1. Tardigrade
  2. Question
  3. Mathematics
  4. The maximum value of ( log x/x) is equal to:

Q. The maximum value of $ \frac{\log x}{x} $ is equal to:

KEAMKEAM 2003 Report Error

Solution:

By Simpson rule taking $ n = 4 $ , the value of the integral $ \int^\limits{1}_{0} \frac{1}{1+x^{2}}dx $ is equal toLet $ y=\frac{\log x}{x} $ On differentiating w.r.t. $ x, $ we get $ \frac{dy}{dx}=\frac{x.\frac{1}{x}-\log x}{{{x}^{2}}} $ $ \Rightarrow $ $ \frac{dy}{dx}=\frac{1-\log x}{{{x}^{2}}} $ Put $ \frac{dy}{dx}=0 $ for maxima or minima $ \therefore $ $ x=e $ Now, $ \frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{{{x}^{2}}\left( -\frac{1}{x} \right)-(1-\log x)2x}{{{({{x}^{2}})}^{2}}} $ $ =\frac{-3+2\log x}{{{x}^{3}}} $ and $ \therefore $ Function is maximum at $ x=e $ . $ \therefore $ $ {{y}_{\max }}=\frac{\log e}{e}=\frac{1}{e} $

Questions from KEAM 2003

1. The number of terms in the expansion of $ {{(a+b+c)}^{10}} $ is:

2. For what value of $ \lambda , $ the system of equations $ x+y+z=6,\text{ }x+2y+3z=10, $ $ x+2y+\lambda z=10 $ is consistent?

3. Let $ f(x) $ be twice differentiable such that $ f(x)=-f(x),f(x)=g(x), $ where $ f(x) $ and $ f(x) $ represent the first and second derivatives of $ f(x) $ respectively. Also, if $ h(x)={{[f(x)]}^{2}}+{{[g(x)]}^{2}} $ and $ h(5)=5, $ then $ h(10) $ is equal to:

4. A straight line through P (1, 2) is such that its intercept between the axes is bisected at P. Its equation is:

5. The radius of any circle touching the lines $ 3x-4y+5=0 $ and $ 6x-8y-9=0 $ is:

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