मराठी

If Y = a Cos (Loge X) + B Sin (Loge X), Then X2 Y2 + Xy1 = (A) 0 (B) Y (C) −Y (D) None of These - Mathematics

Advertisements
Advertisements

प्रश्न

If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =

पर्याय

  • 0

  • y

  • y

  • none of these

MCQ
Advertisements

उत्तर

(c) −y

Here,

\[y = a \cos\left( \log_e x \right) + b \sin\left( \log_e x \right)\]

\[ \Rightarrow y_1 = - a\sin\left( \log_e x \right)\frac{1}{x} + b \cos\left( \log_e x \right)\frac{1}{x}\]

\[ \Rightarrow y_2 = \frac{- a\sin\left( \log_e x \right) + b \cos\left( \log_e x \right)}{x}\]

\[ \Rightarrow y_2 = \frac{- a\cos\left( \log_e x \right) - b \sin\left( \log_e x \right) - \left\{ - a\sin\left( \log_e x \right) + b \cos\left( \log_e x \right) \right\}}{x^2}\]

\[ \Rightarrow x^2 y_2 = - \left\{ a\cos\left( \log_e x \right) + b \sin\left( \log_e x \right) \right\} - \left\{ - a\sin\left( \log_e x \right) + b \cos\left( \log_e x \right) \right\} \]

\[ \Rightarrow x^2 y_2 = - y - x y_1 \]

\[ \Rightarrow x^2 y_2 + x y_1 = - y\]

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 12: Higher Order Derivatives - Exercise 12.3 [पृष्ठ २३]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12
पाठ 12 Higher Order Derivatives
Exercise 12.3 | Q 12 | पृष्ठ २३

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?


Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?


Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?


If \[y = \sqrt{x^2 + a^2}\] prove that  \[y\frac{dy}{dx} - x = 0\] ?


Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?


Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?


Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?


Differentiate 

\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?


If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ? 


Find  \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?

 


If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`


If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?


If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?


If \[y \sqrt{x^2 + 1} = \log \left( \sqrt{x^2 + 1} - x \right)\] ,Show that \[\left( x^2 + 1 \right) \frac{dy}{dx} + xy + 1 = 0\] ?


If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?


Differentiate \[x^{1/x}\]  with respect to x.


Differentiate \[\left( \log x \right)^{\cos x}\] ?


Differentiate \[x^{\tan^{- 1} x }\]  ?


Differentiate  \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?


If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?


If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?

 


\[\text{If y} = 1 + \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{{\beta}/{x}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)} + \frac{{\gamma}/{x^2}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)\left( \frac{1}{x} - \gamma \right)}, \text{ find } \frac{dy}{dx}\] is:

If  \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?

 


Find \[\frac{dy}{dx}\] , when  \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?


If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?


If  \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at  \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?


\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?


Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to  \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?


Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?


Differentiate \[\tan^{- 1} \left( \frac{x - 1}{x + 1} \right)\] with respect to \[\sin^{- 1} \left( 3x - 4 x^3 \right), \text { if }- \frac{1}{2} < x < \frac{1}{2}\] ?


If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .


If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .


If \[y = \frac{\log x}{x}\] show that \[\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}\] ?


If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?


If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?


If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write  \[\frac{d^2 y}{d x^2}\] in terms of y ?


If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is 

 


If y = (sin−1 x)2, then (1 − x2)y2 is equal to

 


f(x) = 3x2 + 6x + 8, x ∈ R


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×