Advanced Mathematics

Let W be a subspace of a vector space V and let v₁, v₂, v₃ ∈ W. Prove then that every linear combination of these vectors is also in W.

Prove that a set of vectors is linearly dependent if and only if atleast one vector in the set is a linear combination of the others.

Find the last digit of 7¹⁰⁰.

Given that 5x = 6 (mod 8), find x.

A tightly stretched flexible string has its ends fixed at x = 0 and x = l. At time t = 0, the string is given a shape defined by f(x) = μx(l - x), μ is a constant and then released. Find the displacement y(x, t) of any point x of the string at any time t > 0.

A binomial variable X satisfies the relation 9P(X = 4) = P(X = 2) when n = 6. Find the value of the parameter p and P(X = 1).

Consider the Markov chain with three states S = (1, 2, 3) that has the following transition matrix.

draw the stable transition diagram of the chain and if we know P(X₁ = 1) = P(X₁ = 2) = ¹⁄₄. Then find P(X₃ = 1, X₂ = 2, X₃ = 1).

Find the expected value of the product of points on n dice.

Find the variance of the number of successes in a series of n independent trials in which the probability of success in the j^th trial is p_j.

Calls in a telephone system arrive randomly at an exchange at the rate of 140 per hour. If there are a very large number of lines available to handle the calls which last an average of 3 minutes, what is the average number of lines in use? Estimate the 90^th and 95^th percentile of number of lines in use.

Describe test of significance and standard error. A machine produced 16 defective articles in a batch of 500. After overhauling it produced 3 defective in a batch of 100. Has the machine improved?

Find the functional for the ordinary differential equation

Describe Rayleigh-Ritz Method.

Test for extremum of the functional

Find the smallest eigen value approximately using Rayleigh-Ritz method y'' + λy = 0, y'(0) = y(1) = 0.

Find the approximate solution of the boundary value problem y'' + y = 1, y'(0) = 0, y(1) = 0.