Consider the system of linear equations in $x, y, z$ :

$$ \begin{aligned} & x+2 y+t z=0 \\ & 6 x+y+5 t z=0 \\ & 3 x+t^2 y+f(t) z=0 \end{aligned} $$

where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t \in \mathbb{R}$, then $f$

$\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)$ is equal to:

Let $\tan A, \tan B$, where $A, B \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, be the roots of the quadratic equation $x^2-2 x-5=0$. Then $20 \sin ^2\left(\frac{A+B}{2}\right)$ is equal to:

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is: