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JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q.53. For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I − EF) is invertible. If G = (I − EF)−1, then which of the following statements is (are) TRUE?       (JEE Advanced 2021)
(a) | FE | = | I − FE| | FGE |
(b) (I − FE)(I + FGE) = I
(c) EFG = GEF
(d) (I − FE)(I − FGE) = I

Ans. a, b, c
 I  EF = G−1  G  GEF = I ..... (i)
and G  EFG = I ..... (ii)
Clearly, GEF = EFG  option (c) is correct.
Also, (I  FE) (I + FGE)
= I  FE + FGE  FEFGE
= I  FE + FGE  F(G  I) E
= I  FE + FGE  FGE + FE
= I  option (b) is correct but option (d) is incorrect.
 (I  FE) (I  FGE) = I  FE  FGE + F(G  I) E
= I  2FE
Now, (I  FE) ( FGE) =  FE
 | I  FE | | FGE | = | FE |
 option (a) is correct.


Q.54. For any 3 × 3 matrix M, let | M | denote the determinant of M. Let
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is(are) TRUE?       (JEE Advanced 2021)
(a) F = PEP and P2 =JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b) | EQ + PFQ−1 | = | EQ | + | PFQ−1 |
(c) | (EF)3 | > | EF |2
(d) Sum of the diagonal entries of P−1EP + F is equal to the sum of diagonal entries of E + P−1FP 

Ans. a, b, d
For option (a)
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Hence, option (a) is correct.
For option (b)
|EQ + PFQ−1| = |EQ| + |PFQ−1| .... (i)
 | E | = 0 and | F | = 0 and | Q |  0
 |EQ| = |E||Q| = 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Let R = EQ + PFQ−1 .... (ii)
⇒ RQ = EQ2 + PF = EQ2 + P2EP = EQ2 + EP [ P2 = I]
= E(Q2 + P)
⇒ |RQ| = |E(Q2 + P)|
⇒ |R||Q| = |E||Q+ P| = 0 [ | E | = O]
⇒ |R| = 0 (as |Q| ≠ 0) ..... (iii)
From Eqs. (ii) and (iii), we get Eq. (i) is true.
Hence, option (b) is correct.
For option (c)
|(EF)3| > |EF|2
i.e. 0 > 0 which is false.
For option (d)
 P2 = I ⇒ P−1 = P
 P−1FP = PFP = PPEPP = E
So, E + P−1FP = E + E = 2E
⇒ Tr(E + P−1FP) = Tr(2E) = 2Tr(E) ..... (iv)
and P−1EP + F
⇒ PEP + F = 2PEP [ F = PEP]
 Tr(2PEP)=2Tr(PEP)=2Tr(F)=2Tr(E) ..... (v)
From Eqs. (iv) and (v) option (d) is also correct.


Q.55. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE∀ n > m and n, m ∈ N. Consider a matrix A = [aij]3 × 3 whereJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       (JEE Main 2021)
(a) (15)× 242 
(b) (15)2 × 234 
(c) (105)2 × 238 
(d) (105)2 × 236

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= (210.218)2
= (105)2 × 238

 

Q.56. Consider the system of linear equations
x + y + 2z = 0
3x  ay + 5z = 1
2x  2y  az = 7
Let S1 be the set of all a ∈ R for which the system is inconsistent and S2 be the set of all aR for which the system has infinitely many solutions. If n(S1) and n(S2) denote the number of elements in S1 and S2 respectively, then       (JEE Main 2021)
(a) n(S1) = 2, n(S2) = 2
(b) n(S1) = 1, n(S2) = 0
(c) n(S1) = 2, n(S2) = 0
(d) n(S1) = 0, n(S2) = 2

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= −1(a2 + 10) − 1(−3a − 10) + 2(−6 + 2a)
= −a2 − 10 + 3a + 10 − 12 + 4a
Δ = −a2 + 7a − 12
Δ = −[a2 − 7a + 12]
Δ = −[(a − 3)(a − 4)]
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= a + 35 − 4 + 14a
15a + 31
Now, Δ= 15a + 31
For inconsistent Δ = 0  a = 3, a = 4 and for a = 3 and 4, Δ1  0
n(S1) = 2
For infinite solution: Δ = 0 and Δ1 = Δ2 = Δ3 = 0
Not possible
 n(S2) = 0


Q.57. If α + β + γ = 2π, then the system of equations
x + (cos γ)y + (cos β)z = 0
(cos γ)x + y + (cos α)z = 0
(cos β)x + (cos α)y + z = 0 has:       (JEE Main 2021)
(a) no solution
(b) infinitely many solution
(c) exactly two solutions
(d) a unique solution

Ans. b
Given α + β + γ = 2π

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

= 1 − cos2α − cos⁡γ(cos⁡γ − cos⁡α cos⁡β) + cos⁡β(cos⁡α cos⁡γ − cos⁡β)

= 1 − cos2α − cos2β − cos2γ + 2cos⁡α cos⁡β cos⁡γ

= sin2α − cos2β − cos⁡γ(cos⁡γ − 2cos⁡α cos⁡β)

= −cos⁡(α + β)cos⁡(α − β) − cos⁡γ(cos⁡(2π − (α − β)) − 2cos⁡α cos⁡β)

= −cos⁡(2π − γ)cos⁡(α − β) − cos⁡γ(cos⁡(α + β) − 2cos⁡α cos⁡β)

= −cos⁡γ cos⁡(α − β) + cos⁡γ(cos⁡α cos⁡β + sin⁡α sin⁡β)

= −cos⁡γ cos⁡(α − β) + cos⁡γ cos⁡(α − β)

= 0

So, the system of equation has infinitely many solutions.


Q.58. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the determinant JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis equal to:       (JEE Main 2021)
(a) a2a6 − a4a8 
(b) a
(c) a1a9 − a3a
(d) a5

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEr = 1, 2, 3, ......, a1, a2, a3, ..... are in G.P.
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, a1a9 − a3a7 = a110 − a110 = 0


Q.59. If the following system of linear equations
2x + y + z = 5
x − y + z = 3
x + y + az = b
has no solution, then:       (JEE Main 2021)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. d
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
forJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEsystem has no solutions.


Q.60. Let [λ] be the greatest integer less than or equal to λ. The set of all values of λ for which the system of linear equations
x + y + z = 4,
3x + 2y + 5z = 3,
9x + 4y + (28 + [λ])z = [λ] has a solution is:       (JEE Main 2021)
(a) R
(b) (−∞, −9) ∪ (−9, ∞)
(c) [−9, −8)
(d) (−∞, −9) ∪ [−8, ∞)

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
if [λ] + 9 ≠ 0 then unique solution
if [λ] + 9 = 0 then D1 = D2 = D3 = 0
so infinite solutions
Hence, λ can be any red number. 


Q.61. Let A(a, 0), B(b, 2b + 1) and C(0, b), b ≠ 0, |b| ≠ 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is:       (JEE Main 2021)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. d
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ a(2b + 1 − b) − 0 + 1(b2 − 0) = ±2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Sum of possible values of 'a' is
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 


Q.62. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval:       (JEE Main 2021)
(a) [68, 69)
(b) [62, 63)
(c) [65, 66)
(d) [60, 61)

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R1  R1  R3 & R2  R2  R3 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
2[x] + 6 + [x] = 192 ⇒ [x] = 62


Q.63. If the matrixJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEsatisfies A(A3 + 3I) = 2I, then the value of K is:       (JEE Main 2021)
(a) 1/2
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c) -1
(d) 1

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
A+ 3IA = 2I
⇒ A4 = 2I − 3A
Also characteristic equation of A is |A−λI|=0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ λ + λ2 − 2k = 0
⇒ A + A2 = 2K . I
⇒ A2 = 2KI − A
⇒ A= 4K2I + A2 − 4AK
Put A2 = 2KI − A
and A4 = 2I − 3A
2I − 3A = 4K2I + 2KI − A − 4AK
⇒ I(2 − 2K − 4K2) = A(2 − 4K)
⇒ −2I(2K2 + K − 1) = 2A(1 − 2K)
⇒ −2I(2K − 1)(K + 1) = 2A(1 − 2K)
⇒ (2K − 1)(2A) − 2I(2K − 1)(K + 1) = 0
⇒ (2K − 1)[2A − 2I(K + 1)] = 0
⇒ K = 1/2


Q.64. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEThen A2025 − A2020 is equal to:       (JEE Main 2021)
(a) A− A 
(b) A
(c) A5 − A 
(d) A6

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.65. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand Q = ATBA, then the inverse of the matrix A Q2021 AT is equal to:        (JEE Main 2021)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Q= ATBAATBA = ATBIBA
⇒ Q2 = ATB2A
Q3 = ATB2AATBA ⇒ Q3 = ATB3A
Similarly: Q2021 = ATB2021A
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 AQ2021 = AT = AATB2021AAT = IB2021I
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.66. Let θ ∈ (0, π/2). If the system of linear equations
(1 + cos2θ)x + sin2θy + 4sin⁡3θz = 0
cos2θx + (1 + sin2θ)y + 4sin⁡3θz = 0
cos2θx + sin2θy + (1 + 4sin⁡3θ)z = 0
has a non-trivial solution, then the value of θ is:        (JEE Main 2021)
(a) 4π/9
(b) 7π/18
(c) π/18
(d) 5π/18

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
C1 → C1 + C2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R1 → R1 − R2, R→ R2 − R3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
or 4sin⁡3θ = −2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
θ = 7π/18


Q.67. Let A and B be two 3 × 3 real matrices such that (A2  B2) is invertible matrix. If A5 = B5 and A3B2 = A2B3, then the value of the determinant of the matrix A3 + B3 is equal to:        (JEE Main 2021) 
(a) 2
(b) 4
(c) 1
(d) 0

Ans. d
C = A2 − B2; | C | ≠ 0
A2 = B5 and A3B2 = A2B2
Now, A5 − A3B2 = B5 − A2B3
⇒ A3 (A2 − B2) + B3 (A2 − B2) = 0
⇒ (A3 + B3(A2 − B2) = 0
⇒ A3 + B3 = 0 (∵|A2 − B2 ≠ 0|)


Q.68. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE If A−1 αI + βA, αβ  R, I is a 2 × 2 identity matrix then 4(α  β) is equal to:         (JEE Main 2021) 
(a) 5
(b) 8/3
(c) 2
(d) 4

Ans. d
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ 4(α − β) = 4(1) = 4


Q.69. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then P50 is:          (JEE Main 2021) 
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.70. The number of distinct real roots of JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE         (JEE Main 2021)
(a) 4 
(b) 1 
(c) 2 
(d) 3 

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Apply: R→ R1 − R& R2 → R2 − R
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ (sin⁡x − cos⁡x)2(sin⁡x + 2cos⁡x) = 0 
∴ x = π/4


Q.71. The values of a and b, for which the system of equations
2x + 3y + 6z = 8
x + 2y + az = 5
3x + 5y + 9z = b
has no solution, are:
        (JEE Main 2021)
(a) a = 3, b ≠ 3 
(b) a ≠ 3, b ≠ 13 
(c) a ≠ 3, b = 3
(d) a = 3, b = 13

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
If a = 3, b  13, no solution.


Q.72. Let A = [aij] be a real matrix of order 3 × 3, such that ai1 + ai2 + ai3 = 1, for i = 1, 2, 3. Then, the sum of all the entries of the matrix A3 is equal to:        (JEE Main 2021)
(a) 2
(b) 1
(c) 3
(d) 9

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 AX = X
Replace X by AX
A2X = AX = X
Replace X by AX
A3X = AX = X
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Sum of all the element = 3


Q.73. The values of λ and μ such that the system of equations x + y + z = 6, 3x + 5y + 5z = 26, x + 2y + λz = μ has no solution, are:        (JEE Main 2021)
(a) λ = 3, μ = 5
(b) λ = 3, μ ≠ 10
(c) λ ≠ 2, μ = 10
(d) λ = 2, μ ≠ 10 

Ans. d
x + y + z = 6 ..... (i)
3x + 5y + 5z = 26 .... (ii)
x + 2y + λz = μ ..... (iii)
5 × (i) − (ii) ⇒ 2x = 4 ⇒ x = 2
 from (i) and (iii)
y + z = 4 ..... (iv)
2y + λz = μ − 2 .....(v)
(v) − 2 × (iv)
⇒ (λ − 2)z = μ − 10
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 For no solution λ = 2 and μ  10.


