Problem 150 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(p\) and \(q\) are positive real numbers such that \(p^{2}+q^{2}\) \(=1\), then the maximum value of \((p+q)\) is \(\quad\) [2007] (A) 2 (B) \(1 / 2\) (C) \(\frac{1}{\sqrt{2}}\) (D) \(\sqrt{2}\)

Step-by-Step Solution

Answer

The maximum value of \(p+q\) is \(\sqrt{2}\), option (D).

Step 1: Understanding the Problem

We need to determine the maximum value of the expression \((p+q)\) given the constraint \(p^2 + q^2 = 1\) and both \(p\) and \(q\) are positive real numbers.

Step 2: Rewrite the Expression in Terms of Angle

Recognize that the constraint \(p^2 + q^2 = 1\) suggests using a trigonometric identity. Assume \(p = \cos(\theta)\) and \(q = \sin(\theta)\) for some angle \(\theta\). This satisfies the equation \(p^2 + q^2 = \cos^2(\theta) + \sin^2(\theta) = 1\).

Step 3: Express \((p + q)\) Using Trigonometric Identities

Substitute \(p = \cos(\theta)\) and \(q = \sin(\theta)\) into the expression \(p + q\), yielding \(p + q = \cos(\theta) + \sin(\theta)\).

Step 4: Use Trigonometric Identity for Maximum Value

The expression \(\cos(\theta) + \sin(\theta)\) can be written as \(\sqrt{2} \sin(\theta + \pi/4)\) using a trigonometric identity. The maximum value of \(\sin(x)\) is 1, so the maximum value of \(\sqrt{2} \sin(\theta + \pi/4)\) is \(\sqrt{2}\).

Step 5: Conclusion

Therefore, the maximum value of \(p+q\) given that \(p^2 + q^2 = 1\) is \(\sqrt{2}\). This indicates that the correct answer is option (D).

Key Concepts

Trigonometric Identities

Trigonometric identities are tools that allow us to simplify expressions involving trigonometric functions. They are especially useful when dealing with equations that contain squared terms like the equation from our problem, where \(p^2 + q^2 = 1\). This specific identity hints that we are dealing with the Pythagorean identity:

• \( \cos^2(\theta) + \sin^2(\theta) = 1 \)

By making the substitutions \(p = \cos(\theta)\) and \(q = \sin(\theta)\), we directly apply this identity. This substitution transforms the constraint into a trigonometric identity, which helps simplify our task of finding the maximum value for \(p+q\).
Furthermore, when faced with expressions like \(\cos(\theta) + \sin(\theta)\), trigonometric identities allow us to rewrite it as \(\sqrt{2} \sin(\theta + \pi/4)\). This form is advantageous because we know the maximum value of \(\sin(x)\) is 1, making it easier to determine the maximum potential of \(p+q\).

Optimization

Optimization is about finding the best solution, often the maximum or minimum, under a given set of conditions. In this problem, we are tasked with maximizing the expression \(p+q\) provided that \(p\) and \(q\) are constrained by their Pythagorean relationship.
By recognizing that the maximum value of \(\sin(x)\) is 1, we can easily conclude that the rewritten expression \(\sqrt{2} \sin(\theta + \pi/4)\) reaches its peak at \(\sqrt{2}\). This conclusion follows from the fact that the trigonometric transformation neatly gives us a pathway from constraints through identity to optimization. It illustrates how a deeper understanding of trigonometric identities directly leads to efficient problem-solving through optimization strategies.
Whenever tasked with such problems, always seek simplification through substitution and identity application. This will help find maximum or minimum values without cumbersome calculations.

Constraints in Mathematics

Constraints in mathematics are predefined limits or conditions, which expressions or variables must satisfy. In this exercise, the constraint is given by \(p^2 + q^2 = 1\). Constraints like these are often the key to solving many mathematical problems, including optimization problems.

• They help limit the scope of possible solutions, making the problem more manageable.
• They inform the methods or techniques that might be viable in finding a solution, such as substituting trigonometric forms in our case.

Mathematical constraints guide our choice of methods. Here, the algebraic constraint transforms into a trigonometric identity, simplifying the path to the solution. Consequently, understanding and identifying constraints is pivotal. It allows you to select the appropriate tools, like trigonometric identities, to explore the solutions effectively.
Always evaluate the given constraints carefully. They hold the power to simplify a seemingly complex problem into a straightforward linear path toward a solution.