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Trigonometry

• Prove that the value of sin squared theta plus cos squared theta equals 1.

Sets and Relations

Let A be the set of all natural numbers less than 10, and B be the set of all even natural numbers less than 10. Find A union B and A intersection B.

Probability Distributions

1. Define a binomial distribution. What is the probability of obtaining exactly 2 heads in 5 flips of a fair coin?

Differential Equations

• Solve the differential equation dy/dx = 3y.

Vectors

If vectors a and b are perpendicular, prove that their dot product is zero.

Three-Dimensional Geometry

• Find the equation of the plane passing through the point (2, 3, -1) and perpendicular to the line with direction ratios 1, -2, and 3.

Probability

A box contains 5 red, 4 blue, and 6 green balls. If two balls are drawn at random, what is the probability that both balls are of the same color?

Linear Programming

A factory produces two types of goods, A and B. Each unit of A requires 2 hours of labor, while each unit of B requires 3 hours of labor. The factory has a maximum of 60 hours of labor available. Formulate this situation as a linear programming problem to maximize production.

Application of Derivatives

Find the maximum and minimum values of the function f(x) = x squared minus 4x plus 7 on the interval [0,3].

Continuity and Differentiability

State the definition of continuity at a point. Is the function f(x) = |x| continuous for all real values of x?