What is the probability of a couple having four sons?

$A$ cup of coffee cools from $90^{\circ} C$ to $80^{\circ} C$ in $t$ minutes,when the room temperature is $20^{\circ} C$. The time taken by a similar cup of coffee to cool from $80^{\circ} C$ to $60^{\circ} C$ at a room temperature same at $20^{\circ} C$ is :

The shortest distance between the line $y - x = 1$ and the curve $x = y^2$ is:

$(a)$ Show that for every natural number $n$ relatively prime to $10$,there exists a natural number $m$ consisting only of digits $1$ such that $n$ divides $m$.
$(b)$ Hence or otherwise,show that every positive rational number can be expressed in the form $\frac{a}{10^b(10^c-1)}$ for some natural numbers $a, b, c$.

In $\triangle ABC$, the midpoints of the sides $AB, BC$, and $CA$ are respectively $(l, 0, 0), (0, m, 0)$, and $(0, 0, n)$. Then, $\frac{AB^2+BC^2+CA^2}{l^2+m^2+n^2}$ is equal to

Let $u = i + j$,$v = i - j$ and $w = i + 2j + 3k$. If $n$ is a unit vector such that $u \cdot n = 0$ and $v \cdot n = 0$,then $|w \cdot n|$ is equal to