JEE  >  Mathematics For JEE  >  JEE Main Previous year questions (2016-20): Matrices and Determinants

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Document Description: JEE Main Previous year questions (2016-20): Matrices and Determinants for JEE 2022 is part of Mathematics For JEE preparation. The notes and questions for JEE Main Previous year questions (2016-20): Matrices and Determinants have been prepared according to the JEE exam syllabus. Information about JEE Main Previous year questions (2016-20): Matrices and Determinants covers topics like and JEE Main Previous year questions (2016-20): Matrices and Determinants Example, for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for JEE Main Previous year questions (2016-20): Matrices and Determinants.

Introduction of JEE Main Previous year questions (2016-20): Matrices and Determinants in English is available as part of our Mathematics For JEE for JEE & JEE Main Previous year questions (2016-20): Matrices and Determinants in Hindi for Mathematics For JEE course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
1 Crore+ students have signed up on EduRev. Have you?

Q.1. If the system of linear equations
2x + 2ay + az = 0
2x + 3by + bz = 0
2x + 4cy + cz = 0,

where a,b,cJEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEare non-zero and distinct; has a non-zero solution, then    (2020)
(1) JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE are in A.P.
(2) a, b, c are in G.P.
(3) a + b + c = 0
(4) a, b, c are in A.P.

Ans. (1) The system of linear equations is
2x + 2ay + az = 0     (1)
2x + 3by + bz = 0    (2)
2x + 4cy + cz = 0     (3)
The system of linear equations has a non- zero solution, then
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(R→ R2 - R1 and R3 → R3 - R1)
⇒ (3b - 2a) (c -a) - (b-a) (4c - 2a) = 0
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEare in A.P.

Q.2. Let A = [aij] and B = [bij] be two 3 × 3 real matrices such that bij = (3)(i + j - 2) aij, where i, j = 1, 2, 3. If the determinant of B is 81, then the determinant of A is    (2020)
(1) 1/3
(2) 1
(3) 1/81
(4) 1/9
Ans. (4)
We have bij = (3)(i + j - 2)aij
Now,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.3. If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + λz = μ

has more than two solutions, then μ - λis  equal to________.    (2020)
Ans. (13.00)
The system of linear equations is
x + y + z = 6    (1)
x + 2y + 3z = 10    (2)
3x + 2y + λz = μ    (3)
The system of equations has more than two solutions, then
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
⇒1(2λ - 6) - 1 (λ - 9) + 1(2 - 6) = 0
⇒ 2λ - 6 - 1 λ - 9 - 4 = 0 ⇒ λ = 1
Now,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
⇒ 6(2 - 3) - 1(10 - 3μ) + 1(20 - 20μ) = 0
⇒ -6 -10 + 3μ + 20 -2μ = 0 ⇒ μ = 14
Hence, μ - λ2 =14 - 1 = 13

Q.4. 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?    (2020)
(1) (4, 3)
(2) (4, 6)
(3) (1, 0)
(4) (3, 4)

Ans. (1)
The system of linear equation is
x + 2y + 3z = 1    (1)
3x + 4y + 5z = μ    (2)
4x + 4y + 4z = δ    (3)
On solving Eqs. (1), (2) and (3), we get
2(3x + 4y + 5z) - 2 (x + 2y + 3z) - 4x + 4y + 4z = 2μ - 2 - δ
⇒ 2μ - 2 - δ = 0 ⇒ δ = 2 (μ - 1)
From the given options only option (4,3) does not satisfies the above equation. So, the system of linear equations is inconsistent for the pair (4,3).

Q.5. If A =JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEand I = JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEthen 10A−1 is equal to    (2020)
(1) A – 4I
(2) 6I – A
(3) A – 6I
(4) 4I – A

Ans. (3)
Given,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE 
Now,JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Therefore,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.6. The system of linear equations    (2020)
λx + 2y + 2z = 5
2λx + 3y + 5z = 8
4x + λy + 6z = 10 has
(1) no solution when λ = 8
(2) a unique solution when λ = -8
(3) no solution when λ = 2
(4) infinitely many solutions when λ = 2

Ans. (3)
The system of linear equations is
λx + 2y + 2z = 5    (1)
2λx + 3y + 5z = 8    (2)
4x + λy + 6z = 10    (3)
Therefore,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
= λ2 + 6λ + 16 = (λ + 8) (2 - λ)
For no solution, D = 0 ⇒ λ = 2.
Now,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, the system of linear equations has no solution for λ = 2.

