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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.

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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?

Matrices and Determinants

If A and B are two 2×2 matrices such that A times B equals B times A, prove that the determinant of A times the determinant of B equals the determinant of B times the determinant of A.

Relations and Functions

Define an equivalence relation. Give an example of an equivalence relation on the set of real numbers.

Organic Chemistry

Describe the mechanism of the addition of hydrogen halides to alkenes. What is Markovnikov’s rule, and how does it apply in this context?

Solutions

Explain Raoult’s law for ideal solutions. How does it apply to a mixture of two volatile liquids?