Problem 83 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 \mathrm{\\{} O n l i n e ~ M a y ~ 1 2 , ~ 2 0 1 2 ] ~ (a) \(a, b, c, d\) are in A.P. (b) \(a b=c d\) (c) \(a c=b d\) (d) \(a, b, c, d\) are in G.P.

Step-by-Step Solution

Answer

(d) \(a, b, c, d\) are in G.P.

Step 1: Analyze the inequality

We start with the inequality \(\left(a^{2}+b^{2}+c^{2}\right) p^{2}-2 p(a b+b c+c d)+\left(b^{2}+c^{2}+d^{2}\right) \leq 0\). This is a quadratic inequality in terms of \(p\) because it has the highest power of \(p\) being \(p^2\).

Step 2: Apply the condition for quadratic inequalities

For a quadratic inequality \(Ap^2 + Bp + C \leq 0\), the discriminant \(B^2 - 4AC\) must be less than or equal to zero for it to hold true for all real \(p\). Identify the coefficients \(A = a^2 + b^2 + c^2\), \(B = -2(ab + bc + cd)\), and \(C = b^2 + c^2 + d^2\), and calculate the discriminant condition.

Step 3: Calculate the discriminant

Substitute the coefficients into the discriminant formula: \(\left(-2(ab+bc+cd)\right)^2 - 4(a^2+b^2+c^2)(b^2+c^2+d^2) \leq 0\). Simplify this to \(4(ab+bc+cd)^2 \leq 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\).

Step 4: Simplify the inequality

Divide the simplified discriminant inequality by 4: \((ab+bc+cd)^2 \leq (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\). Recognize this structure suggests a relationship like the Cauchy-Schwarz inequality.

Step 5: Investigate equality conditions of Cauchy-Schwarz

The Cauchy-Schwarz inequality states that for sequences \(x_i\) and \(y_i\), \((\sum x_i y_i)^2 \leq (\sum x_i^2)(\sum y_i^2)\) with equality if and only if the vectors are proportional, i.e., \(x_i = ky_i\) for some constant \(k\).

Step 6: Apply equality condition to setup equations

For the sequences \((a, b, c)\) and \((b, c, d)\), Cauchy-Schwarz equality implies \(a/b = b/c = c/d\), translating to a common ratio between consecutive terms, suggesting \(a, b, c, d\) must be in geometric progression (G.P.).

Step 7: Conclude with the valid option

The only condition that aligns with the derived equality requirement is option (d) that \(a, b, c, d\) are in G.P., since the analysis of the inequality confirms they are proportional (in same ratio).

Key Concepts

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a fundamental concept in mathematics used to establish relationships between sequences or vectors. In its simplest form, it states that for any sequences or vectors \(x_i\) and \(y_i\), the following inequality holds:

• \( (\sum x_i y_i)^2 \leq (\sum x_i^2)(\sum y_i^2) \)

This can be understood as an expression of the maximum possible correlation between two sequences.

When the inequality achieves equality, the sequences are said to be proportional—meaning there exists a constant \(k\) such that \(x_i = ky_i\) for all \(i\).

In the context of the exercise, we apply the Cauchy-Schwarz inequality to the quadratic expression
\((a^2+b^2+c^2)p^2 - 2p(ab+bc+cd) + (b^2+c^2+d^2) \leq 0\). Recognizing the quadratic inequality structure, we equate the left side to the Cauchy-Schwarz inequality form. Investigating the condition for equality, it points towards proportional relationships, allowing us to infer that the terms \((a, b, c)\) and \((b, c, d)\) must be proportional, leading us to the conclusion of a geometric progression.

Geometric Progression

A geometric progression (G.P.) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. If \(a, b, c,\) and \(d\) are part of a geometric progression, then:

• \( \frac{b}{a} = \frac{c}{b} = \frac{d}{c} \)
• This implies \(b = ar, \) \(c = ar^2, \) \(d = ar^3\) when \(a\) is the first term and \(r\) is the common ratio.

Understanding whether numbers form a geometric progression helps analyze the relationships between them.

In the quadratic inequality from the exercise, by requiring the equality condition of the Cauchy-Schwarz inequality, we find that \((a, b, c, d)\) must have a common ratio. This confirms that they form a geometric progression, which is crucial for solving the inequality condition effectively. Recognizing the type of progression involved can immensely simplify analysis of mathematical expressions.

Discriminant of Quadratic Equation

The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) gives crucial insights into the nature of its roots. It is given by:

• \( \Delta = B^2 - 4AC \)

The sign of the discriminant tells us the character of the roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), it has exactly one real root (a repeated root).
• If \( \Delta < 0 \), there are no real roots (the roots are complex).

In solving quadratic inequalities, particularly in the given exercise, the discriminant helps determine the range of values for which the inequality holds.

In the problem at hand, equating the quadratic inequality to \((a^2+b^2+c^2)p^2 - 2p(ab+bc+cd) + (b^2+c^2+d^2) \leq 0\), we apply the discriminant to check for the conditions under which the inequality is satisfied. The condition \( \Delta \leq 0 \) signifies that the quadratic expression remains non-positive for real values of \(p\), leading us closer to finding the required relationships among variables.