Question

If x is real number in [0, 1], then the value of

$\frac{x}{2}$ [1 + cos^2m (n!πx)] is given by

• A. 2 or 1 according as x is rational or irrational
• B. 1 or 2 according as x is rational or irrational
• C. 1 for all x
• D. 2 or 1 for all x.

Answer 1

Answer: (A)

2 or 1 according as x is rational or irrational

Answer 2

1 + (cos² n! πx)^m

Case I

when x is irrational = 1 + (cos² ∞)^∞

= 1 + (0 to 1)^∞ = 1 + 0 = 1

put x = 0 when x is rational 1 + cos 0 = 2