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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,care nonzero and distinct; has a nonzero solution, then (2020)
(1) 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
(R_{2 }→ R_{2}  R_{1} and R_{3} → R_{3}  R_{1})
⇒ (3b  2a) (c a)  (ba) (4c  2a) = 0
⇒
Hence, are in A.P.
Q.2. Let A = [a_{ij}] and B = [b_{ij}] be two 3 × 3 real matrices such that b_{ij }= (3)^{(i + j  2)} a_{ij}, 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 b_{ij} = (3)^{(i + j  2)}a_{ij}
Now,
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  λ^{2 }is 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
⇒1(2λ  6)  1 (λ  9) + 1(2  6) = 0
⇒ 2λ  6  1 λ  9  4 = 0 ⇒ λ = 1
Now,
⇒ 6(2  3)  1(10  3μ) + 1(20  20μ) = 0
⇒ 6 10 + 3μ + 20 2μ = 0 ⇒ μ = 14
Hence, μ  λ^{2} =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 =and I = then 10A^{−1} is equal to (2020)
(1) A – 4I
(2) 6I – A
(3) A – 6I
(4) 4I – A
Ans. (3)
Given,
Now,
Therefore,
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,
= λ^{2} + 6λ + 16 = (λ + 8) (2  λ)
For no solution, D = 0 ⇒ λ = 2.
Now,
Hence, the system of linear equations has no solution for λ = 2.
Q.7. If the matrices A = , B = adj A and C = 3A, then is 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
(1)
Now,
Hence, from Eq. (1), we get
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
= 7(24 + 4) 6(182) 2(64)
= 140 + 120 + 20 = 0
Also, Δ_{1} = 0, Δ_{2} = 0 and Δ_{3} = 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) =then (2020)
(1) f (−50)= 501
(2) f (−50) = − 1
(3) f (50) =  501
(4) f (50) = 1
Ans. (4)
Given,
R_{1}→R_{1} + R_{3 } 2R_{2}
⇒ f (x) = {(x+3)^{2}  (x+2)(x+4)}
= x^{2 }+9 + 6x  x^{2}  6x  8 = 1
Hence, f (50) = 1
Q.10. Let α be a root of the equation 2x^{2 }+ x + 1 = 0 and the matrix then the matrix A^{31} is equal to (2020)
(1) A
(2) I^{3}
(3) A^{2}
(4) A^{3}
Ans. Given α be a root of the equation x^{2} + x + 1 = 0
α = ω , ω^{2}
Now,
Hence, A^{31} = A^{28} . A^{3} = A^{3}
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 AA^{T} is 3, is ________. (2020)
Ans. (672.00)
The trace of matrix AA^{T }is
Hence, the number of such matrices is
^{9}C_{3} x 2^{3 }= 84 x 8 = 672
Q.12. The system of linear equations (2019)
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a^{2}  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 + (a^{2}  1) z = a + 1 ...(3)
Now,
⇒
(Applying R_{3} → R_{3}  R_{2})
= a^{2}  3
When, A = 0 ⇒ a^{2 } 3 = 0 ⇒ a= √3
If a^{2} = 3, then plane represented by eqn (2) and eqn (3) are parallel.
Hence, the given system of equations is inconsistent.
Q.13. then the matrix A^{50} when θ = π/12, is equal to: (2019)
Ans. (3)
Solution.
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
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.
∴ 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.
Since, the system of equations has infinite many solutions. Hence,
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 nontrivial solution, is:
(1) three
(2) two
(3) four
(4) one
Ans. (2)
Solution.
Since, the system of linear equations has nontrivial solution then determinant of coefficient matrix = 0
sin3θ(21  28)  cos2θ(7 + 7) + 2 (4 + 3) = 0
sin3θ + 2cos2θ  2 = 0
3sinθ  4sin^{3}θ + 2  4sin^{2}θ  2 = 0
4sin^{3}θ + 4sin^{2}θ  3sinθ = 0
sinθ (4sin^{2}θ + 4sinθ  3) = 0
sinθ (4sin^{2}θ + 6sinθ  2sinθ  3) = 0
sinθ [2sinθ (2sinθ  1) + 3 (2sinθ  1)] = 0
sinθ (2sinθ  1) (2sinθ + 3) = 0
Hence, for two values of θ, system of equations has nontrivial solution.
