Matrices and Determinants

 √ 
1 1
√ 3
1. If A = 2
,then : (a) A30 − A25 = 2I (b) A30 + A25 + A = I (c)
− 3 1
A30 + A25 − A = I (d) A30 = A25 Jee Main 2023

2. Let α and β be real numbers.COnsider 3 × 3 matrix A such that A2 = 3A + αI.If

A4 = 21A + βI,then (a) α = 1 (b) α = 4 (c) β = 8 (d) β = −8 Jee Main 2023
 
1 0 0
3. Let A =  0 4 −1 .Then the sum of the diagonal elements of (A + I)11 is equal
0 12 −3
to : (a) 6144 (b) 4094 (c) 4097 (d) 2050 Jee Main 2023

The set of all values of t ∈ R,for which the matrix

4. 
et e−t (sin t − 2 cos t) e−t (−2 sin t − cos t)

 et e−t (2 sin t + cos t) e−t (sin t − 2 cos t)  invertible,is (a) (2k + 1) π , k ∈ Z

2
et e−t cos t e−t sin t
(b) kπ + π4 , k ∈ Z (c) {kπ, k ∈ Z} (d) R Jee Main 2023
" √ #
3 1
   
2 2
1 1 T T 2007 a b
5. Let P = √ ,A = and Q = P AP .If P Q P = ,then
− 12 23 0 1 c d
2a + b − 3c − 4d is equal to (a) 2004 (b) 2005 (c) 2007 (d) 2006 Jee Main 2023

6. If A and B are two non-zero n × n matrices such that A2 + B = A2 B,then (a)

AB = I (b) A2 B = I (c) A2 = I or B = I (d) A2 B = BA2 Jee Main 2023

7. Let A = [aij ]2×2 where aij ̸= 0 for all i, j and A2 = I.Let a be the sum of all
diagonal elements of A and b = |A|,then 3a2 + 4b2 is equal to (a) 7 (b) 14 (c) 3 (d)
4 Jee Main 2023

8. Let P be a square matrix such that P 2 = I − P .For α, β, γ, δ ∈ N,if P α + P β =

γI − 29P and P α − P β = δI − 13P ,then α + β + γ − δ is equal to (a) 18 (b) 40 (c)
24 (d) 22 Jee Main 2023

9. Let S = {M = [aij ], aij ∈ {0, 1, 2}, 1 ≤ i, j ≤ 2} be a sample space and A = {M ∈

50
S : M is invertible } be an event.Then P (A) is equal to (a) 81 (b) 47
81
(c) 49
81
(d)
16
27
Jee Main 2023

10. If P is a 3 × 3 real matrix such that P T = aP + (a − 1)I,where a > 1,then (a) P is

a singular matrix (b) |adjP | > 1 (c) |AdjP | = 12 (d) |AdjP | = 1 Jee Main 2023

11. The number of symmetric matrices of order 3,with all the entries from the set
{0, 1, 2, . . . , 9}, is : (a) 610 (b) 910 (c) 109 (d) 106 Jee Main 2023

12. Let α be a root of the equation (a − c)x2 + (b − a)x +  (c − b) = 0 where a, b, c are

α2 α 1
distinct real numbers such that the matrix  1 1 1  is singular.Then the value
a b c
(a−c)2 (b−a)2 (c−b)2
of (b−a)(c−b) + (a−c)(c−b) + (a−c)(b−a) is (a) 6 (b) 3 (c) 9 (d) 12 Jee Main 2023
13. Let A, B, C be 3 × 3 matrices such that A is symmetric and B and C are skew-
symmetric.Consider the statements
(S1)A13 B 26 − B 26 A13 − B 26 A13 is symmetric
(S2)A26 C 13 − C 13 A26 is symmetric .
Then,
(a) Only S2 is true (b) Only S1 is true (c) both S1 and S2 are false (d) both S1
and S2 are true. Jee Main 2023
1
     
1 51 1 2 −1 −2
14. Let A = .If B = A ,then the sum of all the ele-
0 1 −1 −1 1 1
50
B n is equal to (a) 100 (b) 50 (c) 75 (d) 125Jee Main 2023
P
ments of the matrix
n=1
′
15. Let A be a 2 × 2 matrix with real entries such that A = αA + I,where α ∈
R − {−1, 1}.If det(A2 − A) = 4,then the sum of all possible values of α is equal to
(a) 0 (b) 23 (c) 52 (d) 2 Jee Main 2023
 
1 3 α
16. Let B =  1 2 3  , α > 2 be the adjoint of a matrix A and |A| = 2,then
α α 4 
  α
α −2α α B  −2α  is equal to (a) 16 (b) 32 (c) −16 (d) 0Jee Main 2023
α
   
2 1 1 2
17. Let A be a symmetric matrix such that |A| = 2 and A= .If the
3 23 α β
sum of the diagonal elements of A is s,then αβs2 is equal to —————Jee Main 2023
 
0 1 2
18. Let A =  a 0 3 ,where a, c ∈ R.If A3 = A and the positive values of a belongs
1 c 0
to the interval (n − 1, n] where n ∈ N,then n is equal to ——— Jee Main 2023
 
−1 2
19. Which of the following matrices can not be obtained from the matrix
1 −1
by a single elementary
  row operation
  ?   
0 1 1 −1 −1 2 −1 2
(a) (b) (c) (d) Jee Main 2022
1 −1 −1 2 −2 7 −1 3
20. Let A and B be any two 3 × 3 symmetric and skew-symmetric matrices respec-
tively.Then which of the following is not true ?
(a) A4 −B 4 is a symmetric matrix (b) AB −BA is a symmetric matrix (c) B 5 −A5 is
a skew-symmetric matrix (d) AB+BA is a skew-symmetric matrix.Jee Main 2022
 
1 2
21. Let A = .Let α, β ∈ R be such that αA2 + βA = 2I.Then α + β is
−2 −5
equal to (a) −10 (b) −6 (c) 6 (d) 10 Jee Main 2022

22. Let A = [aij ] be a square matrix of order 3 such that aij =  2j−i ,for all i, j =
10 10
1, 2, 3.Then,the matrix A2 + A3 + · · · + A10 is equal to (a) 3 2−3 A (b) 3 2−1 A
 10   10 
(c) 3 2+1 A (d) 3 2+3 A Jee Main 2022
   10
0 −2
A2k and
P
23. Let A = .If M and N are two matrices given by M =
2 0 k=1
10
A2k−1 then M N 2 is
P
N=
k=1
(a) A non-identity symmetric matrix. (b) A skew-symmetric matrix (c) Neither
symmetric nor skew-symmetric matrix. (d) An identity matrix Jee Main 2022
   
2 −2 −1 2
24. Let A = and B = .Then the number of elements in the set
1 −1 −1 2
{(n, m) : n, m ∈ {1, 2, . . . , 10} and nAn + mB m + I} is ———– Jee Main 2022
  
−1 a
25. Let S = : a, b ∈ {1, 2, . . . , 100} and let Tn = {A ∈ S : An(n+1) =
0 b
100
T
I}.Then the number of elements in Tn is ————– Jee Main 2022
n=1

26. 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 entries of A is a prime number p, 2 < p < 8,then the number of such
matrices A is ———— Jee Main 2022
   
1 a a 1 48 2160
27. Let A =  0 1 b  , a, b ∈ R.If for some n ∈ N, An =  0 1 96 ,then
0 0 1 0 0 1
n + a + b is equal to ——– Jee Main 2022
   
1 −1 β 1
28. Let A = and B = , α, β ∈ R.Let α1 be the value of α which
2 α  1 0
2 2
satisfies (A+B)2 = A2 + and α2 be the value of α which satisfies (A+B)2 =
2 2
B 2 .Then |α1 − α2 | is equal to ————– Jee Main 2022
√
 
1+i 1
29. Let A = where i = −1.Then,the number of elements in the set
−i 0
{n ∈ {1, 2, . . . , 100} : An = A} is ———– Jee Main 2022

30. 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
  49
0 −α
M 2k .If (I −
P
31. Let M = ,where α is a non-zero real number and N =
α 0 k=1
M 2 )N = −2I,then the positive integral value of α is —— Jee Main 2022
 5 3

32. If M = 2
3
2
1 ,then which of the following matrices is equal to M 2022 ?
 − 2
− 2     
3034 3033 3034 −3033 3033 3032
(a) (b) (c)
 −3033 −3032  3033 −3032 −3032 −3031
3032 3031
(d) Jee Advenced 2021
−3031 −3030
 
