In a bank,the principal increases continuously at the rate of $5 \%$ per year. An amount of $Rs. 1000$ is deposited in this bank. How much will it be worth after $10$ years? (Given: $e^{0.5} = 1.648$)

Let $y = y(x)$ be the solution of the differential equation $x dy - y dx = \sqrt{x^2 - y^2} dx$,$x \geq 1$,with $y(1) = 0$. If the area bounded by the lines $x = 1$,$x = e^{\pi}$,$y = 0$ and the curve $y = y(x)$ is $\alpha e^{2\pi} + \beta$,then the value of $10(\alpha + \beta)$ is equal to ....... .

In a bank,the principal increases continuously at the rate of $r \%$ per year. Find the value of $r$ if $Rs \, 100$ doubles itself in $10$ years $\left(\log _{e} 2=0.6931\right)$. (in $\%$)

$A$ body cools according to Newton's law of cooling from $100^{\circ} C$ to $60^{\circ} C$ in $15$ minutes. If the temperature of the surrounding is $20^{\circ} C$,then the temperature of the body after cooling down for one hour is (in $^{\circ} C$)

The equation of a curve passing through $\left( 2, \frac{7}{2} \right)$ and having gradient $1 - \frac{1}{x^2}$ at $(x, y)$ is:

In a culture,the bacteria count is $1,00,000$ initially. The number increases by $10 \%$ in the first $2$ 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?