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Question Number 193759 Answers: 2 Comments: 0

Ques. 5 Prove that if a,b are any elements of a group (G, ∗), then the equation y∗a=b has a unique solution in (G, ∗). Ques. 6 a) Show that the set G of all non-zero complex numbers, is a group under multiplication of complex numbers. b) Show that H={a∈G : a_1 ^2 + a_2 ^2 = 1}, where a_1 = Re a and a_2 = Im a is a subgroup of G.

$$\mathrm{Ques}.\:\mathrm{5}\: \\ $$$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{a},\mathrm{b}\:\mathrm{are}\:\mathrm{any}\:\mathrm{elements}\:\mathrm{of}\:\:\mathrm{a}\: \\ $$$$\mathrm{group}\:\left(\mathrm{G},\:\ast\right),\:\mathrm{then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{y}\ast\mathrm{a}=\mathrm{b} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{in}\:\left(\mathrm{G},\:\ast\right). \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{6}\: \\ $$$$\left.\:\:\:\:\:\mathrm{a}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{set}\:\mathrm{G}\:\mathrm{of}\:\mathrm{all}\:\mathrm{non}-\mathrm{zero}\: \\ $$$$\mathrm{complex}\:\mathrm{numbers},\:\mathrm{is}\:\mathrm{a}\:\mathrm{group}\:\mathrm{under} \\ $$$$\mathrm{multiplication}\:\mathrm{of}\:\mathrm{complex}\:\mathrm{numbers}. \\ $$$$ \\ $$$$\left.\:\:\:\:\:\:\mathrm{b}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{H}=\left\{\mathrm{a}\in\mathrm{G}\::\:\mathrm{a}_{\mathrm{1}} ^{\mathrm{2}} \:+\:\mathrm{a}_{\mathrm{2}} ^{\mathrm{2}} \:=\:\mathrm{1}\right\}, \\ $$$$\mathrm{where}\:\mathrm{a}_{\mathrm{1}} \:=\:\mathrm{Re}\:\mathrm{a}\:\mathrm{and}\:\mathrm{a}_{\mathrm{2}} \:=\:\mathrm{Im}\:\mathrm{a}\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\mathrm{subgroup}\:\mathrm{of}\:\mathrm{G}. \\ $$

Question Number 193757 Answers: 0 Comments: 0

Question Number 193756 Answers: 1 Comments: 0

Question Number 193754 Answers: 1 Comments: 0

Question Number 193750 Answers: 1 Comments: 0

Question Number 193749 Answers: 1 Comments: 0

Question Number 193734 Answers: 1 Comments: 0

I = ∫_0 ^∞ ((x^2 (tan^(−1) (x))^2 )/(x^2 −x+2))dx Help!

$$\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}\right)\right)^{\mathrm{2}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{2}}\mathrm{dx} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 193733 Answers: 1 Comments: 0

Find the ordinary differential equation satisfy by: y = x^n (A + Blogx)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{ordinary}\:\mathrm{differential}\:\mathrm{equation} \\ $$$$\mathrm{satisfy}\:\mathrm{by}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{y}\:\:=\:\:\mathrm{x}^{\mathrm{n}} \left(\mathrm{A}\:\:+\:\:\mathrm{Blogx}\right) \\ $$

Question Number 193726 Answers: 2 Comments: 0

Question Number 193721 Answers: 3 Comments: 0

prove that a^(log_a N) =N

$${prove}\:{that}\:{a}^{{log}_{{a}} {N}} ={N} \\ $$

Question Number 193719 Answers: 1 Comments: 1

Question Number 193715 Answers: 0 Comments: 0

Question Number 193714 Answers: 1 Comments: 3

Question Number 193712 Answers: 0 Comments: 0

Ques. 3 Show that (Z_4 , +) is a group. Hence find the order of the group and of the element 2∈Z_4 , if it exists. Ques. 4 Prove the if a,b are any elements of a group (G, ∗), then the equation y∗a = b has a unique solution in (G, ∗). Help!

