1. Tardigrade
  2. Question
  3. Mathematics
  4. Column I Column II A The least value of 'a' for which the equation, (4/ sin x)+(1/1- sin x)=a has atleast one solution on the interval (0, π / 2) is P 20 B A closed vessel tapers to a point both at its top E and its bottom F and is fixed with EF vertical when the depth of the liquid in it is x cm, the volume of the liquid in it is, x2(15-x) c u. cm. The length E F is Q 13 C If Rolle's theorem is applicable to the function f ( x )=( ln x / x )( x >0) over the interval [a, b] where a, b ∈ I, then the value of (a2+b2) is equal to R 10 S 9

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Q.
Column I Column II
A The least value of 'a' for which the equation, $\frac{4}{\sin x}+\frac{1}{1-\sin x}=a$ has atleast one solution on the interval $(0, \pi / 2)$ is P 20
B A closed vessel tapers to a point both at its top $E$ and its bottom $F$ and is fixed with EF vertical when the depth of the liquid in it is $x cm$, the volume of the liquid in it is, $x^2(15-x) c u$. cm. The length $E F$ is Q 13
C If Rolle's theorem is applicable to the function $f ( x )=\frac{\ln x }{ x }( x >0)$ over the interval $[a, b]$ where $a, b \in I$, then the value of $\left(a^2+b^2\right)$ is equal to R 10
S 9

Application of Derivatives

Solution:

(A) $\frac{ dy }{ dx }=-\frac{4 \cos x }{\sin ^2 x}+\frac{\cos x}{(1-\sin x)^2}=0$ gives $\sin x=\frac{2}{3}$
note that $f ( x ) \rightarrow \infty$ as $x \rightarrow 0^{+}$or $x \rightarrow \frac{\pi^{-}}{2}$ and between two maxima we have a minima.
(B) $\left.\frac{ dv }{ dx }=3 x (10- x )=0 \Rightarrow x =0 ; x =10 ; \frac{ d ^2 v }{ dx ^2}\right]_{ x =10}<0 \Rightarrow v$ is $\max$ at $x =10$ $\Rightarrow EF =10 cm$. Ans.
(C) Clearly, for $f ( a )= f ( b )$ where $a , b \in I$
So, a must be 2 .
image
$\Rightarrow \frac{\ln b}{b}=\frac{\ln 2}{2}$
$\Rightarrow b=4 $
$\text { So, }(a+b)=6 $