If f is an even function from R to R, then f(0) must be equal to 0. ∵ We know that if a function f(x) is even, then f(−x)=f(x) Now, if we assume f(x)=cos‌x f(−x)=cos(−x) =cos‌x‌‌‌‌‌‌‌‌{∵cos(−θ)=(cos‌θ)} =f(x) ∴ f(x)=cos‌x is an even function. Now, f(0)=cos‌0=1≠0 The given statement is false. (b) f:R→R,f(x)=x−[x] ∵ We know that, x=[x]+{x} where {x}= fractional part function ⇒x−[x]={x} ∴f(x)={x} where {x} is a periodic function. ⇒f(x) is periodic function. ∴ The given statement is true. (c) f:R→R is an odd function, then f(0)=0 We know that if f(x) is an odd function, then f(−x)=−f(x) Put x=0, we get f(0)=−f(0) ⇒f(0)+f(0)=0 2f(0)=0 ⇒f(0)=0 The given statement is true. (d) Number of onto functions from {1,2,3,4,5,6} to {1,2} is 62. Let A={1,2,3,4,5,6} and B={1,2} ∵ ⇒n(A)=6 and n(B)=2 ∵ We know that the number of onto function from a set A with m number of elements to set B with n number of elements is