Consider a system of linear equations $a_ix + b_iy + c_iz = d_i$ (where $a_i, b_i, c_i \neq 0$ and $i = 1,2,3$ ) & $(\alpha,\beta,\gamma)$ is its unique solution, then match list-I with list-IIList-IList-II(I) If $a_i = k, d_i = k^2, (k \neq 0)$ and $\alpha + \beta + \gamma = 2$, then $k$ is(P) 1(II) If $a_i = d_i = k \neq 0$, then $\alpha + \beta + \gamma$ is(Q) 2(III) If $a_i = k > 0, d_i = k + 1$, then $\alpha + \beta + \gamma$ can be(R) 0(IV) If $a_i = k (S) 3(T) -1
Consider the matrices: A = \begin{pmatrix} 2 & -5 \\ 3 & m \end{pmatrix}, B = \begin{pmatrix} 20 \\ m \end{pmatrix} and X = \begin{pmatrix} x \\ y \end{pmatrix}. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8 \int_{a}^{b} |A| dm is equal to ________.
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 $M=(a_{ij}), i,j \in \{1, 2, 3\}$, 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
If in the determinant Δ = a1b1c1a2b2c2a3b3c3, A1, B1, C1 etc. be the co-factors of a1, b1, c1 etc., then which of the following relations is incorrect-(A) a1A1 + b1B1 + c1C1 = Δ(B) a2A2 + b2B2 + c2C2 = Δ(C) a3A3 + b3B3 + c3C3 = Δ(D) a1A2 + b1B2 + c1C2 = Δ