Q.7. If the matrices A = JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE, B = adj A and C = 3A, then JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEis equal to    (2020)
(1) 8
(2) 16
(3) 72
(4) 2

Ans. (1)
From the properties of determinants of a square matrix of order n, we have
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE    (1)
Now,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, from Eq. (1), we get
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.8. The following system of linear equations    (2020)
7x + 6y - 2z = 0
3x + 4y + 2z = 0
x - 2y - 6z = 0, has
(1) Infinitely many solutions, (x, y, z) satisfying y = 2z.
(2) no solution.
(3) infinitely many solutions, (x, y, z) satisfying x = 2z.
(4) only the trivial solution.

Ans. (3)
The system of linear equation is
7x + 6y - 2z = 0    (1)
3x + 4y + 2z = 0    (2)
x - 2y - 6z = 0    (3)
Therefore, the coefficient matrix is given by
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
= 7(-24 + 4) -6(-18-2) -2(-6-4)
= -140 + 120 + 20 = 0
Also, Δ1 = 0, Δ2 = 0 and Δ3 = 0
Hence, for the given system of equation infinite many solutions are possible.
From Eqs. (1) and (3), we get
7x + 6y - 2z + 3(x - 2y - 6z) = 0 ⇒ x = 2z

Q.9. Let a – 2b + c = 1. If f(x) =JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEthen    (2020)
(1) f (−50)= 501
(2) f (−50) = − 1
(3) f (50) = - 501
(4) f (50) = 1
Ans. (4)
Given,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
R1→R1 + R- 2R2
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
⇒ f (x) = {(x+3)2 - (x+2)(x+4)}
= x+9 + 6x - x2 - 6x - 8 = 1
Hence, f (50) = 1

Q.10. Let α be a root of the equation 2x+ x + 1 = 0 and the matrix JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEthen the matrix A31 is equal to    (2020)
(1) A
(2) I3
(3) A2
(4) A3
Ans. Given α be a root of the equation x2 + x + 1 = 0
α = ω , ω2
Now,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, A31 = A28 . A3 = A3

Q.11. The number of all 3 × 3 matrices A, with entries from the set {−1, 0, 1} such that the sum of the diagonal elements of AAT is 3, is ________.    (2020)
Ans. (672.00)
The trace of matrix  AAis
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, the number of such matrices is
9C3 x 2= 84 x 8 = 672

Q.12. The system of linear equations     (2019)
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a2 - 1)z = a + 1

(1)  is inconsistent when a = 4
(2) has a unique solution for |a| = √3
(3) has infinitely many solutions for a = 4
(4) is inconsistent when |a| = √3
Ans. (4)
Solution. Since the system of linear equations are
x+y + z = 2    ...(1)
2x + 3y + 2z = 5    ...(2)
2x + 3y + (a2 - 1) z = a + 1    ...(3)
Now, JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(Applying R3 → R3 - R2)
= a2 - 3
When, A = 0 ⇒ a- 3 = 0 ⇒ |a|= √3
If a2 = 3, then plane represented by eqn (2) and eqn (3) are parallel.
Hence, the given system of equations is inconsistent.

Q.13.JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE then the matrix A-50 when θ = π/12, is equal to:    (2019)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.14. If the system of linear equations      (2019)
x - 4y + 7z = g
3y - 5z = h
- 2x + 5y - 9z = k
is consistent, then:

(1) g + 2h + k = 0
(2) g + h + 2k = 0
(3) 2g + h + k = 0
(4) g + h + k = 0
Ans. (3)
Solution.
Consider the system of linear equations
x - 4y + 7z = g    ...(i)
3y - 5z = h    ... (ii)
-2x + 5y - 9z = k    ...(iii)
Multiply equation (i) by 2 and add equation (i), equation (ii) and equation (iii)
  0 = 2g + h + k.  ∴ 2g + h + k = 0
then system of equations is consistent.

Q.15. If JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
then A is:     (2019)
(1) invertible for all t ∈ R.
(2) invertible only if t = π.
(3) not invertible for any t ∈ R.
(4) invertible only if t = π/2.
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
∴ A is invertible.

