Let a, b, c, l, m, n ∈ R such that al + bm + cn = 0, bl + cm + an = 0, cl + am + bn = 0. If a, b & c are distinct & f(x) = ax^3 + bx^2 + cx + 5, then the value of f(1) is
Let $P_1 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, P_3 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_4 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}, P_5 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, P_6 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ and $X = \sum_{k=1}^6 P_k \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{bmatrix} P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?(A) $X - 30I$ is an invertible matrix(B) The sum of diagonal entries of $X$ is 18(C) If $X \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, then $\alpha = 30$(D) $X$ is a symmetric matrix
Let $P_1 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, P_3 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_4 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}, P_5 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, P_6 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ and $X = \sum_{k=1}^6 P_k \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{bmatrix} P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?(A) $X - 30I$ is an invertible matrix(B) The sum of diagonal entries of $X$ is 18(C) If $X \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, then $\alpha = 30$(D) $X$ is a symmetric matrix
Let α and β be the distinct roots of the equation x2 + x - 1 = 0. Consider the set T = {1, α, β}. For a 3 × 3 matrix M = (aij)3×3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3. Match each entry in List-I to the correct entry in List-II.List-I(P) The number of matrices M = (aij)3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is(Q) The number of symmetric matrices M = (aij)3×3 with all entries in T such that Cj = 0 for all j, is(R) Let M = (aij)3×3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set { (x, y, z) ∈ R3 : M(x, y, z)T = 0 } is(S) Let M = (aij)3×3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M isList-II(1) 1(2) 12(3) infinite(4) 6(5) 0
Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1/√3 [[1, 1, 1], [1, α, α2], [1, α2, α4]], then the matrix A31 is equal to:
Let R = { a3bc2d050 : a, b, c, d ∈ {0, 3, 5, 7, 11, 13, 17, 19} }. Then the number of invertible matrices in R is
Let M = \begin{pmatrix} \sin^2 \theta & -1-\sin^2 \theta \\ 1+\cos^2 \theta & \cos^2 \theta \end{pmatrix} = \alpha I + \beta M^{-1}, where \alpha = \alpha(\theta) and \beta = \beta(\theta) are real number, and I is the 2 \times 2 identity matrix. If \alpha^* is the minimum of the set \{\alpha(\theta): \theta \in [0, 2\pi]\} and \beta^* is the minimum of the set \{\beta(\theta): \theta \in [0, 2\pi]\}, then the value of \alpha^* + \beta^* is
Let α be a root of the equation x^2 + x + 1 = 0 and the matrix A = 1/sqrt(3) * [[1, 1, 1], [1, α, α^2], [1, α^2, α^4]], then the matrix A^31 is equal to:
If system of equation a1x + b1y = c1 & a2x + b2y = c2 (where a1, b1, c1, a2, b2, c2 ≠ 0) has infinite solutions, then-(A) a1/a2 = b1/b2 = c1/c2(B) (a1+a2)/(a1-a2) = (b1+b2)/(b1-b2) = (c1+c2)/(c1-c2)(C) the quadratic equations a1x2 + b1x + c1 = 0 & a2x2 + b2x + c2 = 0 have no common root(D) system of equation a12 a2x + b12 b2y = c12 c2 & a1 a22x + b1 b22y = c1 c22 will also have infinite number of solutions
16. Consider the row sums \(R_i = \sum_{j=1}^{n} a_{ij}\) (\(i = 1, 2, \ldots, n\)) and the column sums \(C_j = \sum_{i=1}^{n} a_{ij}\) (\(j = 1, 2, \ldots, n\)). Let \(p\) be the smallest of all these sums \(R_i\) and \(C_j\), i.e., \(p = \min_{i,j}\{R_i, C_j\}\). Show that \(S > n^2/2\), where \(S\) is the sum of all elements of the matrix. What is the value of \(p\) (as a fraction of \(n^2/2\)) in the minimum case?