β 0 1
33. Let β be a real number.Consider the matrix A =  2 1 −2 .If A7 −(β −1)A6 −
3 1 −2
5
βA is a singular matrix,then the value of 9β is —— Jee Advanced 2022
    
1 2 0 2 −1 5
34. Let A + 2B =  6 −3 3  and 2A − B =  2 −1 6 .If T r(A) denotes the
−5 3 1 0 1 2
sum of all diagonal elements of the matrix A,then T r(A) − T r(B) has value equal
to : (a) 1 (b) 2 (c) 3 (d) 0 Jee Main 2022

 1, if i = j
35. Let A = [aij ] be a 3 × 3 matrix,where aij = −x, if |i − j| = 1
2x + 1, otherwise

Let a function f : R → R be defined by f (x) = detA.Then the sum of maximum and
88 20 88
minimum values of f on R is equal to : (a) − 27 (b) 27 (c) 27 (d) − 20
27
Jee Main 2021
       
1 0 50 1 0 1 25 1 0
36. If P = 1 ,then P is : (a) (b) (c)
 2 
1 50 1 0 1 50 1
1 50
(d) Jee Main 2021
0 1
37. Let A and B are 3 × 3 real matrices such that A is symmetric matrix and B is skew-
symmetric matrix.Then the system of linear equations (A2 B 2 − B 2 A2 )X = 0,where
X is a 3 × 1 column matrix of unknown variables and 0 is a 3 × 1 null matrix has :
(a) a unique solution (b) exactly two solutions (c) infinitely many solutions (d) no
solution. Jee Main 2021

38. 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 : (a) 1 (b) 6 (c) 4
(d) 12 Jee Main 2021
 
1 1 1
39. If A =  0 1 1  and M = A + A2 + · · · + A20 ,then the sum of all the elements
0 0 1
of the matrix M is equal to ——– Jee Main 2021
 
1 0 0
40. Let A =  0 1 1 .Then A2025 − A2020 is equal to (a) A5 (b) A6 (c) A5 − A (d)
1 0 0
6
A −A Jee Main 2021

41. 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 the all the elements of A3 is equal to : (a) 9 (b) 3 (c) 1
(d) 2 Jee Main 2021
 
0 2
42. If the matrix A = , satisfies A(A3 + 3I) = 2I,then the value of k is : (a)
k −1
− 21 (b) −1 (c) 1 (d) 12 Jee Main 2021
 
1 −α
43. If for the matrix,A = , AAT = I2 ,then the value of α4 + β 4 is : (a) 3
α β
(b) 4 (c) 2 (d) 1 Jee Main 2021

44. Four dice are thrown simultaneously and the numbers shown on these dice are
recorded in 2 × 2 matrices.The probability that such formed matrices have all dif-
22 45 43 23
ferent entries and are non-singular, is : (a) 81 (b) 162 (c) 162 (d) 81 Jee Main 2021
  
1 −1 0
45. Let A =  0 1 −1  and B = 7A20 − 20A7 + 2I3 .If B = [bij ],then b13 is equal
0 0 1
to —– Jee Main 2021
     
a b α 0
46. Let A = and B = ̸= such that AB = B and a+d = 2021,then
c d β 0
the value of ad − bc is equal to —— Jee Main 2021
 
2 −1
47. Let P = .Then the value of n ∈ N for which P n = 5I2 − 8P is equal to
5 −3
——— Jee Main 2021

48. 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 M T M is seven is ——
——– Jee Main 2021
   
a1 b1
49. Let A = and B = be two 2 × 1 matrices with real entries such
a2 b2 
1 1 −1
that A = XB where X = √3 and k ∈ R.If a21 + a22 = 32 (b21 + b22 ) and
1 k
(k 2 + 1)b2 ̸= −2b1 b2 ,then the value of k is : ——– Jee Main 2021
 
0 1 0
50. Let A =  1 0 0 .Then the number of 3 × 3 matrices B with entries from the
0 0 1
set {1, 2, 3, 4, 5} and satisfying AB = BA is ———– Jee Main 2021
      
0 i a b a b
51. Let S = nN : = , a, b, c, d ∈ R .Then the number
1 0 c d c d
of two digit numbers in the set S is ———- Jee Main 2021
 
1 1 1
52. If A =  0 1 1  and M = A + A2 + A3 + · · · + A20 ,then the sum of all elements
0 0 1
of the matrix M is equal to ——— Jee Main 2021

53. 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 Mian 2021
   
1 0 0 1 0 0
54. If the matrix A =  0 2 0  satisfies the equation A20 +αA19 +βA =  0 4 0 
3 0 −1 0 0 1
for some real numbers α and β,then β − α is equal to ——– Jee Main 2021

55. 
The number
 ofelements in the set 
a b
A= : a, b, c, d ∈ {−1, 0, 1} and (I − A)3 = I − A3 , is Jee Main 2021
0 d

Let a, b, c ∈R be all non-zero and satisfy a3 + b3 + c3 = 2.If the matrix A =

56. 
a b c
 b c a  satisfies AT A = I,then the value of abc can be (a) 3 (b) 1 (c) − 1 (d)
3 3
c a b
2
3
Jee Main 2020
57. Let A be a 2 × 2 real matrix with entries from {0, 1} and |A| = ̸ 0.Consider the
following two statements :
(P) If A ≤ I2 ,then |A| = −1
(Q) If |A| = 1,then tr(A) = 2,
Then (a) (P) is true and (Q) is false (b) both (P) and (Q) are false. (c) both (P)
and (Q) are true. (d) (P) is false and (Q) is true. Jee Main 2020
 
1 1 1
58. Let α be a root of the equation x2 +x+1 = 0 and the matrix A = √13  1 α α2 ,then
1 α2 α4
the matrix A31 is equal to (a) A3 (b) A2 (c) I3 (d) A Jee Main 2020
   
cos θ i sin θ π

a b
59. If A = , θ = 24 and A5 = ,then which of the following
i sin θ cos θ c d
is not true ?
(a) a2 − b2 = 12 (b) a2 − c2 = 1 (c) a2 − d2 = 0 (d) 0 ≤ a2 + b2 ≤ 1Jee Main 2020

60. The number of all 3 × 3 matrices A,with entries from the set {−1, 0, 1} such that
the sum of all diagonal elements of AAT is 3, is —— Jee Main 2020
 
x 1
61. Let A = , x ∈ R and A4 = [aij ],if a11 = 109,then a22 is equal to ————
1 0
————– Jee Main 2020
   
cos α − sin α 0 −1
62. Let A = , (α ∈ R) such that A32 = .Then a value of
sin α cos α 1 0
π π π
α is (a) 16 (b) 0 (c) 32 (d) 64 Jee Main 2019
 
1 0 0
63. Let P =  3 1 0  and Q = [qij ] be two 3×3 matrices such that Q−P 5 = I3 .Then
9 3 1
q21 +q31
q32
is equal to : (a) 10 (b) 135 (c) 15 (d) 9 Jee Main 2019
 
0 2x 2x
64. The total number of matrices A =  2y y −y  , (x, y ∈ R, x ̸= y) for which
1 −1 1
AT A = 3I3 is (a) 6 (b) 2 (c) 3 (d) 4 Jee Main 2019
 
cos θ − sin θ
65. If A = ,then the matrix A−50 where θ = 12 π
, is equal to (a)
sin θ cos θ
" √ # " √ # " √ # " √ #
1 3 3 1 3 1 1 3
√2
− 2 2
−
√2 2 √2 2√ 2
3
(b) 3
(c) (d) Jee Main 2019
2
1
2
1
2 2
− 12 23 − 23 12

66. 
Let A be a symmetric matrix and Bbe a skew-symmetric
  matrix
 such that A +B =
2 3 −4 2 −4 −2 4 −2
,then AB is equal to (a) (b) (c) (d)
 5 −1  1 4 −1 4 −1 −4
4 −2
Jee Main 2019
1 −4

sin4 θ −1 − sin2 θ
 
67. Let M = = αI + βM −1 where α = α(θ) and β = β(θ)
1 + cos2 θ cos4 θ
are real number.If α∗ is the minimum of the set {α(θ) : θ ∈ [0, 2π)} and β ∗ is the
 minimum of the set {β(θ) : θ ∈ [0, 2π)},then the value of α∗ + β ∗ is (a) − 16
37
(b) − 29
16
31 17
(c) − 16 (d) − 16 Jee Advanced 2019

x+1 x x
68. If x x+λ x = 89 (103x + 81),then λ, λ3 are the roots of the equation
x x x + λ2
(a) 4x + 24x − 27 = 0 (b) 4x2 − 24x + 27 = 0 (c) 4x2 + 24x + 27 = 0 (d)
2

