If | a b a α-b b c b α-c 2 1 0 |=0 and α≠(1)/(2), then (

step 1 First, we expand this determinant. We use the nest row. It's two times the determinant of B -C,

If | a b a α-b b c b α-c 2 ...

If $\left|\begin{array}{ccc}a & b & a \alpha-b \\ b & c & b \alpha-c \\ 2 & 1 & 0\end{array}\right|=0$ and $\alpha \neq \frac{1}{2}$, then
(A) $a, b, c$ are in A.P.
(B) $a, b, c$ are in G.P.
(C) $a, b, c$ are in H.P.
(D) None of these

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Key Concepts

Evaluation of Determinant Equations

When the determinant of a matrix is set equal to zero, it implies that the rows or columns are linearly dependent, leading to an equation that relates the variables within the matrix. Solving such an equation reveals algebraic relationships among the parameters, which can then be interpreted in the context of various mathematical sequences or structures.

Sequences and Progressions (AP, GP, HP)

This concept covers various types of sequences, notably arithmetic progression (AP), geometric progression (GP), and harmonic progression (HP). Each progression is defined by a specific relation among its terms—AP by a constant difference, GP by a constant ratio (typically reflected by the condition b² = ac), and HP by the reciprocals forming an arithmetic progression. Recognizing these relationships helps in determining the nature of the sequence formed by a set of parameters.

Matrix Determinants

This concept involves calculating the determinant of a square matrix, which is a scalar value summarizing certain properties of the matrix. Determinants are used to characterize the solvability of linear systems and to analyze the dependency among the rows or columns of a matrix. When a determinant equals zero, it indicates a linear dependency among the matrix’s elements.

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