State TRUE or FALSE justifying your answer with proper reason. a. 2𝑛^2 + 1 = 𝑂(𝑛^2 ) b. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 ) c. 𝑛^2 (1 + √𝑛) = 𝑂(𝑛^2 log 𝑛) d. 3𝑛^2 + √𝑛 = 𝑂(𝑛 + 𝑛√𝑛 + √𝑛) e. √𝑛 log 𝑛 = 𝑂(𝑛)

a.

true

$2𝑛^2 + 1\le 3n^2$

b.

false

$\displaystyle \lim_{n\to \infin} \frac{𝑛^2 (1 + \sqrt𝑛) }{n^2}=\infin$

c.

false

$\displaystyle \lim_{n\to \infin} \frac{𝑛^2 (1 + \sqrt𝑛) }{n^2logn}=\infin$

d.

false

$\displaystyle \lim_{n\to \infin} \frac{ 3𝑛^2 + \sqrt𝑛) }{𝑛 + 𝑛\sqrt𝑛 + \sqrt𝑛}=\infin$

e.

true

$\sqrt𝑛 log 𝑛\le n$