Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(p, q, r \in R^{+}\)and \(27 p q r \geq(p+q+r)^{3}\) and \(3 p+4 q\) \(+5 r=12\) then \(p^{3}+q^{4}+r^{5}\) is equal to (A) 3 (B) 6 (C) 2 (D) None of these

Step-by-Step Solution

Answer

The value is (A) 3.

Step 1: Understand the Given Inequality

The problem states that \(27pqr \geq (p+q+r)^3\) with \(p, q, r \in \mathbb{R}^{+}\), which indicates a relationship of weighted AM-GM inequality. It suggests symmetry might play a role, making us consider cases where \(p = q = r\).

Step 2: Equal Variables Assumption

Assume \(p = q = r = k\) to simplify calculations. Substitute into the constraint equation \(3p + 4q + 5r = 12\) leading to \(12k = 12\) or \(k = 1\).

Step 3: Verify Against Inequality

With \(p = q = r = 1\), we check the inequality: \(27 \times 1 \times 1 \times 1 = 27\), and \( (1 + 1 + 1)^3 = 27\). The inequality holds as both are equal.

Step 4: Calculate Expression Value

Substitute \(p = q = r = 1\) into the expression \(p^3 + q^4 + r^5\). This evaluates to \(1^3 + 1^4 + 1^5 = 1 + 1 + 1 = 3\).

Step 5: Verify Final Solution

Given the symmetry in the setup and validation of assumptions, no alternate solutions lead to a different outcome. Hence \(p^3 + q^4 + r^5 = 3\) must be correct.

Key Concepts

AM-GM Inequality

The AM-GM Inequality is a powerful tool in mathematics that compares arithmetic and geometric means. It asserts that for any list of non-negative numbers, the arithmetic mean (average) is always greater than or equal to the geometric mean. For numbers \(a, b, c\), this can be expressed as:

• Arithmetic Mean: \( \frac{a + b + c}{3} \)
• Geometric Mean: \( (abc)^{1/3} \)

The inequality states: \[ \frac{a + b + c}{3} \geq (abc)^{1/3} \]Equality holds when all the numbers are equal. In our problem, the inequality \(27pqr \geq (p+q+r)^{3}\) hints at this equality being achieved when \(p = q = r\). Therefore, by assuming equal variables, the complexity of the problem is reduced, allowing us to explore potential solutions efficiently.

Symmetry in Equations

Symmetry plays a crucial role in simplifying complex equations. In mathematics, it refers to when parts of an equation can be swapped without affecting the overall solution structure. With equal elements, computations become simpler, making it easier to verify calculations. This exercise shows symmetry in variables by letting \(p = q = r = k\). The equation in question is simplified to \(3p + 4q + 5r = 12\), reducing down to \(12k = 12\), yielding \(k = 1\). The symmetry assumption helps explore balanced cases quickly and validates if solutions like \(p = q = r\) satisfy both the equation and given inequality conditions.

Power Sums

Power sums in mathematics refer to expressions involving sums of powers of variables, like \(p^3 + q^4 + r^5\). These sums provide insights into relationships between the variables and their individual contributions to a total sum. They are useful for identifying distinctive patterns or verifying constraints in equations.In our exercise, after establishing that \(p = q = r = 1\), we calculate the power sum: \[ p^3 + q^4 + r^5 = 1^3 + 1^4 + 1^5 = 1 + 1 + 1 = 3 \]This calculation confirms that the given constraints lead directly to this specific total, supporting the solution that \(p^3 + q^4 + r^5 = 3\). This indicates a harmonious relationship of power among the variables, meeting the original problem's conditions perfectly.

Real Positive Numbers

Real positive numbers are numbers that are greater than zero and do not include any imaginary or negative values. They are significant in problems involving inequalities and equations, as they guarantee certain properties like the positivity needed for AM-GM inequalities.In our given exercise, working with \(p, q, r \in \mathbb{R}^{+}\) ensures that all the arithmetic operations conform to the principles of inequality theorems such as AM-GM etc. This constraint makes it feasible to assume equality among variables without introducing complexities that would arise with negative or zero values, thus maintaining the integrity of mathematical operations throughout the exercise.