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Determinant(Sheet)

Best Approach

Manoj Chauhan Sir (IIT Delhi)

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No. 1 Faculty of Unacademy,

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Maths IIT-JEE ‘Best Approach’ (MC SIR) Determinant

KEY CONCEPTSDETERMINANT

1. The symbola b

a b1 1

2 2is called the determinant of order two.

Its value is given by : D = a1 b2 a2 b1

2. The symbol

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

is called the determinant of order three .

Its value can be found as : D = a1

b c

b c

2 2

3 3 a2

b c

b c1 1

3 3+ a3

b c

b c

1 1

2 2OR

D = a1

b c

b c

2 2

3 3 b1

a c

a c

2 2

3 3+ c1

a b

a b

2 2

3 3....... and so on .

In this manner we can expand a determinant in 6 ways using elements of ;R1 , R2 , R3 or C1 , C2 , C3 .

3. Following examples of short hand writing large expressions are :(i) The lines : a1x + b1y + c1 = 0........ (1)

a2x + b2y + c2 = 0........ (2)a3x + b3y + c3 = 0........ (3)

are concurrent if ,

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

= 0 .

Condition for the consistencyof three simultaneous linear equations in 2 variables.(ii) ax² + 2 hxy + by² + 2 gx + 2 fy + c = 0 represents a pair of straight lines if :

abc + 2 fgh af² bg² ch² = 0 =

a h g

h b f

g f c

(iii) Area of a triangle whose vertices are (xr , yr) ; r = 1 , 2 , 3 is :

D =1

2

x y

x y

x y

1 1

2 2

3 3

1

1

1

If D = 0 then the three points are collinear .

(iv) Equation of a straight line passsing through (x1 , y1) & (x2 , y2) is1yx

1yx

1yx

22

11 = 0

4. MINORS :The minor of a given element of a determinant is the determinant of the elements which remain afterdeleting the row & the column in which the given element stands . For example, the minor of a1 in (Key

Concept 2) isb c

b c

2 2

3 3

& the minor of b2 isa c

a c1 1

3 3.

Hence a determinant of order two will have “4 minors” & a determinant of order three will have“9minors”.

5. COFACTOR :If Mij represents the minor of some typical element then the cofactor is defined as :Cij = (1)i+j . Mij ; Where i & j denotes the row & column in which the particular element lies. Notethat the value of a determinant of order three in terms of ‘Minor’ & ‘Cofactor’ can be written as :D = a11M11 a12M12 + a13M13 OR D = a11C11 + a12C12 + a13C13 & so on .......



6. PROPERTIES OF DETERMINANTS :P 1 : The value of a determinant remains unaltered , if the rows & columns are inter changed.

e.g. if D =

a b c

a b c

a b c

a a a

b b b

c c c

1 1 1

2 2 2

3 3 3

1 2 3

1 2 3

1 2 3

= D

D & D are transpose of each other . If D= D then it is SKEW SYMMETRIC determinant butD= D 2D = 0D = 0 Skew symmetric determinant of third order has the value zero.

P 2: If any two rows (or columns) of a determinant be interchanged , the value of determinant ischanged in sign only. e.g.

Let D =

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

& D =

a b c

a b c

a b c

2 2 2

1 1 1

3 3 3

Then D = D .

P 3: If a determinant has any two rows (or columns) identical , then its value is zero.

e.g. Let D =

a b c

a b c

a b c

1 1 1

1 1 1

3 3 3

then it can be verified that D = 0.

P 4: If all the elements of anyrow (or column) be multiplied by the same number, then the determinant ismultiplied by that number.

e.g. If D =

a b c

a b c

a b c

1 1 1

2 2

3 3 3

2 and D =

Ka Kb Kc

a b c

a b c

1 1 1

2 2

3 3 3

2 Then D= KD

P5: If each element of anyrow (or column) can be expressed as a sum of two terms then the determinantcan be expressed as the sum of two determinants . e.g.

a x b y c z

a b c

a b c

a b c

a b c

a b c

x y z

a b c

a b c

1 1 1

2 2 2

3 3 3

1 1 1

2 2 2

3 3 3

2 2 2

3 3 3

P 6: The value of a determinant is not altered by adding to the elements of any row (or column) thesame multiples of the corresponding elements of any other row

(or column). e.g. Let D =

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

and

D =

232323

222

212121

cncbnbana

cba

cmcbmbama

. Then D= D .

Note : that while applying this property ATLEAST ONE ROW (OR COLUMN) must remainunchanged .

P 7: If by putting x = a the value of a determinant vanishes then (xa) is a factor of the determinant.

7. MULTIPLICATION OF TWO DETERMINANTS :

(i)a b

a bx

l m

l m

a l b l a m b m

a l b l a m b m1 1

2 2

1 1

2 2

1 1 1 2 1 1 1 2

2 1 2 2 2 1 2 2

Similarly two determinants of order three are multiplied.



(ii) If D =

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

0 then , D² =

A B C

A B C

A B C

1 1 1

2 2 2

3 3 3

where AAi , Bi , Ci are cofactors

PROOF : Consider

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

×

A A A

B B B

C C C

1 2 3

1 2 3

1 2 3

=

D

D

D

0 0

0 0

0 0

Note : a1A2 + b1B2 + c1C2 = 0 etc.

therefore , D x

A A A

B B B

C C C

1 2 3

1 2 3

1 2 3

= D3

A A A

B B B

C C C

1 2 3

1 2 3

1 2 3

= D² OR

333

222

111

CBA

CBA

CBA

= D²

8. SYSTEM OF LINEAR EQUATION (IN TWO VARIABLES) :(i) Consistent Equations : Definite & unique solution . [ intersecting lines ](ii) Inconsistent Equation : No solution . [ Parallel line ](iii) Dependent equation : Infinite solutions . [ Identical lines ]

Let a1x + b1y + c1 = 0 & a2x + b2y + c2 = 0 then :

a

a

b

b

c

c1

2

1

2

1

2

Given equations are inconsistent

&a

a

b

b

c

c1

2

1

2

1

2

Given equations are dependent

9. CRAMER'S RULE : [ Simultaneous Equations Involving Three Unknowns ]Let ,a1x + b1y + c1z = d1 ...(I) ; a2x + b2y + c2z = d2 ...(II) ; a3x + b3y + c3z = d3 ...(III)

Then , x =D

D1 , Y =

D

D2 , Z =

D

D3 .

Where D =

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

; D1 =

d b c

d b c

d b c

1 1 1

2 2 2

3 3 3

; D2 =

a d c

a d c

a d c

1 1 1

2 2 2

3 3 3

& D3 =

a b d

a b d

a b d

1 1 1

2 2 2

3 3 3

NOTE :(a) If D 0 and alteast one of D1 , D2 , D3 0 , then the given system of equations are consistent and

have unique non trivial solution .(b) If D 0 & D1 = D2 = D3 = 0 , then the given system of equations are consistent and have trivial

solution only .(c) If D = D1 = D2 = D3 = 0, then the given system of equations are consistent and have infinite solutions.

In case

3333

2222

1111

dzcybxadzcybxa

dzcybxarepresents these parallel planes then also

D = D1 = D2 = D3 = 0 but the system is inconsistent.(d) If D = 0 but atleast one of D1 , D2 , D3 is not zero then the equations are inconsistent and have no

solution .

10. If x , y , z are not all zero , the condition for a1x + b1y + c1z = 0 ; a2x + b2y + c2z = 0 &

a3x + b3y + c3z = 0 to be consistent in x , y , z is that

a b c

a b c

a b c

1 1 1

2 2 2

3 3 3

= 0.

Remember that if a given system of linear equations have Only Zero Solution for all its variablesthen the given equations are said to have TRIVIAL SOLUTION.



PROFICIENCY TEST-01

1.

2

2

2

1 a a

1 b b

1 c c

2. e

10

1 5

log e 5 5

log 10 5 e

3.

19 17 15

9 8 7

1 1 1

4. The value of the determinant

4 6 1

1 1 1

4 11 1

is :


31 37 92

31 58 71

31 105 24

is :

6.

a b b c c a

x y y z z x

p q q r r p

7.

2

2

2

1 a a bc

1 b b ac

1 c c ab

8.

1 1 1

1 1 x 1

1 1 1 y

9. The roots of the equation2

1 4 20

1 2 5

1 2x 5x

= 0 are

10. If a b c, the value of x which satisfies the equation

0 x a x b

x a 0 x c

x b x c 0

= 0, is



11. If a + b + c = 0, then the solution of the equation

a x c b

c b x a

b a c x

= 0 is :

12. If

x 1 3 5

2 x 2 5

2 3 x 4

= 0, then x =

13.

