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Probability

A box contains 5 red, 4 white, and 6 blue balls. Two balls are drawn at random. Find the probability that they are both of the same color.

Linear Programming

A manufacturer produces two types of products A and B, which require 2 hours and 3 hours respectively for production on machine I, and 4 hours and 2 hours respectively on machine II. The machine I has 20 hours and machine II has 18 hours available per week. The profit on each product of type A is ₹6 and on type B is ₹8. Formulate the problem as a linear programming problem to maximize the profit.

Three-Dimensional Geometry

Find the equation of the plane passing through the point (1,−2,3) and perpendicular to the vector .

Vectors

Differential Equations

Application of Integrals

Find the area bounded by the curve:

and the line:

Definite Integrals

Using properties of definite integrals, evaluate

Integrals

Application of Derivatives

A rectangle is inscribed in a semicircle of radius 5. Find the dimensions of the rectangle that will make its area maximum.

Differential Calculus

find the points where f(x)f(x)f(x) has a local maximum or minimum and determine the nature of these points.