Q.74. The value of k ∈ R, for which the following system of linear equations
3x − y + 4z = 3,
x + 2y − 3z = −2
6x + 5y + kz = −3,
has infinitely many solutions, is:        (JEE Main 2021)
(a) 3
(b) −5
(c) 5
(d) −3

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 3(2k + 15) + K + 18  28 = 0
 7k + 35 = 0
 k =  5 

 

Q.75. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE, a ∈ R be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to:        (JEE Main 2021)
(a) 36
(b) 24
(c) 45
(d) 18

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Det (Q) = 9 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
a − 3 = ±6 ⇒ a = 9, −3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
| P | = - 36 or 0
 | 36 + 0 | = 36


Q.76. Define a relation R over a class of n × n real matrices A and B as
"ARB iff there exists a non-singular matrix P such that PAP−1 = B".
Then which of the following is true?        (JEE Main 2021)
(a) R is reflexive, transitive but not symmetric
(b) R is symmetric, transitive but not reflexive
(c) R is reflexive, symmetric but not transitive
(d) R is an equivalence relation

Ans. d
For reflexive relation,
∀(A, A) ∈ R for matrix P.
⇒ A = PAP−1 is true for P = 1
So, R is reflexive relation.
For symmetric relation,
Let (A, B) ∈ R for matrix P.
⇒ A = PBP−1
After pre-multiply by P−1 and post-multiply by P, we get
P−1AP = B
So, (B, A) ∈ R for matrix P−1.
So, R is a symmetric relation.
For transitive relation,
Let ARB and BRC
So, A = PBP−1 and B = PCP−1
Now, A = P(PCP−1)P−1
⇒ A = (P)2C(P−1)⇒ A = (P)2 ⋅ C ⋅ (P2)−1
∴(A, C) ∈ R for matrix P2.
∴ R is transitive relation.
Hence, R is an equivalence relation. 


Q.77. Let the system of linear equations
4x + λy + 2z = 0
2x − y + z = 0
μx + 2y + 3z = 0, λ, μ ∈ R.
has a non-trivial solution. Then which of the following is true?        (JEE Main 2021)
(a) μ = 6, λ ∈ R
(b) λ = 3, μ ∈ R
(c) μ = −6, λ ∈ R
(d) λ = 2, μ ∈ R

Ans. a

 Given, system of linear equations

4x + λy + 2z = 0

2x − y + z = 0

μx + 2y + 3z = 0

For non-trivial solution, Δ = 0

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 4(−3 − 2) − λ(6 − μ) + 2(4 + μ) = 0

⇒ −λ(6 − μ) − 2(6 − μ) = 0

⇒ (6 − μ)(λ + 2) = 0

⇒ λ = −2 and μ ∈ R or μ = 6 and λ ∈ R.


Q.78. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEIf Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) − Tr(B) has value equal to        (JEE Main 2021)
(a) 1
(b) 2
(c) 0
(d) 3

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Similarly, 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Tr(A) − Tr(B) = 1 − (−1) = 2


Q.79. The solutions of the equationJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE        (JEE Main 2021)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. d
By using C1 → C1 − C2 and C3 → C3 − C2 we get
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Expanding by R1 we get
1(1 + cos2x + 4sin⁡2x) − sin2x(−1) = 0
⇒ 2 + 4sin⁡2x = 0
⇒ sin⁡2x = −1/2
⇒ 2x = nπ + (−1)n(−π/6), n ∈ Z
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.80. If x, y, z are in arithmetic progression with common difference d, x ≠ 3d, and the determinant of the matrixJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis zero, then the value of k2 is:        (JEE Main 2021)
(a) 72
(b) 12
(c) 36
(d) 6

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R1 → R+ R3 − 2R2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ (k − 6√2)(4z − 5y) = 0
 k = 6√2 or 4z = 5y (Not possible  x, y, z in A.P.)
So, k2 = 72
 Option (A) 


Q.81. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen a possible value of α is:        (JEE Main 2021)
(a) π/4
(b) π/6
(c) π/2
(d) π/3

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.82. The system of equations kx + y + z = 1, x + ky + z = k and x + y + zk = k2 has no solution if k is equal to:        (JEE Main 2021)
(a) 0
(b) −1
(c) −2
(d) 1

Ans. c

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ k(k2 − 1) − (k − 1) + (1 − k) = 0
⇒ (k − 1)(k2 + k − 1 − 1) = 0
⇒ (k − 1)(k+ k − 2) = 0
⇒ (k − 1)(k − 1)(k + 2) = 0
⇒ k = 1, k = −2
for k = 1 equation identical so k = 2 for no solution. 


Q.83. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Then, the system of linear equations

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       (JEE Main 2021)
(a) Exactly two solutions 
(b) Infinitely many solutions
(c) A unique solution 
(d) No solution

Ans. d
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ 128(x − y) = 8
⇒ x − y = 1/16 .... (1)
and 128(−x + y) = 64 ⇒ x − y = −1/2 .... (2)
 No solution (from eq. (1) & (2))


Q.84. Consider the following system of equations:
x + 2y − 3z = a
2x + 6y − 11z = b
x − 2y + 7z = c,
where a, b and c are real constants. Then the system of equations:       (JEE Main 2021)
(a) has no solution for all a, b and c
(b) has a unique solution when 5a = 2b + c
(c) has infinite number of solutions when 5a = 2b + c
(d) has a unique solution for all a, b and c

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 20  2(25) 3(10)
= 20  50 + 30 = 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 20a  2(7b + 11c) 3(2b  6c)
= 20a  14b  22c + 6b +18c
= 20a  8b  4c
= 4(5a  2b  c)
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 7b + 11c  a(25) 3(2c  b)
= 7b + 11c  25a  6c + 3b
25a + 10b + 5c
5(5a  2b  c)
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 6c + 2b  2(2c  b)  10a
10a + 4b + 2c
2(5a  2b  c)
for infinite solution
D = D1 = D2 = D3 = 0
 5a = 2b + c 


Q.85. The value ofJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       (JEE Main 2021)
(a) −2
(b) 0
(d) (a + 2)(a + 3)(a + 4)
(d) (a + 1)(a + 2)(a + 3)

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R2 → R2 − R1 and R3 → R3 − R1
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 4(a + 2) − 4a − 10
= 4a + 8 − 4a − 10 = −2


Q.86. Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2 is 1, then the possible number of such matrices is:       (JEE Main 2021)
(a) 6
(b) 4
(c) 1
(d) 12

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= a2 + 2b2 + c2 = 1
a = 1, b = 0, c = 0
a = 0, b = 0, c = 1
a = −1, b = 0, c = 0
c = −1, b = 0, a = 0


Q.87. The following system of linear equations
2x + 3y + 2z = 9
3x + 2y + 2z = 9
x − y + 4z = 8       (JEE Main 2021)
(a) does not have any solution
(b) has a solution (α, β, γ) satisfying α + β2 + γ3 = 12
(c) has a unique solution
(d) has infinitely many solutions

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Unique solution: (0, 1, 2) 


Q.88. If for the matrix,JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the value of α4 + β4 is:       (JEE Main 2021)
(a) 3
(b) 2
(c) 1
(d) 4 

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
1 + α2 = 1
α2 = 0
α2 + β2 = 1
β2 = 1
α4 = 0
β4 = 1
α4 + β4 = 1 


Q.89. Let A be a 3 × 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 → 2R2 + 5R3 on 2A, then det(B) is equal to:       (JEE Main 2021)
(a) 64
(b) 16
(c) 128
(d) 80

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R2 → 2R2 + 5R3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R2 → R2 − 5R3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 16 × 4
= 64


Q.90. For the system of linear equations:
x − 2y = 1, x − y + kz = −2, ky + 4z = 6, k ∈ R,
consider the following statements:
(A) The system has unique solution if k ≠ 2, k ≠ −2.
(B) The system has unique solution if k = −2
(C) The system has unique solution if k = 2
(D) The system has no solution if k = 2
(E) The system has infinite number of solutions if k ≠ −2.
Which of the following statements are correct?       (JEE Main 2021)
(a) (B) and (E) only
(b) (C) and (D) only
(c) (A) and (E) only
(d) (A) and (D) only

Ans. d
x − 2y + 0.z = 1
x − y + kz = −2
0.x + ky + 4z = 6
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
For unique solution 4 − k2 ≠ 0
 k  ± 2
For k = 2:
x − 2y + 0.z = 1
x − y + 2z = −2
0.x + 2y + 4z = 6
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Δx = −48 ≠ 0
For k = 2, Δx ≠ 0
 For K = 2; The system has no solution.


