For a vector $$\overline x $$ = [x[0], x[1], ....., x[7]], the 8-point discrete Fourier transform (DFT) is denoted by $$\overline X $$ = DFT($$\overline x $$) = [X[0], X[1], ....., X[7]], where

$$X[k] = \sum\limits_{n = 0}^7 {x[n]\exp \left( { - j{{2\pi } \over 8}nk} \right)} $$.

Here, $$j = \sqrt { - 1} $$. If $$\overline x $$ = [1, 0, 0, 0, 2, 0, 0, 0] and $$\overline y $$ = DFT (DFT($$\overline x $$)), then the value of y[0] is __________ (rounded off to one decimal place).

Mr. X speaks ____________ Japanese ___________ Chinese.

A sum of money is to be distributed among P, Q, R, and S in the proportion 5 : 2 : 4 : 3, respectively.

If R gets Rs. 1000 more than S, what is the share of Q (in Rs.) ?

A trapezium has vertices marked as P, Q, R and S (in that order anticlockwise). The side PQ is parallel to side SR.

Further, it is given that, PQ = 11 cm, QR = 4 cm, RS = 6 cm and SP = 3 cm. What is the shortest distance between PQ and SR (in cm) ?