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

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

The correct answer is (A) A.P.

Step 1: Analyze the Given Inequality

We start by analyzing the given quadratic 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 expression in terms of \(p\). The inequality holds for all real numbers \(p\), indicating that the discriminant of this quadratic equation must be less than or equal to zero.

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by \(B^2 - 4AC\). Here, \(A = a^2 + b^2 + c^2\), \(B = -2(ab + bc + cd)\), and \(C = b^2 + c^2 + d^2\). The discriminant is \([-2(ab + bc + cd)]^2 - 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\).

Step 3: Simplify the Discriminant

Calculate and simplify \(B^2 = 4(ab + bc + cd)^2\) and \(4AC = 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\). The inequality \(4(ab + bc + cd)^2 - 4(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) \leq 0\) must hold. Simplifying gives \((ab + bc + cd)^2 \leq (a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\).

Step 4: Analyze Simplified Expression

Notice that the resulting simplified expression resembles the condition for numbers forming an Arithmetic Progression (A.P.). Specifically, this simplifies under certain symmetrical conditions such as \(b-c = c-b\), suggesting these numbers form an A.P.

Step 5: Verify Arithmetic Progression Condition

To verify, assume \(a, b, c, d\) as members of an arithmetic progression. For these to be in A.P., the condition \(2b = a + c\) and \(2c = b + d\) must hold. Plug in these equalities into the simplified discriminant condition to ensure it resolves to a valid inequality.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference. Here’s how you can identify an arithmetic sequence:

• Each term increases or decreases by the same amount.
• The sequence formula is given by: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.
• For example, the sequence \( 2, 4, 6, 8, ... \) has a common difference of \( 2 \).

Let's apply this to the exercise: given an inequality involving \( a, b, c, \) and \( d \), where the simplification results in \( 2b = a + c \) and \( 2c = b + d \). These are conditions for a sequence to be in A.P., meaning the terms are symmetrical. This symmetry aligns with the definition of arithmetic progressions and confirms that the sequence lies in an A.P.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation provides valuable information about the nature of the roots of the equation. The quadratic equation is generally expressed as \( Ax^2 + Bx + C = 0 \), and its discriminant is calculated using the formula \( B^2 - 4AC \).

Here’s what the discriminant tells us:

• If the discriminant is positive (> 0), the quadratic equation has two distinct real roots.
• If the discriminant is zero (0), there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative (< 0), the quadratic equation has no real roots, indicating two complex roots.

For the original exercise, the discriminant has to be less than or equal to zero, which ensures that the inequality holds for all values of \( p \). The process involves simplifying the quadratic expression and checking that \((ab + bc + cd)^2\) does not exceed \((a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\). This stability under various values of \( p \) confirms a potential relationship among \( a, b, c, \) and \( d \).

Condition for Numbers in Sequence

There are specific mathematical conditions that can validate a sequence as an arithmetic progression or any other type of sequence. Understanding these conditions helps in identifying such properties in given sequences.

The condition for a sequence to be in A.P. is that every consecutive term should have an equal difference. In the context of the exercise, this translates to finding symmetrical relationships that hold across all terms.

Ensuring these conditions requires verifying that equations like \(2b = a + c\) and \(2c = b + d\) are true.

• For arithmetic progression: Use these equalities to set up problems in sequence, checking for consistent differences to prove the progression.
• Apply these findings to identify patterns and confirm whether the numbers strictly follow these conditions, pointing towards an A.P.
• Consistently check if the discriminant condition satisfies the required inequalities in different scenarios.

This rigorous checking confirms the type of progression or highlights other significant sequence conditions related to the given numbers.