Differentiation — Maths STD 12 Science — Question

A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profits from crops X and Y per hectare are estimated as Rs. 10,500 and Rs. 9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at the rate of 20 litres and 10 litres per hectare, respectively. Further not more than 800 litres of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximise the total profit of the society? 

Find the image of the point (1, 3, 4) in the plane 2x - y + z + 3 = 0.

Find the area of the figure bounded by the curves y = |x - 1| and y = 3 - |x|.

Find a point on the parabola y = (x - 4)², where the tangent is parallel to the chord joining (4, 0) and (5, 1).

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

Consider a non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then, R is:

1. Symmetric but not transitive.
2. Transitive but not symmetric.
3. Neither symmetric nor transitive.
4. Both symmetric and transitive.