Problem 118 - 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(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 True Assertion: The function \(y=f(x)\), defined parametrically as \(y=t^{2}+t|t|, x=2 t-|t|, t \in R\), is continuous \mathrm{\\{} f o r ~ a l l ~ r e a l ~ \(x\) Reason: \(f(x)=\left\\{\begin{array}{r}2 x^{2}, x \geq 0 \\ 0, x<0\end{array}\right.\)

Step-by-Step Solution

Answer

(B) Assertion is true, Reason is true; but Reason does not explain Assertion.

Step 1: Understand the Assertion

The assertion states that the function \( y = f(x) \), defined parametrically with \( y = t^2 + t|t| \) and \( x = 2t - |t| \) for \( t \in \mathbb{R} \), is continuous for all real \( x \). A function is continuous if there are no breaks, jumps, or holes in its graph.

Step 2: Understand the Reason

The reason given states the expression of \( f(x) \) in a piecewise form: \( f(x) = \begin{cases} 2x^2, & x \geq 0 \ 0, & x < 0 \end{cases} \). This is meant to justify the continuity of the function by providing an explicit continuous function form.

Step 3: Analyze Parametric Equations

For continuity at \( x \geq 0 \) and \( x < 0 \), plug in the values of \( t \) to get \( y \) in terms of \( x \). From \( x = 2t - |t| \), deduce the value of \( t \). For positive \( t \), \( x = t \); for negative \( t \), \( x = t \). This means \( x = t \) in general, making \( y = t^2 + t|t| = x^2 + x^2 \) for \( t \geq 0 \) and \( y = x^2 \) for \( t < 0 \).

Step 4: Validate Continuity

For \( t \geq 0 \), \( y(x) = 2x^2 \). For \( t < 0 \), turning \( y \) into \( 0 \), seems incorrect since \( y = x^2 \) implies continuity on the entire number line. The correct piecewise form should reflect continuity without a sudden jump from \( 2x^2 \) to \( 0 \).

Step 5: Evaluate Assertion and Reason

The assertion claims continuity, which is true if \( y = x^2 \), correctly interpreted by \( t = x \). The provided reason implies a jump discontinuity which contradicts the assertion's implication. Hence, the reason does not directly justify the assertion.

Key Concepts

Parametric Equations

Parametric equations are a way to express a set of related quantities using an independent parameter. This approach is particularly useful when expressing curves and complex shapes. In a parametric equation setup, each variable of interest, often coordinates like \(x\) and \(y\), is expressed in terms of a third variable, \(t\). For instance:

• The \(x\)-coordinate might be written as \(x = 2t - |t|\).
• The \(y\)-coordinate might be \(y = t^2 + t|t|\).

This specific format helps in visualizing and analyzing curves in a more dynamic way, as you can see how changes in \(t\) directly impact \(x\) and \(y\). This setup makes it easy to articulate functions that are not straight lines or have restrictions on their definitions.
An important aspect to consider with parametric equations is their ability to neatly demonstrate the continuity and smoothness of curves. By investigating how increments in \(t\) affect \(x\) and \(y\), we can observe how a curve evolves in space, and importantly, how it behaves with no breaks or jumps at specific \(t\) values. This leads directly to discussions about continuity and function behavior at the boundaries defined by the parameter \(t\).

Piecewise Functions

Piecewise functions can be best described as functions that have different expressions based on the input value. They might look complicated at first, but they offer a practical way to handle functions that have multiple rules or conditions. In other words, they "piece" together multiple sub-functions to cover the entire domain.

• Consider the function \( f(x) = \begin{cases} 2x^2, & x \geq 0 \ 0, & x < 0 \end{cases} \), which defines two different behaviors:
• For non-negative \(x\), the expression is \(2x^2\).
• For negative \(x\), it is defined as \(0\).

Using piecewise functions lets us handle complex, real-world scenarios where behavior changes depending on conditions. However, to ensure smooth transitions and continuity, we must take care to properly align the endpoints where these separate rules meet. Careless blending might lead to discontinuities—points where the function suddenly jumps or changes direction—leading to difficulties in predicting function behavior.
To truly understand how well a piecewise function works, especially in terms of continuity, it is necessary to examine transitions closely. This means checking what happens as \(x\) approaches the specified boundary conditions, assuring that the function doesn't have unexpected behavior.

Continuity Analysis

Continuity of a function is an essential concept, particularly when examining a graph or a mathematical model for consistency. In simpler terms, a function is continuous if you can sketch its graph without removing your pencil from the paper. This makes analyzing its overall behavior straightforward, as uninterrupted behavior implies predictability.Now, let's delve into the continuity of parametric functions like \(y = t^2 + t|t|\) and \(x = 2t - |t|\). By converting the parametric form into a function \(y = f(x)\), we analyze for continuity by assessing if any gaps or abrupt changes exist. In the given situation:

• At \(t \geq 0\), both \(x = t\) and \(y = 2t^2\), making \(y = 2x^2\).
• At \(t < 0\), initialization shows \(x = t\), resulting in \(y = x^2\).

To confirm continuity, the behavior must seamlessly align at zero, meaning \(t = 0\). If both parametric equations smoothly transition there, the overall function is considered continuous. However, if a mismatch occurs, annotations like "piecewise" may obscure this, as seen in the erroneous reasoning suggesting a jump in value from one piece to another.
Conclusion: analyze carefully by testing these values in practice, ensuring compatible alignment at transitions and no breaks in expected output. This confirms the function behaves as presumed, without unexpected jumps or holes.