The 10th term of `(3-sqrt((17)/4+3sqrt(2)))^(20)` is (a) a irrational number (b) a rational number (c) a positive integer (d) a negative integer

Options

A. an irrational number
B. a rational number
C. a positive integer
D. a negative integer

Answer

a,d

Explanation

We have , `(17)/(4) + 3 sqrt(2) = (1)/(4) (9 + 8 + 12 sqrt(2)) = (1)/(4) (3 + 2 sqrt(2))^(2)`
` rArr 3 - sqrt((17)/(4) + 3sqrt(2)) = 3 - (1)/(2) (3 + 2 sqrt(2)) = ((3)/(2) - sqrt(2))`
` therefore 10th " term in " (3 - sqrt((17)/(4) + 3 sqrt(2)))^(20) ` is
` T_(9+1) = ""^(20)C_(9) ((3)/(2))^(20-9) (-sqrt(2))^(9)`
` = ""^(20)C_(9) (-1)^(9) 3^(11) . 2 ^(-11+ (9)/(2))`
` = - ""^(20)C_(9) 3^(11) 2^(-(13)/(2))`
which is a negative and an irrational number .