JEE Advanced 2014 Paper 2 Mathematics Q17

Mathematics

Problem

Let

z_k=\cos\left(\dfrac{2k\pi}{10}\right)+i\sin\left(\dfrac{2k\pi}{10}\right)

k=1,2,\ldots,9

. Match List I with List II and select the correct option. Column I — P: For each

z_k

there exists a

z_j

such that

z_kz_j=1

.; Q: There exists a

k\in\{1,2,\ldots,9\}

such that

z_1z=z_k

has no solution

z

in the set of complex numbers.; R:

\dfrac{|1-z_1||1-z_2|\cdots|1-z_9|}{10}

equals; S:

1-\displaystyle\sum_{k=1}^{9}\cos\left(\dfrac{2k\pi}{10}\right)

equals Column Ii — 1: True; 2: False; 3:

1

; 4:

2