Problem 177 - A Complete Resource Book in Mathematics for JEE Main

Question

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 \((\mathrm{R})\) is a correct explanation for Assertion (A) (B) Assertion (A) is True, Reason \((\mathrm{R})\) 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 True Assertion: If \(f, g\) and \(h\) be continuous functions on \([0, a]\) such that \(f(x)=f(a-x), g(x)=-g(a-x)\) and \(3 h(x)-4 h(a-x)=5\), then \(\int_{0}^{a} f(x) g(x) h(x) d x=0\) Reason: \(\int_{0}^{a} f(x) g(x) d x=0\)

Step-by-Step Solution

Answer

(B) Assertion (A) is True, Reason (R) is True; Reason (R) is not a correct explanation for Assertion (A)

Step 1: Verify Continuity and Symmetry Conditions

The functions \(f, g,\) and \(h\) are continuous on \([0, a]\). \(f(x) = f(a-x)\) implies \(f\) is an even function over the interval \([0, a]\). \(g(x) = -g(a-x)\) implies \(g\) is an odd function over the interval \([0, a]\). Additionally, \(3h(x) - 4h(a-x) = 5\) is given. We must explore this condition to determine properties of \(h\), although details here will continue in the next step.

Step 2: Analyze \(h\) Function Property

Rewrite the equation \(3h(x) - 4h(a-x) = 5\). To satisfy this equation for all \(x\), solve for each function's symmetry property. If \(x = \frac{a}{2}\), both parts of \(h\) are the same, and this gives insight that potentially includes some symmetry. However, it doesn't by itself make \(h\) odd or even fully - the form of the integral isn't inherently zero like functions \(f\) and \(g\).

Step 3: Examine the Integral of Product

Given the integrand \(f(x)g(x)h(x)\), note that \(f(x)g(x)\) would generally return zero when integrated over \([0, a]\), as they would from an odd and even product property. However, investigating \(h(x)\) might disrupt this zero integral relationship. Substitute: \(f(x)g(x)\) is still even in form and could lead to overall zero if paired each with symmetric \(h(x)\).

Step 4: Revisit the Reason

The reason provides \(\int_0^a f(x)g(x) \, dx = 0\). Given \(f(x)g(x)\) being composed of even and odd function products, this fact holds independently of \(h\). However, the statement's validity as it relates to the overall objective – \(\int_0^a f(x)g(x)h(x) \, dx = 0\) doesn't directly explain the outcome if \(h(x)\) complicates symmetry or nullification.

Step 5: Conclusion and Correct Answer

Combine insights gleaned: the integral of \(f(x)g(x)\) indeed equals zero, confirming its stated reasoning is often true, it does not by itself fully address the given full condition with \(h(x)\). The presence of additional symmetry disrupting \(h(x)\) (and its stated form) remains unexplained justification for nullity. Hence the explanation can't fully depend on the independent relationship.

Key Concepts

Continuous Functions

Continuous functions play a vital role in calculus, especially in integration. A function is said to be continuous on a closed interval \[ [a, b] \] if, loosely speaking, you can draw it without lifting your pen off the paper. For the functions provided in this exercise like \( f(x) \), \( g(x) \), and \( h(x) \), being continuous ensures that we can integrate them easily over the interval \[ [0, a] \]. This property is crucial, as discontinuities could complicate the calculation of an integral.

• Continuous functions are generally easier to handle because they behave predictably, without jumps or breaks.
• This predictability allows for straightforward application of integral rules and theorems, like the Fundamental Theorem of Calculus.

In our problem, the assertion and the reason both assume continuity of the functions involved, setting the stage for valid integration challenges to be tackled analytically.

Even and Odd Functions

Recognizing even and odd functions is important in understanding their integration. An even function satisfies the property \( f(x) = f(-x) \). Thus, these functions are symmetrical with respect to the y-axis. Conversely, an odd function satisfies \( g(x) = -g(-x) \), meaning they are symmetrical around the origin.

• For even functions like \( f(x) = f(a-x) \), this symmetry results in certain simplifications, especially over symmetric intervals like \[ [-a, a] \]. However, note that this exercise deals with the interval \[ [0, a] \], which aligns with considering only half the usual symmetry.
• Odd functions like \( g(x) = -g(a-x) \) have a unique property where their integral over a symmetric interval from \[ [-a, a] \] results in zero due to their sinusoidal symmetry - they cancel out. In the given interval \[ [0, a] \], extra analysis is required to conclude the integration effects.

Understanding these properties helps in predicting outcomes without extensive calculations, especially when dealing with products of even and odd functions, as explored in this problem.

Symmetry in Integration

Symmetry is a powerful tool in integration, allowing for simplifications and quick evaluations. In the exercise, symmetry is key in understanding how the integral of \( f(x)g(x)h(x) \) is tackled. It involves investigating the product of an even and an odd function, alongside the function \( h(x) \), under the integral sign across \[ [0, a] \].

• Even functions under integration on symmetric intervals lead to potentially simplified results since they ensure uniformity on both halves of the intervals.
• Odd functions contribute to zero values when integrated over full symmetric intervals, thanks to their cancelling nature across the center.

Thus, symmetry helps predict the behavior of the complete integral \( \int_{0}^{a} f(x)g(x)h(x) \, dx \). For \( f(x)g(x) \), which combines even and odd properties, the symmetry naturally contributes zero, but any disruptions in \( h(x) \)'s properties can alter this outcome. The problem's critical insight is realizing how \( h(x) \) might disrupt this integral symmetry, even when the individual parts might suggest null results.