4x2 − 24x − 27 = 0 Jee Main 2023

1 + sin2 x cos2 x sin 2x
2
, x ∈ π6 , π3 .If α and β respectively
 
69. Let f (x) = sin x 1 + cos2 x sin 2x
sin2 x cos2 x 1 + sin 2x
√ √
are the maximum and minimum values of f ,then (a) β 2 − 2 α = 19 (b) β 2 + 2 α =
19
√ 4

4
(c) α2 − β 2 = 4 3 (d) α2 + β 2 = 92 Jee Main 2023

1 2k 2k − 1 n
2
n2
P
70. Let Dk = n n + n + 2 .If Dk = 96,then n is equal to ————
n n2 + n n2 + n + 2 k=1

————- Jee Main 2023

   
1 92 −102 112
2 ′
71. Let A =  1  and B =  12 132 −142 ,then the value of A BA is (a)
1 −152 162 172
1224 (b) 1042 (c) 540 (d) 539 Jee Main 2022
 
4 −2
72. Let A = .If A2 + γA + 18I =),then detA is equal to (a) −18 (b) 18 (c)
α β
−50 (d) 50 Jee Main 2022
 
√ 1 0 a
73. Let S = { n : 1 ≤ n ≤ 50 and n is odd }.Let a ∈ S and A =  −1 1 0 .If
P −a 0 1
det(adjA) = 100λ,then λ is equal to (a) 218 (b) 221 (c) 663 (d) 1717
a∈S
. Jee Main 2022
p! (p + 1)! (p + 2)!
74. Let p and p + 2 be prime numbers and let ∆ = (p + 1)! (p + 2)! (p + 3)! ,then
(p + 2)! (p + 3)! (p + 4)!
the sum of the maximum values of α and β such that pα and (p + 2)β divide ∆ is
(a) 0 (b) 1 (c) 2 (d) 4 Jee Main 2022

75. Let |M | denote the determinant of the square matrix M .Let g : 0, π2 → R be the
 
p q
function defined by g(θ) = f (θ) − 1 + f π2 − θ − 1 where

sin π
cos θ + π4 tan θ −
π4

1 sin θ 1
f (θ) = 12 − sin θ 1 sin θ + sin θ − π4
− cos π2
loge π4 .
π π
−1 − sin θ 1 cot θ + 4 loge 4 tan π
Let p(x) be a quadratic polynomial whose roots
√ are the maximum and minimum
values of the function g(θ), and p(2) = 2 − 2.Then which of the following is/are
true ? √   √   √   √ 
(a) p 3+4 2 < 0 (b) p 1+34 2 > 0 (c) p 5 42−1 > 0 (d) p 5−4 2 < 0
. Jee Advanced 2022
 (a + 1)(a + 2) a + 2 1
76. The value of (a + 2)(a + 3) a + 3 1 is (a) −2 (b) 0 (c) (a + 1)(a + 2)(a + 3)
(a + 3)(a + 4) a + 4 1
(d) (a + 2)(a + 3)(a + 4) Jee Main 2021

sin2 x 1 + cos2 x cos 2x

2
77. The maximum value of f (x) = 1 + sin x cos2 x cos 2x , x ∈ R is (a) 34 (b)
2 2
sin x cos x sin 2x
√ √
5 (c) 5 (d) 7 Jee Main 2021
 
[x + 1] [x + 2] [x + 3]
78. Let A =  [x] [x + 3] [x + 3].If detA = 192,then the set of values of x in the
[x] [x + 2] [x + 4]
interval (a) [60, 61) (b) [68, 69) (c) [62, 63) (d) [65, 66) Jee Main 2021

79. Let A be a 3 × 3 matrix with detA = 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 detB is equal
to (a) 64 (b) 16 (c) 128 (d) 80 Jee Main 2021

80. If x, y, z are in arithmetic  progression

√ with common difference d, x ̸= 3d, and the

3 4 √2 x
determinant of the matrix 4 5 2 y  is zero,then the value of k 2 is (a) 6 (b) 72
5 k z
(c) 36 (d) 12 Jee Main 2021
 
0 sin α
and det A2 − 21 I = 0,then the possible value of α is (a) π4


81. If A =
sin α 0
(b) π6 (c) π2 (d) π3 Jee Main 2021

82. If 1, log10 (4x − 2) and log10 4x + 18

5
are in arithmetic progression for a real number
1


2 x − 2 x − 1 x2
x,then the value of the determinant 1 0 x is equal to —————
x 1 0
—– Jee Main 2021
   
a b
83. Let M = A = : a, b, c, d ∈ {±3, ±2, ±1, 0} .Define f : M → Z, as f (A) =
c d
detA,for all A ∈ M .Then the number of A ∈ M such that f (A) = 15 is equal to
——- Jee Main 2021
sin2 x −2 + cos2 x cos 2x
2 2
84. Let f (x) = 2 + sin x cos x cos 2x , x ∈ [0, π].Then the maximum
sin2 x cos2 x 1 + cos 2x
value of f (x) is equal to ——– Jee Main 2021
 
1 2 3
85. For any 3×3 matrix M ,let |M | denote the determinant of M .Let E = 2 3 4 ,P =
    8 13 18
1 0 0 1 3 2
0 0 1 and F = 8 18 13.If Q is a non-singular matrix of order 3 × 3,then
0 1 0 2 4 3
which of the following statement(s) is(are) true ?
  
1 0 0
(a) F = P EP and p2 = 0 1 0 (b) |EQ + P F Q−1 | = |EQ| + |P F Q−1 |
0 0 1
(c) |(EF ) | > |EF | (d) Sum of the diagonal entries of P −1 EP + F is equal to the
3 2

sum of the diagonal entries of E + P −1 F P Jee Advanced 2021

86. For any 3 × 3 matrix M ,let |M | denote the determinant of M .Let I be a 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 statement(s) is(are) true ?
(a) |F E| = |I − F E||F GE| (b) (I − F E)(I + F GE) = I (c) EF G = GEF (d)
(I − F E)(I − F GE) = I Jee Advanced 2021

87. Let a, b, c, d be in arithmetic progression with common difference λ.If

x+a−c x+b x+a
x−1 x+c x + b = 2,then the value of λ2 is equal to —— Jee Main 2021
x−b+d x+d x+c

88. Let A = [aij ] and b = [bij ] be two 3 × 3 real matrices such that bij = 3i+j−2 aji ,where
i, j = 1, 2, 3.If the determinant of B is 81,then the determinant of A is (a) 91 (b) 81 1
1
(c) 3 (d) 3 Jee Main 2020

x − 2 2x − 3 3x − 4
89. If ∆ = 2x − 3 3x − 4 4x − 5 = Ax3 + Bx2 + Cx + D,then B + C is equal to
3x − 5 5x − 8 10x − 17
(a) 9 (b) −1 (c) 1 (d) −3 Jee Main 2020

90. Suppose the vectors x1 , x2 and x3 are the solutions of the system of linear equations,
Ax =b when
 thevector
 b on
 the
 rightside is equal to
 b1 , b2 and b3 respectively.If
1 0 0 1 0
x1 = 1 , x2 = 2 , x3 = 0 , b1 = 0 and b3 = 0,then the determinant of
1 1 1 0 2
1 3
A is equal to (a) 4 (b) 2 (c) 2 (d) 2 Jee Main 2020
 
π cos θ sin θ
91. Let θ = 5 and A = .If B = A + At ,then detB (a) lies in (2, 3) (b) is
− sin θ cos θ
zero (c) is one (d) contains exactly two elements. Jee Main 2020
 
1 2 1
92. If A = {X = (x, y, z)T : P X = 0 and x2 +y 2 +z 2 = 1},where P = −2 3 −4,then
1 9 −1
the set A (a) is singleton (b) contains more than two elements (c) is an empty set
(d) contains exactly two elements. Jee Main 2020

x+a x+2 x+1

93. Let a − 2b + c = 1.If f (x) = x + b x + 3 x + 2 ,then (a) f (−50) = −1 (b)
x+c x+3 x+2
f (50) = 1 (c) f (50) = −501 (d) f (−50) = −501 Jee Main 2020

94. If a+x = b+y = c+z = 1,where a, b, c, x, y, z are non-zero distinct real numbers,then
x a+y x+a
y b + y y + b is equal to (a) y(b−a) (b) y(a−b) (c) y(a−c) (d) 0Jee Main 2020
z c+y z+c
 95. If the minimum and maximum values of the function f : π4 , π2 → R, defined by
 

