The sum of the digits of a two-digit number is $13$. The number obtained by interchanging the digits is $9$ more than the original number. Find the original number.

The solution set of a pair of equations $x+y-1=0$ and $2x+2y-2=0$ is $\ldots \ldots \ldots \ldots$

$A$ shopkeeper sells a saree at $8 \%$ profit and a sweater at $10 \%$ discount,thereby,getting a sum of $Rs. 1008$. If she had sold the saree at $10 \%$ profit and the sweater at $8 \%$ discount,she would have got $Rs. 1028$. Find the cost price of the saree and the list price (price before discount) of the sweater (in $Rs.$).

In a competitive examination,one mark is awarded for each correct answer while $\frac{1}{2}$ mark is deducted for every wrong answer. Jayanti answered $120$ questions and got $90$ marks. How many questions did she answer correctly?

Do the following pair of linear equations have no solution? Justify your answer.
$3x + y - 3 = 0$
$2x + \frac{2}{3}y = 2$

Kushan went to a bookseller's shop and purchased $2$ textbooks of Mathematics $IX$ and $3$ textbooks of Mathematics $X$ for Rs. $250$. His friend Rajan also bought $4$ textbooks of Mathematics $IX$ and $6$ textbooks of Mathematics $X$ of the same kind for Rs. $500$. Represent this situation as a pair of linear equations in two variables.