1 a b

a 1 c

b c 1

14.3 3 3

1 1 1

a b c

a b c

15.

0 a b

a 0 c

b c 0

PROFICIENCY TEST-02

1.

a b c

b c a

c a b

2.

a b a 2b a 3b

a 2b a 3b a 4b

a 4b a 5b a 6b

3.

b c a a

b c a b

c c a b

4. The roots of the equation

1 x 1 1

1 1 x 1 0

1 1 1 x

are

5. One of the roots of the given equation

x a b c

b x c a 0

c a x b

is :



6.

x 1 x 2 x 4

x 3 x 5 x 8

x 7 x 10 x 14

7.

2

2

2

1/ a a bc

1/ b b ca

1/ c c ab

8.

2 2 2 2

2 2 2 2

2 2 2 2

b c a a

b c a b

c c a b

9.

1 x 1 1

1 1 y 1

1 1 1 z

10. If

y z x y

z x z x

x y y z

= k(x + y + z)(x – z)2, then k =

11. If – 9 is a root of the equation

x 3 7

2 x 2 0

7 6 x

then the other two roots are :

12. If a, b, c are unequal what is the condition that the value of the following determinant is zero =

2 3

2 3

2 3

a a a 1

b b b 1

c c c 1

.


1 a b c

1 b c a

1 c a b

is :

14. If a, b and c are non zero numbers, then =

2 2

2 2

2 2

b c bc b c

c a ca c a

a b ab a b

is equal to :

15. If

1 k 3

3 k 2 0

2 3 1

, then the value of k is :



PROFICIENCY TEST-03

1. If =

a b c

x y z

p q r

, then

ka kb kc

kx ky kz

kp kq kr

=

2. If

2

2

2

x x x 1 x 2

2x 3x 1 3x 3x 3

x 2x 3 2x 1 2x 1

=Ax – 12, then the value ofAis :

3. A root of the equation

3 x 6 3

6 3 x 3

3 3 6 x

= 0 is :

4.

2 2

2 2

sin x cos x 1

cos x sin x 1

10 12 2

5. If Dp =2

3

p 15 8

p 35 0

p 25 10

, then D1 + D2 + D3 + D4 + D5 =

6. If

24 1 3 2 x 3

2 1 1 x 2 1

, then x =

7. If a, b, c are inA.P., then the value of

x 2 x 3 x a

x 4 x 5 x b

x 6 x 7 x c

is :

8. If =

x y z

p q r

a b c

, then

x 2y z

2p 4q 2r

a 2b c

equals

9. If

a b c

m n p

x y z

= k, then

6a 2b 2c

3m n p

3x y z

10. If

1 1 1

2 2 2

3 3 3

a b c

a b c

a b c

= 5 ; then the value of2 3 3 2 2 3 3 2 2 3 3 2

3 1 1 3 3 1 1 3 3 1 1 3

1 2 2 1 1 2 2 1 1 2 2 1

b c b c c a c a a b a b



is :



11. If

2 2 2

2 2 2

2 2 2

(b c) a a

b (c a) b

c c (a b)

= kabc(a + b + c)3, then the vlaue of k is :

12. IfA, B, C be the angles of a triangle, then

1 cosC cosB

cosC 1 cos A

cos B cosA 1

13.

x x 2 x x 2

x x 2 x x 2

x x 2 x x 2

(a a ) (a a ) 1

(b b ) (b b ) 1

(c c ) (c c ) 1


1 cos( ) cos

cos( ) 1 cos

cos cos 1

is :

15. If

y z x z x y

y z z x y x

z y z x x y

= kxyz, then the value of k is :



EXERCISE–I

1. (a) Prove that the value of the determinant

9i54i43

2i548i35

i43

2i357

is real.

(b) On which one of the parameter out of a, p, d or x, the value of the determinant

1 2a a

p d x px p d x

p d x px p d x

cos( ) cos cos( )

sin( ) sin sin( )

does not depend.

2. Without expanding as far as possible, prove that

(a)13312a1a211a2a2a2

= (a 1)3 (b)333 zyx

zyx111

= [(xy) (yz) (zx) (x+y+z)]

3. If

zz1z

yy1yxx1x

23

23

23

= 0 and x , y , z are all different then , prove that xyz = 1.

4. Using properties of determinants or otherwise evaluate440198891988940894018

.

5. Prove thatbacc2c2

b2acbb2a2a2cba

= (a + b + c)3 .

6. If D =acbbaccba

and D =cbbaacaccbbabaaccb

then prove that D= 2 D.

7. Prove that22

22

22

ba1a2b2a2ba1ab2b2ab2ba1

= (1 + a² + b²)3.

8. Prove thatcabba

acbcabccba

= (a + b + c) (a² + b² + c²).

9. Show that the value of the determinant)RCtan()RBtan()RAtan()QCtan()QBtan()QAtan()PCtan()PBtan()PAtan(

vanishes for all values of

A, B, C, P, Q & R where A + B + C + P + Q + R = 0

10. Prove that

1

1

1

24

24

24

= 64( )( )( )( ) ( ) ( )



11. For a fixed positive integer n, if D=)!4n()!3n()!2n()!3n()!2n()!1n()!2n()!1n(!n

thenshow thatD

n( !)34

is divisible

by n.

12. Solve for x

(a)17x108x55x35x44x33x24x33x22x

= 0. (b)64x327x28x16x39x24x4x33x22x

= 0.

13. If a + b + c = 0 , solve for x :xcab

axbcbcxa

= 0.

14. If a2 + b2 + c2 = 1 then show that the value of the determinant

cos)ba(c)cos1(bc)cos1(ac

)cos1(cbcos)ac(b)cos1(ab

)cos1(ca)cos1(bacos)cb(a

222

222

222

simplifies to cos2.

15. If p + q + r = 0 , prove that

qapcrbpbraqcrcqbpa

= pqracbbaccba

.

16. If a , b , c are all different &1ccc1bbb1aaa

43

43

43

= 0, then prove that, abc(ab + bc + ca) = a + b + c.

17. Show that,

2

2

2

cbcacbcbabacaba

is divisible by 2 and find the other factor..

18. Prove that :111cba

cba4

)1c()1b()1a(

)1c()1b()1a(cba 222

222

222

222

.

19. In a ABC, determine condition under which

111

tantantantantantan

cotcotcot

2B

2A

2A

2C

2C

2B

2C

2B

2A

= 0

20. Prove that :

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

a p a q a r

b p b q b r

c p c q c r

ap aq ar

bp bq br

cp cq cr

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

1 1 1

1 1 1

1 1 1

21. Prove that2

332

232

13

232

222

212

231

221

211

)ba()ba()ba(

)ba()ba()ba(

)ba()ba()ba(

= 2(a1a2)(a2a3)(a3 a1)(b1b2)(b2b3)(b3b1)

22. Prove that

2)()()()()()(2

2= 0.



23. If ax1² + by1² + cz12 = ax2

2 + by22 + cz2

2 = ax32 + by3

2 + cz32 = d

and ax2x3 + by2y3 + cz2z3 = ax3x1 + by3y1 + cz3z1 = ax1x2 + by1y2 + cz1z2 = f,

then prove that

333

222

111

zyx

zyx

zyx

= (d f)

2/1

abc

f2d

(a , b , c 0)

24. If Sr = r + r + r then show that

432

321

210

SSS

SSS

SSS

= ( )2 ( )2 ( )2 .

25. If u = ax2 + 2 bxy+ cy2 , u = ax2 + 2 bxy + cy2. Prove that

ybxabyaxuu

y

1ycxbybxa

cybxbyax

cbacba

xxyy 22

.

EXERCISE–II

1. Solve the following using Cramer’s rule and state whether consistent or not.

(a)03z2yx01zyx206zyx

(b)

0y2x6zyx31zy2x

(c)5z5y3x2

7z5yx33z5y7x7

2. For what value of K do the following system of equations possess a non trivial (i.e. not all zero)solution over the set of rationals Q ?x + K y + 3 z = 0 , 3 x + K y 2 z = 0 , 2 x + 3 y 4 z = 0.For that value of K , find all the solutions of the system.

3. The system of equationsx + y + z = – 1x + y + z = – 1x + y + z = – 1

has no solution. Find.

4. If the equations a(y + z) = x, b(z + x) = y, c(x + y) = z have nontrivial solutions, then find the value of

c1

1

b1

1

a1

1

.

