\(\underline{\text { Assertion - Reason Type Questions: }}\) Each question has four choices (a), (b), (c) and (d) out of which only one is correct. Write (a), (b), (c) and (d) according to the following rules. (a) Statement- 1 is True, Statement-2 is True, Statement- 2 is a correct explanation for Statement-1. (b) Statement-1 is True, Statement- 2 is True, Statement-2 is not a correct explanation for Statement-1. (c) Statement- 1 is True, Statement- 2 is False. (d) Statement- 1 is False, Statement- 2 is True. Statement \(-2:\) The differential equation \(\mathrm{y}^{\prime}=(\mathrm{y} / 2 \mathrm{x})\) is variable separable. Statement-1: Curve satisfying the differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) passing through \((2,1)\) is a parabola with Focus \([(1 / 4), \underline{0}]\). Statement- 2 : The differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) is variable separable.

Step by step solution

01

Verify Statement-2

Determine if the given differential equation is indeed variable separable. The given differential equation is: \[\frac{dy}{dx} = \frac{y}{2x}\] To check if it is variable separable, we can try to rewrite the equation as a product of functions of x and y: \[\frac{dy}{y} = \frac{dx}{2x}\] Since we can do this, we can confirm that statement-2 is true and the given differential equation is variable separable.

02

Solve the differential equation

Now that we know the equation is variable separable, we can proceed with solving the differential equation by integrating both sides: \[\int{\frac{dy}{y}} = \int{\frac{dx}{2x}}\] This gives: \[\ln{y} = \frac{1}{2}\ln{x} + C\] To find the constant C, we use the given point (2,1) through which the curve passes: \[\ln{1} = \frac{1}{2}\ln{2} + C\] Since \(\ln{1}=0\), we have \(C = -\frac{1}{2}\ln{2}\). Now, we can rewrite the equation as: \[y = x^{\frac{1}{2}}e^{-\frac{1}{2}\ln{2}}\] By applying the properties of logarithms and exponentials, we can simplify this as: \[y = \sqrt{\frac{x}{2}}\]

03

Verify Statement-1

Now that we have the curve equation, let's check if it is a parabola with the given focus: \[y = \sqrt{\frac{x}{2}}\] To represent a parabola with focus at \(\left(\frac{1}{4}, 0\right)\), the equation must satisfy the definition of a parabola: the distance from a point on the curve to the focus is equal to its distance to the directrix. In the case of a parabola with the given focus, the directrix will be a vertical line located to the left of the focus. The curve we found does not follow this parabola definition, as it is not symmetric and does not have the same distances to the focus and directrix. Therefore, statement-1 is false. To summarize: - Statement-1 is False - Statement-2 is True Thus, the correct answer for this exercise is: (d) Statement-1 is False, Statement-2 is True.

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