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

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Q.1. Let β be a real number. Consider the matrix
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
If A7 − (β − 1)A6 − βA5 is a singular matrix, then the value of  is _________.       (JEE Advanced 2022)

Ans.  3
A7 − (β − 1)A− βA5 is a singular matrix. So determinant of this matrix equal to zero.

∴ |A7 − (β − 1)A− βA5| = 0

⇒ |A5(A2 − (β − 1)A − βI)| = 0

⇒ |A5||(A2 − βA + A − βI)| = 0

⇒ |A|5|A(A + I) − β(A + I)| = 0

⇒ |A|5|(A − βI)(A + I)| = 0

⇒ |A|5|A − βI||A + I| = 0

Now given,
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
|A| = 2 − 3 = −1
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= (β + 1)(−2 + 2) + 1(2 − 6)

= −4

∴ We get |A| ≠ 0 and |A + I| ≠ 0

∴ |A|5|A − βI||A + I| = 0 is possible only when |A − βI| = 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 2 − 3 − 3β
∴ 2 − 3 + 3β = 0
⇒ 3β = 1
⇒ 9β = 3


Q.2. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then which of the following matrices is equal to M2022?       (JEE Advanced 2022)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. a


Q.3. Let p, q, r be nonzero real numbers that are, respectively, the 10th,100th  and 1000th  terms of a harmonic progression. Consider the system of linear equations
x + y + z = 1
10x + 100y + 1000z = 0
qrx + pry + pqz = 0

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
The correct option is:       (JEE Advanced 2022)
(a) (I) → (T); (II) → (R); (III) → (S); (IV) → (T)
(b) (I) → (Q); (II) → (S); (III) → (S); (IV) → (R)
(c) (I) → (Q); (II) → (R); (III) → (P); (IV) → (R)
(d) (I) → (T); (II) → (S); (III) → (P); (IV) → (T)

Ans. b


Q.4. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE. For k ∈ N, if X′ AX = 33, then k is equal to _______.       (JEE Main 2022)

Ans. 10


Q.5. Let p and p + 2 be prime numbers and let
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Then the sum of the maximum values of α and β, such that pα and (p + 2)β divide Δ, is __________.
      (JEE Main 2022)

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

= 2(p!) . ((p + 1)!) . ((p + 2)!)

= 2(p + 1) . (p!)2 . ((p + 2)!)

= 2(p + 1)2 . (p!)3 . ((p + 2)!)

∴ Maximum value of α is 3 and β is 1.

∴ α + β = 4


Q.6. The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is __________.       (JEE Main 2022)

Ans. 282

In a 3 × 3 order matrix there are 9 entries.

These nine entries are zero or one.

The sum of positive prime entries are 2, 3, 5 or 7.

Total possible matrices =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

= 34 + 84 + 126 + 36
= 282


Q.7. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE, α, β ∈ R. Let α1 be the value of α which satisfiesJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand α2 be the value of α which satisfies (A + B)2 = B2. Then |α1−α2| is equal to ___________.       (JEE Main 2022)

Ans. 2
(A + B)= A2 + B2 + AB + BA
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

By (1) we get
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ α = 1 β = 0 ⇒ α1 = 1

Similarly if A+ AB + BA = 0 then
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ β = 0 and α = −1 ⇒ α2 = −1
∴ |α1 − α2| = |2| = 2


Q.8. Consider a matrixJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere α, β, γ are three distinct natural numbers. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the number of such 3 - tuples (α, β, γ) is ____________.       (JEE Main 2022)

Ans. 42 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ det(A) = (α + β + γ)(α − β)(β − γ)(γ − α)

Also, det(adj(adj(adj(adj(A)))))

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ α + β + γ = 12
⇒ (α, β, γ) distinct natural triplets
= 11C− 1 − 3C2(4) = 55 − 1 − 12
= 42


Q.9. Let S be the set containing all 3 × 3 matrices with entries from {−1, 0, 1}. The total number of matrices A ∈ S such that the sum of all the diagonal elements of ATA is 6 is ____________.       (JEE Main 2022)

Ans. 5376
Sum of all diagonal elements is equal to sum of square of each element of the matrix. 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
then tr(A . AT)

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

∵ ai, bi, c∈ {−1, 0, 1} for i = 1, 2, 3

∴ Exactly three of them are zero and rest are 1 or −1.

