One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${R_2}$$. The profits per unit by selling product $${P_1}$$ and $${P_2}$$ are Rs. $$2000$$ and Rs. $$3000$$ respectively. The manufacturer has $$90$$ $$kg$$ of resource $${R_1}$$ and $$100$$ $$kg$$ of resource $${R_2}$$.

The unit worth of resource $${R_2}$$. i.e. dual price of resource $${R_2}$$ in Rs. per $$kg$$ is

One unit of product $${P_1}$$ requires $$3$$ $$kg$$ of resource $${R_1}$$ and $$1$$ $$kg$$ of resource $${R_2}$$. One unit of product $${P_2}$$ requires $$2$$ $$kg$$ of resource $${R_1}$$ and $$2$$ $$kg$$ of resource $${R_2}$$. The profits per unit by selling product $${P_1}$$ and $${P_2}$$ are Rs. $$2000$$ and Rs. $$3000$$ respectively. The manufacturer has $$90$$ $$kg$$ of resource $${R_1}$$ and $$100$$ $$kg$$ of resource $${R_2}$$.

The manufacturer can make a maximum profit of Rs.

Match the following criteria of material failure, under biaxial stresses $${\sigma _1}$$and $${\sigma _2}$$ and yield stress $${\sigma _y}$$ with their corresponding graphic representations:
$$P.$$ Maximum normal stress criterion
$$Q.$$ Maximum distortion energy criterion
$$R.$$ Maximum shear stress criterion

Two identical ball bearings $$P$$ and $$Q$$ are operating at loads $$30kN$$ and $$45kN$$ respectively. The ratio of the life of bearing $$P$$ to the life of bearing $$Q$$ is