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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

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 =
(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)

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

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. Let∀ n > m and n, m ∈ N. Consider a matrix A = [aij]3 × 3 where       (JEE Main 2021)
(a) (15)× 242
(b) (15)2 × 234
(c) (105)2 × 238
(d) (105)2 × 236

Ans. c

= (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

= −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)]

= 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

= 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. Ifthen the determinant is equal to:       (JEE Main 2021)
(a) a2a6 − a4a8
(b) a
(c) a1a9 − a3a
(d) a5

Ans. c
r = 1, 2, 3, ......, a1, a2, a3, ..... are in G.P.

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)
(b)
(c)
(d)

Ans. d

forsystem 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

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)
(b)
(c)
(d)

Ans. d

⇒ a(2b + 1 − b) − 0 + 1(b2 − 0) = ±2

Sum of possible values of 'a' is

Q.62. Letwhere [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

R1  R1  R3 & R2  R2  R3

2[x] + 6 + [x] = 192 ⇒ [x] = 62

Q.63. If the matrixsatisfies A(A3 + 3I) = 2I, then the value of K is:       (JEE Main 2021)
(a) 1/2
(b)
(c) -1
(d) 1

Ans. a

A+ 3IA = 2I
⇒ A4 = 2I − 3A
Also characteristic equation of A is |A−λI|=0

⇒ 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. LetThen A2025 − A2020 is equal to:       (JEE Main 2021)
(a) A− A
(b) A
(c) A5 − A
(d) A6

Ans. a

Q.65. Ifand Q = ATBA, then the inverse of the matrix A Q2021 AT is equal to:        (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. b

Q= ATBAATBA = ATBIBA
⇒ Q2 = ATB2A
Q3 = ATB2AATBA ⇒ Q3 = ATB3A
Similarly: Q2021 = ATB2021A

AQ2021 = AT = AATB2021AAT = IB2021I

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

C1 → C1 + C2

R1 → R1 − R2, R→ R2 − R3

or 4sin⁡3θ = −2

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. Let 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

∴ 4(α − β) = 4(1) = 4

Q.69. If then P50 is:          (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. a

Q.70. The number of distinct real roots of          (JEE Main 2021)
(a) 4
(b) 1
(c) 2
(d) 3

Ans. b

Apply: R→ R1 − R& R2 → R2 − R

⇒ (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

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

AX = X
Replace X by AX
A2X = AX = X
Replace X by AX
A3X = AX = X

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

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

3(2k + 15) + K + 18  28 = 0
7k + 35 = 0
k =  5

Q.75. Let, 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

Det (Q) = 9

a − 3 = ±6 ⇒ a = 9, −3

| 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

⇒ 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. LetIf 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

Similarly,

Tr(A) − Tr(B) = 1 − (−1) = 2

Q.79. The solutions of the equation        (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. d
By using C1 → C1 − C2 and C3 → C3 − C2 we get

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

Q.80. If x, y, z are in arithmetic progression with common difference d, x ≠ 3d, and the determinant of the matrixis zero, then the value of k2 is:        (JEE Main 2021)
(a) 72
(b) 12
(c) 36
(d) 6

Ans. a

R1 → R+ R3 − 2R2

⇒ (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. Ifthen a possible value of α is:        (JEE Main 2021)
(a) π/4
(b) π/6
(c) π/2
(d) π/3

Ans. a

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

⇒ 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. Let Then, the system of linear equations

(JEE Main 2021)
(a) Exactly two solutions
(b) Infinitely many solutions
(c) A unique solution
(d) No solution

Ans. d

⇒ 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

= 20  2(25) 3(10)
= 20  50 + 30 = 0

= 20a  2(7b + 11c) 3(2b  6c)
= 20a  14b  22c + 6b +18c
= 20a  8b  4c
= 4(5a  2b  c)

= 7b + 11c  a(25) 3(2c  b)
= 7b + 11c  25a  6c + 3b
25a + 10b + 5c
5(5a  2b  c)

= 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 of       (JEE Main 2021)
(a) −2
(b) 0
(d) (a + 2)(a + 3)(a + 4)
(d) (a + 1)(a + 2)(a + 3)

Ans. a

R2 → R2 − R1 and R3 → R3 − R1

= 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

= 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

Unique solution: (0, 1, 2)

Q.88. If for the matrix,then the value of α4 + β4 is:       (JEE Main 2021)
(a) 3
(b) 2
(c) 1
(d) 4

Ans. c

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

R2 → 2R2 + 5R3

R2 → R2 − 5R3

= 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

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

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

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

3(4 + 4) + 2(−2 + 2) − k(4 + 4) = 0
⇒ k = 3

10(4 + 4) + 2(−6 + 10m) − 3(12 + 20m) ≠ 0
80 − 12 + 20m − 36 − 60m ≠ 0
40m ≠ 32 ⇒ m ≠ 4/5

3(−6 + 10m) − 10(−2 + 2) − 3(10m − 6) ≠ 0
−18 + 30m − 30m + 18 ≠ 0 ⇒ 0

3(−20m − 12) + 2(10m − 6) + 10(4 + 4) − 40m + 32 ≠ 0 ⇒ m ≠ 4/5

Q.93. The number of elements in the setwhere 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

⇒ 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
= 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. If 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

So, required sum

= 60 + 420 + 105 + 35 × 41 = 2020

Q.97. LetThen the maximum value of f(x) is equal to ______________.          (JEE Main 2021)

Ans. 6

= −2(cos2x) + 2(2 + 2cos⁡2x + sin2x)
= 4 + 4cos⁡2x − 2(cos2x − sin2x)

⇒ 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

Δ = 3(2 + α) = 0
α = 2

1(1 + 2β2(1 + 4)  (β  2) = 0
β  7 = 0
β = 7
α + β = 5

Q.99. LetDefine 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. Let 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

∵ AB = BA

⇒ d = b, e = a, f = c, g = h

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,

then det(3Adj(2A−1)) is equal to _____________.          (JEE Main 2021)

Ans. 108

|A| = 4

Q.102. Let a, b, c, d in arithmetic progression with common difference λ. If  then value of λ2 is equal to ________________.          (JEE Main 2021)

Ans. 1

C2 → C2 − C3

R2 → R2 − R1, R→ R3 − R1

⇒ 1(4λ− 4λ2 + 2λ) = 2
⇒ λ2 = 1

Q.103. Letand 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

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 =Then the value of n ∈ N for which Pn = 5I − 8P is equal to ____________.         (JEE Main 2021)

Ans. 6

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. If(4x+185) are in arithmetic progression for a real number x, then the value of the determinantis equal to:         (JEE Main 2021)

Ans. 2

(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

= 3(−2) − 1(0 − 4) + 4(1 − 0)
= −6 + 4 + 4 = 2

Q.106. Letsuch that AB = B and a + d = 2021, then the value of ad − bc is equal to ___________.         (JEE Main 2021)

Ans. 2020

AB = B

α(a − 1) = −bβ and cα = β(1 − d)

−bc = (a − 1)(1 − d)
−bc = a − ad − 1 + d
ad − bc = a + d − 1
= 2021 - 1 = 2020

Q.107. Ifthen the value of det(A4) + det(A10  (Adj(2A))10) is equal to _____________.         (JEE Main 2021)

Ans. 16

= 16 + 0 = 16

Q.108. Letbe two 2 × 1 matrices with real entries such that A = XB, whereIf then the value of k is __________.         (JEE Main 2021)

Ans. 1
XB = A

Comparing both sides, We get

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

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 == 63 × 4 = 252
Case IV: one 2, five 1, rest 0= 63 × 8 = 504
Total cases = 9 + 252 + 504 + 1 = 766

Q.110.

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

= (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 matrixsatisfies the equation

for some real numbers α and β, then β − α is equal to ___________.         (JEE Main 2021)

Ans. 4

⇒ α + β = 0 and 220 + α219 + 2β = 4
⇒ 220 + α(219 − 2) = 4

⇒ 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

k (4  4)  1 ( 12  4) + 2 ( 6  2)
16  16 = 0
Also, D= D2 = D3 = 0

k(8 + 6)  1( 12  4) + 2(9 + 4) = 0
2k + 16 + 26 = 0
2k = 42
k = 21

Q.113. Letwhere 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

∴ |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. Ifand

then 13(a2 + b2) is equal to         (JEE Main 2021)

Ans. 13

⇒ (1 + A)(I − A)−1

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

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 ==504
Total = 540

Q.116. Letwhere α  R. Suppose Q = [ qij] is a matrix satisfying PQ = kl3 for some non-zero k  R.
Ifthen a2 + k2 is equal to ______.          (JEE Main 2021)

Ans. 17

⇒ 2(3α + 4) = 5 + 3α
3α = −3 ⇒ α = −1

⇒ (20 + 12α) = 2k ⇒ 8 = 2k ⇒ k = 4

The document JEE Main Previous year questions (2021-22): Matrices and Determinants - 2 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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