Total number of possible matrices 9C3 × 26

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 5376


Q.10. The number of matricesJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere a, b, c, d ∈ {−1, 0, 1, 2, 3,……, 10}, such that A = A−1, is ___________.       (JEE Main 2022)

Ans. 50
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
For A−1 must exist ad − bc ≠ 0 ...... (i)

and A = A−1 ⇒ A= I

∴ a2 + bc = d2 + bc = 1 ...... (ii)

and b(a + d) = c(a + d) = 0 ...... (iii)

Case I: When a = d = 0, then possible values of (b, c) are (1, 1), (−1, 1) and (1, −1) and (−1, 1).

Total four matrices are possible.

Case II: When a = −d then (a, d) be (1, −1) or (−1, 1).

Then total possible values of (b, c) are (12 + 11) × 2 = 46.

∴ Total possible matrices = 46 + 4 = 50.


Q.11. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEa, b ∈ R. If for some n ∈ N,JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen n + a + b is equal to ____________.       (JEE Main 2022)

Ans. 24
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

B3 = 0

∴ An = (1 + B)n = nC0I + nC1B + nC2B2 + nC3B3 +....

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

On comparing we get na = 48, nb = 96 and
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ a = 4, n = 12 and b = 8
n + a + b = 24


Q.12. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

then the number of elements in the set {n ∈ {1, 2,…, 100} : An + (ωB)n = A + B} is equal to ____________.       (JEE Main 2022)

Ans. 17


Q.13. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand let T= {A ∈ S : An(n+1) = I}. Then the number of elements inJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis ___________.       (JEE Main 2022)

Ans. 100
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ Tn = {A ∈ S; An(n+1) = I}
 b must be equal to 1
 In this case A2 will become identity matrix and a can take any value from 1 to 100
 Total number of common element will be 100. 


Q.14. Let A be a 3 × 3 matrix having entries from the set {−1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ___________.       (JEE Main 2022)

Ans. 414
Case-I:
1 → 7 times  

and −1 → 2 times
number of possible matrix =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Case-II:
1 → 6 times,
−1 → 1 times
and 0 → 2 times
number of possible matrix =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Case-III:
1 → 5 times,
and 0 → 4 times
number of possible matrix =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence total number of all such matrix A = 414


Q.15. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE. Then the number of elements in the set {(n, m) : n, m ∈ {1, 2, .........., 10} and nAn + mBm = I} is ____________.       (JEE Main 2022)

Ans1
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ AK = A, K ∈ I

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

So, BK = B, K ∈ I

nA+ mB= nA + mB

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

So, 2n − m = 1, −n + m = 0, 2m − n = 1

So, (m, n) = (1, 1)


Q.16. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere α is a non-zero real number anJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the positive integral value of α is ____________.       (JEE Main 2022)

Ans. 1


Q.17. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhere i = √−1. Then, the number of elements in the set { n ∈ {1, 2, ......, 100} : A= A } is ____________.       (JEE Main 2022)

Ans. 25
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
So A5 = A, A9 = A and so on.

Clearly n = 1, 5, 9, ......, 97

Number of values of n = 25


Q.18. If the system of linear equations
2x − 3y = γ + 5,
αx + 5y = β + 1
where αβγ  R has infinitely many solutions then the value
of | 9α + 3β + 5γ | is equal to ____________.       (JEE Main 2022)

Ans. 58

If 2x − 3y = γ + 5 and αx + 5y = β + 1 have infinitely many solutions then

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

⇒ α =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand 3β + 5γ = −28

So |9α + 3β + 5γ| = |−30 − 28| = 58


Q.19. Let A be a matrix of order 2 × 2, whose entries are from the set {0, 1, 2, 3, 4, 5}. If the sum of all the entries of A is a prime number p, 2 < p < 8, then the number of such matrices A is ___________.       (JEE Main 2022)

Ans. 180

∵ Sum of all entries of matrix A must be prime p such that 2 < p < 8 then sum of entries may be 3, 5 or 7.

If sum is 3 then possible entries are (0, 0, 0, 3), (0, 0, 1, 2) or (0, 1, 1, 1).

∴ Total number of matrices = 4 + 4 + 12 = 20

If sum of 5 then possible entries are

(0, 0, 0, 5), (0, 0, 1, 4), (0, 0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) and (1, 1, 1, 2).

∴ Total number of matrices = 4 + 12 + 12 + 12 + 12 + 4 = 56

If sum is 7 then possible entries are

(0, 0, 2, 5), (0, 0, 3, 4), (0, 1, 1, 5), (0, 3, 3, 1), (0, 2, 2, 3), (1, 1, 1, 4), (1, 2, 2, 2), (1, 1, 2, 3) and (0, 1, 2, 4).

Total number of matrices with sum 7 = 104

∴ Total number of required matrices

= 20 + 56 + 104

= 180


Q.20. The positive value of the determinant of the matrix A, whoseJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis _____________.       (JEE Main 2022)

Ans. 14
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= (14)3(3 − 2(−5) − 1(−1))
|A|4 = (14)4 ⇒ |A| = 14


Q.21. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEY = αI + βX + γXand Z = α2I − αβX + (β2 − αγ)X2, α, β, γ ∈ R. IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen (α − β + γ)2 is equal to ____________.       (JEE Main 2022)

Ans. 100
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 Y . Y−1 = I

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

∴ α = 5, β = 10, γ =15

∴ (α − β + γ)2 = 100


Q.22. If the system of equations
x + y + z = 6
2x + 5y + αz = β
x + 2y + 3z = 14
has infinitely many solutions, then α + β is equal to
       (JEE Main 2022)
(a) 8
(b) 36
(c) 44
(d) 48

Ans. c
Given,

x + y + z = 6 ...... (1)

2x + 5y + αz = β ..... (2)

x + 2y + 3z = 14 ...... (3)

System of equation have infinite many solutions.

∴ Δx = Δy = Δz = 0 and
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
C1 → C1 − C3

C2 → C2 − C3
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ −2 + α + 10 − 2α = 0

⇒ 8 − α = 0

⇒ α = 8

Now, x + y + z = 6

2x + 5y + 8z = β

x + 2y + 3z = 14
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

C→ C1 − 6C3

C2 → C2 − C3

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

⇒ −β + 48 − 12 = 0

⇒ β = 36


Q.23. Which of the following matrices can NOT be obtained from the matrixJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEby a single elementary row operation?       (JEE Main 2022)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

For option A:

R1 → R+ R2

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

∴ Option A can be obtained.

For option B:

R↔ R2

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

∴ Option B can be obtained.

Option C:

Not possible by a single elementary row operation.

Option D:

R2 → R2 + 2R1

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

∴ Option D can be obtained.


Q.24. Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then       (JEE Main 2022)
(a) the system of linear equations AX = 0 has a unique solution
(b) the system of linear equations AX = 0 has infinitely many solutions
(c) B is an invertible matrix
(d) adj(A) is an invertible matrix

Ans. b
AB is zero matrix

⇒ |A| = |B| = 0

So neither A nor B is invertible

If |A| = 0

⇒ |adjA| = 0 so adjA

AX = 0 is homogeneous system and |A| = 0

So, it is having infinitely many solutions


Q.25. Let A and B be any two 3 × 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?       (JEE Main 2022)
(a) A4 − B4 is a smmetric matrix
(b) AB − BA is a symmetric matrix
(c) B− A5 is a skew-symmetric matrix
(d) AB + BA is a skew-symmetric matrix

Ans. c
(A) M = A4 − B4

MT = (A4 − B4)T = (AT)4 − (BT)4

= A4 − (−B)4 = A− B= M

(B) M = AB − BA

M= (AB − BA)T = (AB)T − (BA)T

= BTA− ATBT

= −BA − A(−B)

= AB − BA = M

(C) M = B− A5

MT = (BT)− (AT)= −(B5 + A5) ≠ −M

(D) M = AB + BA

MT = (AB)+ (BA)T
= BTAT + ATBT = −BA − AB = −M


Q.26. Let the matrixJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand the matrix B0 = A49 + 2A98. If Bn = Adj(Bn−1) for all n ≥ 1, then det(B4) is equal to:       (JEE Main 2022)
(a) 328
(b) 330
(c) 332
(d) 336

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

Now B0 = A49 + 2A98 = (A3)16 . A + 2(A3)32 . A2

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

|B0| = 9

Since, Bn = Adj|Bn−1| ⇒ |Bn| = |Bn−1|2

Hence |B4| = |B3|= |B2|4 = |B1|8 = |B0|16
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.27. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
If A2 + γA + 18I = O, then det(A) is equal to _____________.       (JEE Main 2022)
(a) −18
(b) 18
(c) −50
(d) 50

Ans. b
Characteristic equation of A is given by

|A − λI| = 0

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

⇒ λ2 − (4 + β)λ + (4β + 2α) = 0

So, A2 − (4 + β)A + (4β + 2α)I = 0
|A| = 4β + 2α = 18


Q.28. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEELet α, β ∈ R be such that αA2 + βA = 2I. Then α + β is equal to       (JEE Main 2022)
(a) −10
(b) −6
(c) 6
(d) 10

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

On Comparing

8α = 2β, −3α + β = 2, 21α − 5β = 2

⇒ α = 2, β = 8

So, α + β = 10


Q.29. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the value of A′BA is:       (JEE Main 2022)
(a) 1224
(b) 1042
(c) 540
(d) 539

Ans. d
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= [92 + 122 − 152 − 102 + 132 + 162 + 112 − 14+ 172]
= [(92 − 102) + (112 + 122) + (132 − 142) + (162 − 152) + 172]
= [−19 + 265 + (−27) + 31 + 289]
= [585 − 46] = [539]


Q.30. Let A be a 2 × 2 matrix with det (A) = − 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be:       (JEE Main 2022)
(a) −1
(b) 2
(c) 1
(d) −√2

Ans. b
|(A + I)(adj A + I)| = 4

⇒ |A adj A + A + adj A + I| = 4

⇒ |(A)I + A + adj A + I| = 4

|A| = −1 ⇒ |A + adj A| = 4

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ a + d = ±2


Q.31. If the system of linear equations.
8x + y + 4z = −2
x + y + z = 0
λx − 3y = μ
has infinitely many solutions, then the distance of the point (λ, μ ,JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE) from the plane 8x + y + 4z + 2 = 0 is:       (JEE Main 2022)
(a) 3√5
(b) 4
(c) 26/9
(d) 10/3

Ans. d
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 8(3) − 1(−λ) + 4(−3 − λ)

= 24 + λ − 12 − 4λ

= 12 − 3λ

So for λ = 4, it is having infinitely many solutions.
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= −2(3) − 1(−μ) + 4(−μ)

= −6 − 3μ = 0

For μ = −2

Distance of (4, −2, −1/2) from 8x + y + 4z + 2 = 0
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.32. The number of real values of λ, such that the system of linear equations
2x − 3y + 5z = 9
x + 3y − z = −18
3x − y + (λ2 − | λ |)z = 16
has no solutions, is
       (JEE Main 2022)
(a) 0
(b) 1
(c) 2
(d) 4

Ans. c

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 9λ2 − 9|λ| − 43

= 9|λ|− 9|λ| − 43

Δ = 0 for 2 values of |λ| out of which one is −ve and other is +ve

So, 2 values of λ satisfy the system of equations to obtain no solution.


Q.33. The number of θ ∈ (0, 4π) for which the system of linear equations
3(sin⁡3θ)x − y + z = 2
3(cos⁡2θ)x + 4y + 3z = 3
6x + 7y + 7z = 9
has no solution, is:       (JEE Main 2022)
(a) 6
(b) 7
(c) 8
(d) 9

Ans. b
Given,

3(sin⁡3θ)x − y + z = 2

3(cos⁡2θ)x + 4y + 3z = 3

6x + 7y + 7z = 9

For no solutions determinant of coefficient will be = 0

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

⇒ 3sin⁡3θ(28 − 21) + 1(21cos⁡2θ − 18) + 1(21cos⁡2θ − 24) = 0

⇒ 21sin⁡3θ + 42cos⁡2θ − 42 = 0

⇒ sin⁡3θ + 2cos⁡2θ − 2 = 0

⇒ 3sin⁡θ − 4sin3θ + 2(1 − 2sin2θ) − 2 = 0

⇒ 3sin⁡θ − 4sin3θ − 4sin2θ = 0

⇒ 4sin3θ + 4sin2θ − 3sin⁡θ = 0

⇒ sin⁡θ(4sin2θ + 4sin⁡θ − 3) = 0

∴ sin⁡θ = 0

⇒ θ = π, 2π, 3π when θ ∈ (0, 4π)

or,

4sin2θ + 4sin⁡θ − 3 = 0

⇒ 4sin2θ + 6sin⁡θ − 2sin⁡θ − 3 = 0

⇒ 2sin⁡θ(2sin⁡θ + 3) − 1(2sin⁡θ + 3) = 0

⇒ (2sin⁡θ − 1)(2sin⁡θ + 3) = 0

∴ sin⁡θ = 1/2

or,

sin⁡θ =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE[not possible as sin ∈ [−1, 1]]

∴ sin⁡θ = 1/2

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

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

∴ Total 7 values of θ possible.