5. Given x = cy+ bz ; y = az + cx ; z = bx + ay where x , y , z are not all zero , prove thata² + b² + c² + 2 abc = 1.

6. Given a =zy

x

; b =

xz

y

; c =

yx

z

where x, y, z are not all zero, prove that: 1 + ab + bc + ca = 0.

7. If sin q cos q and x, y, z satisfy the equationsx cos p – y sin p + z = cos q + 1x sin p + y cos p + z = 1 – sin qx cos(p + q) – y sin (p + q) + z = 2

then find the value of x2 + y2 + z2.

8. Investigate for what values of , the simultaneous equations x + y + z = 6;x + 2 y + 3 z = 10 & x + 2 y + z = have; (a) A unique solution.(b) An infinite number of solutions. (c) No solution.



9. For what values of p , the equations : x + y+ z = 1 ; x + 2 y + 4 z = p &x + 4y+ 10z = p2 have a solution? Solve them completely in each case.

10. Solve the equations : K x + 2 y 2 z = 1, 4 x + 2 K y z = 2, 6 x + 6 y + K z = 3considering specially the case when K = 2.

11. Let a, b, c, d are distinct numbers to be chosen from the set {1, 2, 3, 4, 5}. If the least possible positive

solution for x to the system of equations

2dycx1byax can be expressed in the form

q

pwhere p and q

are relativelyprime, then find the value of (p + q).

12. If bc + qr = ca + rp = ab + pq = 1 show thatrccrqbbqpaap

= 0.

13. If the following system of equations (a t)x + by + cz = 0 , bx + (c t)y + az = 0 andcx +ay+(b t)z = 0 has nontrivial solutions for different values of t , then show that we can expressproduct of these values of t in the form of determinant.

14. Show that the system of equations3x – y + 4z = 3, x + 2y – 3z = –2 and 6x + 5y + z = – 3

has atleast one solution for any real number . Find the set of solutions of = –5.

15. Solve the system of equations ;

0cxccyz

0bxbbyz

0axaayz

32

32

32



EXERCISE–III

1. (a) If f(x) =

1 x x 1

2x x(x 1) (x 1)x

3x(x 1) x(x 1)(x 2) (x 1)x(x 1)

then f(100) is equal to :

(A) 0 (B) 1 (C) 100 (D) –100

(b) Let a, b, c, d be real numbers on G.P. If u, v, w satisfy the system of equations,

u + 2v + 3w = 6

4u + 5v + 6w = 12

6u + 9v = 4

then show that the roots of the equation,

1 1 1

u v w

x2 + [(b – c)2 + (c – a)2 + (d – b)2] x + u + v + w = 0

and 20x2 + 10 (a – d)2 x – 9 = 0

are reciprocals of each other. [JEE 1999]

2. If the system of equations x – Ky – z = 0, Kx – y – z = 0 and x + y – z = 0 has a non zero solution, then

the possible values of K are

(A) –1, 2 (B) 1, 2 (C) 0, 1 (D) –1, 1

[JEE 2000]

3. Prove that for all values of ,

34

32

32

34

32

32

2sincossin

2sincossin2sincossin

= 0 [JEE 2000 ]

4. Find the real values of r for which the followingsystem of linear equations has anon-trivial solution.Also

find thenon-trivial solutions : [REE 2000 ]

2 r x 2 y + 3 z = 0

x + r y + 2 z = 0

2 x + r z = 0

5. Solve for x the equation

a a

n x nx n x

n x nx n x

2 1

1 1

1 1

sin( ) sin sin( )

cos( ) cos cos( )

= 0 [REE 2001]

6. Test the consistencyand solve them whenconsistent, the followingsystemof equations for all values of

x + y + z = 1

x + 3y – 2z =

3x + ( + 2)y – 3z = 2 + 1 [REE 2001]



7. Let a, b, c be real numbers with a2 + b2 + c2 = 1 . Show that the equation

ax by c bx ay cx a

bx ay ax by c cy b

cx a cy b ax by c

= 0

represents a straight line. [JEE 2001]

8. If a > 0 and discriminant of ax2 + 2bx + c is –ve, then [AIEEE 2002]

a b ax b

b c bx c

ax b bx c 0

is equal to :

(A) +ve (B) (ac – b2)(ax2 + 2bx + c)

(C) –ve (D) 0

9. The number of values of k for which the system of equations [JEE 2002]

(k + 1)x + 8y = 4k

kx + (k + 3)y = 3k – 1

has infinitelymanysolutions is

(A) 0 (B) 1 (C) 2 (D)inifinite

10. If the system of linear equations [AIEEE 2003]

x + 2ay + az = 0; x + 3by + bz = 0; x + 4cy + cz = 0; has a non-zero solution, then a, b, c.

(A) Satisfy a + 2b + 3c = 0 (B)Are inA.P.

(C)Are in G.P. (D)Are in H.P.

11. If a1, a2, a3,.....,an..... are in G.P., then the value of the determinant [AIEEE 2004]

n n 1 n 2

n 3 n 4 n 5

n 6 n 7 n 8

loga log a loga

loga loga log a

loga loga loga

, is

(A) –2 (B) 1 (C) 2 (D) 0

12. The value of for which the system of equations 2x – y – z = 12, x – 2y + z = –4, x + y + z = 4 has no

solution is [JEE 2004]

(A) 3 (B) –3 (C) 2 (D) –2

13. The system of equations [AIEEE 2005]

x + y + z = – 1

x + y + z = – 1

x + y + z = – 1

has infinite solutions, if is

(A) –2 (B) Either –2 or 1 (C) not –2 (D) 1



14. If a2 + b2 + c2 = –2 and f(x) =

2 2 2

2 2 2

2 2 2

1 a x (1 b )x (1 c )x

(1 a )x 1 b x (1 c )x

(1 a )x (1 b )x 1 c x

, [AIEEE 2005]

then f(x) is a polynomial of degree

(A) 1 (B) 0 (C) 3 (D) 2

15. If D =

1 1 1

1 1 x 1

1 1 1 y

for x 0, y 0, then D is [AIEEE 2007]

(A) Divisible byx but not y (B) Divisible byybut not x

(C) Divisible byneither x nor y (D) Divisible byboth x and y

16. Let A =

5 5

0 5

0 0 5

. If |A2| = 25, then || equals. [AIEEE 2007]

(A) 1/5 (B) 5 (C) 52 (D) 1

17. Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that

x = cy + bz, y = az + cx, and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal to [AIEEE 2008]

(A) 2 (B) –1 (C) 0 (D) 1

18.(a) Consider three points P = cos),sin( , Q = sin),cos( and

R = )sin(),cos( , where 0 < , , < /4 [JEE 2008]

(A) P lies on the line segment RQ (B) Q lies on the line segment PR

(C) R lies on the line segment QP (D) P, Q, R are non collinear

(b) Consider the system of equations

x – 2y + 3z = –1

– x + y – 2z = k

x – 3y + 4z = 1

STATEMENT-1 : The system of equations has no solution for k 3.

STATEMENT-2 : The determinant141k21131

0, for k 3.

(A) Statement-1 is True, Statement-2 is True ; statement-2 is a correct explanation for statement-1

(B) Statement-1 is True, Statement-2 isTrue ; statement-2 is NOT a correct explanation for statement-1

(C) Statement-1 is True, Statement-2 is False

(D) Statement-1 is False, Statement-2 is True



19. Let a, b, c be such that b(a + c) 0 if [AIEEE 2009]

n 2 n 1 n

a a 1 a 1 a 1 b 1 c 1

b b 1 b 1 a 1 b 1 c 1

c c 1 c 1 ( 1) a ( 1) b ( 1) c

= 0 , then the value of n is :

(A)Anyeven integer (B)Anyodd integer

(C)Anyinteger (D) Zero

20. Consider the system of linear equations : [AIEEE 2010]

x1 + 2x2 + x3 = 3

2x1 + 3x2 + x3 = 3

3x1 + 5x2 + 2x3 = 1

Then system has

(A) Exactly3 solutions (B)Aunique solution

(C) No solution (D) Infinitenumber of solutions

21. The number of values of k for which the linear equation 4x + ky + 2z = 0, kx + 4y + z = 0 and

2x + 2y + z = 0 posses a non-zero solution is [AIEEE 2011]

(A) 2 (B) 1 (C) zero (D) 3

22. The number of values of k, for which the system of equations : [JEE Main 2013]

(k + 1)x + 8y = 4k

kx + (k + 3)y = 3k – 1

has no solution, is :

(A) 3 (B)infinite (C) 1 (D) 2

23. If , 0, and f(n) = n + n and

3 1 f (1) 1 f (2)

1 f (1) 1 f (2) 1 f (3)

1 f (2) 1 f (3) 1 f (4)

= K(1 – )2(1 – )2(a – )2, then K

is equal to [JEE Main 2014]

(A) –1 (B) (C)1

(D) 1

24. The set of all values of for which the system of linear equations : [JEE Main 2015]

2x1 – 2x2 + x3 = x1

2x1 – 3x2 + 2x3 = x2

– x1 + 2x2 = x3

has anon-trivial solution,

(A) contains more than two elements (B) is an empty set

(C) is a singleton (D) contains two elements



25. Which of the following values of a satisfy the equation

2 2 2

2 2 2

2 2 2

(1 ) (1 2 ) (1 3 )

(2 ) (2 2 ) (2 3 )

(3 ) (3 2 ) (3 3 )

= – 648?