Q.16. If the system of equations      (2019)
x + y + z = 5
x + 2y + 3z = 9
x + 3y + αz = β

has infinitely many solutions, then β - α equals:
(1) 21
(2) 8
(3) 18
(4) 5

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Since, the system of equations has infinite many solutions. Hence,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.17. The number of values of θ ∈ (0, π) for which the system of linear equations       (2019)
x + 3y + 7z = 0
- x + 4y + 7z = 0
(sin 3θ)x + (cos 2θ)y + 2z = 0
has a non-trivial solution, is:
(1) three
(2) two
(3) four 
(4) one

Ans. (2)
Solution.
Since, the system of linear equations has non-trivial solution then determinant of coefficient matrix = 0
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
sin3θ(21 - 28) - cos2θ(7 + 7) + 2 (4 + 3) = 0
sin3θ + 2cos2θ - 2 = 0
3sinθ - 4sin3θ + 2 - 4sin2θ - 2 = 0
4sin3θ + 4sin2θ - 3sinθ = 0
sinθ (4sin2θ + 4sinθ - 3) = 0
sinθ (4sin2θ + 6sinθ - 2sinθ - 3) = 0
sinθ [2sinθ (2sinθ - 1) + 3 (2sinθ - 1)] = 0
sinθ (2sinθ - 1) (2sinθ + 3) = 0
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE  JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, for two values of θ, system of equations has non-trivial solution.

Q.18. JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE where b > 0. Then the minimum value of JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE     (2019)
(1) 2√3
(2) -2√3
(3) -√3
(4) √3
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
= 2(2b2 + 2 - b2)- b(2b - b)+ 1 (b2 - b- 1)
= 2b2 + 4 - b2 - 1 = b2 + 3

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Using A.M≥G.M,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.19. If the system of linear equations       (2019)
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:
(1) b - c + a = 0
(2) b - c - a = 0
(3) a + b + c = 0
(4) b + c- a = 0

Ans. (2)
Solution.
∵ System of equations has more than one solution
∴ Δ = Δ1 = Δ2 = Δ3 = 0 for infinite solution
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.20. 
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(1) 1/√5
(2) 1/√3
(3) 1/√2
(4) 1/√6

Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.21. 
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
= (a + b + c) (x + a + b + c)2, x ≠ 0 and a + b + c ≠ 0, then x is equal to:     (2019)
(1) abc
(2) -(a + b + c)
(3) 2(a + b + c)
(4) -2(a + b + c)

Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.22. Let A and B be two invertible matrices of order 3 x 3. If det (ABAT) = 8 and det (AB-1) = 8, then det (BA-1 BT) is equal to:     (2019)
(1) 1/4
(2) 1
(3) 1/16
(4) 16

Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.23. An ordered pair (α, β) for which the system of linear equations
(1 + α)x + βy + z = 2
ax + (1 + β)y + z = 3
αx+ βy+ 2z = 2
has a unique solution, is :     (2019)

(1) (2, 4)
(2) (-3, 1)
(3) (-4, 2)
(4) (1, -3)

Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.24.  
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE matrices such that Q - P5 = I3. ThenJEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE is equal to:     (2019)
(1) 10
(2) 135
(3) 15
(4) 9
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.25. 
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE then for all JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE det (A) lies in the interval:     (2019)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Ans. (4)
Solution.

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.26. The set of all values of λ for which the system of linear equations     (2019)
x - 2y - 2z = λx
x + 2y + z = λy
- x - y = λ2
has a non-trivial solution:
(1) is a singleton
(2) contains exactly two elements
(3) is an empty set
(4) contains more than two elements

Ans. (1)
Solution.
Consider the given system of linear equations
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Now, for a non-trivial solution, the determinant of coefficient matrix is zero.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
⇒ (1 - λ)3 = 0
λ = 1

Q.27. The greatest value of c ∈ R for which the system of linear equations     (2019)
x - cy - cz = 0
cx - y + cz = 0
cx + cy - z = 0

has a non-trivial solution, is:
(1) -1
(2) 1/2
(3) 2
(4) 0

Ans. (2)
Solution.
If the system of equations has non-trivial solutions, then the determinant of coefficient matrix is zero
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, the greatest value of c is 1/2 for which the system of linear equations has non-trivial solution.

Q.28.JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE , (α ∈ R) such that JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE.
Then a value of α is:     (2019)
(1) π/32
(2) 0
(3) 
π/64
(4) π/16
Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.29. Let the numbers 2, b, c be in an A.P. and JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE , If det(A)∈ [2, 16], then c lies in the interval:     (2019)
(1) [2,3)
(2) (2 + 23/4, 4)
(3) [4,6]
(4) [3,2 + 23/4]
Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.30. If the system of linear equations
x - 2y + kz = 1
2x + y + z = 2
3x - y - kz = 3

has a solution (x, y, z), z = ≠ 0, then (x, y) lies on the straight line whose equation is:     (2019)
(1) 3x - 4y - 1 = 0
(2) 4x - 3y - 4 = 0
(3) 4x - 3y - 1 = 0
(4) 3x - 4y - 4 = 0