Q.18. where b > 0. Then the minimum value of (2019)
(1) 2√3
(2) 2√3
(3) √3
(4) √3
Ans. (1)
Solution.
= 2(2b^{2} + 2  b^{2}) b(2b  b)+ 1 (b^{2}  b^{2 } 1)
= 2b^{2} + 4  b^{2}  1 = b^{2} + 3
Using A.M≥G.M,
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 nonzero 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 = Δ_{1} = Δ_{2} = Δ_{3} = 0 for infinite solution
Q.20.
(1) 1/√5
(2) 1/√3
(3) 1/√2
(4) 1/√6
Ans. (3)
Solution.
Q.21.
= (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.
Q.22. Let A and B be two invertible matrices of order 3 x 3. If det (ABA^{T}) = 8 and det (AB^{1}) = 8, then det (BA^{1} B^{T}) is equal to: (2019)
(1) 1/4
(2) 1
(3) 1/16
(4) 16
Ans. (3)
Solution.
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.
Q.24.
matrices such that Q  P^{5} = I_{3}. Then is equal to: (2019)
(1) 10
(2) 135
(3) 15
(4) 9
Ans. (1)
Solution.
Q.25.
then for all det (A) lies in the interval: (2019)
Ans. (4)
Solution.
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 nontrivial 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
Now, for a nontrivial solution, the determinant of coefficient matrix is zero.
⇒ (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 nontrivial solution, is:
(1) 1
(2) 1/2
(3) 2
(4) 0
Ans. (2)
Solution.
If the system of equations has nontrivial solutions, then the determinant of coefficient matrix is zero
Hence, the greatest value of c is 1/2 for which the system of linear equations has nontrivial solution.
Q.28. , (α ∈ R) such that .
Then a value of α is: (2019)
(1) π/32
(2) 0
(3) π/64
(4) π/16
Ans. (3)
Solution.
Q.29. Let the numbers 2, b, c be in an A.P. and , If det(A)∈ [2, 16], then c lies in the interval: (2019)
(1) [2,3)
(2) (2 + 2^{3/4}, 4)
(3) [4,6]
(4) [3,2 + 2^{3/4}]
Ans. (3)
Solution.
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,
∴ System of equation has infinite many solutions.
Q.31.
then the inverse of (2019)
Ans. (2)
Solution.
Q.32. Let α and β be the roots of the equation x^{2} + x + 1 = 0. Then for y ≠ 0 in R,
(2019)
(1) y(y^{2}  1)
(2) y(y^{2}  3)
(3) y^{3}
(4) y^{3}  1
Ans. (3)
Solution.
Let α = ω and β = ω^{2} are roots of x^{2} + x + 1 = 0
Q.33. The total number of matrices (x, y ∈ R, x ≠ y) for which A^{T}A = 3I_{3} is: (2019)
(1) 2
(2) 3
(3) 6
(4) 4
Ans. (4)
Solution.
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 nontrivial solution (x, y, z), then (2019)
(1) 3/4
(2) 1/2
(4) 4
Ans. (2)
Solution.
Given system of equations has a nontrivial solution.
∴ equations are 2x + 3y  z = 0 ...(i)
2x  y + z = 0 ... (ii)
2x + 9y  4z = 0 ...(iii)
Q.35.
(2019)
(1) Δ_{1}  Δ_{2} = 2x^{3}
(2) Δ_{1}  Δ_{2} = x(cos2θ  cos4θ)
(3) Δ_{1} x Δ_{2} = 2(x^{3} + x  1)
(4) Δ_{1} + Δ_{2} = 2x^{3 }
Ans. (4)
Solution.
= x ( x^{2}  1)  sin θ ( x sin θ  cos θ)+ cos θ ( sin θ + x cos θ)
=  x^{3}  x + x sin^{2}θ + sin θ cos θ  cos θ sin θ + x cos^{2}θ
= x^{3 } x + x = x^{3}
Similarly, Δ_{2} =  x^{3} Then, Δ_{1} + Δ_{2} =  2x^{3}
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
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) λ^{2} + 3λ  4 = 0
(2) λ^{2}  3λ  4 = 0
(3) λ^{2} + λ  6 = 0
(4) λ^{2}  λ  6 = 0
Ans. (4)
Solution. ∵ system of equations has infinitely many solutions.
