Let a, λ, μ ∈ R. Consider the system of linear equationsax + 2y = λ3x - 2y = μWhich of the following statement(s) is(are) correct?(A) if a = -3, then the system has infinitely many solutions for all values of λ and μ(B) if a ≠ -3, then the system has a a unique solution for all values of λ and μ(C) if λ + μ = 0, then the system has infinitely many solutions for a = -3(D) if λ + μ ≠ 0, then the system has no solution for a = -3
System of linear equations in x, y, z have infinite solutions which2x + y + z = 1x - 2y + z = 23x - y + 2z = 3(A) can be written as (-3λ -1, λ, 5λ + 3) ∀ λ ∈ R(B) can be written as (3λ -1, -λ, -5λ + 3) ∀ λ ∈ R(C) are such that every solution satisfy x - 3y + 1 = 0(D) are such that none of them satisfy 5x + 3z = 1
Let α, β and γ be real numbers. consider the following system of linear equationsx + 2y + z = 7x + αz = 112x - 3y + βz = γMatch each entry in List-I to the correct entries in List-IIList-IList-II(P) If β = 1/2(7α - 3) and γ = 28, then the system has(1) a unique solution(Q) If β = 1/2(7α - 3) and γ ≠ 28, then the system has(2) no solution(R) If β ≠ 1/2(7α - 3) where α = 1 and γ ≠ 28, then the system has(3) infinitely many solutions(S) If β ≠ 1/2(7α - 3) where α = 1 and γ = 28, then the system has(4) x = 11, y = -2 and z = 0 as a solution(5) x = -15, y = 4 and z = 0 as a solution
Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is
Column-I(A) Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form 1abω1cω2ω1, where each of a, b and c is either ω or ω2. Then the number of distinct matrices in the set S is-(B) Let M be 3 × 3 matrix satisfying M100=-123,M1-10=11-1 and M111=0012. Then the sum of the diagonal entries of M is(C) The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system Axyz=100 has exactly two distinct solutions, is(D) Let k be a positive real number and let A=2k-12k2k2k1-2k-2k2k-1 and B=02k-1k1-2k02k-k-2k0. If det(adj A) + det(adj B) = 106, then [k] is equal to [Note: adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].Column-II(P) 0(Q) 4(R) 9(S) 2
Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right) \neq 0, \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Then, among the statements: (I) Trace $(R) = 0$ (II) If trace $(\text{adj}(\text{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.
Let $M=(a_{ij}), i,j \in \{1, 2, 3\}$, be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j+1$ is divisible by $i$, otherwise $a_{ij} = 0$. Then which of the following statements is (are) true ?(A) $M$ is invertible(B) There exists a nonzero column matrix $\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ such that $M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} -a_1 \\ -a_2 \\ -a_3 \end{pmatrix}$(C) The set $\{X \in \mathbb{R}^3 : MX=0\} \neq \{0\}$, where $0 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$(D) The matrix $(M - 2I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix