Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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

where a,b,cPrevious year Questions (2016-20) - Matrices and Determinants Notes | EduRevare non-zero and distinct; has a non-zero solution, then    (2020)
(1) Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev 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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(R→ R2 - R1 and R3 → R3 - R1)
⇒ (3b - 2a) (c -a) - (b-a) (4c - 2a) = 0
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevare 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,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
⇒1(2λ - 6) - 1 (λ - 9) + 1(2 - 6) = 0
⇒ 2λ - 6 - 1 λ - 9 - 4 = 0 ⇒ λ = 1
Now,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
⇒ 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 =Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevand I = Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevthen 10A−1 is equal to    (2020)
(1) A – 4I
(2) 6I – A
(3) A – 6I
(4) 4I – A

Ans. (3)
Given,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev 
Now,Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Therefore,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
= λ2 + 6λ + 16 = (λ + 8) (2 - λ)
For no solution, D = 0 ⇒ λ = 2.
Now,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, the system of linear equations has no solution for λ = 2.

Q.7. If the matrices A = Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev, B = adj A and C = 3A, then Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevis 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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev    (1)
Now,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, from Eq. (1), we get
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
= 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) =Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevthen    (2020)
(1) f (−50)= 501
(2) f (−50) = − 1
(3) f (50) = - 501
(4) f (50) = 1
Ans. (4)
Given,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
R1→R1 + R- 2R2
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
⇒ 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 Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevthen 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,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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, Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(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.Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev then the matrix A-50 when θ = π/12, is equal to:    (2019)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Ans. (3)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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 Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
∴ 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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Since, the system of equations has infinite many solutions. Hence,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev  Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, for two values of θ, system of equations has non-trivial solution.

Q.18. Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev where b > 0. Then the minimum value of Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev     (2019)
(1) 2√3
(2) -2√3
(3) -√3
(4) √3
Ans. (1)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
= 2(2b2 + 2 - b2)- b(2b - b)+ 1 (b2 - b- 1)
= 2b2 + 4 - b2 - 1 = b2 + 3

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Using A.M≥G.M,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.20. 
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(1) 1/√5
(2) 1/√3
(3) 1/√2
(4) 1/√6

Ans. (3)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.21. 
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
= (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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.24.  
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev matrices such that Q - P5 = I3. ThenPrevious year Questions (2016-20) - Matrices and Determinants Notes | EduRev is equal to:     (2019)
(1) 10
(2) 135
(3) 15
(4) 9
Ans. (1)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.25. 
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev then for all Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev det (A) lies in the interval:     (2019)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Ans. (4)
Solution.

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Now, for a non-trivial solution, the determinant of coefficient matrix is zero.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
⇒ (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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, the greatest value of c is 1/2 for which the system of linear equations has non-trivial solution.

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

Q.29. Let the numbers 2, b, c be in an A.P. and Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev , 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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
∴ System of equation has infinite many solutions.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.31.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev then the inverse of Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev     (2019)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Ans. (2)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.32. Let α and β be the roots of the equation x2 + x + 1 = 0. Then for y ≠ 0 in R,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev      (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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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

Ans. (4)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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 Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev     (2019)
(1) 3/4
(2) 1/2
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(4) -4
Ans. (2)
Solution.
Given system of equations has a non-trivial solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

∴ equations are 2x + 3y - z = 0    ...(i)
2x - y + z = 0    ... (ii)
2x + 9y - 4z = 0    ...(iii)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.35.  
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev       (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.

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
= 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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
∴ for λ = 3, system of equations has infinitely many solutions.

Q.38. The sum of the real roots of the equation       (2019)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(1) 6
(2) 0
(3) 1
(4) -4
Ans. (2)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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 Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevthen AB is equal to:      (2019)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Ans. (2)
Solution.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
On comparing each term,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.40. If Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev 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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.41. If Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev = ( 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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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.

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
hence equations are x + 11y + 3z = 0
3x + 11y - 2z = 0
and 2x + 4y - 3z = 0
let z = t
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Therefore, set S = equal to R-{0}

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

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
∴ c = 0, 2a + 3b = 0, a = 2c + 3d  a = 3d
∴a2 = 9d2 = 36
|3A| = 108
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, option 3 is the answer.

Q.45. Let A = Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevand 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:

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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
:
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev⇒ –(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.Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev,then adj (3A2 + 12A) is equal to     (2017)
(1)Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(2) Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(3)Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(4)Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Ans. 
(3)
Solution.

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRevPrevious year Questions (2016-20) - Matrices and Determinants Notes | EduRev
= (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
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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.
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

Q.50. If A = Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev and A adj A = A AT, then 5a + b is equal to:    (2016)
(1) -1
(2) 5
(3) 4
(4) 13
Ans.
(2)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
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,
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev 

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

Q.53. If P = Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev then PT Q2015 P is    (2016)
(1) Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(2) Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(3) Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
(4) Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Ans.
(3)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
⇒ An = nA - (n-1)I
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev

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)
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Hence, statement 1 is true
Now A3 - 2A2 - 3A + I = A(A2) - 2A2 - 3A + I
Previous year Questions (2016-20) - Matrices and Determinants Notes | EduRev
Statement 2 is also correct

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

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