Problem 6 - A Complete Resource Book in Mathematics for JEE Main

Question

The least positive non-integral solution of the equation \(\sin \pi\left(x^{2}+x\right)=\sin \pi x^{2}\) is (A) rational (B) irrational of the form \(\sqrt{p}\) (C) irrational of the form \(\frac{\sqrt{p}-1}{4}\), where \(p\) is an odd integer (D) irrational of the form \(\frac{\sqrt{p}+1}{4}\), where \(p\) is an even integer

Step-by-Step Solution

Answer

The least non-integral solution is \(\frac{-1 + \sqrt{5}}{2}\), an irrational of the form."

Step 1: Analyze the Equation

We start by analyzing the given equation \(\sin \pi\left(x^{2}+x\right) = \sin \pi x^{2}\). For sine functions, the equation \(\sin A = \sin B\) holds when \(A = B + 2k\pi\) or \(A = \pi - B + 2k\pi\) for some integer \(k\).

Step 2: Set Up Equalities

Based on the sine properties, set up the equalities: \(\pi(x^2 + x) = \pi x^2 + 2k\pi\) and \(\pi(x^2 + x) = \pi - \pi x^2 + 2k\pi\). Simplifying gives us two cases: \(x = 2k\) and \(x^2 + x = 1 - 2k\).

Step 3: Simplify and Solve Case 2

For the second case, \(x^2 + x = 1 - 2k\), first consider the simplest integer \(k = 0\), making the equation \(x^2 + x = 1\). Solve the quadratic equation \(x^2 + x - 1 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 4: Calculate Roots

The quadratic equation is \(x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{5}}{2}\). We have two solutions: \(x = \frac{-1 + \sqrt{5}}{2}\) and \(x = \frac{-1 - \sqrt{5}}{2}\).

Step 5: Identify Positive Non-Integral Solution

From the solutions found, \(x = \frac{-1 + \sqrt{5}}{2}\) is the positive non-integral solution, provided \(\sqrt{5} > 1\). This is an irrational number of the form \(\frac{\sqrt{p}+1}{2}\) where \(p = 5\).

Step 6: Match the Solution to Options

Check the form of \(\frac{-1 + \sqrt{5}}{2}\) against the given options. None exactly match; however, observe it nearly matches option D's requirement: form \(\frac{\sqrt{p}+1}{4}\), which isn't directly the solution but informs about closest understanding.

Key Concepts

Quadratic Formula

In solving quadratic equations, one of the most powerful tools we use is the quadratic formula. This formula enables you to find roots of any quadratic equation in the form of \(ax^2 + bx + c = 0\). The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows us to determine the solutions of the equation without factoring.Here's a breakdown of what each part does:

• The "\(-b\)" shifts the value by the formula's midpoint based on the coefficient of \(x\).
• "\(b^2 - 4ac\)" under the square root is called the discriminant. It tells us the nature of the roots:
  • If positive, two distinct real roots exist.
  • If zero, one real root exists.
  • If negative, no real roots exist; instead, two complex roots arise.
• The "\(\pm\)" symbol indicates two potential roots, derived from adding and subtracting the square root part.

Understanding and applying the quadratic formula efficiently requires familiarity with these components and practice with manipulating quadratic expressions.

Sine Function Properties

The properties of the sine function are crucial for solving trigonometric equations. The sine function, \(\sin(\theta)\), has a periodicity of \(2\pi\), meaning it repeats every \(2\pi\) units. This feature is leveraged when equations like \(\sin A = \sin B\) arise:

• The identities \(A = B + 2k\pi\) or \(A = \pi - B + 2k\pi\) for some integer \(k\) exploit this periodicity to solve such equations.
• "\(A = B + 2k\pi\)" indicates one wave after another aligns at equivalent points.
• "\(A = \pi - B + 2k\pi\)" uses the symmetry property of sine, reflecting the graph at the midpoint, or \(\pi\).

Understanding these properties is essential, especially as sine is foundational in oscillatory and wave-like scenarios encountered in higher math and physics.

Irrational Numbers

Irrational numbers form an important class of real numbers that cannot be expressed as simple fractions. Take, for instance, the number \(\sqrt{5}\): while \(5\) is a perfect square, \(\sqrt{5}\) generates an endless non-repeating decimal, making it an irrational number.Irrational numbers are often discovered in solutions involving square roots and other radicals. They play crucial roles in diverse mathematical contexts, including:

• Geometry, where irrational lengths arise frequently (e.g., diagonals in squares).
• Algebra, especially in quadratic equations like \(x = \frac{-1 + \sqrt{5}}{2}\) where solutions surface as irrational numbers.

Despite their "intractable" nature in terms of exact representations, irrational numbers are vital for completeness in the set of real numbers, enabling a robust numerical framework.