− sin2 θ −1 − sin2 θ 1
f (θ) = − cos2 θ −1 − cos2 θ 1 are m and M respectively,then the ordered pair
12 10 −2
√
(m, M ) is equal to (a) (0, 2 2) (b) (0, 4) (c) (−4, 4) (d) (−4, 0) Jee Main 2020

96. Let m and M be respectively the minimum and maximum values of

cos2 x 1 + sin2 x sin 2x
2 2
1 + cos x sin x sin 2x .Then the ordered pair (m, M ) is equal to (a)
cos2 x sin2 x 1 + sin 2x
(1, 3) (b) (−3, −1) (c) (−4, −1) (d) (−3, 3) Jee Main 2019

x −6 −1
97. The sum of the real roots of the equation 2 −3x x − 3 = 0, is equal (a) 6 (b)
−3 2x x + 2
1 (c) 0 (d) −4 Jee Main 2019

x sin θ cos θ x sin 2θ cos 2θ

98. If ∆1 = − sin θ −x 1 and ∆2 = − sin 2θ −x 1 , x ̸= 0;then for all

π

cos θ 1 x cos 2θ 1 x
θ ∈ 0, 2
(a) ∆1 − ∆2 = x(cos 2θ − cos 4θ) (b) ∆1 + ∆2 = −2x3 (c) ∆1 − ∆2 = −2x3 (d)
∆1 + ∆2 = −2(x3 + x − 1) Jee Main 2019

a−b−c 2a 2a
99. If 2b b−c−a 2b = (a + b + c)(x + a + b + c)2 and a + b + c ̸= 0,then
2c 2c c−a−b
x is equal to (a) abc (b) −(a + b + c) (c) 2(a + b + c) (d) −2(a + b +c)Jee Main 2019
 
−2 4+d sin θ − 2
100. Let d ∈ R, and A = 1 sin θ + 2 d  , θ ∈ [0, 2π].If the
5 2 sin θ − d − sin θ + 2 + 2d
√
minimum
√ value of detA is 8,then a value of d is (a) −5 (b) −7 (c) 2( 2 + 1) (d)
2( 2 + 2) Jee Main 2019
 
2 b 1 √
101. Let A =  b b2 + 1 b  where b > 0.Then the minimum value of detA b
is (a) 2 3
1 b 2
√ √ √
(b) −2 3 (c) − 3 (d) 3 Jee Main 2019
 
1 sin θ 1
sin θ then for all θ ∈ 3π 5π


102. If A = − sin θ 1 4
, 4
, detA lies in the interval
−1 − sin θ 1
(a) 1, 52 (b) 52 , 4 (c) 0, 23 (d) 23 , 3
 
 
Jee Main 2019
 
0 2q r
103. Let A = p q −r.If AAT = I3 ,then |p| is
p −q r
(a) 5 (b) √13 (c) √12 (d) √16
√1
Jee Main 2019
 1 + cos2 θ sin2 θ 4 cos 6θ
π 2

2
104. A value of theta ∈ 0, 3
,for which cos θ 1 + sin θ 4 cos 6θ = 0, is (a)
cos2 θ sin2 θ 1 + 4 cos 6θ
7π π π 7π
24
(b) 18
(c) 9
(d) 36
Jee Main 2019
105. Let a1 , a2 , . . . , a30 be in G.P. with ai > 0 for i = 1, 2, . . . , 10 and S be the set of pairs
loge ar1 ak2 loge ar2 ak3 loge ar3 ak4
(r, k); r, k ∈ N for which log3 ar4 ak5 loge ar5 ak6 loge ar6 ak7 = 0.Then the number of
loge ar7 ak8 loge ar8 ak9 loge ar9 ak10
elements in S is (a) 4 (b) infinitely many (c) 2 (d) 10 Jee Main 2019
106. Let α and β be the roots of the equation x2 + x + 1 = 0.Then for y ̸= 0 in
y+1 α β
R, α y+β 1 is equal to (a) y 3 (b) y 3 − 1 (c) y(y 2 − 1) (d) y(y 2 −
β 1 y+α
3) Jee Main 2019
 
1 1 1
107. Let the number 2, b, c be in A.P. and A = 2 b c .If detA ∈ [2, 16],then c lies
2 2
  h4 b c i 3 3
in the interval (a) [2, 3) (b) 2 + 2 4 , 4 (c) 3, 2 + 2 4 (d) [4, 6] Jee Main 2019

108. Let α and β be the roots of the equation x2 + x + 1 = 0.Then for y ̸= 0 in

y+1 α β
R, α y+β 1 is equal to (a) y 3 (b) y 3 − 1 (c) y(y 2 − 1) (d) y(y 2 −
β 1 y+α
3) Jee Main 2019
 n n

n 2
P P
k ck k  n
 = 0,holds for some positive integer n.Then P n ck

k=0 k=0
109. Suppose det  n
 n n k+1
n
ck 3k
P P 
ck k k=0
k=0 k=0
equals —– Jee Advanced 2019
 
m n
110. Let A = , d = |A| ̸= 0, |A − detA| = 0.Then (a) (1 + d)2 = (m + q)2 (b)
p q
1 + d2 = (m + q)2 (c) (1 + d)2 = m2 + q 2 (d) 1 + d2 = m2 + q 2 Jee Main 2023
 
2 1 0
111. Let A = 1 2 −1.If |adj(adj(adj2A))| = (16)n ,then n is equal to (a) 8 (b) 9
0 −1 2
(c) 10 (d) 12 Jee Main 2023
" #
√1 √3
 
1 −i
112. Let A = 10 10
and B = .If M = AT BA,then the inverse of the
− √310 √110 0 1
       
2023 T 1 −2023i 1 0 1 0 1 2023i
matrix AM A is (a) (b) (c) (d)
0 1 −2023i 1 2023i 1 0 1
. Jee Main 2023
 
5! 6! 7!
1 
113. If A = 5!6!7! 6! 7! 8!,then adj(adj(2A)) is equal to (a) 28 (b) 212 (c) 220 (d)
7! 8! 9!
1
26 Jee Main 2023
114. If A is a 3 × 3 matrix and |A| = 2,then |3adj(|3A|A2 )| is equal to (a) 311 .610 (b)
312 .610 (c) 310 .611 (d) 312 .611 Jee Main 2023

115. Let the determinant of a square matrix A of order m be m − n,where m and n

satisfy 4m + n = 22 and 17m + 4n = 93.If (nadj(adj(mA))) = 3a 5b 6c ,then a + b + c
is equal to (a) 96 (b) 101 (c) 109 (d) 84 Jee Main 2023
 
1 2 3
116. Let for A = α 3 1 , |A| = 2.If |2adj(2adj(2A))| = 32n ,then 3n + α is equal to
1 1 2
(a) 10 (b) 9 (c) 12 (d) 11 Jee Main 2023
 
1 5
117. If A = , A−1 = αA + βI and α + β = −2,then 4α2 + β 2 + λ2 is equal to (a)
λ 10
12 (b) 10 (c) 19 (d) 14 Jee Main 2023
 
1 logx y logx z
118. Let x, y, z > 1 and A =  logy x 2 logy x .Then |adj(adjA2 )| is equal to (a)
logz x logz y 3
64 (b) 28 (c) 48 (d) 24 Jee Main 2023

119. Let A be 3 matrix such that |adj(adj(adjA))| = 124 .Then |A−1 adjA| is equal
√a 3 ×√
to (a) 2 3 (b) 6 (c) 12 (d) 1 Jee Main 2023

120. Let A be a n × n matrix such that |A| = 2.If the determinant of the matrix
adj(2.adj(2A−1 )) is 284 ,then n is equal to —— Jee Main 2023
  
 a 3 b 
121. Let R =  c 2 d : a, b, c, d ∈ {0, 3, 5, 7, 11, 13, 17, 19} .Then the number of
0 5 0
 
invertible matrices in R is —— Jee Advanced 2023

122. Let M = (aij ), i, j ∈ {1, 2, 3} be the 3×3 matrix such that aij = 1 if j +1 is divisible
by i,otherwise aij = 0.Then which of the following statement(s) is(are)
 true ?
a1
(a) M is invertible. (b) There exists a non-zero column matrix a2  such that
    a3  
a1 −a1 0
M a2  = −a2  (c) The matrix {X ∈ R3 : M X = 0} = ̸ {0},where 0 = 0
a3 −a3 0
(d) The matrix (M − 2I) is invertible matrix. Jee Main 2023