Q.34. Let A and B be two square matrices of order 2. If det(A) = 2, det(B) = 3 and det((det5(detA)B)A2) = 2a3b5c for some a, b, c, ∈ N, then a + b + c is equal to:       (JEE Main 2022)
(a) 10
(b) 12
(c) 13
(d) 14

Ans. b
Given,

det(A) = 2,

det(B) = 3

and det((det(5(detA)B))A2) = 2a3b5c

⇒ |det(5(detA)B)A2| = 2a3b5c

⇒ ||5(detA)B)A2| = 2a3b5c

⇒ ||5|A|B|A2| = 2a3b5c

⇒ ||5 . 2 . B|A2| = 2a . 3b . 5c

⇒ ||10B|A2| = 2a . 3b . 5c

⇒ |10. |B|A2| = 2a . 3b . 5c

As |k . A| = kn|A|

⇒ |100 × 3A2| = 2a . 3b . 5c

⇒ (300). |A2| = 2a . 3b . 5c

⇒(300)2 . |A|2 = 2a . 3b . 5c

⇒(300)2 . 22 = 2a . 3b . 5c

⇒ 9 × 100 × 100 × 22 = 2a . 3b . 5c

⇒ 32 × 2× 52 × 22 × 5× 22 = 2a . 3b . 5c

⇒ 26 . 32 . 54 = 2a . 3b . 5c

Comparing both sides, we get

a = 6, b = 2, c = 4

∴ a + b + c = 6 + 2 + 4 = 12


Q.35. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEα ∈ C. Then the absolute value of the sum of all values of α for which det(AB) = 0 is:       (JEE Main 2022)
(a) 3
(b) 4
(c) 2
(d) 5

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

|AB| = 0

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

⇒ (4α + 4)(α2 + 9 + 2α − 6) = 0

⇒ (4α + 4)(α2 + 2α + 3) = 0

∴ α− = −1

or α2 + 2α + 3 = 0

α1 + α2 = −2

∴ Sum of all values of α = −1 − 2 = −3

∴ Absolute value of α = |−3| = 3


Q.36. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
then the sum of all elements of the matrix B is       (JEE Main 2022)
(a) −5
(b) −6
(c) −7
(d) −8

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
and
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now let, 
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now sum of elements = −1 − 5 − 1 + 0 = −7


Q.37. Let A = [aij] be a square matrix of order 3 such that aij = 2j−i, for all i, j = 1, 2, 3. Then, the matrix A2 + A3 + ...... + A10 is equal to:        (JEE Main 2022)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. a

Given, aij = 2j−i

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 3A

Similarly, A3 = 32A

A4 = 33A

∴ A2 + A3 +......+ A10

= 3A + 32A + 33A + ...... + 39A

= A(3 + 32 + 33 +...... + 39)

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


Q.38. If the system of linear equations
2x + y − z = 7
x − 3y + 2z = 1
x + 4y + δz = k, where δ, k ∈ R has infinitely many solutions, then δ + k is equal to:        (JEE Main 2022)
(a) −3
(b) 3
(c) 6
(d) 9 

Ans. b
2x + y − z = 7

x − 3y + 2z = 1

x + 4y + δz = k

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

δ = −3

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

⇒ 6 − k = 0 ⇒ k=6
δ + k = −3 + 6 = 3


Q.39. Let A be a matrix of order 3 × 3 and det (A) = 2. Then det (det (A) adj (5 adj (A3))) is equal to _____________.        (JEE Main 2022)
(a) 512 × 106 
(b) 256 × 106 
(c) 1024 × 106 
(d) 256 × 1011

