If p, q, r are in AP, then the value of 2 Δ^2-3 Δ+1 where, Δ=| 6 7 p 7 8

step 1 So in the given question we have P, Q and R, the numbers P, Q and R, and they are told to be in

If p, q, r are in AP, then the value of 2 Δ^2-3 Δ+1 where,...

If $p, q, r$ are in AP, then the value of $2 \Delta^{2}-3 \Delta+1$ where, $\Delta=\left|\begin{array}{ccc}6 & 7 & p \\ 7 & 8 & q \\ 8 & 9 & r\end{array}\right|$ is
(a) $\left(p-2 q+r^{2}\right)$
(b) 1
(c) 0
(d) $\mathrm{p}-\mathrm{q}+\mathrm{r}$

Key Concepts

Determinant Properties and Row Operations

Properties of determinants, such as their behavior under row operations, play a critical role in simplifying determinant expressions. Recognizing how operations like row addition or factor extraction affect the value of a determinant can facilitate the evaluation of more complicated expressions or reveal underlying simplifications, especially when combined with conditions like those found in arithmetic progressions.

Algebraic Manipulation and Expression Simplification

Effective algebraic manipulation involves recognizing patterns and applying identities to simplify expressions. This is particularly relevant when a problem includes conditions (such as the arithmetic progression relationship among variables) that allow parts of the expression to cancel out or simplify significantly, ultimately leading to a more tractable solution.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive terms is constant. This property often leads to a relation among the involved variables (for instance, in a three-term AP, the middle term is the average of the first and third terms), which can be crucial when substituting values or simplifying expressions in problems that involve sequences.

Matrix Determinants

A matrix determinant is a scalar value computed from a square matrix that provides useful information about the matrix, such as whether it is invertible or the volume scaling factor under the associated linear transformation. In the case of a 3×3 matrix, the determinant can be calculated by using the rule of Sarrus or cofactor expansion, and understanding this process is important for handling algebraic expressions with matrices.

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