123. Let A be a 3 × 3 invertible matrix.If |adj(24A)| = |adj(3adj(2A))|,then |A|2 is equal

to (a) 66 (b) 212 (c) 26 (d) 1 Jee Main 2022
 
2 −1
124. Let A = .If 5B = 1 −5 c1 (adjA) +5 c2 (adjA)2 − · · · −5 c5 (adjA)5 ,then the
0 2
sum of all elements of the matrix B is (a) −5 (b) −6 (c) −7 (d) −8Jee Main 2022
1
125. Let A and B be two 3×3 matrices such that AB = I and |A| = 8
then |adj(Badj(2A))|
is equal to (a) 16 (b) 32 (c) 64 (d) 128 Jee Main 2022

126. Let A be matrix of order 3×3 and detA = 2,Then det(det(A)adj(5adj(A3 ))) is equal
to (a) 512 × 106 (b) 256 × 106 (c) 1024 × 106 (d) 256 × 1011 Jee Main 2022
127. Let A be a 2 × 2 matrix with detA = −1 and det((A + I)(adjA√ + I)) = 4.Then the
sum of the diagonal elements of A can be (a) −1 (b) 2 (c) 1 (d) − 2Jee Main 2022
 
0 1 0
128. Let the matrix A = 0 0 1 and the matrix B0 = A49 + 2A98 .If Bn = adj(Bn−1 )
1 0 0
for all n ≥ 1,then det(B4 ) is equal to (a) 328 (b) 330 (c) 332 (d) 336 Jee Main 2022
 
α β γ
129. Consider a matrix A =  α2 β2 γ 2  where α, β, γ are three distinct
β+γ γ+α α+β
adj(adj(adj(adjA)))
natural number.If (α−β)16 (β−γ)16 (γ−α)16 = 232 × 316 then the number of such 3-tuples
(α, β, γ) is —— Jee Main 2022

130. The positive  value of the determinant

 of the matrix A,whose
14 28 −14
adj(adjA) = −14 14 28  is ——– Jee Main 2022
28 −14 14
   
1 −1 2 3
131. Let X = 1 and A =  0 1 6 .For k ∈ N, if X t Ak X = 33,then k is equal to
1 0 0 −1
—— Jee Main 2022
 
0 1 0
132. Let X = 0 0 1 , Y = αI+βX+γX 2 and Z = α2 I−αβX+(β 2 −αγ)X 2 , α, β, γ ∈
0 0 0
1 2 1

5
− 5 5
R.If Y Z =  0 51 − 25 ,then (α − β + γ)2 is equal to ——- Jee Main 2022
1
0 0 5
 
a b
133. The number of matrices A = ,where a, b, c, d ∈ {−1, 0, 1, 2, . . . , 10},such that
c d
A = A−1 , is —– Jee Main 2022
 
1 2
134. Let A = .If A−1 = αI + βA, α, β ∈ R,then 4(α − β) is equal to (a) 2 (b) 5
−1 4
(c) 4 (d) 38 Jee Main 2021
!
√1 √2
 
1 0
135. If A = 5 5
,B = and Q + AT BA,then the inverse of the matrix
− √25 √15 i 1
!
√1 −2021
   
2021 T 1 0 5 1 0
AQ A is equal to (a) (b) √1
(c) (d)
2021i 1 2021 5
−2021i 1
 
1 −2021i
Jee Main 2021
0 1
   
−30 20 56 2 7 ω2
136. Let P =  90 140 112 and A = −1 −ω 1 .If the determinant of the
120 60 14 0 −ω −ω + 1
−1 2 2
matrix (P AP − I3 ) is αω ,then the value of α is equal to ——- Jee Main 2021
  
2 3
137. Let A = , a ∈ R be written as P + Q where P is a symmetric matrix and Q is
a 0
a skew symmetric matrix.If det(Q) = 9,then the modulus of the sum of all possible
value of the determinant of P is equal to (a) 18 (b) 36 (c) 24 (d) 45Jee Main 2021

138. Let A and B be two 3 × 3 real matrices such that A2 − B 2 is invertible matrix.If
A5 = B 5 and A3 B 2 = A2 B 3 ,then the value of the determinant of the matrix A3 + B 3
is equal to (a) 2 (b) 1 (c) 0 (d) 4 Jee Main 2021
1
R2 xn
139. Let Jn,m = xm −1
dx for all n > m and m, n ∈ N.Consider a matrix A = [aij ]3×3
0
J6+i,3 − Ji+3,3 , i ≤ j
where aij = .Then |adjA−1 | is (a) 152 × 234 (b) 152 × 242 (c)
0, i>j
1052 × 238 (d) 1052 × 236 Jee Main 2021

0
− tan 2θ

  
a −b
140. If A = and (I2 + A)(I2 − A)−1 = ,then 13(a2 + b2 ) is
tan 2θ 0 b a
equal to ——– Jee Main 2021
 
2 3
141. If A = ,then the value of det(A4 ) + det(A10 − (adj(2A)10 ) is equal to —–
0 −1
Jee Main 2021

142. Let A be a 3 × 3 real matrix.If det(2adj(2adj(adj(2A)))) = 241 ,then the value of

det(A2 ) equals —– Jee Main 2021

(−1)j−i if i < j
143. Let A = {aij } be a 3×3 matrix,where aij = 3 if i = j ,then det(3adj(2A−1 ))
i+j
(−1) if i > j

is equal to ——- Jee Main 2021
 
3 −1 −2
144. Let P = 2 0 α  where α ∈ R.Suppose Q = [qij ] is a matrix satisfying
3 −5 0
2
P Q = kI3 for some non-zero k ∈ R.If q23 = − k8 and |Q| = k2 ,then α2 + k 2 is equal
to —– Jee Main 2021
 
2 −1 1
145. Let A be a 3 × 3 matrix such that adjA = −1 0 2  and B = adj(adjA).If
1 −2 −1
−1 T
|A| =
λ and |(B
) | = 1µ,then the ordered pair,(|λ|, µ) is equal to (a) (3, 81) (b)
1 1


9, 9 (c) 3, 81 (d) 9, 81 Jee Main 2020
 
1 1 2
146. If the matrices A = 1 3 4 , B = adjA and C = 3A,then |adjB| |C|
is equal to (a)
1 −1 3
72 (b) 8 (c) 16 (d) 2 Jee Main 2020
 
2 2
147. If A = ,then 10A−1 is equal to (a) 4I − A (b) A − 6I (c) A − 4I (d)
9 4
6I − A Jee Main 2020
148. Let M be a 3 × 3 invertible matrix with real entries.If M −1 = adj(adjM ),then which
of the following statements is/are always true ? (a) M = I (b) detM = 1 (c) M 2 = I
(d) (adjM )2 = I Jee Advanced 2020
         
1 1 1 2 1 n−1 1 n−1 1 78 1 n
149. If ... = ,then the inverse of is
0 1 0  1 0 1 0 1  0 1 0 1
1 −13 1 0 1 −12 1 0
(a) (b) (c) (d) Jee Main 2019
0 1 12 1 0 1 13 1

e−t cos t e−t sin t

 t 
e
150. If A = et −e−t cos t − et sin t −et sin t + e−t cos t.Then A is (a) invertible only
et 2e−t sin t −2et cos t
π
if t = 2 (b) not invertible for any t ∈ R (c) invertible for all t ∈ R (d) invertible
only if t = π Jee Main 2019
 
5 2α 1
151. If B =  0 2 1  is the inverse of a 3 × 3 matrix A,then the sum of all values
α 3 −1
of α for which detA + 1 = 0, is (a) 0 (b) 2 (c) 1 (d) −1 Jee Main 2019

152. Let A and B be two invertible matrices of order 3 × 3.If det(ABAT ) = 8 and
det(AB −1 ) = 8,then det(BA−1 B T ) is equal to (a) 41 (b) 1 (c) 16
1
(d) 16Jee Main 2019
   
0 1 a −1 1 −1
153. Let M = 1 2 3 and adjM =  8 −6 2  where a and b are real num-
3 b 1 −5 3 −1
bers.Which of the following options is/are correct ?
(a)a  + b = 3 (b) det(adj(M 2 )) = 81 (c) (adjM )−1 + adjM −1 = −M (d) if
α 1
M β = 2,then α − β + γ = 3
   Jee Advanced 2019
γ 3
       
1 0 0 1 0 0 0 1 0 0 1 0
154. Let P1 = I = 0 1 0 , P2 = 0 0 1 , P3 = 1 0 0 , P4 = 0 0 1,
 0 0 1  0 1 0 0 0 1 
 1 0 0
0 0 1 0 0 1 6 2 1 3
Pk 1 0 2 PkT .Then which of the
P
P5 = 1 0 0 , P6 = 0 1 0 and X =
    
