`152(sin30^(@)+2cos^(@)45^(@)+3sin30^(@)+4cos2^(2)45+ . . . .+17sin30^(@)+18cos^(2)45^(@)` का मान क्या होगा?

Options

A. An interger but not perfect square
B. a rational number but not an interger
C. a perfect square of an integer
D. irrational

Answer

c

Explanation

`152(sin30^(@)+2cos^(2)45^(@)+3sin30^(@).........+17sin30^(@)+18cos^(2)45^(@))`
`=152[(1)/(2)+2((1)/sqrt(2))^(2)+3xx(1)/(2)........17xx(1)/(2)+18((1)/sqrt(2))^(2)]`
`rArr 152[(1)/(2)+1+1(1)/(2)+...........+ 8(1)/(2)+9]`
`rArr` This is in A.P. where `a=(1)/(2),d=(1)/(2),n= 18`
`rArr 15 2[(18)/(2)(2xx(1) /(2)+(18-1)(1)/(2))]`
`rArr 152 [(18)/(2)(1+(17)/(2))]`
`rArr 152xx9xx(19)/(2)rArr12996`
`rarrsqrt(12996)`
=114