a) Let M be a matching in a graph G such that G has no M-augmenting path. Show that M is a maximum matching. Let G be a bipartite graph with bipartition (X,Y). For SCX, let Ej be the set of edges in G incident on some vertex in S, and let E2 be the set of edges in G incident with some vertex in N(S). Is it true in general that E¡ C E2? Why? Prove that if G is a graph with no isolated vertices, then a'(G)+B'(G) = n(G) b) d) Which of the following is true ? Justify. i) Every tree has a perfect matching. ii) Every tree has at most one perfact matching.

Q: Solve. ET + 16y-2y² - 32 = 0

A: We have given an equation  y3+16y-2y2-32=0 yy2+16-2y2+16=0 y2+16y-2=0 y2+16=0 or y-2=0 y2=-16 or y=2…

Q: Solve for b. 20=b+a (10

A: Given equation 20 = b + a

Q: FILD 3xdx x+2 IND

A: Solution:- We have to find ∫023xx2+2dx it is definite integration. =∫023xx2+2dx=3∫02xx2+2dx   let…

Q: x+ y+ z = -4 2x+ 4y+ 2z -x+ 8y- 3z - 16 %3D -32 %3D

A:

Q: i 12-4x) de

A:

Q: Solve for X. مل 3 5 1 -4 - -9 7 = -3X - 5 -6 1 2 -9 -4 8

A:

Q: solve for A =  2 4 0 0     4 2 0 0 0 0 2 4 0 0 4 2

A:

Q: 6. Solve for x. 1S- 28 6x +7

A:

Q: Зх + 9 6x P R

A: AAS criteria: If two angles and an opposite side of one triangle are congruent to the two angles and…

Q: For AY 1 1. Find b. 4 b 2 1370

A:

Q: 7. solve the following x+2 X-2

A:

Q: 4 + (In 8x)2 xp

A: Given,          ∫4+ln8x2xdx

Q: 4. Find tricingeo. FincX 3.2 2.7

A:

Q: Solve for a a %3D 2m²

A: Solve for a

Q: (x, 5), (2, 6) m = 3/4

A: Let the equation of line be y=mx + b  ...(i) We have m=3/4 = 0.75

Q: -3+M =10 9

A: Given: -3+m9=10

Q: (d) a₁ = 2, an+1 = 1 3- an

A:

Q: 4. Compute 23 2 3 4 3 2 0

A:

Q: Factori 1-800/2+12 4√4

A: Let,

Q: v-10 3 三 v² – 5v+4 v-1 v-4

A:

Q: 11. Solve 6x + - 5 = 3x - 3 10

A:

Q: 3. Solve 25a +b+c = 25 64a + 8b + c = 71 144a +12b+c =155

A: In order to solve the system of 3 equations in 3 variables, we first write the equation in the…

Q: 2x x + 2 12 8

A:

Q: -5 -6 = 2X – 5 -4 -5 3 -6 6 3 X =

A: Given: The matrix equation is, 6-59-4-53=2X-521-6-663 The objective is to find the matrix X.

Q: Find + 7a +3 dæ + C

A:

Q: - 4x, =-10 X, +3x, = 2 3x, + 5x, +8x, =-6 2x 8.

A:

Q: Solve. √?−4=√?+2

A: Given: √?−4=√?+2 we assume that variables under the square root are different because if they are…

Q: dt 10 A 3 +t+c B) 128+금2. 8t + - t2 D +C + 20

A:

Q: Solve for b. a+b b = 0 L 00 X G

A: Given expression  a + b4 = L

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.