Let $x, y, z$ be positive reals. Which of the following implies $x=y=z$ ?
$I.$ $x^3+y^3+z^3=3 x y z$
$II.$ $x^3+y^2 z+y z^2=3 x y z$
$III.$ $x^3+y^2 z+z^2 x=3 x y z$
$IV.$ $(x+y+z)^3=27 x y z$
$I, IV$ only
$I, II$ and $IV$ only
$I, II$ and $III$ only
All of them
A man standing on a railway platform noticed that a train took $21\, s$ to cross the platform (this means the time elapsed from the moment the engine enters the platform till the last compartment leaves the platform) which is $88\,m$ long, and that it took $9 s$ to pass him. Assuming that the train was moving with uniform speed, what is the length of the train in meters?
The sum of the roots of the equation $x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0$, is :
If $a, b, c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$, then $(3 a+5 b-8 c)^2+(-8 a+3 b+5 c)^2$ $+(5 a-8 b+3 c)^2$ is equal to
Leela and Madan pooled their music $CD's$ and sold them. They got as many rupees for each $CD$ as the total number of $CD's$ they sold. They share the money as follows: Leela first takes $10$ rupees, then Madan takes $10$ rupees and they continue taking $10$ rupees alternately till Madan is left out with less than $10$ rupees to take. Find the amount that is left out for Madan at the end, with justification.
If $|x - 2| + |x - 3| = 7$, then $x =$