Ans. a
|A| = 2

||A| = adj(5adjA3)|

= |25|A|adj(adjA3)|

= 253|A|3 . |adjA3|2

= 253 . 23 . |A3|4
= 25. 23 . 212 = 10. 512


Q.40. If the system of linear equations
2x + 3y − z = −2
x + y + z = 4
x − y + |λ|z = 4λ − 4
where, λ ∈ R, has no solution, then        (JEE Main 2022)
(a) λ = 7
(b) λ = −7
(c) λ = 8
(d) λ= 1

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
But at λ = 7, Dx = Dy = Dz = 0

P1: 2x + 3y − z = −2

P2: x + y + z = 4

P3: x − y + |λ|z = 4λ − 4

So clearly 5P2 − 2P1 = P3, so at λ = 7, system of equation is having infinite solutions.

So λ = −7 is correct answer.


Q.41. Let A and B be two 3 × 3 matrices such that AB = I and |A| = 1/8. Then |adj(Badj(2A))| is equal to        (JEE Main 2022)
(a) 16
(b) 32
(c) 64
(d) 128

Ans. c
A and B are two matrices of order 3 × 3.
and AB = I,
|A| = 1/8
Now, |A||B| = 1
|B| = 8
∴ |adj(B(adj(2A))| = |B(adj(2A))|2
= |B|2|adj(2A)|2
= 26|2A|2×2
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.42. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEThen the sum of the squares of all the values of a, for which 2f′(10) − f′(5) + 100 = 0, is        (JEE Main 2022)
(a) 117
(b) 106
(c) 125
(d) 136

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
f(x) = a(a2 + ax) + 1(a2x + ax2)

= a(x + a)2

f′(x) = 2a(x + a)

Now, 2f′(10) − f′(5) + 100 = 0

⇒ 2 . 2a(10 + a) − 2a(5 + a) + 100 = 0

⇒ 2a(a + 15) + 100 = 0

⇒ a2 + 15a + 50 = 0

⇒ a = −10, −5

∴ Sum of squares of values of a = 125.


Q.43. Let the system of linear equations
x + 2y + z = 2,
αx + 3y − z = α,
−αx + y + 2z = −α
be inconsistent. Then α is equal to:        (JEE Main 2022)
(a) 5/2

(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c) 7/2
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. d
x + 2y + z = 2,
αx + 3y − z = α,
−αx + y + 2z = −α
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 7 + 2α

Δ = 0 ⇒ α =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 

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

∴ For no solution α =JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q.44. If the system of equations
αx + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = β
has infinitely many solutions, then the ordered pair (α, β) is equal to:        (JEE Main 2022)
(a) (1, −3)
(b) (−1, 3)
(c) (1, 3)
(d) (−1, −3)

Ans. c
Given system of equations

αx + y + z = 5

x + 2y + 3z = 4, has infinite solution

x + 3y + 5z = β

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

⇒ α = 1

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

⇒ 5(1) − 1(20 − 3β) + 1(12 − 2β) = 0

⇒ β = 3

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

⇒ −2β + 6 = 0

⇒ β = 3

Similarly,

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

∴ (α, β) = (1, 3)


Q.45. The ordered pair (a, b), for which the system of linear equations
3x − 2y + z = b
5x − 8y + 9z = 3
2x + y + az = −1
has no solution, is:
        (JEE Main 2022)
(a)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. c
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now 3 (equation (1)) − (equation (2)) − 2 (equation (3)) is

3(3x − 2y + z − b) − (5x − 8y + 9z − 3) − 2(2x + y + az + 1) = 0

⇒ −3b + 3 − 2 = 0 ⇒ b = 1/3

So for no solution a = −3 and b ≠ 1/3


Q.46. Let A be a 3 × 3 invertible matrix. If |adj (24A)| = |adj (3 adj (2A))|, then |A|2 is equal to:        (JEE Main 2022)
(a) 66
(b) 212
(c) 26
(d) 1

Ans. c
We know, |adjA| = |A|n−1

Now, |adj24A| = |adj3(adj2A)|

⇒ |24A|3−1 = |3adj2A|3−1

⇒ |24A|2 = |3adj2A|2

Also, we know, |KA| = Kn|A|

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

⇒ (24)6|A|2 = 36 . (|2A|3−1)2

⇒ (24)6|A|2 = 36 . |2A|4

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

⇒ 36 . 86 . |A|2 = 36 . 84 . |A|4

⇒ 82 = |A|2

⇒ |A|2 = 64 = 26


Q.47. The system of equations
−kx + 3y − 14z = 25
−15x + 4y − kz = 3
−4x + y + 3z = 4
is consistent for all k in the set        (JEE Main 2022)
(a) R
(b) R − {−11, 13}
(c) R − {13}
(d) R − {−11, 11}

Ans. d
The system may be inconsistent if

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

Hence if system is consistent then k ∈ R − {11, −11}.


