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

Q.1. Let β be a real number. Consider the matrix

If A7 − (β − 1)A6 − βA5 is a singular matrix, then the value of 9β is _________.       (JEE Advanced 2022)

Ans.  3
A⁷ − (β − 1)A⁶ − βA⁵ is a singular matrix. So determinant of this matrix equal to zero.

∴ |A⁷ − (β − 1)A⁶ − 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,

|A| = 2 − 3 = −1

= (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

= 2 − 3 − 3β
∴ 2 − 3 + 3β = 0
⇒ 3β = 1
⇒ 9β = 3

Q.2. If then which of the following matrices is equal to M²⁰²²?       (JEE Advanced 2022)
(a)
(b)
(c)
(d)

Ans. a

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

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. Let. For k ∈ N, if X′ A^k X = 33, then k is equal to _______.       (JEE Main 2022)

Ans. 10

Q.5. Let p and p + 2 be prime numbers and let

Then the sum of the maximum values of α and β, such that p^α and (p + 2)^β divide Δ, is __________.       (JEE Main 2022)

Ans. 4

= 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 =

= 34 + 84 + 126 + 36
= 282

Q.7. Let, α, β ∈ R. Let α₁ be the value of α which satisfiesand 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)² = A² + B² + AB + BA

By (1) we get

∴ α = 1 β = 0 ⇒ α₁ = 1

Similarly if A² + AB + BA = 0 then

⇒ β = 0 and α = −1 ⇒ α₂ = −1
∴ |α₁ − α₂| = |2| = 2

Q.8. Consider a matrixwhere α, β, γ are three distinct natural numbers. Ifthen the number of such 3 - tuples (α, β, γ) is ____________.       (JEE Main 2022)

Ans. 42 

∴ det(A) = (α + β + γ)(α − β)(β − γ)(γ − α)

Also, det(adj(adj(adj(adj(A)))))

⇒ α + β + γ = 12
⇒ (α, β, γ) distinct natural triplets
= ¹¹C₂ − 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 A^TA 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. 

then t_r(A . A^T)

∵ a_i, b_i, c_i ∈ {−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

= 5376

Q.10. The number of matriceswhere a, b, c, d ∈ {−1, 0, 1, 2, 3,……, 10}, such that A = A⁻¹, is ___________.       (JEE Main 2022)

Ans. 50

For A⁻¹ must exist ad − bc ≠ 0 ...... (i)

and A = A⁻¹ ⇒ A² = 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. Leta, b ∈ R. If for some n ∈ N,then n + a + b is equal to ____________.       (JEE Main 2022)

Ans. 24

B³ = 0

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

On comparing we get na = 48, nb = 96 and

⇒ a = 4, n = 12 and b = 8
n + a + b = 24

Q.12. Let

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. Letand let T_n = {A ∈ S : Aⁿ⁽ⁿ⁺¹⁾ = I}. Then the number of elements inis ___________.       (JEE Main 2022)

Ans. 100


∴ T_n = {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 =
Case-II:
1 → 6 times,
−1 → 1 times
and 0 → 2 times
number of possible matrix =
Case-III:
1 → 5 times,
and 0 → 4 times
number of possible matrix =
Hence total number of all such matrix A = 414

Q.15. Let. Then the number of elements in the set {(n, m) : n, m ∈ {1, 2, .........., 10} and nAⁿ + mB^m = I} is ____________.       (JEE Main 2022)

Ans. 1

⇒ A^K = A, K ∈ I

So, B^K = B, K ∈ I

nAⁿ + mB^m = nA + mB

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

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

Q.16. Letwhere α is a non-zero real number anthen the positive integral value of α is ____________.       (JEE Main 2022)

Ans. 1

Q.17. Letwhere i = √−1. Then, the number of elements in the set { n ∈ {1, 2, ......, 100} : Aⁿ = A } is ____________.       (JEE Main 2022)

Ans. 25

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 α, β, γ ∈ 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

⇒ α =and 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, whoseis _____________.       (JEE Main 2022)

Ans. 14

= (14)³(3 − 2(−5) − 1(−1))
|A|⁴ = (14)⁴ ⇒ |A| = 14

Q.21. LetY = αI + βX + γX² and Z = α²I − αβX + (β² − αγ)X², α, β, γ ∈ R. Ifthen (α − β + γ)² is equal to ____________.       (JEE Main 2022)

Ans. 100


 Y . Y⁻¹ = I

∴ 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.

