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Explain solution RD Sharma class 12 chapter 13 Differentials Errors and Approximations exercise multiple choice question 10 maths

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Answer: n:1

Hint:  Here we use the concept of relative error

            \frac{d y}{y}=\left(\frac{d y}{d x}\right) \Delta x

Given: y=x^{n}



Let’s differentiate

        \frac{d y}{d x}=n x^{n-1}

Approximate error in y is

        \begin{aligned} d y &=\left(\frac{d y}{d x}\right) \Delta x \\\\ &=n x^{n-1} \times \Delta x \end{aligned}

Relative error in y is  \frac{d y}{y}=\frac{n}{x} \Delta x


Approximate error  x is dx

            \begin{aligned} &=\left(\frac{d x}{d y}\right) \Delta y \\\\ &=\frac{1}{n x^{n-1}} \Delta y \end{aligned}

Relative error in  x  is \frac{d x}{x}=\frac{1}{n x^{n}} \Delta y

Required solution,

            \frac{\frac{n}{x} \times \Delta x}{\frac{1}{n x^{n}} \times \Delta y}=n^{2} x^{n-1} \frac{\Delta x}{\Delta y}

            \begin{aligned} &=\frac{x}{y} \times \frac{n \times x^{n-1} \times \Delta x}{\Delta x} \\\\ &=\frac{n \times x^{n}}{x^{n}} \\\\ &=\frac{n}{1} \end{aligned}

So, ratio is n:1



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