Q.91. Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A2B2 − B2A2) X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has:       (JEE Main 2021)
(a) no solution
(b) exactly two solutions
(c) infinitely many solutions
(d) a unique solution

Ans. c
AT = A, BT = B
Let A2B2  B2A2 = P
PT = (A2B2  B2A2)T = (A2B2)T  (B2A2)T
= (B2)T (A2)T  (A2)T (B2)T
= B2A2  A2B2
 P is skew-symmetric matrix
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 ay + bz = 0 ..... (1)
ax + cz = 0 .... (2)
bx cy = 0 ..... (3)
From equation 1, 2, 3
Δ = 0 & Δ1 = Δ2 = Δ3 = 0
 equation have infinite number of solution


Q.92. The system of linear equations
3x - 2y - kz = 10
2x - 4y - 2z = 6
x + 2y - z = 5m
is inconsistent if:       (JEE Main 2021)
(a) k ≠ 3, m ∈ R
(b) k = 3, m ≠ 4/5
(c) k = 3, m = 4/5
(d) k ≠ 3, m ≠ 4/5

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
3(4 + 4) + 2(−2 + 2) − k(4 + 4) = 0
⇒ k = 3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
10(4 + 4) + 2(−6 + 10m) − 3(12 + 20m) ≠ 0
80 − 12 + 20m − 36 − 60m ≠ 0
40m ≠ 32 ⇒ m ≠ 4/5
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
3(−6 + 10m) − 10(−2 + 2) − 3(10m − 6) ≠ 0
−18 + 30m − 30m + 18 ≠ 0 ⇒ 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
3(−20m − 12) + 2(10m − 6) + 10(4 + 4) − 40m + 32 ≠ 0 ⇒ m ≠ 4/5


Q.93. The number of elements in the setJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere I is 2 × 2 identity matrix, is:       (JEE Main 2021)

Ans. 8
(I − A)3 = I3 − A3 − 3A(I − A) = I − A3
⇒ 3A(I − A) = 0 or A2 = A

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ a2 = a, b(a + d − 1) = 0, d2 = d
If b  0, a + d = 1  4 ways
If b = 0, a = 0, 1 & d = 0, 1  4 ways
 Total 8 matrices 


Q.94. If the system of linear equations
2x + y  z = 3
  z α
3x + 3y + βz = 3
has infinitely many solution, then α + β  αβ is equal to _____________.        (JEE Main 2021)

Ans. 5
× (i)  (ii)  (iii) gives :
 (1 + β)z = 3  α
For infinitely many solution
β + 1 = 0 = 3  α  (αβ) = (3, 1)
Hence, α + β  αβ = 5 


Q.95. Let A be a 3 × 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 241, then the value of det(A2) equal __________.         (JEE Main 2021)

Ans. 4
adj (2A) = 22 adjA
 adj(adj (2A)) = adj(4 adjA) = 16 adj (adj A)
= 16 | A | A
 adj (32 | A | A) = (32 | A |)2 adj A
12(32| A |)2 |adj A | = 23 (32 | A |)6 | adj A |
23 . 230 | A |6 . | A |2 = 241
| A |8 = 28  | A | = ±2
| A |2 = | A |2 = 4 


Q.96. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and M = A + A2 + A3 + ....... + A20, then the sum of all the elements of the matrix M is equal to _____________.          (JEE Main 2021)

Ans. 2020
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
So, required sum 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 60 + 420 + 105 + 35 × 41 = 2020


Q.97. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEThen the maximum value of f(x) is equal to ______________.          (JEE Main 2021)

Ans. 6
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= −2(cos2x) + 2(2 + 2cos⁡2x + sin2x)
= 4 + 4cos⁡2x − 2(cos2x − sin2x)