∴ for λ = 3, system of equations has infinitely many solutions.
Q.38. The sum of the real roots of the equation (2019)
(1) 6
(2) 0
(3) 1
(4) 4
Ans. (2)
Solution.
On expanding,
x ( 3x^{2}  6x  2x^{2} + 6x)  6 ( 3x + 9  2x  4)  (4x  9x) = 0
⇒ x ( 5x^{2})  6 ( 5x + 5)  4x + 9x = 0
⇒ x^{3}  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 skewsymmetrix matrix such that then AB is equal to: (2019)
Ans. (2)
Solution.
On comparing each term,
Q.40. If 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.
Q.41. If = ( 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.
B = 5
Q.42. If the system of linear equations
x + ky + 3z = 0
3x + ky  2z = 0
2x + 4y  3z = 0
has a nonzero solution (x, y, z), then xz/y^{2} is equal to: (2018)
(1) 10
(2) 10
(3) 30
(4) 30
Ans: (2)
Solution.
hence equations are x + 11y + 3z = 0
3x + 11y  2z = 0
and 2x + 4y  3z = 0
let z = t
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.
Therefore, set S = equal to R{0}
Q.44. Let A be a matrix such that A ∙ is a scalar matrix and 3A = 108. Then A^{2} equals: (2018)
(1)
(2)
(3)
(4)
Ans: (3)
Solution:
∴ c = 0, 2a + 3b = 0, a = 2c + 3d a = 3d
∴a^{2} = 9d^{2} = 36
3A = 108
Hence, option 3 is the answer.
Q.45. Let A = and B = A^{20}. 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:
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:
⇒ –(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.,then adj (3A^{2} + 12A) is equal to (2017)
(1)
(2)
(3)
(4)
Ans. (3)
Solution.
= (2 – 2λ λ + λ^{2})  12
f (λ)= λ^{2}  3λ  10
∵ A satisfies f (λ)
∴ A^{2} – 3A –10I = 0
A^{2} – 3A = 10I
3A^{2} – 9A = 30I
3A^{2} + 12A = 30I + 21A
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.
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 = I_{3}, where B' is the transpose of B and I_{3 }is 3×3 identity matrix. Then: (2017)
(1) 10A + 5B = 3I_{3}
(2) 3A + 6B = 2I_{3}
(3) 5A + 10B = 2I_{3}
(4) B + 2A = I_{3}
Ans. (1)
Solution.
Q.50. If A = and A adj A = A A^{T}, then 5a + b is equal to: (2016)
(1) 1
(2) 5
(3) 4
(4) 13
Ans. (2)
Equate, 10a + 3b = 25a^{2} + b^{2}
& 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 nontrivial 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,
Q.52. The number of distinct real roots of the equation, in the interval is: (2016)
(1) 4
(2) 1
(3) 2
(4) 3
Ans. (3)
Solution.
tanx = 1 ⇒ x = π/4
Q.53. If P = then P^{T} Q^{2015} P is (2016)
(1)
(2)
(3)
(4)
Ans. (3)
⇒ A^{n} = nA  (n1)I
⇒
Q.54. Let A be a 3 × 3 matrix such that A^{2}  5A + 7I = 0.
Statement  I: A^{1} = 1/7(5IA).
Statement  II : The polynomial A^{3}  2A^{2}  3A + I can be reduced to 5(A  4I).
Then (2016)
(1) StatementI is false, but StatementII is true.
(2) Both the statements are false.
(3) Both the statements are true.
(4) StatementI is true, but StatementII is false.
Ans. (3)
Hence, statement 1 is true
Now A^{3}  2A^{2}  3A + I = A(A^{2})  2A^{2}  3A + I
Statement 2 is also correct
Q.55. If A = then the determinant of the matrix (A^{2016}  2A^{2015}  A^{2014}) is (2016)
(1) 2014
(2) 2016
(3) 175
(4) 25
Ans. (4)
⇒ A^{2016}  2A^{2015}  A^{2014} = A^{2014} A^{2}  2A  I = 1 (100 + 75) = 25
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99 videos291 docs212 tests
