A certain coin is tossed with probability of showing head being `'p'`. Let `'q'` denotes the probability that when the coin is tossed four times the number of heads obtained is even. Then
(a) there is no value of `p`, if `q=(1)/(4)`
(b) there is exactly two value of `p`, if `q=(3)/(4)`
(c) there are exactly three value of `p`, if `q=(3)/(5)`
(d) there are exactly four value of `p`, if `q=(4)/(5)`

Options

A. (a) there is no value of `p`, if `q=(1)/(4)`
B. (b) there is exactly one value of `p`, if `q=(3)/(4)`
C. (c) there are exactly three value of `p`, if `q=(3)/(5)`
D. (d) there are exactly four value of `p`, if `q=(4)/(5)`

Answer

A, C

Explanation

`(a,c)` We have `q=P(0H or 2H or 4H)`
`:.q=p^(4)+^(4)C_(2)p^(2)(1-p)^(2)+(1-p)^(4)`
`=8p^(4)-16p^(3)+12p^(2)-4p+1`
`=((2p-1)^(4)+1)/(2)`
Now put the values and check.