Straight Lines ...

Consider the following statements:
$$S_1: \lim_\limits{x \to 0} \dfrac{[x]}{x}$$ is an indeterminate form (where [.] denotes greatest integer function).
$$S_2: \lim_\limits{x\to\infty}\dfrac{sin(3^x)}{3^x}=0$$
$$S_3: \lim_\limits{x \to \infty}\sqrt{\dfrac{x- sinx}{x+cos^2x}}$$ does not exist.
$$S_4:  \lim_\limits{n\to \infty}\dfrac{(n+2)!+(n+1)!}{(n+3)! }(n \in N=0$$
State, in order, whether $$S_1, S_2, S_3, S_4$$ are true or false

The number of $$n$$ digit numbers which consists of the digits $$1$$ & $$2$$ only if each digits is to be used atleast once, is equal to $$510$$  then $$n$$ is equal to

Observe the pattern carefully
$$11\times11=121$$
$$111\times111=12321$$
$$1111\times1111=\,?$$

The value of $$\displaystyle\sum^{10}_{r=0}(r)$$ $$^{20}C_r$$ is equal to?

How many Pythagorean triangles are there with integer sides and one leg is Ramanujan Number 1729 .

Find the missing number, if a certain rule is followed

Direction: In a given question, one number is missing in the series. you have to understand the pattern of the series and insert the number 
JN    28   27    GP
CE   12   45    TU
LR     ?     ?     MS

Fill in the missing values.

If $$24 + 37 = 7,12 + 18 = 3$$, then 54+21=?

A library has $$'a'$$ copies of one book, $$'b'$$ copies of each of two books, $$'c'$$ copies of each of three books, and single copy each of $$'d'$$. The total number of ways in which these books can be arranged in a row is