The rate of increase of the population of a city is proportional to the population present at that instant. In the period of $40$ years,the population increased from $30,000$ to $40,000$. At any time $t$,the population is given by $P(t) = (a)(b)^{\frac{t}{40}}$. Then the values of $a$ and $b$ are respectively:

In a culture,the bacteria count is $1,00,000$. The number increases by $10 \%$ in $2$ hours. In how many hours will the count reach $2,00,000$,if the rate of growth of bacteria is proportional to the number present?

If the population grows at the rate of $8 \%$ per year,then the time taken for the population to be doubled is $\quad$ (given $\log 2=0.6912$ ) (in $years$)

The rate of increase of bacteria in a culture is proportional to the number of bacteria present at that instant. It is found that the number doubles in $6 \text{ hours}$. The number of bacteria becomes $n$ times the initial number at the end of $18 \text{ hours}$. Find $n$.

If a curve passes through the origin,such that the length of the subnormal is equal to one more than the square of the ordinate,then:

Let $\Gamma$ denote a curve $y = y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_p$. If $PY_p$ has length $1$ for each point $P$ on $\Gamma$,then which of the following options is/are correct?
$(1)$ $y=\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}$
$(2)$ $xy^{\prime}+\sqrt{1-x^2}=0$
$(3)$ $y=-\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}$
$(4)$ $xy^{\prime}-\sqrt{1-x^2}=0$