[IITAdvance 2015]

(A) – 4 (B) 9 (C) – 9 (D) 4

26. The system of linear equations [JEE Main 2016]

x + y – z = 0

x – y – z = 0

x + y – z = 0

has a non-trivial solution for :

(A) infinitelymanyvalues of (B) exactly one value of

(C) exactly two values of (D) exactly three values of

27. The total number of distinct xR for which

2 3

2 3

2 3

x x 1 x

2x 4x 1 8x

3x 9x 1 27x

= 10 is : [IITAdvance 2016]

28. LetR. Consider the system of linear equations [IITAdvance 2016]

x + 2y =

3x – 2y =

Which of the following statement(s) is(are) correct ?

(A) If= –3, then the system has infinitely many solutions for all values of and.

(B) If–3, then the system has a unique solutions for all values of and.

(C) If += 0, then the system has infinitely many solutions for= – 3

(D) If + 0, then the system has no solution for= –3

29. If S is the set of distinct values of 'b' for which the following system of linear equations

x + y + z = 1

x + ay + z = 1

ax + by + z = 0

has no solution, then S is [JEE Main 2017]

(A) a finite set containing two or more elements

(B) a singleton

(C) an empty set

(D) an infinite set

30. For a real number, if the system

2

2

1 x 1

1 y 1

z 11

of linear equation, has infinitely many

solutions, then 1 + + 2 = [JEE Adv. 2017]



31. If the system of linear equations x + ky + 3z = 0 3x + ky – 2z = 0 2x + 4y – 3z = 0 has a non-zero

solution (x, y, z), then 2

xz

yis equal to : [JEE Main 2018]

(A) 30 (B) –10 (C) 10 (D) –30

32. If 2x – 4 2x 2x

2x x – 4 2x = A + Bx x – A

2x 2x x – 4

, then the ordered pair (A, B) is equal to :

(A) (4, 5) (B) (–4, –5) (C) (–4, 3) (D) (–4, 5)

[JEE Main 2018]

33. Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {–1, 0, 1}. Then, the

maximum possible value of the determinant of P is _____ . [JEE Adv. 2018]

34. Let S be the of all column matrices

1

2

3

b

b

b

such that b1, b

2, b

3R and the system of equations (in real

variables) [JEE Adv. 2018]–x + 2y + 5z = b

1

2x – 4y + 3z = b2

x – 2y + 2z = b3

has at leastone solution. Then, which of the following system(s) (in real variables) has (have)at least one

solution of each

1

2

3

b

b

b

S ?

(A) x + 2y + 3z = b1, 4y + 5z = b

2and x + 2y + 6z = b

3

(B) x + y + 3z = b1, 5x + 2y + 6z = b

2and –2x – y – 3z = b

3

(C) –x + 2y – 5z = b1, 2x – 4y + 10z = b

2and x – 2y + 5z = b

3

(D) x + 2y + 5z = b1, 2x + 3z = b

2and x + 4y – 5z = b

3

35. If 1

x sin cos

sin x 1

cos 1 x

and 2

x sin 2 cos 2

sin 2 x 1

cos 2 1 x

, x 0; then for all 0,2

:

(A) 1

– 2

= x(cos2 – cos4) (B) 1

+ 2

= –2x3 [JEE Main 2019](C)

1–

2= –2x3 (D)

1+

2= –2(x3 + x – 1)

36. If the system of linear equations [JEE Main 2019]x + y + z = 5x + 2y + 2z = 6x + 3y + z = µ, (,R), has infinitelymany solutions, then the value of + µ is :(A) 12 (B) 10 (C) 9 (D) 7



37. If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution

(x, y, z), thenx y z

ky z x is equal to : [JEE Main 2019]

(A)4

(B) –4 (C)

1

2(D)

1

4

38. Let and be the roots of the equation x2 + x + 1 = 0. Then for y 0 in R,

y 1

y 1

1 y

is equal

to [JEE Main 2019]

(A) y3 (B) y3 – 1 (C) y (y2 –1) (D) y (y2– 3)

39. If the system of linear equation [JEE Main 2019]x – 2y + kz = 12x + y + z = 23x – y – kz = 3has a solution (x, y, z), z 0, then (x, y) lies on the straight line whose equation is :(A) 3x – 4y – 1 = 0 (B) 3x – 4y – 4 = 0 (C) 4x – 3y – 4 = 0 (D) 4x – 3y – 1 = 0

40. Let the numbers 2, b, c be in an A.P. and2 2

1 1 1

A 2 b c

4 b c

. If det(A) [2,16], then c lies in the

interval : [JEE Main 2019](A) [2, 3) (B) (2 + 23/4, 4) (C) [3, 2 + 23/4] (D) [4, 6]

41. The greatest value of cR for which the system of linear equations : [JEE Main 2019]x – cy – cz = 0cx – y + cz = 0cx + cy – z = 0has a non-trivial solution, is :

(A)1

2(B) –1 (C) 0 (D) 2

42. If

1 sin 1

A sin 1 sin ;

1 sin 1

then for all3 5

,4 4

, det (A) lies in the interval:

[JEE Main 2019]

(A)5

1,2

(B)5

,42

(C)3

0,2

(D)3

,32



43. An ordered pair for which the system of linear equations [JEE Main 2019]

1 x y z 2

x 1 y z 3

x y 2z 2

has a unique solution, is :(A) (2, 4) (B) (3, 1) (C) (4, 2) (D) (1, 3)

44. If

a b c 2a 2a

2b b c a 2b

2c 2c c a b

= (a + b + c) (x + a + b + c)2, x 0 and a + b + c 0, then x is equal

to : [JEE Main 2019](A) abc (B) – (a + b + c) (C) 2(a + b + c) (D) –2(a + b + c)

45. If the system of linear equations [JEE Main 2019]2x + 2y + 3z = a3x – y + 5z = bx – 3y + 2z = cWhere a, b, c are non zero real numbers, has more than one solution. then:(A) b – c + a = 0 (B) b – c – a = 0 (C) a + b + c = 0 (D) b + c – a = 0

46. The number of value of 0, for which the system of linear equations

x + 3y + 7z = 0– x + 4y + 7z = 0(sin 3)x + (cos 2) y + 2z = 0 has a non - trivial solution, is: [JEE Main 2019]

(A) three (B) two (C) four (D) one

47. Let 2

2 b 1

A b b 1 b

1 b 2

where b > 0. Then the minimum value of det A

bis: [JEE Main 2019]

(A) 2 3 (B) 2 3 (C) 3 (D) 3

48. If the system of equations [JEE Main 2019]x + y + z = 5x + 2y + 3z = 9

x + 3y +z =

has infinitelymanysolutions, then equals:

(A) 21 (B) 8 (C) 18 (D) 5



49. Let d R and [JEE Main 2019]

2 4 d (sin ) 2

A 1 (sin ) 2 d , [0,2 ]

5 (2sin ) d ( sin ) 2 2d

If the minimum value of det (A) is 8, then a value of d is: [JEE Main 2019]

(A) –7 (B) 2 2 2 (C) –5 (D) 2 2 1

50. If the system of linear equations

x – 4y + 7z = g

3y – 5z = h

–2x + 5y - 9z = k

is consistent, then: [JEE Main 2019]

(A) g + h + k = 0 (B) 2g + h + k = 0 (C) g + h + 2k = 0 (D) g + 2h + k = 0

51. Let a1, a

2, a

3, ......., a

10be in G.P. with a

i> 0 for i = 1, 2, ...., 10 and S be the set of pairs (r,k), r, k N

(the set of natural numbers) for which [JEE Main 2019]

r k r k r ke 1 2 e 2 3 e 3 4

r k r k r ke 4 5 e 5 6 e 6 7

r k r k r ke 7 8 e 8 9 e 9 10

log a a log a a log a a

log a a log a a log a a 0

log a a log a a log a a

. Then the number of elements in S, is :

(A) infinitelymany (B) 4 (C) 10 (D) 2

52. The set of all values of for which the system of linear equation. [JEE Main 2019]

x – 2y – 2z = x

x + 2y + z = y

–x – y = z

has a non - trivial solution.

