC
1
, we get
D
= {n )(n+
l) (n+2) [{2n ; .6) -
(2n+
4)]
8/11/2019 JEE Questions Determinants
7/10
D= (n )(ll-t-1)
(n 1 2) [2]
Divide both
non-trivial '"luti
on
Jr..
. )
.= cos2a -rslu2a
{we know
,- y u +
1
/
:> usinO + bwoH
i
/a
'-r /J)
-- . 1 )
Again
when).= 1,
ros2ct +sin 2n = 1
or ~
c o s 2 c t + { s i
n 2 a =
I_
.. 2
?
=>
c o s 2 o : - n / 4 )
- c o s ~ / 4
2o.
-
1t
l 4 ~ 2 n 1 t ni 4
=
>
2a
=2
n7t-1tl 4 < 1t/
4.2a + nl
4
+
n/ 4
a = nn
or
""
+
"/-1.
cos(
A -P)
cos(A-QJ ros{A
R)
12
cos(ll-f ')
cos(B-Ql
cos(lJ-1(> (gi,en)
:cos
(C - ') cos(C
-Q) co;(C
-Ril
'Eos
AcosP+sinA\i
nP
o > ( - ~ -Q
=> J.=
_-BcosP-r
smBsinf' cos(fl-QJ
osc
c.,, f '+ ' lnCsm
P c o s C - ~
)
cm(A-R)
cos(B-R)
cos(C-R
)
]c
osAco;f'
cn>(A-Qj co.'(A
-R)I
' w s / J c o s f ' cos(B-( ) eo ; B
-R)
cosCcusl'
~ < J > ( C ' - Q ) w
o(C-R)j
js
in4sin/'
cos(A-QJ cos(
+
l,;nBsinl' cus(B-QI
cos(E
~ > J I
I C > t n P
wo
(C-(J) co
siC
co'A cos(A-Q)
co '( ,f-R
)
= tl= cos f
cosB
cos(B-Q)
cos( IJ
II)
co ll
sin II
sin
B
co
>C
sin
(
'
inC,
j '
iiiA W>A c
~ , i n l '
Q c o s l l i s m B
ru>B c
j ' inC
co>C co
8/11/2019 JEE Questions Determinants
8/10
Le
a>O.d>O
and le
t
'
1
(a+dl
a(a ~ d )
'
(a+d)(a+2dj
(a+d)(a
+2d) (u+
2d)(at3d)
,(a+2d)
(a+2d)(a
+3d)
(a+
3d)(a+4dl
'
km
g
tommo rl fro
tll
u(a+d)(a+2d)
-cccc_''c
cc--c'
,.
-;-
(a
+
d)( a
+
2J
){cr ... 3d
)
'
rom R,.
-
from R,
( a + 2 d ) { a
) ( a 14d)
a (a td l '( a
2-d
)
3
(a+3d) '("+4J
)
'
(a+d)(
a3d)
(at3d)
(a+4d)
a .
2d )
(a+
3d
(a+4dl
( a
~ d ) I
(a-,2d
ll
0---,,---:---,_---
.
a_(a
+
d)
- (a + 2Jj
(" + Jd)" (a +
4d)
(a+d)(a+
2d) (a+2
dl a
e r ~ L l . ~
(at2d)(a+J
dl (a+3d) (a+d)
'
a
. J d ) { a ~ 4 d )
(a+4d)
(a+2d)j
A pp
ly ingR
2
->
R
2
- R
1
,R
1
->
R,
- R
,
l
(a+d)(
a+ M ) (a+2
d)
=
(a+2d)(2d
)
d
d
:
(a+
3d )(2d) d d
Applying
R,
-
>
R3 - R,.
we gel
( a + d J (
a ~ 2 d )
(a+2d) a
(a+
2d 2
d d d
E ? . p > ~ . ~ d m ~
Jlt>ngR
3
. ""gel
, \ ' 2d ' l : 2d
:
.'1
'
=
(2d
2
)(d)( a - 2-d
- a 1= 4d
4
TherdOrc.
14
smcea,b
,carop ',q'' andr'
1
hmnsofHY
I I
.
=; - -,-.
-.arcmA.P.
a
b e
_ _ ~ A + ( p - i ) D -
;
I
- = A - r
q - I ) U ~
;
I
- - A ~ v
- I D
'
I
be ca
, I
Ll .= p q
' I
_
..
I
a b
~ a h c l p
q r
(
uoing
A + p -1 \
D
A+
q-IJD
A+(r-11/J
I
Applying R
1
---->-
R
1
-(A--
D)
R -
DR,, we gd
U 0
I
=abc p q
~ I t
pqr
O
A pp
lyingR, ---->-R -R 2R,,ceg
d
2=
2ax-l 2w:+bt l i
f ( x ) ~
h
_,
0
12': '
2m-liJ2a.t -11
h--rl
lb
il
(Usin
g(', ->Co
-C
j ( c ) ~ 2 a x - >
b
In t
egrating,
w ~
gel f
x
)- uc' + bx
+
c
whe r
e c
is ""
arbilra r
) cons tant S i n ~
h o s x i m ~
x ~ 5 1 2 .
I
2 ) ~ 0
5a b=O
A
lso.
j
(0)=2 c=2
d
/ 1
) -1
-
a+h+c-1
Solving for
(I )and
2)
fur a
b
l 'e
ge\
a = l l ~ , o
~ - 5 1 4
/ ( r ) - _
_ ~ c - ~ X I 2
' '
I
hus,
.
. .
8/11/2019 JEE Questions Determinants
9/10
>
inW
1
' 0
4
:r
'
'
'
inl
2e-
I
' '
'
A
pp l
y iug
R,
---
.
R, +
1?
3
'
B
J
r.
c o
s -
;
sm
20
'
'
~ ,
il l
,
3
I
~ 2 o
m H
cos
(::t
- n
,3 )
e
"/3
an
d ru
>
le +
2
1t
I 3 +O
O\
(6 .
2"
/
J
)
~ :
r o s
-
-
J
co s
3
-
_1
1
Z.
+a-
2
]
ro
2
-o+
2
; 1
l
'
'
.
J
-
1cn
; a
-
cos(
21t 1
3)
'
I J
2c
osa.
l- ;:
;
=
-ro
se
'
~
- ~
,
- h
.
, ~
~
,
an
u
'Ell
1
-
sm
( ~
-
-
' '
""+
+ =
- -
l
.,
4n
..., 1
7: ,
-2
sin
-
J
2
3
J
(
20+
4"
-2
+4"
''1
x c o
s ~
3
_]
.
'
'
-2
sin::
U.cl
'>(;-
r 1 ;-rl_
lJ
-
-- C
'
2
>C
,
a
' ~ b
' -
c '
'
ov
-1-h
,-
b l-
-e -
w ;
'
I
b-e
y
ay+
fn:
b v
~
=
h
1
c
y
c
x ~ a
b+C
f
- ~ -
c .
l
b
+
C)