The region satisfying the inequalities $y - x \geq 2$, $x + y \leq 5$, $x \geq 0$, and $y \geq 0$ is represented by:

$A$ manufacturer has $600 \text{ litres}$ of a $12 \%$ solution of acid. How many litres of a $30 \%$ acid solution must be added to it so that the acid content in the resulting mixture will be more than $15 \%$ but less than $18 \%$?

Solve the following system of inequalities: $\frac{2x+1}{7x-1} > 5$ and $\frac{x+7}{x-8} > 2$.

There are $30$ questions in a multiple-choice test. $A$ student gets $1$ mark for each unattempted question,$0$ mark for each wrong answer,and $4$ marks for each correct answer. If a student answered $x$ questions correctly and scored $60$,then the number of possible values of $x$ is:

Let $2\sin^2 x + 3\sin x - 2 > 0$ and $x^2 - x - 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

$A$ triangle is formed by the $X$-axis,$Y$-axis,and the line $3x + 4y = 60$. The number of points $P(a, b)$ which lie strictly inside the triangle,where $a$ is an integer and $b$ is a multiple of $a$,is $...........$