Problem 1 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f(x)=x^{3}+x^{2}+100 x+7 \sin x\), then the equation \(\frac{1}{y-f(1)}+\frac{2}{y-f(2)}+\frac{3}{y-f(3)}=0\) has (A) one real root (B) two real roots (C) more than two real roots (D) no real root

Step-by-Step Solution

Answer

(B) two real roots

Step 1: Evaluate function values

First, evaluate the function \(f(x) = x^3 + x^2 + 100x + 7\sin x\) at specific points: \(f(1)\), \(f(2)\), and \(f(3)\). Calculate each separately. \[f(1) = 1^3 + 1^2 + 100 \times 1 + 7\sin(1) = 102 + 7\sin(1)\]\[f(2) = 2^3 + 2^2 + 100 \times 2 + 7\sin(2) = 108 + 7\sin(2)\]\[f(3) = 3^3 + 3^2 + 100 \times 3 + 7\sin(3) = 136 + 7\sin(3)\]

Step 2: Simplify the given equation

Rewrite the given equation using the evaluated function values: \[\frac{1}{y - (102 + 7\sin(1))} + \frac{2}{y - (108 + 7\sin(2))} + \frac{3}{y - (136 + 7\sin(3))} = 0\]

Step 3: Consider the asymptotes and signs

Analyze the behavior as \(y\) approaches the critical values \(102 + 7\sin(1)\), \(108 + 7\sin(2)\), and \(136 + 7\sin(3)\). Notice that the expression will approach infinity or a change in sign will occur when \(y\) equals these function values. Observe how the signs of each term change as \(y\) varies between and outside these points.

Step 4: Determine the existence of real roots

By analyzing the signs and behavior around critical points from asymptotic behavior, infer that there are sign changes between intervals. Consequently, there exist values of \(y\) for which the equation holds. This indicates the real roots exist. As there are three critical values resulting in \(\infty\) behavior, the changes confirm the presence of exactly two intervals where the equation equals zero.

Key Concepts

Real Roots Determination

In mathematics, determining the real roots of an equation is fundamental, especially for polynomial equations. For an equation to have a real root, it means there is a real number solution where the equation equals zero.
Understanding the behavior of an equation, particularly when it involves fractions, requires careful observation of sign changes and where the function crosses the x-axis or y-axis.
In the given problem, we handle a fractional expression of the form \(\frac{1}{a} + \frac{2}{b} + \frac{3}{c} = 0\).
For real roots, we must look for conditions under which the sum of such terms becomes zero.

• Each denominator represents a linear function of \(y\) that can potentially change signs as \(y\) changes.
• Critical points occur when each denominator approaches zero, which causes the fractions to shoot towards infinity.

As described in the steps, it's essential to determine how many times this sum transitions from positive to negative between critical points to find real roots. By examining these intervals, the solution suggests that two real roots exist due to the distinct intervals where signs change.

Asymptotic Behavior Analysis

Asymptotic behavior plays a crucial role in understanding complex functions, especially where denominators approach zero, leading towards negative or positive infinity.
In the context of the given exercise, the asymptotes occur at each critical function value computed in the earlier step: \(f(1), f(2),\) and \(f(3)\).
This results from approaching these values, causing fractions to become very large, signaling the potential for real root discovery.

• Asymptotic behavior is analyzed by observing what happens as \(y\) approaches these values, where each denominator comes close to zero.
• At these points, the contribution of each term is significant, mainly pointing allies or alongside infinity.

Therefore, in the original equation \(\frac{1}{y-f(1)}+\frac{2}{y-f(2)}+\frac{3}{y-f(3)}=0\), analyzing asymptotes helps predict behavior at these transitions and understand why or how the roots are spaced.
Hence, the performance between these asymptotic points reveals valuable information on the existence and location of real roots.

Polynomial Evaluation

Polynomial evaluation is crucial for understanding polynomial equations, like the one in this exercise.
Here, the polynomial \(f(x) = x^3 + x^2 + 100x + 7\sin x\) is evaluated at specific integer points \(x = 1, 2,\) and \(3\).
Evaluation is performed by substituting each value into the polynomial.

• For \(f(1)\), \(f(2)\), and \(f(3)\), we involve straightforward substitution and arithmetic operations.
• This method simplifies expression handling in the equation that involves the function \(f(x)\).

The process yields evaluated results that contribute directly to analyzing the entire original equation, particularly how the expressions behave asymptotically. Polynomial evaluation assists in simplifying otherwise complex expressions by reducing them into specific numeric relationships in the context of the roots determination problem.
This approach allows us to further infer polynomial characteristics and thoroughly understand equation behavior in critical intervals.