∴ Δ_x = Δ_y = Δ_z = 0 and

C₁ → C₁ − C₃

C₂ → C₂ − C₃

⇒ −2 + α + 10 − 2α = 0

⇒ 8 − α = 0

⇒ α = 8

Now, x + y + z = 6

2x + 5y + 8z = β

x + 2y + 3z = 14

C₁ → C₁ − 6C₃

C₂ → C₂ − C₃

⇒ −β + 48 − 12 = 0

⇒ β = 36

Q.23. Which of the following matrices can NOT be obtained from the matrixby a single elementary row operation?       (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. c
Given matrix

For option A:

R₁ → R₁ + R₂

∴ Option A can be obtained.

For option B:

R₁ ↔ R₂

∴ Option B can be obtained.

Option C:

Not possible by a single elementary row operation.

Option D:

R₂ → R₂ + 2R₁

∴ 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) B⁵ − A⁵ is a skew-symmetric matrix
(d) AB + BA is a skew-symmetric matrix

Ans. c
(A) M = A⁴ − B⁴

M^T = (A⁴ − B⁴)^T = (A^T)⁴ − (B^T)⁴

= A⁴ − (−B)⁴ = A⁴ − B⁴ = M

(B) M = AB − BA

M^T = (A^B − B^A)^T = (AB)^T − (BA)^T

= B^TA^T − A^TB^T

= −BA − A(−B)

= AB − BA = M

(C) M = B⁵ − A⁵

M^T = (B^T)⁵ − (A^T)⁵ = −(B⁵ + A⁵) ≠ −M

(D) M = AB + BA

M^T = (AB)^T + (BA)^T
= B^TA^T + A^TB^T = −BA − AB = −M

Q.26. Let the matrixand the matrix B₀ = A⁴⁹ + 2A⁹⁸. If B_n = Adj(B_n−1) for all n ≥ 1, then det(B₄) is equal to:       (JEE Main 2022)
(a) 3²⁸
(b) 3³⁰
(c) 3³²
(d) 3³⁶

Ans. c

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

|B₀| = 9

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

Hence |B₄| = |B₃|² = |B₂|⁴ = |B₁|⁸ = |B₀|¹⁶

Q.27. Let
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

⇒ λ² − (4 + β)λ + (4β + 2α) = 0

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

Q.28. LetLet 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

On Comparing

8α = 2β, −3α + β = 2, 21α − 5β = 2

⇒ α = 2, β = 8

So, α + β = 10

Q.29. Letthen the value of A′BA is:       (JEE Main 2022)
(a) 1224
(b) 1042
(c) 540
(d) 539

Ans. d


= [9² + 12² − 15² − 10² + 13² + 16² + 11² − 14² + 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

⇒ 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 ,) 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

= 8(3) − 1(−λ) + 4(−3 − λ)

= 24 + λ − 12 − 4λ

= 12 − 3λ

So for λ = 4, it is having infinitely many solutions.

= −2(3) − 1(−μ) + 4(−μ)

= −6 − 3μ = 0

For μ = −2

Distance of (4, −2, −1/2) from 8x + y + 4z + 2 = 0

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

= 9λ² − 9|λ| − 43

= 9|λ|² − 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

⇒ 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 =[not possible as sin ∈ [−1, 1]]

∴ sin⁡θ = 1/2

∴ 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) = 2^a3^b5^c 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) = 2^a3^b5^c

