The rate at which the population of a city increases varies as the population present. Within the period of $30$ years,the population grew from $20$ lakhs to $40$ lakhs. Then,the population after a further period of $15$ years will be (Take $\sqrt{2} = 1.41$) (in $lakhs$)

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?
$(1)$ $y=\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\sqrt{1-x^2}$
$(2)$ $xy^{\prime}+\sqrt{1-x^2}=0$
$(3)$ $y=-\ln\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\sqrt{1-x^2}$
$(4)$ $xy^{\prime}-\sqrt{1-x^2}=0$

$A$ radioactive substance has a half-life of $h \ days$. Its initial decay rate is given by (Note that at $t = 0, M = m_0$):

$A$ spherical metal ball at $80^{\circ} C$ cools in $5 \text{ minutes}$ to $60^{\circ} C$ in a surrounding temperature of $20^{\circ} C$. The temperature of the ball after $20 \text{ minutes}$ is approximately: (in $^{\circ} C$)

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:

Number of straight lines which satisfy the differential equation $\frac{dy}{dx} + x \left( \frac{dy}{dx} \right)^2 - y = 0$ is: