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Probability

A bag contains 4 white, 5 red, and 6 black balls. If three balls are drawn at random, what is the probability that they are all of the same color?

Three-Dimensional Geometry

Find the equation of a plane passing through the point (1, -2, 3) and perpendicular to the line with direction ratios 2, -1, and 4.

Vectors

If two vectors a and b satisfy a . b = 0, show that the vectors are perpendicular to each other.

Integrals

Evaluate the area of the region bounded by the lines y = x, y = 0, and x = 2.

Integrals

Evaluate the area of the region bounded by the lines y = x, y = 0, and x = 2.

Application of Derivatives

A 10-meter ladder is leaning against a vertical wall. The bottom of the ladder is pulled along the ground away from the wall at a speed of 1 meter per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 meters from the wall?

Continuity and Differentiability

Prove that the function f(x) = |x| is continuous for all real values of x but not differentiable at x = 0.

Determinants

If A is a 3×3 matrix such that det(A) = 5, find the value of det(3A).

Matrices

If A and B are two square matrices of order 3 such that AB = BA = I, where I is the identity matrix, show that B is the inverse of A.

Hydrocarbons

11. Explain Markovnikov’s rule with an example. Why is this rule important in the addition reactions of alkenes?