(A) contains more than two elements (B) is a singleton

(C) is an empty set (D) contains exactly two element

53. The system of linear equations x + y + z = 2, 2x + 3y + 2z = 5, 2x + 3y + (a2 – 1)z = a + 1

[JEE Main 2019]

(A) is inconsistent when a = 4

(B) has a unique solution for | a | 3

(C) has infinitelymanysolutions for a = 4

(D) is inconsistent when | a | 3



54. A value of (, /3), for which [JEE Main 2019]

2 2

2 2

2 2

1 cos sin 4cos6

cos 1 sin 4cos6 0

cos sin 1 4cos6

, is :

(A)7

24

(B)

18

(C)

9

(D)

7

36

55. If [x] denotes the greatest integer x, then the system of linear equations [JEE Main 2019][sin] x + [–cos]y = 0; [cot] x + y = 0 :

(A)have infinitelymanysolutions if2 7

, ,2 3 6

(B) have infinitelymanysolutions if2

,2 3

and has a unique solution if

7,

6

(C) has aunique solution if2

,2 3

and have infinitelymanysolutions if

7,

6

(D) has a unique solution if2 7

, ,2 3 6

56. The sum of the real roots of the equation :

x 6 1

2 3x x 3 0

3 2x x 2

, is equal to : [JEE Main 2019]

(A) 6 (B) 1 (C) 0 (D) –4

57. Suppose

n nn 2

kk 0 k 0

n nn n k

k kk 0 k 0

k C k

det 0

C k C 3

, holds for some positive integer n. Then

nnk

k 0

C

k 1 equals

[JEEAdvanced 2019]

58. If the system of linear equations :2x + 2ay + az = 02x + 3by + bz = 0

2x + 4cy + cz = 0, where a, b, cR are non-zero and distinct ; has a non-zero solution, then :

[JEE Main 2020]

(A)1 1 1

, ,a b c

are in A.P.. (B) a, b, c are in A.P.

(C) a, b, c are in G.P. (D) a + b + c = 0



59. If the system of linear equations,x + y + z = 6x + 2y + 3z = 103x + 2y + z = has more than two solutions, then is equal to ________. [JEE Main 2020]

60. For which of the following ordered pairs (µ, ), the system of linear equations

x + 2y + 3z = 1

3x + 4y + 5z = µ

4x + 4y + 4z =

is inconsistent ? [JEE Main 2020]

(A) (4, 6) (B) (3, 4) (C) (1, 0) (D) (4, 3)

61. The number of all 3 × 3 matricesA, with entries from the set {–1, 0, 1} such that the sum of the diagonalelements ofAAT is 3, is _______. [JEE Main 2020]

62. The system of linear equations

x + 2y + 2z = 5

2x + 3y + 5z = 8

4x + y + 6z = 10 has : [JEE Main 2020]

(A) No solution when = 2 (B) infinitelymanysolutions when= 2

(C) no solution when = 8 (D) a unique solution when = – 8

63. Let a – 2b + c = 1. If ƒ(x) =

x a x 2 x 1

x b x 3 x 2

x c x 4 x 3

, then : [JEE Main 2020]

(A) ƒ(– 50) = 501 (B) ƒ(–50) = – 1 (C) ƒ(50) = 1 (D) ƒ(50) = –501

64. Let S be the set of all R for which the system of linear equations2x – y + 2z = 2x – 2y + z = –4x + y + z = 4has no solution. Then the set S [JEE Main 2020](A) is an empty set. (B) is a singleton.(C) contains more than two elements. (D) contains exactly two elements.

65. If =

x 2 2x 3 3x 4

2x 3 3x 4 4x 5

3x 5 5x 8 10x 17

= Ax3 + Bx2 + Cx + D, then B + C is equal to: [JEE Main 2020]

(A) 1 (B)-1 (C) –3 (D) 9



66. If the system of equationsx – 2y + 3z = 92x + y + z = bx – 7y + az = 24, has infinitely manysolutions, then a-b is equal to......... [JEE Main 2020]

67. If the system of equationsx + y + z = 22x + 4y – z = 63x + 2y + z = has infinitelymanysolutions, then [JEE Main 2020]

(A) 2 5 (B) 2 14 (C) 2 14 (D) 2 5

68. If a + x = b + y = c + z + 1, where a, b, c, x, y, z are non-zero distinct real numbers, then

x a y x a

y b y y b

z c y z c

is equal to: [JEE Main 2020]

(A) y(a – b) (B) 0 (C) y(b – a) (D) y(a – c)

69. Let R. The system of linear equations2x

1– 4x

2+ x

3= 1

x1– 6x

2+ x

3= 2

x1– 10x

2+ 4x

3= 3

is inconsistent for: [JEE Main 2020](A) exactly two values of (B) exactlyone negative value of.(C) every value of . (D) exactly one positive value of.

70. The values of and µ for which the system of linear equations

x + y + z = 2

x + 2y + 3z = 5

x + 3y + z = µ

has infinitelymanysolutions are, respectively: [JEE Main 2020]

(A) 6 and 8 (B) 5 and 8 (C) 5 and 7 (D) 4 and 9

71. The sum of distinct values of for which the system of equations

( 1)x (3 1)y 2 z 0

( 1)x (4 2)y ( 3)z 0

2x (3 1)y 3( 1)z 0 ,

has non-zero solutions,is ______ [JEE Main 2020]



ANSWER KEY

DETERMINANT

PROFICIENCY TEST-01

1. (a – b)(b – c)(c – a) 2. 0 3. 0 4. –5 5. 0

6. 0 7. 0 8. xy 9. x = 2, –1 10. x = 0

11. x = 0,3

2(a2 + b2 + c2) 12. 1, 9 13. 1 + a2 + b2 + c2

14. (a – b)(b – c)(c –a)(a + b + c) 15. 0

PROFICIENCY TEST-02

1. –(a3 + b3 + c3 – 3abc) 2. 0 3. 4abc 4. 0, –3 5. – (a + b + c)

6. –2 7. 0 8. 4a2b2c2 9. xyz1 1 1

1x y z

10. 1

11. 2, 7 12. abc = –1 13. 0

14. abc(a – b)(b – c)(c – a)(a + b + c – ab – bc – ca) 15.33

8

PROFICIENCY TEST-03

1. k3 2. 24 3. 0 4. 0 5. –28000 6. 6

7. 0 8. 4 9. 6k 10. 25 11. 2 12. 0

13. 0 14. 0 15. 8

EXERCISE–I

1. (b) p

4. 1 11. (ab ab) (bc bc) (ca ca) 12. (a) x = 1 or x = 2; (b) x = 4

13. x = 0 or x = ± 3

22 2 2a b c 17. 2 ( a2 + b2 + c2 + )

19. Triangle ABC is isosceles.

EXERCISE–II

1. (a) x = 1 , y = 2 , z = 3 ; consistent (b) x = 2 , y = 1 , z = 1 ; consistent

(c) inconsistent

2. K =33

2, x : y : z = -

15

2: 1 : – 3 3. – 2 4. 2 7. 2



8. (a) 3 (b) = 3, =10 (c) = 3, 10

9. x = 1 + 2K , y = 3K , z = K, when p = 1 ; x = 2K, y = 1 3K , z = K when p = 2 ; where K R

10. If K2, 15K2K2

1

)2K(6

z

3K2

y

)6K(2

x2

,

If K= 2, then x = , y =2

21 and z = 0 where R

11. 19

13.

a b c

b c a

c a b

14. If –5 then x =7

4; y = –

9

7and z = 0 ;

If = 5 then x =4 5

7

K; y =

7

9K13 and z = K where K R

15. x = (a + b + c) , y = ab + bc + ca , z = abc

EXERCISE–III

1. (a) A2. D

4. r = 2 ; x = k; y =k

2; z = k where k R {0}

5. x = n, n I

6. If= 5, system is consistent with infinite solution given byz = K,

y =1

2(3K + 4) and x = –

1

2(5K + 2) where K R

If 5, system is consistent with unique solution given by z =1

3(1 – ); x =

1

3( + 2) and y = 0.