0 1 0 1 0 0 k=1 3 2 1
following options is/are correct ?
(a) X−30I
 is  an invertible matrix. (b) The sum of the diagonal entries of X is 18 (c)
1 1
If X 1 = α 1,then α = 30. (d) X is a symmetric matrix. Jee Advenced 2019
  
1 1
   
1 1 1 2 x x
155. Let x ∈ R and let P = 0 2 2 , Q =  0 4 0  and R = P QP −1 .Then which
0 0 3 x x 6
of the following options is/are correct ?    
α 0
(a) For x = 1,there exists a unit vector αî + β ĵ + γ k̂ for which R β  = 0. (b)
γ 0
  
2 x x
There exists a real number x such that P Q = QP . (c) detR = det  0 4 0  +8,for
    x x 5
1 1
all x ∈ R. (d) For x = 0,if R a = 6 a,then a + b = 5
   Jee Advanced 2019
b b
156. If the system of equations 2x + y − z = 5, 2x − 5y + λz = µ, x + 2y − 5z = 7 has
infinitely many solutions,then (λ + µ)2 + (λ − µ)2 is equal to (a) 916 (b) 912 (c) 920
(d) 904 Jee Main 2023
 
2 10 8

157. If a point P (α, β, γ) satisfying (α, β, γ) 9 3 8 = 0 0 0 lies on the plane
8 4 8
2x+4y+3z = 5,then 6α+9β+7γ is equal to (a) −1 (b) 11 5
(C) 45 (d) 11Jee Main 2023

158. Let the system of linear equations x + y + kz = 2, 2x + 3y − z = 1, 3x + 4y + 2z = k

have infinitely many solutions.Then the system (k + 1)x + (2k − 1)y = 7, (2k +
1)x + (k + 5)y = 10 has (a) infinitely many solutions (b) unique solution satisfying
x − y = 1 (c) no solution (d) unique solution satisfying x + y = 1 Jee Main 2023

159. For the system of linear equations x + y + z = 6, αx + βy + 7z = 3, x + 2y + 3z = 14

which of the following is not true ?
(a) If α = β = 7,then the system has no solution.
(b) If α = β and α ̸= 7 then the system has unique solution.
(c) There is a unique point (α, β) on the line x + 2y + 18 = 0 for which the system
has infinitely many solutions.
(d) For every point (α, β) ̸= (7, 7) on the line x−2y +7 = 0, the system has infinitely
many solutions. Jee Main 2023

160. Consider the following system of equations αx + 2y + z = 1, 2αx + 3y + z = 1, 3x +

αy + 2z = β.For some α, β ∈ R,then which of the following is not correct ?
(a) It has no solution if α = −1 and β ̸= 2
(b) It has no solution for α = −1 and for all β ∈ R
(c) It has no solution for α = 3 and for all β ≠= 2
(d) It has a solution for all α ̸= −1 and β = 2 Jee Main 2023

161. Let N denote the number that turn up when a fair die is rolled.If the probability
that the system of equations x + y + z = 1, 2x + N y + 2z = 2, 3x + 3y + N z = 3 has
a unique solution is k6 ,then the sum of value of k and all possible values of N is (a)
18 (b) 19 (c) 20 (d) 21 Jee Main 2023

162. If the system of equations x + 2y + 3z = 3, 4x + 3y − 4z = 4, 8x + 4y − λz = 9
+ µ

has infinitely

many solutions,then the
ordered pair (λ, µ) is equal to (a) 72
5 5
, 21 (b)
72 21 72 21 72 21


− 5 , − 5 (c) 5 , − 5 (d) − 5 , 5 Jee Main 2023

the set of all values of θ ∈ √

163. Let S be √ [−π, π] for which the system of linear equations
x + y + 3z = 0, −x + tan θy + 7z = 0, x + y + tan θz = 0 has non-trivial
120
P
solution.Then π θ is equal to (a) 40 (b) 10 (c) 20 (d) 30 Jee Main 2023
θ∈S

164. For the system of equations 2x + 4y + 2az = b, x + 2y + 3z = 4, 2x − 5y + 2z = 8

which of the following is not correct ?
 (a) It has infinitely many solutions if a = 3, b = 6 (b) It has unique solution if
a = b = 6 (c) It has a unique solution if a = b = 8 (d) It has infinitely many
solution if a = 3, b = 8 Jee Main 2023

165. For the system of equations x + y + z = 6, x + 2y + αz = 10, x + 3y + 5z = β,which

one of the following is not true ?
(a) system has a unique solution for α = 3, β ̸= 14.(b) System has no solution for
α = 3, β = 4 (c) system has a unique solution for α = −3, β = 14 (d) system has
infinitely many solutions for α = 3, β = 14 Jee Main 2023

166. If the system of linear equations 7x + 11y + αz = 13, 5x + 4y + 7y = β, 175x + 194y +

57z = 361 has infinitely many solutions,then α + β + 2 is equal to (a) 4 (b) 3 (c) 5
(d) 6 Jee Main 2023

167. Let S denote the set of all real values of λ such that the system
P of 2equations λx +
y + z = 1, x + λy + z = 1, x + y + λz = 1 is inconsistent,then (|λ| + |λ|) is equal
λ∈S
to (a) 2 (b) 12 (c) 4 (d) 6 Jee Main 2023

168. For the system of linear equations 2x − y + 3z = 5.3x + 2y − z = 7, 4x + 5y + αz = β

which of the following is not correct ?
(a) The system has infinitely many solutions for α = −5 and β = 9
(b) The system has a unique solution for α ̸= −5 and β = 8 (c) The system has
infinitely many solutions for α = −6 and β = 9 (d) The system is inconsistent for
α = −5 and β = 8 Jee Main 2023

169. Let S1 and S2 be respectively the sets of all a ∈ R{0} for which the system of linear
equations ax + 2zy − 3az = 1, (2a + 1)x + (2a + 3)y + (a + 1)z = 2, (3a + 5)x + (a +
5)y + (a + 2)z = 3 has unique solution and infinitely many solutions.Then
(a) n(S1 ) = 2 and S2 is an infinite set. (b) S1 is an infinite set and n(S2 ) = 2 (c)
S1 = ϕ and S2 = R − {0} (d) S1 = R − {0} and S2 = ϕ Jee Main 2023

170. Let the system of linear equations −x+2y −9z = 7, −x+3y +7z = 9, −2x+y +5z =
8, −3x + y + 13z = λ has a unique solution x = α, y = β, z = γ.Then the distance
of the point (α, β, γ) from the plane 2x − 2y + z = λ is (a) 9 (b) 11 (c) 13 (d)
7 Jee Main 2023

171. If the system of equations x+y+az = b.2x+5y+2z = 6, x+2y+3z = 3 has infinitely

many solutions,then 2a + 3b is equal to (a) 23 (b) 28 (c) 25 (d) 20Jee Main 2023

172. For the system of linear equations αx+y +z = 1, x+αy +z = 1, x+y +γz = β,which
one of the following statements is not correct ? (a) It has infinitely many solutions
if α = 2 and β = −1
(b) It has no solution if α = −2 and β = 1
(c) x + y + z = 43 if α = −2 and β = 1
(d) It has infinitely many solutions if α = 1 and β = 1 Jee Main 2023

173. For α, β ∈ R,suppose the system of linear equations x − y + z = 5, 2x + 2y + αz =

8, 3x − y + 4z = β has infinitely many solutions.Then α and β are the roots of (a)
x2 − 10x + 16 = 0 (b) x2 + 18x + 56 = 0 (c) x2 − 18x + 56 = 0 (d) x2 + 14x + 24 =
0 Jee Main 2023
 P− 3y + 3z =
174. Let S be the set of values of λ,for which the system of equations 6λx
4λ2 , 2x + 6λx + 4z = 1, 3x + 2y + 3λz = λ has no solution.Then 12 |λ| is equal
λ∈S
to —— Jee Main 2023

175. Let α, β and γ be real numbers.Consider the following system of linear equations
x + 2y + z = 7, x + αz = 11, 2x − 3y + βz = γ.Match List-I with List-II:
List-I List-II
1
A. If β = 2 (7α − 3) and γ = 28, I. a unique solution.
then the system has
B. If β = 12 (7α − 3) and γ ̸= 28 II. no solution.
then the system has
C. If β ̸= 21 (7α − 3) where α = 1 and γ ̸= 28 III. infinitely many solutions.
then the system has
1
D. If β ̸= 2 (7α − 3) where α = 1 and γ = 28 IV. x = 11, y = −2 and z = 0
then the system has as a solution.
V. x = −15, y = 4 and z = 0
as a solution.
Choose the correct answer from the options given below :
(a) A → III; B → II; C → I; D → IV (b) A → III; B → II; C → V ; D → IV
(c) A → II; B → I; C → IV ; D → V (d) A → II; B → I; C → I; D →
III Jee Advanced 2023

