JEE MAIN - Mathematics (2021 - 26th February Morning Shift - No. 13)
The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria is increased by 20% in 2 hours. If the population of bacteria is 2000 after $${k \over {{{\log }_e}\left( {{6 \over 5}} \right)}}$$ hours, then $${\left( {{k \over {{{\log }_e}2}}} \right)^2}$$ is equal to :
16
8
2
4
Explanation
$${{dx} \over {dt}} \propto x$$
$${{dx} \over {dt}} = \lambda x$$
$$\int\limits_{1000}^x {{{dx} \over x} = \int\limits_0^t {\lambda dt} } $$
$$\ln x - \ln 1000 = \lambda t$$
$$\ln \left( {{x \over {1000}}} \right) = \lambda t$$
Put t = 2, x = 1200
$$\ln \left( {{{12} \over {10}}} \right) = 2\lambda \Rightarrow \lambda = {1 \over 2}\ln {6 \over 5}$$
Now, $$\ln \left( {{x \over {1000}}} \right) = {t \over 2}\ln \left( {{6 \over 5}} \right)$$
$$x = 1000{e^{{t \over 2}\ln \left( {{6 \over 5}} \right)}}$$
$$x = 2000$$ at $$t = {k \over {\ln \left( {{6 \over 5}} \right)}}$$
$$ \Rightarrow 2000 = 1000{e^{{k \over {2\ln (6/5)}} \times \ln (6/5)}}$$
$$ \Rightarrow 2 = {e^{k/2}}$$
$$ \Rightarrow \ln 2 = {k \over 2}$$
$$ \Rightarrow {k \over {\ln 2}} = 2$$
$$ \Rightarrow {\left( {{k \over {\ln 2}}} \right)^2} = 4$$
$${{dx} \over {dt}} = \lambda x$$
$$\int\limits_{1000}^x {{{dx} \over x} = \int\limits_0^t {\lambda dt} } $$
$$\ln x - \ln 1000 = \lambda t$$
$$\ln \left( {{x \over {1000}}} \right) = \lambda t$$
Put t = 2, x = 1200
$$\ln \left( {{{12} \over {10}}} \right) = 2\lambda \Rightarrow \lambda = {1 \over 2}\ln {6 \over 5}$$
Now, $$\ln \left( {{x \over {1000}}} \right) = {t \over 2}\ln \left( {{6 \over 5}} \right)$$
$$x = 1000{e^{{t \over 2}\ln \left( {{6 \over 5}} \right)}}$$
$$x = 2000$$ at $$t = {k \over {\ln \left( {{6 \over 5}} \right)}}$$
$$ \Rightarrow 2000 = 1000{e^{{k \over {2\ln (6/5)}} \times \ln (6/5)}}$$
$$ \Rightarrow 2 = {e^{k/2}}$$
$$ \Rightarrow \ln 2 = {k \over 2}$$
$$ \Rightarrow {k \over {\ln 2}} = 2$$
$$ \Rightarrow {\left( {{k \over {\ln 2}}} \right)^2} = 4$$
Comments (0)