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ f(x)max = 4 + 2 = 6 


Q.98. For real numbers α and β, consider the following system of linear equations:
x + y − z = 2, x + 2y + αz = 1, 2x − y + z = β. If the system has infinite solutions, then α + β is equal to ______________. 
         (JEE Main 2021)

Ans. 5
For infinite solutions
Δ = Δ1 = Δ2 = Δ3 = 0 

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Δ = 3(2 + α) = 0
 α = 2 

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
1(1 + 2β2(1 + 4)  (β  2) = 0
β  7 = 0
β = 7
 α + β = 5


Q.99. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEDefine f : M → Z, as f(A) = det(A), for all A ∈ M, where z is set of all integers. Then the number of A∈M such that f(A) = 15 is equal to _____________.          (JEE Main 2021)

Ans. 16
| A | = ad  bc = 15
where a, b, c, d ∈ {±3, ±2, ±1, 0}
Case I ad = 9 & bc = 6
For ad possible pairs are (3, 3), (3, 3)
For bc possible pairs are (3, 2), (3, 2), (2, 3), (2, 3)
So total matrix = 2 × 4 = 8
Case II ad = 6 & bc = 9
Similarly total matrix = 2 × 4 = 8
 Total such matrices are = 16 


Q.100. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Then the number of 3 × 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________.          (JEE Main 2021)

Ans. 3125
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∵ AB = BA
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ d = b, e = a, f = c, g = h

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
No. of ways of selecting a, b, c, g, i
= 5 × 5 × 5 × 5 × 5
= 5= 3125
 No. of matrices B = 3125 


Q.101. Let A = {aij} be a 3 × 3 matrix,
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
then det(3Adj(2A−1)) is equal to _____________.          (JEE Main 2021)

Ans. 108
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
|A| = 4
det(3adj(2A−1))
= 33|adj(2a−1)|
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.102. Let a, b, c, d in arithmetic progression with common difference λ. If JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then value of λ2 is equal to ________________.          (JEE Main 2021)

Ans. 1
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

C2 → C2 − C3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
R2 → R2 − R1, R→ R3 − R1
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ 1(4λ− 4λ2 + 2λ) = 2
⇒ λ2 = 1


Q.103. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand B = 7A20  20A7 + 2I, where I is an identity matrix of order 3 × 3. If B = [bij], then b13 is equal to _____________.         (JEE Main 2021)

Ans. 910
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
B = 7A20 − 20A7 + 2I
= 7(I + C)20 + 20(I + C)7 + 2I
= 7(I + 20C + 20C2C2) − 20(I + 7C + 7C2C2) + 2I
So b13 = 7 × 20C2C2 − 20 × 7C2 = 910


Q.104. Let I be an identity matrix of order 2 × 2 and P =JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEThen the value of n ∈ N for which Pn = 5I − 8P is equal to ____________.         (JEE Main 2021) 

Ans. 6
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 λ2 + λ  1 = 0
 P2 + P  I = 0
 P2 = I  P
 P4 = I + P2  2P
 P4 = 2I  3P
Now, P4 . P2 = (2I  3P)(I  P) = 2I  5P + 3P2
 P6 = 5I  8P
So n = 6 


Q.105. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(4x+185) are in arithmetic progression for a real number x, then the value of the determinantJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis equal to:         (JEE Main 2021) 

Ans. 2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(4x)+ 4 − 4.4x = 10.4x + 36
(4x)2 − 14.4x − 32 = 0
(4x)2 + 2.4x − 16.4x − 32 = 0
4x(4x + 2) − 16.(4x + 2) = 0
(4x + 2)(4x − 16) = 0
4x = -2 (Not Possible)
Or 4x = 16
⇒ x = 2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 3(−2) − 1(0 − 4) + 4(1 − 0)
= −6 + 4 + 4 = 2


Q.106. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEsuch that AB = B and a + d = 2021, then the value of ad − bc is equal to ___________.         (JEE Main 2021) 

Ans. 2020
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
AB = B
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
α(a − 1) = −bβ and cα = β(1 − d)
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
−bc = (a − 1)(1 − d)
−bc = a − ad − 1 + d
ad − bc = a + d − 1
= 2021 - 1 = 2020


Q.107. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the value of det(A4) + det(A10  (Adj(2A))10) is equal to _____________.         (JEE Main 2021) 