Ans. (2)
Solution.
Given system of linear equations,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
∴ System of equation has infinite many solutions.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.31.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE then the inverse of JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE     (2019)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.32. Let α and β be the roots of the equation x2 + x + 1 = 0. Then for y ≠ 0 in R,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE      (2019)
(1) y(y2 - 1)
(2) y(y2 - 3)
(3) y3
(4) y3 - 1

Ans. (3)
Solution.
Let α = ω and β = ω2 are roots of x2 + x + 1 = 0
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.33. The total number of matrices JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE (x, y ∈ R, x ≠ y) for which ATA = 3I3 is:     (2019)
(1) 2
(2) 3
(3) 6
(4) 4

Ans. (4)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Number of combinations of (x, y) = 2 x 2 = 4

Q.34. 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 JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE     (2019)
(1) 3/4
(2) 1/2
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(4) -4
Ans. (2)
Solution.
Given system of equations has a non-trivial solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

∴ equations are 2x + 3y - z = 0    ...(i)
2x - y + z = 0    ... (ii)
2x + 9y - 4z = 0    ...(iii)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.35.  
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE       (2019)
(1) Δ1 - Δ2 = -2x3
(2) Δ1 - Δ2 = x(cos2θ - cos4θ)
(3) Δ1 x Δ2 = -2(x3 + x - 1)
(4) Δ1 + Δ2 = -2x

Ans. (4)
Solution.

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
= x (- x2 - 1) - sin θ (- x sin θ - cos θ)+ cos θ (- sin θ + x cos θ)
= - x3 - x + x sin2θ + sin θ cos θ - cos θ sin θ + x cos2θ
= -x- x + x = -x3
Similarly, Δ2 = - x3   Then, Δ1 + Δ2 = - 2x3

Q.36. 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 :       (2019)
(1) 12
(2) 9
(3) 7 
(4) 10

Ans. (4)
Solution.
Given system of linear equations: x + y + z = 5; x + 2y + 2z = 6 and x + 3y + λz = μ have infinite solution
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.37. 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:      (2019)
(1) λ2 + 3λ - 4 = 0
(2) λ2 - 3λ - 4 = 0
(3) λ2 + λ - 6 = 0
(4) λ2 - λ - 6 = 0
Ans. (4)
Solution. ∵ system of equations has infinitely many solutions.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
∴ for λ = 3, system of equations has infinitely many solutions.

Q.38. The sum of the real roots of the equation       (2019)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(1) 6
(2) 0
(3) 1
(4) -4
Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
On expanding,
x (- 3x2 - 6x - 2x2 + 6x) - 6 (- 3x + 9 - 2x - 4) - (4x - 9x) = 0
⇒ x (- 5x2) - 6 (- 5x + 5) - 4x + 9x = 0
⇒ x3 - 7x + 6 = 0
∵ all the roots are real.
∴ sum of real roots = 0/1 = 0

Q.39. If A is a symmetric matrix and B is a skew-symmetrix matrix such that JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEthen AB is equal to:      (2019)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
On comparing each term,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.40. If JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE is the inverse of a 3 x 3 matrix A, then the sum of all values of α for which delta (A) + 1 = 0, is:       (2019)
(1) 0
(2) -1
(3) 1 
(4) 2
Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.41. If JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE = ( A + Bx)( x- A)2 , then the ordered pair (A, B) is equal to:     (2018)
(1) (-4, -5)
(2) (-4, 3)
(3) (-4, 5)
(4) (4, 5)

Ans. (3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
B = 5

Q.42. If the system of linear equations
x + ky + 3z = 0
3x + ky - 2z = 0
2x + 4y - 3z = 0
has a non-zero solution (x, y, z), then xz/y2 is equal to:    (2018)
(1) -10
(2) 10
(3) -30
(4) 30
Ans: 
(2)
Solution.

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
hence equations are x + 11y + 3z = 0
3x + 11y - 2z = 0
and 2x + 4y - 3z = 0
let z = t
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.43. Let S be the set of all real values of k for which the system of linear equations      (2018)
x+ y + z = 2
2x + y - z = 3
3x + 2y + kz = 4
has a unique solution.  

Then S is :
(1) S equal to {0}
(2) equal to R-{0}
(3) an empty set
(4) equal to R

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Therefore, set S = equal to R-{0}

Q.44. Let A be a matrix such that A ∙ JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE is a scalar matrix and |3A| = 108. Then A2 equals:     (2018)
(1)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(2)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Ans:
(3)
Solution:

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
∴ c = 0, 2a + 3b = 0, a = 2c + 3d  a = 3d
∴a2 = 9d2 = 36
|3A| = 108
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, option 3 is the answer.

