Problem 103 - A Complete Resource Book in Mathematics for JEE Main

Question

Instructions: In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is TrueAssertion: Let \(f(x)\) be a polynomial function satisfying \(f(x) . f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right) .\) If \(f(4)=65\) and \(l_{1}, l_{2}\), \(l_{3}\) are in G.P., then \(f^{\prime}\left(l_{1}\right), f^{\prime}\left(l_{2}\right), f^{\prime}\left(l_{3}\right)\), are also in G.P. Reason: \(f(x)=\pm x^{n}+1\)

Step-by-Step Solution

Answer

B; The reason supports but doesn't explain the assertion sufficiently.

Step 1: Understand the Assertion

The assertion states that if a polynomial function \( f(x) \) satisfies the equation \( f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \) and given \( f(4) = 65 \), then the derivatives \( f'(l_1), f'(l_2), f'(l_3) \) are in geometric progression (G.P.) when \( l_1, l_2, l_3 \) are in G.P.

Step 2: Analyze the Reason

The reason provided is that the function satisfies \( f(x) = \pm x^n + 1 \), suggesting a specific form for the polynomial. This implies that the polynomial might be of a specific structure such that the derivatives would naturally fall in G.P.

Step 3: Evaluate Given Equation

Plug the given condition \( f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \) into the supposed form \( f(x) = \pm x^n + 1 \) to check if the assertion holds. Simplifying gives specific instances which satisfy the equation.

Step 4: Check Consistency with Assertion

Verify if the polynomial form leads to the derivatives \( f'(l_1), f'(l_2), f'(l_3) \) being in G.P. If the solution to the equation holds for \( \pm x^n + 1 \), derivatives maintain proportionality as per G.P., the assertion seems plausible.

Step 5: Validate the Reason as an Explanation

Analyze whether \( f(x) = \pm x^n + 1 \) inherently explains and supports the assertion's condition due to its algebraic properties. Given this aligns well but doesn't directly compel \( f'(l_1), f'(l_2), f'(l_3) \) in G.P., reason isn't fully explanatory.

Key Concepts

Understanding Assertion and Reason Questions

Assertion and reason questions are often asked in exams to test the depth of a student's understanding and ability to link facts logically. In these questions, the assertion is a statement that is often a fact or a given condition, while the reason explains why the assertion is true or provides additional information.

• Assertion: It’s important to understand what these questions intend. They check if you can not only recognize if the assertion and reason are true, but also if the reason justly explains the assertion.
• Options: Typically, these come with options like:
  • (A) Both are true and reason correctly explains the assertion.
  • (B) Both are true but reason does not explain the assertion.
  • (C) Assertion is true, reason is false.
  • (D) Assertion is false, reason is true.

For the student, it's essential to thoroughly analyze both statements and their relation. Understanding both the logic and the underlying principles of the assertion and reason aids in making the right distinction.

Geometric Progression (G.P.)

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

• Formula: If the first term is denoted by \( a \) and the common ratio by \( r \), the nth term of a G.P. can be expressed by:\[ a_n = a \, r^{n-1} \]
• Characteristics:
  • Each term is a product of the previous term and the constant factor \( r \).
  • The sequence follows a pattern that grows or decreases at a consistent rate.

This concept is crucial, as it helps identify patterns and proportionality in various mathematical contexts, such as polynomials, where derivative terms might maintain such progression under certain conditions.

Derivatives of Polynomials

Derivatives measure how a function changes as its input changes—specifically, they indicate the function's rate of change at a given point. When it comes to polynomial functions, derivatives can be found using standard rules.

• Basic Rule: The power rule states that the derivative of \( x^n \) is \( n \cdot x^{n-1} \).
• Example: Given a polynomial function \( f(x) = ax^n + bx^{n-1} + \, ... + c \), the derivative \( f'(x) \) is computed as:\[ f'(x) = n \, ax^{n-1} + (n-1) \, bx^{n-2} + \, ... \]

Understanding how to derive these functions helps in analyzing changes in the function values, especially in contexts like identifying whether derivatives of given points form a pattern like G.P. as mentioned initially.

Polynomial Equations

Polynomial equations involve expressions set equal to zero, featuring variables raised to positive integer powers. These equations are fundamental to algebra and frequently explored using various methods.

• Degree of Polynomial: The highest power of the variable indicates the degree, which in turn influences the nature and number of solutions.
• Solving Polynomials: Techniques include factorization, substitution, or using the quadratic formula where applicable.

In more complex scenarios, polynomials need to be evaluated at specific points or simplified to understand their properties, such as those required in the original assertion and reason exercise where equation form influences derivative behavior.