jee questions determinants

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


2/10

/

J

e

t

e

r

m

m

a

m

s

I

0

0

'

a

1

1

"

"

a

:

-

b

c

l

1

1

"

"

"

I

"

"

I

I

"

0

I

b

.

b

-

1

-

1

1

i

'

c

-

a

b

l

1

1

c

'

I

1

1

'

,

I

.

'

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

'

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


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)]


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


8/10

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,

.

. .


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)

jee questions determinants

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