The cost and revenue functions of a product are given by $C(x) = 20x + 4000$ and $R(x) = 60x + 2000$ respectively,where $x$ is the number of items produced and sold. The value of $x$ to earn profit is

The region represented by the inequations $2x + 3y \leqslant 18$,$x + y \geqslant 10$,$x \geqslant 0$,$y \geqslant 0$ is

Let $a$ and $b$ be two positive real numbers such that $a+2b \leq 1$. Let $A_1$ and $A_2$ be respectively the areas of circles with radii $ab^3$ and $b^2$. Then,the maximum possible value of $\frac{A_1}{A_2}$ is

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:

If $x$ is real and satisfies $x + 2 > \sqrt{x + 4}$,then

Find all pairs of consecutive odd natural numbers,both of which are larger than $10$,such that their sum is less than $40$.