Problem 32 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c, d\) and \(p\) are distinct real numbers such that \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2 p(a b+b c+c d)+\left(b^{2}+c^{2}+d^{2}\right)\) \(\leq 0\) then \(a, b, c, d\) are in (A) A.P. (B) G.P. (C) H.P. (D) \(a b=c d\)

Step-by-Step Solution

Answer

A. a, b, c, d are in A.P.

Step 1: Examine the Quadratic Inequality Structure

Notice that the expression \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2 p(a b+b c+c d)+\left(b^{2}+c^{2}+d^{2}\right)\) is quadratic in terms of \(p\). We can rewrite it as \(Ap^2 - 2Bp + C \leq 0\), where \(A = a^2 + b^2 + c^2\), \(B = ab + bc + cd\), and \(C = b^2 + c^2 + d^2\).

Step 2: Use Discriminant Condition for Non-Positive Quadratic

For the quadratic expression \(Ap^2 - 2Bp + C \leq 0\) to hold for any value of \(p\), the discriminant of this quadratic equation must be non-positive. The discriminant \(D\) is given by \((2B)^2 - 4AC\). This translates to \(4B^2 - 4AC \leq 0\). Simplifying gives \(B^2 \leq AC\).

Step 3: Substitute Expressions for \(A\), \(B\), and \(C\)

Substitute \(A = a^2 + b^2 + c^2\), \(B = ab + bc + cd\), and \(C = b^2 + c^2 + d^2\) into the discriminant condition. Rearrange and expand the terms to compare them successfully: \((ab + bc + cd)^2 \leq (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\).

Step 4: Analyze the Expression Structure

By comparing both sides of the inequality, notice that this form resembles the conditions required for sequences, particularly an arithmetic progression (A.P.), because differences between consecutive terms sum up to form the middle term condition \( (b - a) + (c - b) + (d - c) = 0 \).

Step 5: Conclude Based on Sequence Properties

Given the analysis, since the quadratic reduces to a constant sequence structure and maintains equality conditions satisfied with distinct numbers and differences being identical, \(a, b, c, d\) must form an arithmetic sequence.

Key Concepts

Arithmetic Progression

Arithmetic progression, often abbreviated as A.P., is a sequence of numbers in which the difference between consecutive terms is always the same. This constant difference is what defines the progression as 'arithmetic'. For example, in the sequence 2, 5, 8, 11, the difference between each term is 3.
An important property of an arithmetic progression is that the sum of terms can be calculated using the formula for the sum of an arithmetic series. However, in our context, we focus on identifying A.P. through their characteristics:

• The constant difference will be the same: within a sequence like a-d-c-b, (b-a) = (c-b) = (d-c).
• The middle term of any three consecutive terms is the average of its neighbors, ensuring the differences remain consistent.
• When given conditions like the ones from a quadratic inequality, it can be useful to check if the sequence maintains the same differences, which is a key indicator of A.P.

This concept is essential in solving the given problem. By recognizing an arithmetic sequence's structure, it helps determine that the sequence of terms meets the problem's requirements, allowing for insightful deductions and correct answers.

Discriminant Condition

The discriminant of a quadratic equation is a crucial factor in determining the nature of its roots. It's often symbolized by "D" and calculated as below:
For any quadratic equation of the form \(ax^2 + bx + c\), the discriminant is given by \(b^2 - 4ac\).
Evaluating the discriminant helps us understand root characteristics:

• If \(D > 0\), the quadratic has two distinct real roots.
• If \(D = 0\), the quadratic has exactly one real root (a repeated root).
• If \(D < 0\), the quadratic has no real roots; instead, it has complex or imaginary roots.

In the context of our given inequality, the discriminant condition is manipulated to ensure an inequality holds for all values of the variable, in this case, \(p\). By setting up \((2B)^2 - 4AC \leq 0\), we ensure that the quadratic expression does not have any positive roots, indicating that the terms in the inequality maintain a particular sequence property, leading us to the conclusion about the progression.

Sequence Properties

Sequences are ordered lists of numbers following specific rules for the arrangement of terms. When considering properties of sequences in algebra, recognizing these rules is essential for solving related problems.
Several sequence types are significant to inequality problems:

• Arithmetic Progression (A.P.) - As discussed, A.P.'s maintain a constant difference between terms.
• Geometric Progression (G.P.) - Each term is a constant multiple of the previous term, like 2, 4, 8 where each is multiplied by 2.
• Harmonic Progression (H.P.) - Formed by reciprocals of an A.P.; it appears in contexts involving rates and times.

In our problem, identifying the required sequence property was essential. By examining the quadratic inequality and transforming it based on sequence properties, it becomes clear which type of progression fits. Furthermore, the relevant sequence characteristics provide the groundwork for validating required mathematical conditions and solving these complex algebraic structures effectively.