Q.48. LetJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEIf M and N are two matrices given byJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen MN2 is:        (JEE Main 2022)
(a) a non-identity symmetric matrix
(b) a skew-symmetric matrix
(c) neither symmetric nor skew-symmetric matrix
(d) an identity matrix

Ans. a
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
M = A2 + A4 + A6 + ..... + A20

= −4I + 16I − 64I + ..... upto 10 terms

= −I [4 − 16 + 64 .... + upto 10 terms]
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
N = A1 + A3 + A5 + .... + A19

= A − 4A + 16A +  ..... upto 10 terms

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

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

(MN2)T = (KI)T = KI

∴ A is correct


Q.49. Let A be a 3 × 3 real matrix such that

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
If X = (x1, x2, x3)T and I is an identity matrix of order 3, then the systemJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEhas        (JEE Main 2022)
(a) no solution 
(b) infinitely many solutions
(c) unique solution 
(d) exactly two solutions 

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

a = −2, b = 3, c = 1, d = −1, e = 2, f = 1, g = −1, h = 1, i = 2

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

⇒ −4x1 + 3x2 + x3 = 4 ..... (i)

−x1 + x3 = 1 ...... (ii)

−x1 + x= 1 ...... (iii)

So 3(iii) + (ii) = (i)

∴ Infinite solution


Q.50. Let the system of linear equations
x + y + αz = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x, y, z). If (α, x), (y, α) and (x, −y) are collinear points, then the sum of absolute values of all possible values of α is         (JEE Main 2022)
(a) 4 

(b) 3
(c) 2
(d) 1

Ans. c
Given system of equations

x + y + az = 2 ..... (i)

3x + y + z = 4 ..... (ii)

x + 2z = 1 ..... (iii)

Solving (i), (ii) and (iii), we get

x = 1, y = 1, z = 0 (and for unique solution a ≠ −3)

Now, (α, 1), (1, α) and (1, −1) are collinear

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

⇒ α(α + 1) − 1(0) + 1(−1 − α) = 0

⇒ α2 − 1 = 0

∴ α = ±1

∴ Sum of absolute values of α = 1 + 1 = 2


Q.51. Let S = {√n : 1 ≤ n ≤ 50 and n is odd}.
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 IfJEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen λ is equal to:         (JEE Main 2022)
(a) 218
(b) 221
(c) 663
(d) 1717

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

|adjA| = |A|n−1

Here, n = order of matrix.

Here, n = 3

∴ |adjA| = |A|3−1 = |A|2

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

= 1(1 − 0) − 0 + a(0 − (−a))

= a2 + 1

∴ |adjA| = |A|= (a2 + 1)2

JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= (12 + 1)+ (3 + 1)2 + (5 + 1)2 + .... + (49 + 1)2
= 2+ 42 + 62 + .... + 502
= 22(12 + 22 + 32 + .... + 252)
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 ∴ 100K = 100.221
⇒ K = 221


Q.52. The number of values of α for which the system of equations:
x + y + z = α
αx + 2αy + 3z = −1
x + 3αy + 5z = 4
is inconsistent, is         (JEE Main 2022)
(a) 0
(b) 1
(c) 2
(d) 3

Ans. b
JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 1(10α − 9α) − 1(5α − 3) + 1(3α2 − 2α)
= α − 5α + 3 + 3α2 − 2α
= 3α− 6α + 3
For inconsistency Δ = 0 i.e. α = 1
Now check for α = 1
x + y + z = 1...(i)
x + 2y + 3z = −1...(ii)
x + 3y + 5z = 4...(iii)
By (ii) ×2− (i) ×1
x + 3y + 5z = −3
so equations are inconsistent for α = 1

The document JEE Main Previous year questions (2021-22): Matrices and Determinants - 1 - 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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