AP EAPCET 24-Aug-2021 Shift 1 Paper

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 Here n = 2 and m = 6
So, total number of onto function are
26−{2C1(2−1)6+‌2C2(2−2)6]
=64−[2+0]
=64−2=62
∴ The given statement is true.