176. The number of real values λ,such that the system of linear equations 2x − 3y + 5z =
9, x+3y −z = −18, 3x−y +(λ2 −|λ|)z = 16 is (a) 0 (b) 1 (c) 2 (d) 4Jee Main 2022

177. 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 (a) (1, −3) (b)
(−1, 3) (c) (1, 3) (d) (−1, −3) Jee Main 2022

178. Let the system of linear equations x + 2y + z = 2, αx + 3y − z = α, x + y + 2z = −α

be inconsistent,Then α is equal to (a) 25 (b) − 25 (c) 72 (d) − 72 Jee Main 2022

179. The number of real values λ,such that the system of linear equations 2x − 3y + 5z =
9, x+3y −z = −18, 3x−y +(λ2 −|λ|)z = 16 is (a) 0 (b) 1 (c) 2 (d) 4Jee Main 2022

180. 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 (a) 4 (b) 3 (c) 2 (d) 1Jee Main 2022

181. 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 (a) λ = 7 (b) λ = −7 (c) λ = 8 (d)
λ2 = 1 Jee Main 2022

182. If the system of equations 8x + y + 4z = −2, x + y + z = 0, λx
− 3y = µ has

1
infinitely many solutions,then the distance of the point λ, µ, − 2 from the plane
√
8x + y + 4z + 2 = 0 is (a) 3 5 (b) 4 (c) 26
9
(d) 10
3
Jee Main 2022

183. Let A and B be two 3×3 non-zero real matrices such that AB is a zero matrix.Then
(a) The system of linear equation AX = 0 has a unique solution. (b) The system of
linear equation AX = 0 has infinitely many solutions. (c) B is an invertible matrix.
(d) adjA is an invertible matrix. Jee Main 2022
184. 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 (a) 8 (b) 36 (c) 44 (d) 48 Jee Main 2022

185. The system of equations −kx + 3y − 14z = 25, −15x + 4y − kz = 3, −4x + y +

3z = 4 is consistent for all k in the set (a) R (b) R − {−11, 13} (c) R − {13} (d)
R − {−11, 11} Jee Main 2022

186. 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 (a) 3, 13 (b) −3, 13 (c)
−3, − 31 (d) 3, − 13


Jee Main 2022

187. 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 (a) 6 (b) 7 (c) 8 (d)
9 Jee Main 2022
       
1 1 1 −1
188. Let A be a 3 × 3 real matrix such that A 1 = 1 ; A 0 =  0  and
    0 0 1 1
0 1
A 0 = 1. If X = (x1 , x2 , x3 )T and I is an identity matrix of order 3,then
1 2  
4
the system (A − 2I)X = 1 has (a) no solution (b) infinitely many solutions (c)
1
unique solution (d) exactly two solutions. Jee Main 2022

189. 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 (a) 0 (b) 1 (c) 2 (d) 3Jee Main 2022

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

191. If the system of linear equations 2x−3y = γ +5, α+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

192. Let p, q, r be non-zero 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
List-I List-II
q
A. If r = 10 then the p. x = 0, y = 10 9
, z = − 19
system of linear equation has as a solution
p
B. If r ̸= 100,then the q. x = 109
, y = − 19 , z = 0
system of linear equation has as a solution
C. If pq ̸= 10,then the r. infinitely many solutions.
system of linear equation has
D. If pq = 10,then the s. no solution.
system of linear equation has
(a) A → (p); B → (s); C → (p); D → (s) (b) A → (p); B → (p); C → (t); D → (r)
(c) A → (q); B → (p); C → (t); D → (s) (d) A → (q); B → (s); C → (s); D →
(r) Jee Advanced 2022
193. The following system of linear equations 2x+3y+2z = 9, 3x+2y+2z = 9, x−y+4z =
8 (a) has a unique solution. (b) has a solution (α, β, γ) satisfying α + β 2 + γ 2 = 12
(c) has infinitely many solutions (d) does not have any solution. Jee Main 2021

194. 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 a 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 ?
(a) (B) and (E) only. (b) (C) and (D) only. (c) (A) and (E) only. (d) (A) and (D)
only. Jee Main 2021

195. The system of equations kx + y + z = 1, x + ky + z = k, x + y + kz = k 2 has no

solution if k is equal to (a) 0 (b) −1 (c) −2 (d) 1 Jee Main 2021

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

197. 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 (a) a ̸= 3, b = 3 (b) a ̸= 3, b ̸= 13 (c)
a = 3, b ̸= 13 (d) a = 3, b = 13 Jee Main 2021

198. Two fair dice are thrown.The numbers on them are taken as λ and µ, and a system
of linear equations x+y +z = 5, x+2y +3z = µ and x+3y +λz = 1 is constructed.If
p is the probability that the system has a unique solution and q is the probability
that the system has no solution,then (a) p = 56 and q = 36 5
(b) p = 16 and q = 36
1
(c)
1 5 5 1
6
and q = 36
(d) p = 6
and q = 36
Jee Main 2021

199. Let θ ∈ 0, π2 .If the system of linear equations (1 + cos2 θ) x + sin2 θy + 4 sin 3θz =

0, cos2 θx + 1 + sin2 θ y + 4 sin 3θz = 0, cos2 θx + sin2 θ + (1 + 4 sin 3θ) z = 0 has a

non-trivial solution,then the value of θ is : (a) 4π

9
(b) 7π
18
π
(c) 18 (d) 5π
18
Jee Main 2021

200. The system of linear equations 3x − 2y − kz = 10, 2x − 4y − 2z = 6, x + 2y − z = 5m

is inconsistent if (a) k ̸= 3, m ̸= 45 (b) k = 3, m ̸= 45 (c) k ̸= 3, m ∈ R (d)
k = 3, m = 54 Jee Main 2021

201. 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 values of a ∈ R 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 (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 Jee Main 2021

202. Consider the following system of equations: x + 2y − 3z = a, 2x + 6y − 11z =

b, x − 2y + 7z = c where a, b and c real constants.Then the system of equations: (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 Jee Main 2021
203. 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 (a) infinitely many solutions. (b)
exactly two solutions (c) no solution (d) a unique solution. Jee Main 2021

204. Let α, β, γ be real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c ∈ R and a, b ̸=

0).If the system of equations ( in u, v, w ) given by αu+βv +γw = 0; βu+γv +αw =
2
0; γu + αu + βw = 0 has non-trivial solution,then the value of ab is (a) 5 (b) 3 (c)
1 (d) 0 Jee Main 2021

205. 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 ?
(a) µ = 6, λ ∈ R (b) µ = −6, λ ∈ R (c) λ = 2, µ ∈ R (d) λ = 3, µ ∈ RJee Main 2021

206. 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 : (a) (−∞, −9)∪(−9, ∞)
(b) [−9, −8) (c) R (d) (−∞, −9) ∪ [−8, ∞) Jee Main 2021
     
i −i 8 x 8
207. Let A = .Then the system of linear equation A = has (a)
−i i y 64
infinitely many solutions. (b) no solution (c) exactly two solutions (d) a unique
solution. Jee Main 2021

208. 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 : (a) λ = 2, µ ̸= 10 (b) λ = 3, µ ̸= 10 (c)
λ = 3, µ = 5 (d) λ ̸= 2, µ = 10 Jee Main 2021

209. If the following system of linear equations 2x + y + z = 5, x − y + z = 3 and

x + y + az = b has no solution,then : (a) a ̸= − 13 , b = 73 (b) a ̸= 31 , b = 37 (c)
a = 31 , b ̸= 73 (d) a = − 13 , b ̸= 37 Jee Main 2021

210. 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
 
x y z
211. Let A = y z x,wher x, y and z are real numbers such that x + y + z > 0 and
z x y
xyz = 2.If A2 = I3 ,then the value of x3 + y 3 + z 3 is —– Jee Main 2021

212. If the system of lineat equations 2x + y − z = 3, x − y − z = α, 3x + 3y + βz = 3

has infinitely many solution,then α + β − αβ is equal to —- Jee Main 2021

213. 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

214. Let α, β and γ be real numbers such that the system of linear equations
 x+2y+3z
 =
α 2 γ
α, 4x + 5y + 6z = β, 7x + 8y + 9z = γ − 1 is inconsistent.Let M =  β 1 0 .Then
−1 0 1
the value of |M | is —– Jee Advanced 2021

