Consider the following statements I.` n (sin^2 (67 1/2^@) -sin^2 (22 1/2^@)) gt 1` for all positive integers `n geq 2`.II. If `x` is any positive real number, then `nx gt 1` for all positive integers `n geq 2`.Which of the above statement(s) is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

A

Explanation

Statement 1 :
Given `n(sin^(2)67(1^(@))/(2)-sin^(2)22(1^(@))/(2))`
or `n(sin^(2)67(135^(@))/(2)-sin^(2)22(45^(@))/(2))`
`n=[2"sin" (((135^(@))/(2)+(45^(@))/(2)))/(2)."cos "((135^(@))/(2)-(45^(@))/(2))/(2)]`
`=n[2"sin"(((135^(@))/(2)+(45^(@))/(2)))/(2)."sin"(((135^(@))/(2)-(45^(@))/(2)))/(2)]`
`[2."cos"(((135^(@))/(2)+(45^(@))/(2)))/(2)."sin"(((135^(@)-45^(@))/(2)))/(2)]`
`=n[2.sin((90^(@))/(2)).cos((45^(@))/(2))]`
`[2.cos((90^(@))/(2)).sin((45^(@))/(2))]`
`=2n(2"sin"(45^(@))/(2)."cos"(45^(@))/(2))(sin45^(@).cos45^(@))`
`=2n. sin(2xx(45^(@))/(2))((1)/(sqrt(2))xx(1)/(sqrt(2)))`
`=2n.sin45xx(1)/(2)=n.(1)/(sqrt(2))=(n)/(sqrt(2))`
`therefore(n)/(sqrt(2))gt1` for all positive integers `nl=ge2`.
`therefore` Statement 1 is true
Statement 2
`nx1gt1,AAnge2`
`rArrngt(1)/(x),AAnge2`
`x in(0,oo)`, then we take x =1
`ngt1`, but n is always greater or equal to 2 for all x positive real number .
`therefore` Statement 2 is false .