$$\mathrm{Ques}.\:\mathrm{3}\: \\ $$$$\:\:\:\:\:\mathrm{Show}\:\mathrm{that}\:\left(\mathbb{Z}_{\mathrm{4}} ,\:+\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{group}.\:\mathrm{Hence} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{order}\:\mathrm{of}\:\mathrm{the}\:\mathrm{group}\:\mathrm{and}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{element}\:\mathrm{2}\in\mathbb{Z}_{\mathrm{4}} ,\:\mathrm{if}\:\mathrm{it}\:\mathrm{exists}. \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{4} \\ $$$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{if}\:\mathrm{a},\mathrm{b}\:\mathrm{are}\:\mathrm{any}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{a}\: \\ $$$$\mathrm{group}\:\left(\mathrm{G},\:\ast\right),\:\mathrm{then}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{y}\ast\mathrm{a}\:=\:\mathrm{b} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{in}\:\left(\mathrm{G},\:\ast\right). \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 193707 Answers: 1 Comments: 0

Question Number 193706 Answers: 0 Comments: 0

Question Number 193691 Answers: 1 Comments: 1

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Question Number 193688 Answers: 1 Comments: 0

Question Number 193687 Answers: 0 Comments: 2

Question Number 193686 Answers: 1 Comments: 0

Question Number 193685 Answers: 2 Comments: 0

(f(x))^2 −4xf(x)+3=0 f^(−1) (3)=?

$$\:\:\:\:\:\:\:\left(\mathrm{f}\left(\mathrm{x}\right)\right)^{\mathrm{2}} −\mathrm{4xf}\left(\mathrm{x}\right)+\mathrm{3}=\mathrm{0} \\ $$$$\:\:\:\:\mathrm{f}^{−\mathrm{1}} \left(\mathrm{3}\right)=?\: \\ $$

Question Number 193711 Answers: 1 Comments: 0

Ques. 1 Let G be a group and b a fixed element of G. Prove that the map G into G given by x→bx is bijective Ques. 2 Let G be a group and g be an element of G. Prove that a) (g^(−1) )^(−1) =g b) g^m g^n = g^(m+n) Help!

$$\mathrm{Ques}.\:\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{Let}\:\mathrm{G}\:\mathrm{be}\:\mathrm{a}\:\mathrm{group}\:\mathrm{and}\:\mathrm{b}\:\mathrm{a}\:\mathrm{fixed}\:\mathrm{element} \\ $$$$\mathrm{of}\:\mathrm{G}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{map}\:\mathrm{G}\:\mathrm{into}\:\mathrm{G}\:\mathrm{given} \\ $$$$\mathrm{by}\:\mathrm{x}\rightarrow\mathrm{bx}\:\mathrm{is}\:\mathrm{bijective} \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{2}\: \\ $$$$\:\:\:\:\:\mathrm{Let}\:\mathrm{G}\:\mathrm{be}\:\mathrm{a}\:\mathrm{group}\:\mathrm{and}\:\mathrm{g}\:\mathrm{be}\:\mathrm{an}\:\mathrm{element}\: \\ $$$$\mathrm{of}\:\mathrm{G}.\:\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\left.\mathrm{a}\right)\:\left(\mathrm{g}^{−\mathrm{1}} \right)^{−\mathrm{1}} =\mathrm{g} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{g}^{\mathrm{m}} \mathrm{g}^{\mathrm{n}} \:=\:\mathrm{g}^{\mathrm{m}+\mathrm{n}} \: \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 193677 Answers: 1 Comments: 0

lim_(x→1) ((2arctan^2 x−(π/8))/(x^2 −1))=...

$${li}\underset{{x}\rightarrow\mathrm{1}} {{m}}\:\frac{\mathrm{2}{arctan}^{\mathrm{2}} {x}−\frac{\pi}{\mathrm{8}}}{{x}^{\mathrm{2}} −\mathrm{1}}=... \\ $$

Question Number 193676 Answers: 1 Comments: 0

Solve arcsin(2x)+arcsin(x(√3))=arcsin(x)

$${Solve} \\ $$$${arcsin}\left(\mathrm{2}{x}\right)+{arcsin}\left({x}\sqrt{\mathrm{3}}\right)={arcsin}\left({x}\right) \\ $$

Question Number 193671 Answers: 1 Comments: 0

Question Number 193670 Answers: 1 Comments: 1

show that ∫_c e^(1/z^2 ) dz = 0 when ∣z∣ <1

$${show}\:{that}\:\int_{{c}} \:{e}^{\frac{\mathrm{1}}{{z}^{\mathrm{2}} }} \:{dz}\:=\:\mathrm{0}\:{when}\:\mid{z}\mid\:<\mathrm{1} \\ $$

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