Ans. 16
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
|A10 − adj(2A)10| = 0
 det(A4) + det(A10  (Adj(2A))10)
= 16 + 0 = 16 


Q.108. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEbe two 2 × 1 matrices with real entries such that A = XB, whereJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEIf JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the value of k is __________.         (JEE Main 2021) 

Ans. 1
XB = A
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Comparing both sides, We get 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 k2 = 1
 k = ± 1 ......(1)
and 2/3(k − 1)=0  k = 1 ....(2)
From (1) and (2),
k = 1 


Q.109. The total number of 3 × 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to _____________.         (JEE Main 2021) 

Ans. 766
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 
Tr(AAT) = x2 + y2 + z2 + a2 + b2 + c+ d2 + e2 + f2 = 9
Case-I: Nine ones = 1 case
Case-II: 8 zeroes and one entry is 3 = 9!/8!=9 cases
Case-III: Two 2’s, one 1’s and 6 zeroes =JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE= 63 × 4 = 252
Case IV: one 2, five 1, rest 0JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE= 63 × 8 = 504
 Total cases = 9 + 252 + 504 + 1 = 766


Q.110. JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
and I3 be the identity matrix of order 3. If the
determinant of the matrix (P−1API3)2 is αω2, then the value of α is equal to ______________.         (JEE Main 2021) 

Ans. 36
 |P−1AP−I|2
= |(P−1AP − I)(P−1AP−1)2|
= |P−1APP−1AP − 2P−1AP + I|
= |P−1A2P − 2P−1AP + P−1IP|
= |P−1(A2 − 2A + I)P|
= |P−1(A − I)2P|
= |P−1||A − I|2|P|
= |A − I|2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= (1(ω(ω + 1) + ω) − 7ω + ω2.ω)2
= (ω2 + 2ω − 7ω + 1)2
= (ω2 − 5ω + 1)2
= (−6ω)2
= 36ω2
∴ αω2 = 36ω2
⇒ α = 36


Q.111. If the matrixJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEsatisfies the equation 

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEfor some real numbers α and β, then β − α is equal to ___________.         (JEE Main 2021) 

Ans. 4
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ α + β = 0 and 220 + α219 + 2β = 4
⇒ 220 + α(219 − 2) = 4
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ β = 2
∴ β − α = 4


Q.112. If the system of equations
kx + y + 2z = 1
3x − y − 2z = 2
−2x −2y −4z = 3
has infinitely many solutions, then k is equal to __________. 
        (JEE Main 2021)

Ans. 21
D = 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 k (4  4)  1 ( 12  4) + 2 ( 6  2)
 16  16 = 0
Also, D= D2 = D3 = 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 k(8 + 6)  1( 12  4) + 2(9 + 4) = 0
  2k + 16 + 26 = 0
 2k = 42
 k = 21


Q.113. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If A2 = I3, then the value of x3 + y3 + z3 is ____________.         (JEE Main 2021)

Ans. 7
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ |A| = (x3 + y3 + z3 − 3xyz)
Given A2 = I3
|A2| = 1
∴ (x3 + y3 + z3 − 3xyz)2 = 1
⇒ x3 + y3 + z3 − 3xyz = 1 only as (x + y + z > 0)
⇒ x3 + y3 + z3 = 6 + 1 = 7


Q.114. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen 13(a2 + b2) is equal to         (JEE Main 2021)

Ans. 13 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ (1 + A)(I − A)−1
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 a2 + b2 = 1
 13(a2 + b2) = 13


Q.115. Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is ________.         (JEE Main 2021)

Ans. 540
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
a2 + b2 + c2 + d+ e2 + f2 + g2 + h2 + i2 = 7
Case I: Seven (1's) and two (0's)
Number of such matrices = 9C2 = 36
Case II: One (2) and three (1's) and five (0's)
Number of such matrices =JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE=504
 Total = 540


Q.116. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere α  R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k  R.
IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen a2 + k2 is equal to ______.          (JEE Main 2021)

Ans. 17
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ 2(3α + 4) = 5 + 3α
3α = −3 ⇒ α = −1

JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ (20 + 12α) = 2k ⇒ 8 = 2k ⇒ k = 4

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