⇒ |det(5(detA)B)A²| = 2^a3^b5^c

⇒ ||5(detA)B)A²| = 2^a3^b5^c

⇒ ||5|A|B|A²| = 2^a3^b5^c

⇒ ||5 . 2 . B|A²| = 2^a . 3^b . 5^c

⇒ ||10B|A²| = 2^a . 3^b . 5^c

⇒ |10² . |B|A²| = 2^a . 3^b . 5^c

As |k . A| = kn|A|

⇒ |100 × 3A²| = 2^a . 3^b . 5^c

⇒ (300)² . |A²| = 2^a . 3^b . 5^c

⇒(300)² . |A|² = 2^a . 3^b . 5^c

⇒(300)² . 2² = 2^a . 3^b . 5^c

⇒ 9 × 100 × 100 × 2² = 2^a . 3^b . 5^c

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

⇒ 2⁶ . 3² . 5⁴ = 2^a . 3^b . 5^c

Comparing both sides, we get

a = 6, b = 2, c = 4

∴ a + b + c = 6 + 2 + 4 = 12

Q.35. Letα ∈ 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, 


Given,

|AB| = 0

⇒ (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. Let
then the sum of all elements of the matrix B is       (JEE Main 2022)
(a) −5
(b) −6
(c) −7
(d) −8

Ans. c

and





Now let, 


Now sum of elements = −1 − 5 − 1 + 0 = −7

Q.37. Let A = [a_ij] be a square matrix of order 3 such that a_ij = 2^j−i, for all i, j = 1, 2, 3. Then, the matrix A² + A³ + ...... + A¹⁰ is equal to:        (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. a

Given, a_ij = 2^j−i

= 3A

Similarly, A³ = 3²A

A⁴ = 3³A

∴ A² + A³ +......+ A¹⁰

= 3A + 3²A + 3³A + ...... + 39A

= A(3 + 3² + 3³ +...... + 3⁹)

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

δ = −3

⇒ 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
= 25³ . 2³ . 2¹² = 10⁶ . 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) λ² = 1

Ans. b

But at λ = 7, D_x = D_y = D_z = 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

Q.42. LetThen 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

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)
(c) 7/2
(d)

Ans. d
x + 2y + z = 2,
αx + 3y − z = α,
−αx + y + 2z = −α

= 7 + 2α

Δ = 0 ⇒ α = 

∴ For no solution α =

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

⇒ α = 1

⇒ 5(1) − 1(20 − 3β) + 1(12 − 2β) = 0

⇒ β = 3

⇒ −2β + 6 = 0

⇒ β = 3

Similarly,

∴ (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)
(b)
(c)
(d)

Ans. c

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|

⇒ (24)⁶|A|² = 36 . (|2A|³⁻¹)²

⇒ (24)⁶|A|² = 36 . |2A|⁴

⇒ 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

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

Q.48. LetIf M and N are two matrices given bythen 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


M = A² + A⁴ + A⁶ + ..... + A²⁰

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

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

N = A¹ + A³ + A⁵ + .... + A¹⁹

= A − 4A + 16A +  ..... upto 10 terms

(MN²)^T = (KI)^T = KI

∴ A is correct

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

If X = (x₁, x₂, x₃)^T and I is an identity matrix of order 3, then the systemhas:         (JEE Main 2022)
(a) no solution 
(b) infinitely many solutions
(c) unique solution 
(d) exactly two solutions 

Ans. b



Solving will get

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

⇒ −4x₁ + 3x₂ + x₃ = 4 ..... (i)

−x₁ + x₃ = 1 ...... (ii)

−x₁ + x₂ = 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

⇒ 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}.

 Ifthen λ is equal to:         (JEE Main 2022)
(a) 218
(b) 221
(c) 663
(d) 1717

Ans. b

We know,

|adjA| = |A|ⁿ⁻¹

Here, n = order of matrix.

Here, n = 3

∴ |adjA| = |A|³⁻¹ = |A|²

= 1(1 − 0) − 0 + a(0 − (−a))

= a² + 1

∴ |adjA| = |A|² = (a² + 1)²

= (1² + 1)² + (3 + 1)² + (5 + 1)² + .... + (49 + 1)²
= 2² + 4² + 6² + .... + 50²
= 2²(1² + 2² + 3² + .... + 25²)

 ∴ 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

= 1(10α − 9α) − 1(5α − 3) + 1(3α² − 2α)
= α − 5α + 3 + 3α² − 2α
= 3α² − 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