Q.45. Let A = JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEand B = A20. Then the sum of the elements of the first column of B is:     (2018)
(1) 210
(2) 211
(3) 251
(4) 231
Ans: 
(4)
Solution:

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Sum of the elements of first column = 231

Q.46. If S is the set of distinct values of b for which the following system of linear equations    (2017)
x +y+ z = 1
x +ay+ z = 1
ax + by+ z = 0
has no solution, then S is
(1) A singleton
(2) An empty set
(3) An infinite set
(4) A finite set containing two or more elements
Ans. 
(1)
Solution
:
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE⇒ –(1 – a)2 = 0
⇒ a = 1
For a = 1
Eq. (1) & (2) are identical i.e.,x + y + z = 1
To have no solution with x + by + z = 0
b = 1

Q.47.JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE,then adj (3A2 + 12A) is equal to     (2017)
(1)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(3)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(4)JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Ans. 
(3)
Solution.

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEEJEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
= (2 – 2λ- λ + λ2) - 12
f (λ)= λ2 - 3λ - 10
∵ A satisfies f (λ)
∴ A2 – 3A –10I = 0
A2 – 3A = 10I
3A2 – 9A = 30I
3A2 + 12A = 30I + 21A
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.48. The number of real values of λ for which the system of linear equations
2x + 4y – λz = 0
4x + λy + 2z = 0
λx + 2y + 2z = 0
has infinitely many solutions, is:    (2017)
(1) 3
(2) 1
(3) 2
(4) 0

Ans. (2)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
It will give only one real value of λ

Q.49. For two 3 × 3 matrices A and B, let A+B = 2B' and 3A + 2B = I3, where B' is the transpose of B and Iis 3×3 identity matrix. Then:     (2017)
(1) 10A + 5B = 3I3

(2) 3A + 6B = 2I3
(3) 5A + 10B = 2I3
(4) B + 2A = I
3
Ans. (1)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.50. If A = JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE and A adj A = A AT, then 5a + b is equal to:    (2016)
(1) -1
(2) 5
(3) 4
(4) 13
Ans.
(2)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Equate,  10a + 3b = 25a2 + b2
& 10a + 3b = 13
& 15a - 2b = 0
a/2 = b/15 = k (let) 
Solving a = 2/5, b = 3 
So, 5a + b = 5 x 2/5 + 3 = 5 

Q.51. The system of linear equations
x + λy - z = 0
λx - y - z = 0
x + y - λz = 0
has a non-trivial solution for:    (2016)
(1) infinitely many values of λ
(2) exactly one value of λ
(3) exactly two values of λ
(4) exactly three values of  λ
Ans.
(4)
For trivial solution,
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE 

Q.52. The number of distinct real roots of the equation, JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE in the interval JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE is:    (2016)
(1) 4
(2) 1
(3) 2
(4) 3
Ans.
(3)
Solution.
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE   tanx = 1 ⇒ x = π/4
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.53. If P = JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE then PT Q2015 P is    (2016)
(1) JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(2) JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(3) JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
(4) JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Ans.
(3)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
⇒ An = nA - (n-1)I
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

Q.54. Let A be a 3 × 3 matrix such that A2 - 5A + 7I = 0.
Statement - I: A-1 = 1/7(5I-A).
Statement - II : The polynomial A3 - 2A2 - 3A + I can be reduced to 5(A - 4I).
Then    (2016)
(1) Statement-I is false, but Statement-II is true.
(2) Both the statements are false.
(3) Both the statements are true.
(4) Statement-I is true, but Statement-II is false.
Ans.
(3)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Hence, statement 1 is true
Now A3 - 2A2 - 3A + I = A(A2) - 2A2 - 3A + I
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
Statement 2 is also correct

Q.55. If A = JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE  then the determinant of the matrix (A2016 - 2A2015 - A2014) is    (2016)
(1) 2014
(2) 2016
(3) -175
(4) -25
Ans.
(4)
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE
⇒ A2016 - 2A2015 - A2014| = |A|2014 |A2 - 2A - I| = 1JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE  (-100 + 75) = -25

The document JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE is a part of the JEE Course Mathematics For JEE.
All you need of JEE at this link: JEE
130 videos|359 docs|306 tests
Download as PDF

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

Exam

,

mock tests for examination

,

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

,

MCQs

,

Semester Notes

,

shortcuts and tricks

,

Important questions

,

Objective type Questions

,

Sample Paper

,

pdf

,

past year papers

,

ppt

,

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

,

Viva Questions

,

JEE Main Previous year questions (2016-20): Matrices and Determinants Notes | Study Mathematics For JEE - JEE

,

practice quizzes

,

Previous Year Questions with Solutions

,

study material

,

Free

,

Extra Questions

,

video lectures

,

Summary

;