9. Consider the two-parameter family of linear systemsab- (2) YY.adYdtIn the ab-plane, identify all regions where this system possesses a saddle, a sink, aspiral sink, and so on. [Hint: Draw a picture of the ab-plane and shade each point(a, b) of the plane a different color depending on the type of linear system for thatchoice (a, b) of the parameters.]

Answered: Image /qna-images/answer/eed93a25-eced-4094-8d5e-4e51b14f4b57.jpg

Answered: 9. Consider the two-parameter family of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the condition for the two lines to be coplanar. a1 + bjx + C7x + d, =0 = az + b2x + C2x + d2 az + b3x + C3x + d3 =0 = a4 + bạx + C4X + d4 I know that the general line passing through the first lines can be represented as a, + b;x + C,x + d, +k(a2 + b2x + C2x + d2), for real number k Also, the general line passing through the second lines can be represented as az + b3x + C3x + dz +d(a4 + bạx + C4x + d4), for real number r How do we then, prove the required result? Thank you!
The point(s) of intersection between the y-axis and the plane 2x-y-z=-6 is/are: O {(x, y, z): x = 2t-3, y=-t, z = t} O (-3,0,0) O {(x, y, z): x = 0, y =z=-6} O (0,-1,1) None of the given options IRIS is sharing your screen. Stop sharing Hide
3. Express the following regions in the stated coordinate systems. You do not need to justify your answer but showing your though process may help with partial marks. Your answer should be a set in the following form: {(V1, V2, V3) : a ≤ v₁ ≤ b, f(v1) ≤ v2 ≤ g(v1), Ø(V1, V2) ≤ v3 ≤ (v₁, v₂)} where (v₁, V2, V3) are the three variables of the coordinate system in questions in any order, a, b R are constants, f, g are functions of v₁ and 6, are functions of v₁, V2. (a) The region outside the cylinder x² + y² ≤ r² and inside the sphere x² + y² + z² r< R. Express this in the cylindrical and spherical coordinate systems. - R² assuming (b) The region in the first octant (x, y, z ≥ 0) below the plane x + 2y + 3z = 6. Express this in the cylindrical, spherical and Cartesian coordinate system.
11. Write the equation of the plane that contains the three points (1,0,1), (1,0,2), and (0,1,1). (This will use both of the preceding problems.)
(a) Find the locus of a point, the sum of squares of whose distances from the planes 6 and x + y + z = X z = 0, x 2y + z 1 is 36 - (b) A ray of light passing through the point 4(1,2,3), strikes the plane x + y + z = 12 at 8 and on reflection passes through point C(3, 5, 9). Find the coordinates of point B (c) Find the radius of the circular section in which the sphere vec | r | = r. (i+j+ k) = 3√√3 5 cut by the plane
The line of intersection of the planes 2x +y – z= 4 and 4x – y+ 3z = 2 has equation %3D O a) (x, y, z) = (1, 2, 0) + t(1, –5, 1) O b) (x, y, z) = (1, –2,0) + t(2, –10, –6) O c) None of these O d) (x, y, z) = (1, 2, 0) + t(1, –5, -3)
1. Is L = {1 − 2x + 3x^2, x − 2x^2, 2 − 5x + 8x^2} linearly independent? If yes, show why. If not, show a dependence relationship. Is L = {1 + 2x + 2x^2, x − 2x^2, −2 − 3x − 5x^2} linearly independent? If yes, show why. If not, show a dependence relationship.
Let ∆ABC have vertices at A = (0,0), B = (3,4), and C = (12,5). Let ∆HFK have vertices at H = (6,0), F = (8,1.5), and K = (12,2.5). Is ∆ABC similar to ∆HFK ? Which similarity theorem would you use to check whether ∆ABC and ∆HFK are similar? Determine whether ∆ABC and ∆HFK are similar.
2 Determine if the statements below are True or False. If it's True, explain why. If it's False explain why not, or simply give an example demonstrating why it's false. A correct choice of "True" or "False" with no explanation will not receive any credit. 2.1 If (x, y, z} is a linearly independent subset of R", then {x, x + y,x+ y+ z} is also linearly independent. 2.2 If (x, y} and {z, w} are both linearly independent sets in R", then {x, y,z, w} is also linearly independent.

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