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Let G be a connected graph and let X be the set of vertices of G of odd degree. suppose that ∣X∣=2k, where k≥1 show that there are k edge-disjoint trail Q_1 , Q_2 ,...,Q_k in G such that E(G)=E(Q_1 )∪E(Q_2 )∪....∪E(Q_k )

$${Let}\:{G}\:{be}\:{a}\:{connected}\:{graph}\:{and}\:{let}\:{X}\:{be}\:{the}\:{set}\:{of}\:{vertices}\:{of}\:{G}\:{of}\:{odd}\:{degree}.\:{suppose}\:{that}\:\mid{X}\mid=\mathrm{2}{k},\:{where}\:{k}\geqslant\mathrm{1}\: \\ $$$${show}\:{that}\:{there}\:{are}\:{k}\:{edge}-{disjoint}\:{trail}\:{Q}_{\mathrm{1}} ,\:{Q}_{\mathrm{2}} ,...,{Q}_{{k}} \:{in}\:{G}\:{such}\:{that} \\ $$$${E}\left({G}\right)={E}\left({Q}_{\mathrm{1}} \right)\cup{E}\left({Q}_{\mathrm{2}} \right)\cup....\cup{E}\left({Q}_{{k}} \right) \\ $$

Question Number 94418 Answers: 3 Comments: 0

1.Find value of 𝛉 in Mean Value theorem f(x+h)= f(x)+hf^( ′) (x+θh), if f(x)=(1/x) 2.If (√(1−x^2 )) +(√(1−y^2 )) = k(x−y) prove that (dy/dx) = ((√(1−y^2 ))/(√(1−x^2 ))). 3. Solve the differential equation: x^2 (d^2 y/dx^2 )−3x(dy/dx)+3y = x^2 (2x−1).

$$\:\mathrm{1}.\mathrm{F}\boldsymbol{\mathrm{ind}}\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\theta}\:\mathrm{in}\:\mathrm{Mean}\:\mathrm{Value}\:\mathrm{theorem} \\ $$$$\:\mathrm{f}\left(\mathrm{x}+\mathrm{h}\right)=\:\mathrm{f}\left(\mathrm{x}\right)+\mathrm{hf}^{\:'} \left(\mathrm{x}+\theta\mathrm{h}\right),\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{x}} \\ $$$$\:\:\mathrm{2}.\mathrm{If}\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\:\:\:+\sqrt{\mathrm{1}−\mathrm{y}^{\mathrm{2}} }\:\:\:=\:\mathrm{k}\left(\mathrm{x}−\mathrm{y}\right)\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\sqrt{\mathrm{1}−\mathrm{y}^{\mathrm{2}} }}{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}. \\ $$$$\:\mathrm{3}.\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$$\:\:\:\mathrm{x}^{\mathrm{2}} \frac{\mathrm{d}^{\mathrm{2}} \mathrm{y}}{\mathrm{dx}^{\mathrm{2}} }−\mathrm{3x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{3y}\:=\:\mathrm{x}^{\mathrm{2}} \left(\mathrm{2x}−\mathrm{1}\right). \\ $$

Question Number 94416 Answers: 0 Comments: 0

A direct similitude of center,Ω, transforms point A into point A′ and point B into point B′. Prove that there existe a direct similitude of center,Ω, which transforms A and B into A′ and B′.

$$\mathrm{A}\:\mathrm{direct}\:\mathrm{similitude}\:\mathrm{of}\:\mathrm{center},\Omega,\:\mathrm{transforms}\:\mathrm{point}\:\mathrm{A}\:\mathrm{into}\:\mathrm{point}\:\mathrm{A}' \\ $$$$\mathrm{and}\:\mathrm{point}\:\mathrm{B}\:\mathrm{into}\:\mathrm{point}\:\mathrm{B}'.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{existe}\:\mathrm{a} \\ $$$$\mathrm{direct}\:\mathrm{similitude}\:\mathrm{of}\:\mathrm{center},\Omega,\:\mathrm{which}\:\mathrm{transforms}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{into}\:\mathrm{A}'\:\mathrm{and}\:\mathrm{B}'. \\ $$

Question Number 94397 Answers: 1 Comments: 0