8. C 9. B 10. D 11. D 12. D 13. D 14. D

15. D 16. A 17. D 18.(a)- D, (b)- A 19. B 20. C

21. A 22. C 23. D 24. D 25. B, C 26. D 27. 2

28. B,C,D 29. B 30. 1 31. C 32. D 33. 4

34. A,C,D 35. B 36. B 37. C 38. A 39. C 40. D

41. A 42. D 43. A 44. D 45. B 46. B 47. A

48. B 49. C 50. B 51. A 52. B 53. D 54. C

55. B 56. C 57. 6.20 58. A 59. 13 60. D 61. 672

62. A 63. C 64. D 65. C 66. 5 67. B 68. A

69. B 70. B 71. 3



QUESTION BANK

PART - A [SINGLE CORRECT]Q.1 to Q.70 has four choices (A), (B), (C), (D) out of which only one is correct and carry 3 markseach. There is NEGATIVE marking. 1 mark will be deducted for each wrong answer.

Q.1 The value of the determinant

a a

nx n x n x

nx n x n x

2 1

1 2

1 2

cos( ) cos( ) cos( )

sin ( ) sin ( ) sin ( )

is independent of :

(A) n (B) a (C) x (D) a , n and x

Q.2 If A =

1111

and det. (An – I) = 1 – n, n N then the value of , is

(A) 1 (B) 2 (C) 3 (D) 4

Q.3 Ais an involutary matrix given by A=

433434110

then the inverse of2

Awill be

(A) 2A (B)2

A 1

(C)2

A(D)A2

Q.4 If , & are real numbers , then D =

1

1

1

cos( ) cos( )

cos( ) cos( )

cos( ) cos( )

=

(A) 1 (B) cos cos cos(C) cos + cos + cos (D) zero

Q.5 IfA=

cossinsincos

,AA–1 is given by

(A) –A (B)AT (C) –AT (D)A

Q.6 If the product of n matrices

1011

1021

1031 ..........

10n1 is equal to the matrix

103781

then the value of n is equal to(A) 26 (B) 27 (C) 377 (D) 378

Q.7 LetA=

zyxrqpcba

and suppose that det.(A) = 2 then the det.(B) equals, where B =

rc2z4qb2y4pa2x4

(A) det(B) = – 2 (B) det(B) = – 8 (C) det(B) = – 16 (D) det(B) = 8

Q.8 Let 1 =

1ar2ar

1aq2aq

1ap2ap

2

2

2

and 2 =1)pr(aarp1)rq(aaqr1)qp(aapq

then

(A) 1 = 2 (B) 2 = 21 (C) 1 = 22 (D) 1 + 22 = 0



Q.9 If an idempotent matrix is also skew symmetric then it must be(A) an involutarymatrix (B) an identitymatrix(C) an orthogonal matrix (D) anull matrix.

Q.10 Let the matrixAand B be defined asA=

1223

and B =

3713

then the value of Det.(2A9B–1), is

(A) 2 (B) 1 (C) – 1 (D) – 2

Q.11 Let f (x) =

1 4 2

1 4 2

1 4 2

2 2

2 2

2 2

sin cos sin

sin cos sin

sin cos sin

x x x

x x x

x x x

, then the maximum value of f (x) =

(A) 2 (B) 4 (C) 6 (D) 8

Q.12 Which of the following statements is not correct(A) (AB)T = ATBT (B) (AT)T = A(C) (A + B)T = BT + AT (D) (kA)T = kAT (k is a scalar)

Q.13 IfAand B are invertible matrices, which one of the following statements is not correct(A) Adj. A = |A| A–1 (B) det (A–1) = |det (A)|–1

(C) (A + B)–1 = B–1 + A–1 (D) (AB)–1 = B–1A–1

Q.14 Tr (A) of a 3 x 3 matrix A = (aij) is defined by the relation Tr(A) = a11 + a22 + a33 (i.e. Tr(A) is sum ofthe main diagonal elements). Which of the following statement cannot always hold?(A) Tr (kA) = kTr(A) (k is a scalar) (B) Tr(A + B) = Tr (A) + Tr(B)(C) Tr(I3) = 3 (D) Tr(A

2) = ( Tr(A) )2

Q.15 If px4 + qx3 + rx2 + sx + t

x x x x

x x x

x x x

2 3 1 3

1 2 3

3 4 3

then t =

(A) 33 (B) 0 (C) 21 (D) none

Q.16 IfAand B are different matrices satisfyingA3 = B3 andA2B = B2A, then(A) det (A2 + B2) must be zero.(B) det (A – B) must be zero.(C) det (A2 + B2) as well as det (A – B) must be zero.(D) At least one of det (A2 + B2) or det (A – B) must be zero.

Q.17 The system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + z = has no solution if(A) = 5, = 13 (B) = 5 (C) = 5, 13 (D) = 13

Q.18 If D =

a ab ac

ba b bc

ca cb c

2

2

2

1

1

1

then D =

(A) 1 + a2 + b2 + c2 (B) a2 + b2 + c2 (C) (a + b + c)2 (D) ab + bc + ca



Q.19 Identify the incorrect statement in respect of two square matricesAand B conformable for sum andproduct.(A) tr(A + B) = tr(A) + tr(B) (B) tr(A) = tr(A), R(C) tr(A

T) = tr(A) (D) tr(AB) tr(BA)

Q.20 The determinant

cos ( ) sin ( ) cos

sin cos sin

cos sin cos

2

is :

(A) 0 (B) independent of (C) independent of (D) independent of & both

Q.21 Which of the following statements is incorrect for a square matrixA. ( |A| 0)

(A) IfAis a diagonal matrix,A–1 will also be a diagonal matrix(B) IfAis a symmetric matrix,A–1 will also be a symmetric matrix(C) IfA–1 =A Ais an idempotent matrix(D) IfA–1 =A Ais an involutary matrix

Q.22 Let0 =

333232

232221

131211

aaa

aaa

aaa

and let1 denote the determinant formed bythe cofactors of elements of0

and 2 denote the determinant formed by the cofactor at 1 and so on n denotes the determinantformed by the cofactors atn – 1 then the determinant value ofn is

(A) n20 (B)

n20 (C)

2n0 (D) 2

0

Q.23 If A =

2

2

sincossin

cossincos; B =

2

2

sincossin

cossincos

are such that,AB is a null matrix, then which of the following should necessarily be an odd integral

multiple of2

.

(A) (B) (C) – (D) +


cos ( ) cos ( ) cos ( )

cos ( ) cos ( ) cos ( )

sin ( ) sin ( ) sin ( )

x y y z z x

x y y z z x

x y y z z x

=

(A) 2 sin (xy) sin(yz) sin (zx) (B) 2 sin (xy) sin (yz) sin (zx)(C) 2 cos (x y) cos (y z) cos (z x) (D) 2 cos (x y) cos (y z) cos (z x)

Q.25 For a given matrix A =

cossinsincos

which of the following statement holds good?

(A) A = A–1 R (B) A is symmetric, for = (2n + 1)2

, In

(C) A is an orthogonal matrix for R (D) A is a skew symmetric, for = n ; n I




1

1

1

a x a y a z

b x b y b z

c x c y c z

=

(A) (1 + a + b + c) (1 + x + y + z) 3 (ax + by + cz)(B) a (x + y) + b (y+ z) + c (z + x) (xy+ yz + zx)(C) x (a + b) + y (b + c) + z (c + a) (ab + bc + ca)(D) none of these

Q.27 IfAis matrix such that A2 +A+ 2I = O, then which of the following is INCORRECT ?