215. Let α, β and γ be real numbers such that the system of linear equations x+2y+3z =
α, 4x+5y +6z = β, 7x+8y +9z = γ −1 is inconsistent.Let P be the plane containing
all those (α, β, γ) for which the above system of linear equations is consistent, and
 D be the square of the distance of the point (0, 1, 0) from the plane P .The value of
D is — Jee Advanced 2021
216. If the system of linear equations x + y + 3z = 0, x + 3y + k 2 z = 0, 3x + y + 3z = 0
has a non-zero solution (x, y, z) for some k ∈ R,then x + yz is equal to (a) 3 (b) 9
(c) −3 (d) −9 Jee Main 2020
217. Let λ ∈ R.The system of linear equations 2x1 − 4x2 + λx3 = 2, λx1 − 10x2 + 4x3 = 3
is inconsistent for (a) exactly two values of λ (b) exactly one positive values of λ (c)
every value of λ (d) exactly one negative values of λ Jee Main 2020
218. If the system of equations x + y + z = 2, 2x + 4y − z = 6, 3x + 2y + λz = µ has
infinitely many solutions,then (a) λ + 2µ = 14 (b) 2λ + µ = 14 (c) 2λ − µ = 5 (d)
λ − 2µ = −5 Jee Main 2020
219. If the system of linear equations 2x+2ay+az = 0, 2x+3by+bz = 0, 2x+4cy+cz = 0
where a, b, c ∈ R are non-zero and distinct;has a non-zero solution,then (a) a, b, c are
in A.P. (b) a1 , 1b , 1c are in A.P. (c) a + b + c = 0 (d) a, b, c are in G.P. Jee Main 2020
220. For which of the following ordered pairs (µ, δ),the system of linear equations x +
2y + 3z = 1, 3x + 4y + 5z = µ, 4x + 4y + 4z = δ is inconsistent ? (a) (3, 4) (b) (1, 0)
(c) (4, 3) (d) (4, 6) Jee Main 2020
221. The system of linear equations λx+2y +2z = 5, 2λx+3y +5z = 8, 4x+λy +6z = 10
has (a) no solution when λ = 8 (b) infinitely many solution when λ = 2 (c) no
solution when λ = 2 (d) a unique solution when λ = −8 Jee Main 2020
222. If for some α and β in R,the intersection of the following three planes x + 4y − 2z =
1, x + 7y − 5z = β, x + 5y + αz = 5 is a line in R3 ,then α + β is equal to (a) 0 (b) 2
(c) 10 (d) −10 Jee Main 2020
223. Let S be the set of all λ ∈ R for which the system of linear equations 2x − y + 2z =
2, x − 2y + λz = −4, x + λy + z = 4 has no solution.Then the set S (a) contains
more than two elements. (b) is a singleton (c) is an empty set.(d) contains exactly
two elements. Jee Main 2020
224. The following system of linear equations 7x + 6y − 2z = 0, 3x + 4y + 2z = 0, x − 2y −
6z = 0, has (a) infinitely many solutions, (x, y, z) satisfying y = 2z (b) infinitely
many solution, (x, y, z) satisfying x = 2z (c) only the trivial solution. (d) None of
these. Jee Main 2020
225. The value of λ and µ for which the system of linear equations x + y + z = 2, x +
2y + 3z = 5, x + 3y + λz = µ has infinitely many solutions are,respectively: (a) 6
and 8 (b) 5 and 7 (c) 5 and 8 (d) 4 and 9 Jee Main 2020
226. If the system of equations x + 2y + 3z = 9, 2x + y + z = b, x − 7y + az = 24 has
infinitely many solutions,then a − b is equal to —– Jee Main 2020
227. The sum of distinct values of λ for which the system of equations (λ − 1)x + (3λ +
1)y + 2λz = 0, (λ − 1)x + (4λ − 2)y + (λ + 3)z = 0, 2x + (3λ + 1)y + 3(λ − 1)z = 0
has a non-zero solutions, is —– Jee Main 2020
228. If the equation a plane P ,passing through the intersection of the plane x+4y−z+7 =
0 and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b ∈ Z,then the distance of
the point (3, 2, −1) from the plane P is —— Jee Main 2020
229. If the system of linear equations, x + y + z = 6, x + 2y + 3z = 10, 3x + 2y + λz = µ
has more than two solutions,the µ − λ2 is equal to —— Jee Main 2020

230. Let S be the set of all integer solutions (x, y, z), of the system of equations x −
2y + 5z = 0, −2x + 4y + z = 0, −7x + 14y + 9z = 0 such that 15 ≤ x2 + y 2 + z 2 ≤
150.Then,the number of elements in the set S is equal to —- Jee Main 2020

231. If the system of linear equations 2x+2y+3z = a, 3x−y+5z = b, x−3y+2z = c, where

a, b, c are non-zero real numbers,has more than one solution,then (a) b − c + a = 0
(b) b − c − a = 0 (c) a + b + c = 0 (d) b + c − a = 0 Jee Main 2019

232. An ordered pair (α, β) for which the system of linear equations (1 + α)x + βy + z =
2, αx + (1 + β)y + z = 3, αx + βy + 2z = 2 has a unique solution, is (a) (2, 4) (b)
(−3, 1) (c) (−4, 2) (d) (1, −3) Jee Main 2019

233. If the system of equations 2x + 3y − z = 0, x + ky − 2z = 0 and 2x − y + z = 0 has

a non-trivial solution (x, y, z),then xy + yz + xz + k is equal to (a) 34 (b) −4 (c) 21 (d)
− 14 Jee Main 2019

234. If the system of linear equation x − 4y + 7z = g, 3y − 5z = h, −2x + 5y − 9z = k

is consistent,then: (a) g + h + k = 0 (b) 2g + h + k = 0 (c) g + h + 2k = 0 (d)
g + 2h + k = 0 Jee Main 2019

235. The number of values of θ ∈ (0, π) for which the system of linear equations x + 3y +
7z = 0, −x + 4y + 7z = 0, sin 3θx + cos 2θy + 2z = 0 has a non-trivial solution, is
(a) three (b) two (c) four (d) one Jee Main 2019

236. The system of linear equations [sin θ]x + [− cos θ]y
= 0 and [cot θ]x + y = 0
π 2π
(a) have infinitely many solutions if θ ∈ ,
2 3
and has a unique solution if θ ∈
7π


π, 6 .
(b) have infinitely many solution if θ ∈
π2 , 2π 7π



3
∪ π, 6
.
π 2π
(c) has a unique solution if θ ∈ ,
2 3
and have infinitely many solutions if θ ∈
7π


π, 6
(d) has a unique solution if θ ∈ π2 , 2π 7π



3
∪ π, 6
Jee Man 2019

237. The system of linear equation x + y + z = 2, 2x + 3y + +2z = 5, 2x + 3y + (a2 − 1)z =

a + 1, √
(a) is inconsistent when a = 4 (b) has a unique solution for
√ |a| = 3 (c) has infinitely
many solutions for a = 4 (d) is inconsistent when |a| = 3 Jee Main 2019

238. If the system of linear equations x − 2y + kz = 1, 2x + y + z = 2, 3x − y + kz = 3

has a solution (x, y, z) ̸= 0,then (x, y) lies on the straight line whose equation is
(a) 3x − 4y − 1 = 0 (b) 4x − 3y − 4 = 0 (c) 4x − 3y − 1 = 0 (d) 3x − 4y − 4 =
0 Jee Main 2019

239. If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + αz = β has

infinitely many solutions,then β − α equals (a) 21 (b) 8 (c) 18 (d) 5Jee Main 2019

240. If the system of linear equations x + y + z = 5, x + 2y + 2z = 6, x + 3y + λz =

µ(λ, µ ∈ R), has infinitely many solutions then the value of λ + µ is (a) 12 (b) 7 (c)
10 (d) 9 Jee Main 2019
241. The set of all values of λ for which the system of linear equations x − 2y − 2z =
λx, x + 2y + z = λy, −x − y− = λz (a) is a singleton. (b) contains exactly two
elements.(c) is an empty set. (d) contains more than two elements. Jee Main 2019

242. Let λ be a real number for which the system of linear equations x + y + z =
6, 4x + λy − λz = λ − 2, 3x + 2y − 4z = −5 has infinitely many solutions.Then λ
is a root of the quadratic equation. (a) λ2 − 3λ − 4 = 0 (b) λ2 − λ − 6 = 0 (c)
λ2 + 3λ − 4 = 0 (d) λ2 + λ − 6 = 0 Jee Main 2019

243. The greatest value of c ∈ R for which the system of linear equations x − cy − cz =
0, cx − y + cz = 0, cx + cy − z = 0 has a non-trivial solution, is (a) −1 (b) 12 (c) 2
(d) 0 Jee Main 2019