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Vectors

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

11/11/2024

Mathematics

12th (Science)

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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11/11/2024

Mathematics

12th (Science)

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?

11/11/2024

Mathematics

12th (Science)

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.

11/11/2024

Mathematics

12th (Science)

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].

11/11/2024

Mathematics

12th (Science)

Continuity and Differentiability

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

11/11/2024

Mathematics

12th (Science)

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.

11/11/2024

Mathematics

12th (Science)

Relations and Functions

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

11/11/2024

Mathematics

12th (Science)

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?

11/11/2024

Chemistry

12th (Science)

Solutions

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

11/11/2024

Chemistry

12th (Science)