(A)Ais non-singular (B)A O (C)Ais symmetric (D) A–1 = –2

1(A + I)

(Where I is unit matrix of order 2 and O is null matrix of order 2 )

Q.28 Let A + 2B =

135336021

and 2A – B =

210612512

then Tr (A) – Tr(B) has the value equal to(A) 0 (B) 1 (C) 2 (D) none

Q.29 Let D1 =babadcdcbaba

and D2 =cbaca

dbdbcaca

then the value of2

1

D

Dwhere b 0 and

ad bc, is(A) – 2 (B) 0 (C) – 2b (D) 2b

Q.30 The number of solutions of the matrix equation X2 =I other than I, is(A) 0 (B) 1 (C) 2 (D) more than 2(where I is the 2 × 2 unit matrix )

Q.31 If a2 + b2 + c2 = – 2 and f (x) =

xc1x)b1(x)a1(

x)c1(xb1x)a1(

x)c1(x)b1(xa1

222

222

222

then f (x) is a polynomial of degree

(A) 0 (B) 1 (C) 2 (D) 3

Q.32 GivenA=

2231

; I =

1001

. IfA– I is a singular matrix then

(A) (B) 2 – 3 – 4 = 0 (C) 2 + 3 + 4 = 0 (D) 2 – 3 – 6 = 0

Q.33 The values of for which the following equationssinx – cosy + ( + 1)z = 0 ; cosx + siny – z = 0; x + ( + 1)y + cos z = 0have non trivial solution, is(A) = n, R – {0} (B) = 2n, is any rational number

(C) = (2n + 1), R+, n I (D) = (2n + 1)2

, R, n I



Q.34 Let A =

1sin1sin1sin

1sin1

, where 0 < 2, then

(A) Det (A) = 0 (B) Det A (0, ) (C) Det (A) [2, 4] (D) Det A [2, )Q.35 The system of equations :

2x cos2 + y sin2 – 2sin = 0x sin2 + 2y sin2 = – 2 cosx sin – y cos = 0 , for all values of , can(A) have a unique non - trivial solution (B) not have a solution(C)have infinite solutions (D) have a trivial solution

Q.36 LetA, B, C, D be (not necessarily square) real matrices such thatAT = BCD; BT = CDA; CT = DAB and DT = ABC

for the matrix S =ABCD, consider the two statements.I S3 = SII S2 = S4

(A) II is true but not I (B) I is true but not II(C) both I and II are true (D) both I and II are false.

Q.37 Number of value of 'a' for which the system of equations,a2 x + (2 a) y = 4 + a2

a x + (2 a 1) y = a5 2 possess no solution is(A) 0 (B) 1 (C) 2 (D)infinite

Q.38 A=

1xtan

xtan1then let us define a function f (x) = det. (ATA–1) then which of the following can

not be the value of

timesn

)x(f...........ffff is (n 2)

(A) f n(x) (B) 1 (C) f n – 1(x) (D) n f (x)

Q.39 Number of triplets of a, b & c for which the system of equations,ax by = 2a b and (c + 1) x + cy = 10 a + 3 b

has infinitely many solutions and x = 1, y = 3 is one of the solutions, is :(A) exactly one (B) exactly two(C) exactly three (D) infinitely many

Q.40 There are two possible values of A in the solution of the matrix equation

1

A451A2

C2A2B5A

=

FED14

where A, B, C, D, E, F are real numbers. The absolute

value of the difference of these two solutions, is

(A)3

8(B)

3

11(C)

3

1(D)

3

19

Q.41 The following system of equations 3x – 7y + 5z = 3; 3x + y + 5z = 7 and 2x + 3y + 5z = 5 are(A) consistentwith trivial solution (B) consistent withunique non trivial solution(C) consistentwith infinite solution (D) inconsistentwith no solution



Q.42 If everyelement of a square non singular matrixAis multiplied byk and the new matrix is denoted byBthen | A–1| and | B–1| are related as

(A) | A–1| = k | B–1| (B) | A–1| =k

1| B–1| (C) | A–1| = kn | B–1| (D) | A–1| = k–n | B–1|

where n is order of matrices.

Q.43 The number of real values of x satisfying1x126x172x71x3x41x21x22x3x

= 0 is

(A) 3 (B) 0 (C) more than 3 (D) 1

Q.44 Let A =

111312

111

and 10B =

321

05224

. If B is the inverse of matrix A, then is

(A) – 2 (B) – 1 (C) 2 (D) 5

*Q.45 If D(x) =

2 3

2 3

2 3

x 1 (x 1) x

x 1 x (x 1)

x (x 1) (x 1)

then the coefficient of x in D(x) is

(A) 5 (B) –2 (C) 6 (D) 0

*Q.46 Matrix AsatisfiesA2 = 2A– I where I is the identity matrix then for n 2,An is equal to (nN)(A) nA – I (B) 2n – 1A – (n – 1)I (C) nA – (n – 1)I (D) 2n – 1A – I

*Q.47 The set of equationsx – y + (cos) z = 03x + y + 2z = 0(cos)x + y + 2z = 0

0 < < 2 , has nontrivial solution(s)(A) for no value of and (B) for all values of and (C) for all values of and only two values of (D) for onlyone value of and all values of

Q.48 LetA=

120502321

and b =

130

. Which of the following is true?

(A)Ax = b has a unique solution. (B)Ax = b has exactly three solutions.(C)Ax = b has infinitelymanysolutions. (D)Ax = b is inconsistent.

Q.49 If a, b, c are real then the value of determinant1cbcac

bc1bab

acab1a

2

2

2

= 1 if

(A) a + b + c = 0 (B) a + b + c = 1 (C) a + b + c = –1 (D) a = b = c = 0



*Q.50 If A, B and C are n × n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of thedet(A2BC–1) is equal to

(A)5

6(B)

5

12(C)

5

18(D)

5

24

*Q.51 Let three matrices A =

1412

; B =

3243

and C =

3243

then

tr(A) + tr

2

ABC+ tr

4

)BC(A 2

+ tr

8

)BC(A 3

+ ....... + =

(A) 6 (B) 9 (C) 12 (D) none

Q.52 The equationx32x21x

x51x31x2

)x2()x1()x1( 222

+3x22x3x21

x2x3)x1(

1x1x2)x1(2

2

= 0

(A) has no real solution (B) has 4 real solutions(C) has two real and two non-real solutions (D) has infinite number of solutions , real or non-real

Q.53 A is a 2 × 2 matrix such that A

11

=

21

and AA2

11

=

01

. The sum of the elements of A, is

(A) –1 (B) 0 (C) 2 (D) 5

Q.54 The number ofpositive integral solutions

1 2 1

3 2

2 2 1

= 0 is

(A) 0 (B) 2 (C) 3 (D) 1

Q.55 In a square matrix A of order 3 the elements, ai i's are the sum of the roots of the equationx2 – (a + b)x + ab = 0; ai , i + 1's are the product of the roots, ai , i – 1's are all unity and the rest of theelements are all zero. The value of the det. (A) is equal to(A) 0 (B) (a + b)3 (C) a3 – b3 (D) (a2 + b2)(a + b)

Q.56 Let Dk is the k × k matrix with 0's in the main diagonal, unity as the element of 1st row and th)k(f

column and k for all other entries. If f (x) = x – {x} where {x} denotes the fractional part function thenthe value of det. (D2) + det. (D3) equals(A) 32 (B) 34 (C) 36 (D) none

Q.57 If x = a + 2b satisfies the cubic (a, bR) f (x)=

xabb

bxab

bbxa

=0, then its other two roots are

(A) real and different (B) real and coincident(C) imaginary (D) such that one is real and other imaginary



Q.58 For a matrixA=

101r21

, the value of

50

1r

101r21 is equal to

(A)

101001

(B)

1049501

(C)

1050501

(D)

1025001

Q.59 Let A =

222

222

222

yxz1)xyz(2)yzx(2

)xyz(2xzy1)zxy(2

)yzx(2)zxy(2zyx1

then det. A is equal to

(A) (1 + xy + yz + zx)3 (B) (1 + x2 + y2 + z2)3

(C) (xy + yz + zx)3 (D) (1 + x3 + y3 + z3)2

Q.60 Consider a matrixA() =

sincoscossin

then

(A)A() is symmetric (B)A() is skew symmetric

(C) A–1() = A( – ) (D) A2() = A

2

2

Q.61 LetA=

2

1/3

x 0

dy0

dx

and B =

2/3

1/3

y 0

0 x

. Equation tr(AB) =dy

dxis a differential equation of order

'm' and degree 'n' then (m + n) is equal to

(A) 2 (B) 3 (C) 4 (D) 5

Q.62 If f(x) satisfies the equation

f (x 2) f (x 5) f (x 2)

5 4 5

10 12 30

= 0 x R

Then the value of f(2009) is equal to(A) f(0) (B) f(7) (C) f(11) (D) f(17)

Q.63 Let g(x) =

f (x c) f (x 2c) f (x 3c)

f (c) f (2c) f (3c)

f '(c) f '(2c) f '(3c)

, where c is constant thenx 0

g(x)lim

xis equal to

(A) 0 (B) 1 (C) –1 (D) f(c)



Q.64 Let 1, 1, be the distinct non zero real roots of the equation ax2 + bx + c = 0, where a, b, cR,

a 0. Let Sn = n + n, n 0 and =

0 1 2

1 2 3

2 3 4

1 S 1 S 1 S

1 S 1 S 1 S

1 S 1 S 1 S

, then

(A) 0 (B) > 0 (C) < 0 (D) = 0

Q.65 Let A and B be 3 × 3 matrices such that AB + A + B = 0Statement-1 AB = BAStatement-2 PP–1 = I = P–1 P for every matrix P which is invertible.

