The origin belongs to a region between the lines $x+2y-5=0$ and $3x-4y+5=0$. The number of points in that region of the form $((\alpha-1)^2, \alpha)$,where $\alpha \in \mathbb{Z}$,is

The solution set of the constraints $|x-y| \leqslant 1, x, y \geqslant 0$ is

How many litres of water will have to be added to $1125$ litres of the $45 \%$ solution of acid so that the resulting mixture will contain more than $25 \%$ but less than $30 \%$ acid content?

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:

The feasible region represented by the inequations $2x + 3y \leq 18$,$x + y \geq 10$,$x \geq 0$,$y \geq 0$ is

To receive grade $A$ in a course,one must obtain an average of $90$ marks or more in five examinations (each of $100$ marks). If Sunita's marks in the first four examinations are $87, 92, 94,$ and $95,$ find the minimum marks that Sunita must obtain in the fifth examination to get grade $A$ in the course.