Engineering Mathematics and Numerical Analysis — Practice

Question 1

A 2 by 2 matrix has trace 7 and determinant 10. What are its eigenvalues?

Accepted Answer

Two numbers with sum 7 and product 10: 2 and 5.

Question 2

Solve y'' + 4y' + 4y = 0.

Accepted Answer

The auxiliary equation (m+2)^2 = 0 has equal roots -2, so y = (C1 + C2 x) e^(-2x).

Question 3

State the final value theorem and a condition for its use.

Accepted Answer

The limit of f(t) as t tends to infinity equals the limit of sF(s) as s tends to 0, provided every pole of sF(s) lies in the left half of the s-plane.

Question 4

Evaluate the integral of x^2 from 0 to 2 using Simpson's 1/3 rule with h = 1.

Accepted Answer

h/3 [y0 + 4 y1 + y2] = (1/3)[0 + 4(1) + 4] = 8/3, which is exact because the function is a quadratic.

Question 5

For y' = x + y with y(0) = 1 and h = 0.1, find y(0.2) by Euler's method.

Accepted Answer

y1 = 1 + 0.1(0 + 1) = 1.1. y2 = 1.1 + 0.1(0.1 + 1.1) = 1.22.

Question 6

Use Newton-Raphson on f(x) = x^2 - 5 starting at x = 2 to find the next two iterates.

Accepted Answer

x1 = 2 - (4-5)/(2*2) = 2.25. x2 = 2.25 - (5.0625-5)/(2*2.25) = 2.25 - 0.0139 = 2.2361, close to the square root of 5.

Question 7

A component fails with probability 0.1. In a sample of 10, what is the mean and variance of the number of failures?

Accepted Answer

Binomial with n = 10 and p = 0.1: mean np = 1 and variance np(1-p) = 0.9.

Question 8

Which coefficients vanish for an odd periodic function?

Accepted Answer

All the cosine coefficients a_n, including a_0, leaving only sine terms.

Practice — Engineering Mathematics and Numerical Analysis

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