(A) Statement-1 is true ; Statement-2 is true and Statement-2 is correct explanation for Statement-1(B) Statement-1 is true; Statement-2 is trueandStatement-2 is NOT correct explanation forStatement-1(C) Statement-1 is true ; Statement-2 is false(D) Statement-1 is false ; Statement-2 is true

Q.66 Let Abe any 3 × 2 matrixStatement-1 Inverse ofAAT does not existStatement-2 AAT is a singular matrix(A) Statement-1 is true ; Statement-2 is true and Statement-2 is correct explanation for Statement-1(B) Statement-1 is true; Statement-2 is trueandStatement-2 is NOT correct explanation forStatement-1(C) Statement-1 is true ; Statement-2 is false(D) Statement-1 is false ; Statement-2 is true

Paragraph for question nos. 67 & 68Consider the system of equations

x + y – z = b12x + z = b2x – y + az = b3

Q.67 If this system is denoted byAX = B then for a = 1, the value ofA–1

(A)

11123121

21021(B)

11123121

21021

(C)

11123121

21021

(D)

11123121

21021

Q.68 If B = 0 then the value of 'a' for whichAX = B have non trivial solution, is(A) 2 (B) 3 (C) 1 (D) none

Paragraph for question nos. 69 to 71IfAis a symmetric and B skew symmetric matrix andA+ B is non singular and C = (A+ B)–1(A– B)then

Q.69 CT(A + B)C =(A) A + B (B) A – B (C) A (D) B



Q.70 CT(A – B)C =(A) A + B (B) A – B (C) A (D) B

Q.71 CTAC(A) A + B (B) A – B (C) A (D) B

Paragraph for question nos. 72 to 74

If A0 =

321431422

and B0 =

344101334

Bn = adj(Bn – 1), nN and I is an identity matrix of order 3 then answer the following questions.

Q.72 det. (A0 + 20A 2

0B + 30A + 4

0A 40B + ....... 10 terms) is equal to

(A) 1000 (B) – 800 (C) 0 (D) – 8000

Q.73 B1 + B2 + ........ + B49 is equal to(A) B0 (B) 7B0 (C) 49B0 (D) 49I

Q.74 For a variable matrix X the equation A0X = B0 will have(A)unique solution (B) infinitesolution(C) finitelymanysolution (D) no solution

Paragraph for question nos. 75 & 76

LetAis a matrix of order 3 × 3 and aij is its element of ith row and jth column. tr is arithmetic mean of

elements of rth row and aij + ajk + aki = 0 holds for all 1 i, j, k 3 then answer the following questions.

Q.75 3j,i1

jia is not equal to

(A) t1 + t2 + t3 (B) zero (C) (det(A))2 (D) t1t2t3

Q.76 Ais(A)non singular (B) symmetric(C) skew symmetric (D) neither symmetric nor skew symmetric

[MULTIPLE CORRECT ANSWERS TYPE]

Q.77 to Q.89 has four choices (A), (B), (C), (D) out of which one or more than one is/are correct

and carry 4 marks each. There is NO NEGATIVE marking. Marks will be awarded only if all the

correct alternatives are selected.

Q.77 LetAand B are two square idempotent matrices such that AB ± BA is a null matrix, then the value of

the det. (A – B) can be equal

(A) – 1 (B) 1 (C) 0 (D) 2

Q.78 Asquare matrixAwith elements from the set of real numbers is said to be orthogonal ifA=A–1. IfAis

an orthogonal matrix, then

(A)A is orthogonal (B)A–1 is orthogonal

(C) Adj. A = A (D) |A–1| = 1

Q.79 The set of equations x – y + 3z = 2 , 2x – y + z = 4 , x – 2y + z = 3 has

(A) unique soluton only for= 0 (B) unique solution for 8

(C) infinite number of solutions for= 8 (D) no solution for= 8



Q.80 If A=

dcba

(where bc 0) satisfies the equations x2 + k = 0, then

(A) a + d = 0 (B) k = –|A| (C) k = |A| (D) none of these


a a b c bc

b b c a ca

c c a b ab

2 2 2

2 2 2

2 2 2

( )

( )

( )

is divisible by :

(A) a + b + c (B) (a + b) (b + c) (c + a)(C) a2 + b2 + c2 (D) (a b) (b c) (c a)

Q.82 LetA=

122212221

, then then the correct statement is

(A) A2 – 4A – 5I3 = 0 (B) A–1 =1

5(A – 4I3)

(C)A3 is not invertible (D)A2 is invertible

Q.83 D is a 3 × 3 diagonal matrix. Which of the following statements is not true ?

(A) D = D (B)AD = DA for every matrix Aof order 3 x 3

(C) D–1 if exists is a scalar matrix (D) none of these

Q.84 The value of lying between

4&

2and 0 AA

2and satisfying the equation

1 2 4

1 2 4

1 2 4

2 2

2 2

2 2

sin cos sin

sin cos sin

sin cos sin

A A

A A

A A

= 0 are :

(A) A =

4, =

8(B) A =

3

8

=

(C) A =

5, =

8(D) A =

6, =

3

8

Q.85 Given the matricesA and B as A =

1411

and B =

2211

.

The two matrices X andYare such that XA= B andAY= B then which of the following hold(s) true ?

(A) X =

2211

3

1(B) Y =

0403

3

1(C) det. X = det. Y (D) 3(X + Y) =

2214

Q.86 If the system of equations, a2 x by = a2 b & bx b2 y = 2 + 4 b possess an infinite number of

solutions then the possible values of 'a' and 'b' are

(A) a = 1 , b = 1 (B) a = 1 , b = 2

(C) a = 1 , b = 1 (D) a = 1 , b = 2



Q.87 IfAand B are two 3 × 3 matrices such that their productAB is a null matrix then

(A) det. A 0 B must be a null matrix.

(B) det. B 0 A must bea null matrix.

(C) If none ofAand B are null matrices then atleast one of the two matrices must be singular.

(D) If neither det.A nor det. B is zero then the given statement is not possible.

Q.88 If there are three square matrixA, B, C of same order satisfying the equationA2 =A–1 and let B = n2A

& C =)2n(

2A

then which of the following statements are true ?

(A) det. (B – C) = 0 (B) (B + C)(B – C) = 0

(C) B must be equal to C (D) none

Q.89 If p, q, r, s are inA.P. and f(x) =

p x q x p r x

q x r x x

r x s x s q x

sin sin sin

sin sin sin

sin sin sin

1 such that0

2

f (x)dx = – 4 then

the common difference of theA.P. can be :

(A) 1 (B)1

2(C) 1 (D) 2



ANSWERS

Q.1 A Q.2 B Q.3 A Q.4 D Q.5 B Q.6 B Q.7 C

Q.8 D Q.9 D Q.10 D Q.11 C Q.12 A Q.13 C Q.14 D

Q.15 C Q.16 D Q.17 C Q.18 A Q.19 D Q.20 B Q.21 C

Q.22 B Q.23 C Q.24 B Q.25 C Q.26 A Q.27 C Q.28 C

Q.29 A Q.30 D Q.31 C Q.32 D Q.33 D Q.34 C Q.35 D

Q.36 C Q.37 C Q.38 D Q.39 B Q.40 B Q.41 B Q.42 C

Q.43 C Q.44 B Q.45 A Q.46 C Q.47 A Q.48 A Q.49 D

Q.50 B Q.51 A Q.52 D Q.53 D Q.54 D Q.55 D Q.56 B

Q.57 B Q.58 D Q.59 B Q.60 C Q.61 C Q.62 A Q.63 A

Q.64 B Q.65 A Q.66 A Q.67 D Q.68 A Q.69 A Q.70 B

Q.71 C Q.72 C Q.73 C Q.74 D Q.75 D Q.76 C Q

[MULTIPLE CORRECT ANSWERS TYPE]

Q.77 ABC Q.78 AB Q.79 BD Q.80 AC

Q.81 ACD Q.82 ABD Q.83 BC Q.84 ABCD

Q.85 CD Q.86 ABCD Q.87 ABCD Q.88 ABC

Q.89 AC

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