Problem 132 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: If \(a, b, c\) are distinct positive real numbers and \(a^{2}+b^{2}+c^{2}=1\), then \(a b+b c+c a\) is less than 1 . Reason: A.M. > G.M. for unequal numbers

Step-by-Step Solution

Answer

Yes, \(ab + bc + ca < 1\) for distinct positive unequal numbers.

Step 1: Understanding the Assertion

We are given that \( a, b, c \) are distinct positive real numbers such that \( a^2 + b^2 + c^2 = 1 \). The statement claims that the product \( ab + bc + ca \) must be less than 1.

Step 2: Understanding the Reason

The Reason provided is that the Arithmetic Mean (A.M.) is greater than the Geometric Mean (G.M.) for unequal numbers. This inequality can be used here due to the distinct nature of numbers \( a, b, c \).

Step 3: Applying the Cauchy-Schwarz Inequality

We apply the Cauchy-Schwarz inequality in the form:\[(a^2 + b^2 + c^2)(1 + 1 + 1) \geq (a + b + c)^2\]Given \( a^2 + b^2 + c^2 = 1 \), it follows:\[3 \cdot 1 \geq (a + b + c)^2 \]Thus,\[(a + b + c)^2 \leq 3\]

Step 4: Expanding and Rearranging

Expand the expression \((a+b+c)^2 \leq 3\), which results in:\[a^2 + b^2 + c^2 + 2(ab + bc + ca) \leq 3\]Substituting \( a^2 + b^2 + c^2 = 1 \), we get:\[1 + 2(ab+bc+ca) \leq 3\]

Step 5: Solving for \( ab + bc + ca \)

Rearrange the inequality obtained previously:\[2(ab + bc + ca) \leq 2\]Divide both sides by 2:\[ab + bc + ca \leq 1\]

Step 6: Confirming the Inequality

The steps show that \( ab + bc + ca \leq 1 \). However, to satisfy the assertion, \( ab + bc + ca < 1 \), it holds for distinct positive unequal numbers as strict inequality from A.M > G.M.

Key Concepts

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a fundamental concept in mathematics. It states that for any real or complex numbers, the square of the sum is less than or equal to the sum of the squares. In formal terms,

• For two sequences of real numbers \( (a_1, a_2, \ldots, a_n) \) and \( (b_1, b_2, \ldots, b_n) \), the inequality is:\[(a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2) \geq (a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2\]

In our exercise, we used this inequality to compare sums and products of distinct numbers. The Cauchy-Schwarz Inequality helps us understand how variables interact when summed and is closely related to the magnitude of vectors in geometry.
This inequality serves as a powerful tool when dealing with symmetric expressions, like the one presented in the problem.

Arithmetic Mean-Geometric Mean Inequality

The Arithmetic Mean-Geometric Mean (A.M.-G.M.) Inequality is another cornerstone in inequality problems, often abbreviated to "A.M. \> G.M." This inequality states that, for a given set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean.

• For example, given numbers \( a, b, c \), the A.M.-G.M. inequality is written as:\[\frac{a+b+c}{3} \geq \sqrt[3]{abc}\]

A critical feature of this inequality is its use in proving relations where numbers are distinct (unequal in this context). Since in our original problem \( a, b, \text{and } c \) are distinct, the inequality holds as a strict inequality, which plays a vital role in concluding \( ab + bc + ca < 1\). This logical framework is indispensable in various mathematical proofs, especially those concerning optimization and analysis.

Symmetric Inequalities

Symmetric inequalities are expressions where the inequality remains unchanged under any permutation of its variables. This means that if you swap any variables, the inequality still holds true.
Symmetric inequalities are common in problems involving conditions on sums and products and often require a strategic blend of mathematical concepts to solve.

• In our exercise, the relationship \( ab + bc + ca \) embodies symmetry with respect to variables \( a, b, \text{and } c \).

When dealing with symmetric inequalities, algebraic manipulation, such as expanding and rearranging expressions, becomes useful. Recognizing symmetry can simplify the proof and provide insights into how different components relate and affirm the inequality's consistency.

JEE Mathematics Problems

The Joint Entrance Examination (JEE) comprises problems that test the depth of high school mathematics understanding, including inequalities.
This exam requires students to be adept at spotting and utilizing mathematical principles like the Cauchy-Schwarz and A.M.-G.M. inequalities.

• In preparing for JEE, mastering these concepts helps students tackle complex problems efficiently and accelerate their problem-solving skills.

Problems posed in the JEE often present multi-step challenges, where students must recognize which tools to apply and understand deeply how these different mathematical elements interact.
Consistently practicing inequalities and understanding their geometric and algebraic interpretations furnish students with the versatility needed to excel in exams and beyond.