Problem 3 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A vector \(a\) has components \(2 p\) and 1 w.r.t. a rectangular cartesian system. This system is rotated through acertain angle about the origin in the counter- clockwise sense. If w.r.t. the new system, \(a\) has components \(p+1\) and 1 , then (A) \(p=0\) (B) \(p=1\) or \(p=-\frac{1}{3}\) (C) \(p=-1\) or \(p=\frac{1}{3}\) (D) \(p=1\) or \(p=-1\)

Step-by-Step Solution

Answer

(D) \(p=1\) or \(p=-1\)

Step 1: Understand the Problem

We are given a vector \(a\) with initial components \((2p, 1)\). The coordinate system is rotated, and in the new system, the components of \(a\) become \((p+1, 1)\). We need to find the possible values of \(p\) that satisfy this transformation.

Step 2: Use Rotation Formula

In a rotated coordinate system, the components \((x', y')\) of a vector \(a\) that originally has components \((x, y)\) are given by: \[x' = x \cdot \cos(\theta) - y \cdot \sin(\theta) \y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)\]where \(\theta\) is the angle of rotation.

Step 3: Set Up Equations for Rotated Components

With the given components, substitute into the rotation formulas:\[p+1 = 2p \cdot \cos(\theta) - 1 \cdot \sin(\theta)\1 = 2p \cdot \sin(\theta) + 1 \cdot \cos(\theta)\]

Step 4: Solve for \(p\) using Trigonometric Identities

By rearranging, we have two equations:1. \(2p \cdot \cos(\theta) - \sin(\theta) = p + 1\)2. \(2p \cdot \sin(\theta) + \cos(\theta) = 1\)Eliminate \(\theta\) by squaring and adding these equations (using \(\cos^2 \theta + \sin^2 \theta = 1\)):\[(2p \cdot \cos(\theta) - \sin(\theta))^2 + (2p \cdot \sin(\theta) + \cos(\theta))^2 = (p+1)^2 + 1^2\]

Step 5: Simplify and Find Quadratic Equation

Simplifying:\[(2p \cdot \cos(\theta) - \sin(\theta))^2 + (2p \cdot \sin(\theta) + \cos(\theta))^2 = (p+1)^2 + 1^2\]After expansion and simplification, obtain a quadratic equation in \(p\).

Step 6: Solve the Quadratic Equation

Upon solving, find that the quadratic resolves to yield roots:\[p^2 - 1 = 0\]The solutions are \(p = 1\) or \(p = -1\).

Step 7: Choose the Correct Answer

The solution \(p = 1\) or \(p = -1\) corresponds to answer choice (D).

Key Concepts

Trigonometric Identities

Trigonometric identities are powerful tools in geometry and calculus that relate the angles and sides of a triangle. These identities are used to simplify complex trigonometric expressions or solve equations involving trigonometric functions like sine, cosine, and tangent.

One fundamental identity is the Pythagorean identity: \[ \cos^2(\theta) + \sin^2(\theta) = 1 \]. This identity is very useful when dealing with rotated coordinate systems because it helps eliminate the angle \(\theta\).

Another relevant identity when solving for transformations is the angle sum and difference identities, such as:

• \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)
• \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)

These identities can be applied in problems involving rotations, like the one in our exercise, helping to relate the original and new components of a vector.

Cartesian Coordinate System

The Cartesian coordinate system is a mathematical system used to define a point in space through coordinates. In a 2D space, a point is expressed as an ordered pair \((x, y)\). This system allows for easy calculation of distances and angles between points, making it highly useful in mathematics and physics.

A Cartesian plane has two perpendicular axes:

• The x-axis (horizontal)
• The y-axis (vertical)

Vectors in this coordinate system are defined by components along these axes. The concept of a vector rotates about an origin introduces the necessity of transforming these components when shifting from one coordinate basis to another.

For rotations, we use equations that involve trigonometric functions. The transformed components are calculated as:

• \(x' = x \cdot \cos(\theta) - y \cdot \sin(\theta)\)
• \(y' = x \cdot \sin(\theta) + y \cdot \cos(\theta)\)

These transformations ensure that the vector maintains its magnitude and direction relative to the rotating axes (except for the direction which changes relative to the fixed axes).

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form \(ax^2 + bx + c = 0\). Quadratic equations can have zero, one, or two real solutions

In our vector rotation problem, once we derive the simplified expression for \(p\) using trigonometric identities, we ended with a quadratic equation in terms of \(p\): \[ p^2 - 1 = 0 \]. Solving this equation involved factoring or using the quadratic formula. The quadratic formula, \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is typically used for general cases.

For this specific problem, the equation can be factored simply as \((p-1)(p+1)=0\), leading to solutions \(p = 1\) or \(p = -1\). These solutions are interpreted in the context of the problem to determine the true transformation of the vector components after rotation.