If Y = Xn−1 Log X Then X2 Y2 + (3 − 2n) Xy1 Is Equal To (A) −(N

English0CBSECommerce (English Medium) Class 12Question PapersQuestion Papers2512Textbook Solutions20187MCQ Online Mock Tests42Important Solutions19860Concept Notes & Videos260Time Tables24SyllabusIf Y = Xn−1 Log X Then X2 Y2 + (3 − 2n) Xy1 is Equal to (A) −(N − 1)2 Y (B) (N − 1)2y - Mathematics

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Question

If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to

Options

• −(n − 1)2 y

• (n − 1)2y

• −n2y

• n2y

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SolutionShow Solution

(a) −(n − 1)2 y

Here,

\[y = x^{n - 1} \log x\]

\[ \Rightarrow y_1 = \left( n - 1 \right) x^{n - 2} \log x + \frac{x^{n - 1}}{x}\]

\[ \Rightarrow y_1 = \frac{\left( n - 1 \right) x^{n - 1} \log x + x^{n - 1}}{x}\]

\[ \Rightarrow x y_1 = \left( n - 1 \right)y + x^{n - 1} \]

\[ \Rightarrow x y_2 + y_1 = \left( n - 1 \right) y_1 + \left( n - 1 \right) x^{n - 2} \]

\[ \Rightarrow x y_2 + y_1 = \left( n - 1 \right) y_1 + \frac{\left( n - 1 \right) x^{n - 1}}{x}\]

\[ \Rightarrow x^2 y_2 + x y_1 = x\left( n - 1 \right) y_1 + \left( n - 1 \right) x^{n - 1} \]

\[ \Rightarrow x^2 y_2 + x y_1 = x\left( n - 1 \right) y_1 + \left( n - 1 \right)\left\{ x y_1 - \left( n - 1 \right)y \right\}\]

\[ \Rightarrow x^2 y_2 + x y_1 = x\left( n - 1 \right) y_1 + \left( n - 1 \right)x y_1 - \left( n - 1 \right)^2 y\]

\[ \Rightarrow x^2 y_2 + x y_1 = 2x\left( n - 1 \right) y_1 - \left( n - 1 \right)^2 y\]

\[ \Rightarrow x^2 y_2 + x y_1 - 2x\left( n - 1 \right) y_1 = - \left( n - 1 \right)^2 y\]

\[ \Rightarrow x^2 y_2 + x y_1 \left( 1 - 2n + 2 \right) = - \left( n - 1 \right)^2 y\]

\[ \Rightarrow x^2 y_2 + \left( 3 - 2n \right)x y_1 = - \left( n - 1 \right)^2 y\]

shaalaa.comSimple Problems on Applications of Derivatives Report Error Is there an error in this question or solution?Q 24Q 23Q 25Chapter 12: Higher Order Derivatives - Exercise 12.3 [Page 24]

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RD Sharma Mathematics [English] Class 12Chapter 12 Higher Order DerivativesExercise 12.3 | Q 24 | Page 24

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