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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
A curve passes through the point $\left(1, \dfrac{\pi}{6}\right)$.
Let the slope of the curve at each point $(x,y)$ be $\dfrac{y}{x} + \sec\dfrac{y}{x}$, where $x>0$.
Then the equation of the curve is:
Solution
Given $\dfrac{dy}{dx} = \dfrac{y}{x} + \sec\dfrac{y}{x}.$ Let $\dfrac{y}{x} = v \Rightarrow y = vx \Rightarrow \dfrac{dy}{dx} = v + x\dfrac{dv}{dx}.$ Substitute: $v + x\dfrac{dv}{dx} = v + \sec v \Rightarrow x\dfrac{dv}{dx} = \sec v.$ Integrate: $\int \cos v\,dv = \int \dfrac{dx}{x} \Rightarrow \sin v = \log x + C.$ At $(x,y) = (1, \pi/6)$ ⇒ $v = y/x = \pi/6$. $\sin(\pi/6) = 1/2 = \log 1 + C \Rightarrow C = 1/2.$ Hence equation: $\sin\dfrac{y}{x} = \log x + \dfrac{1}{2}.$Online Test Series, Information About Examination,
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