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Question Number 193875 Answers: 2 Comments: 0

a,b,c,d,e,f, are + real numbers prove: (a/(b+c))+(b/(c+d))+(c/(d+e))+(d/(e+f))+(e/(f+a))+(f/(a+b))≥3

$${a},{b},{c},{d},{e},{f},\:{are}\:+\:{real}\:{numbers} \\ $$$${prove}: \\ $$$$\frac{{a}}{{b}+{c}}+\frac{{b}}{{c}+{d}}+\frac{{c}}{{d}+{e}}+\frac{{d}}{{e}+{f}}+\frac{{e}}{{f}+{a}}+\frac{{f}}{{a}+{b}}\geqslant\mathrm{3} \\ $$

Question Number 193874 Answers: 1 Comments: 0

Question Number 193871 Answers: 1 Comments: 1

Ques. 8 Find the signum (sign or sgn) of the permutation θ=(12345678). Hint : for any permutation β, take sgn β = {_(−1 if β is odd) ^(1 if β is even) Ques. 9 Prove that ∣S_n ∣ = n!. Ques. 10 Provd that for b∈S_n , sgn b = sgn b^(−1) .

$$\mathrm{Ques}.\:\mathrm{8}\: \\ $$$$\:\:\:\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{signum}\:\left(\mathrm{sign}\:\mathrm{or}\:\mathrm{sgn}\right)\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{permutation}\:\theta=\left(\mathrm{12345678}\right). \\ $$$$\mathrm{Hint}\::\:\mathrm{for}\:\mathrm{any}\:\mathrm{permutation}\:\beta,\:\mathrm{take} \\ $$$$\mathrm{sgn}\:\beta\:=\:\left\{_{−\mathrm{1}\:\:\:\:\:\:\:\mathrm{if}\:\beta\:\mathrm{is}\:\mathrm{odd}} ^{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{if}\:\beta\:\mathrm{is}\:\mathrm{even}} \right. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{9} \\ $$$$\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}\:\mid\mathrm{S}_{\mathrm{n}} \mid\:=\:\mathrm{n}!. \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{10} \\ $$$$\:\:\:\:\:\:\mathrm{Provd}\:\mathrm{that}\:\mathrm{for}\:\mathrm{b}\in\mathrm{S}_{\mathrm{n}} ,\:\mathrm{sgn}\:\mathrm{b}\:=\:\mathrm{sgn}\:\mathrm{b}^{−\mathrm{1}} \:. \\ $$

Question Number 193866 Answers: 1 Comments: 0

prove Σ_(i=1) ^(+∞) (1/n^i )=(1/(n−1)) n∈N^∗ and if n>0∧ n∈R is it right?

$${prove} \\ $$$$\:\:\underset{{i}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{i}} }=\frac{\mathrm{1}}{{n}−\mathrm{1}}\:\:\:\:\:\:{n}\in\mathbb{N}^{\ast} \\ $$$${and}\:{if}\:{n}>\mathrm{0}\wedge\:{n}\in\mathbb{R} \\ $$$${is}\:{it}\:{right}? \\ $$

Question Number 193864 Answers: 1 Comments: 0

Question Number 193863 Answers: 2 Comments: 0

lim_(x→0) ((1−(1/2)x^2 −cos ((x/(1−x^2 ))))/x^4 ) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}^{\mathrm{2}} −\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\right)}{\mathrm{x}^{\mathrm{4}} }\:=? \\ $$

Question Number 193853 Answers: 0 Comments: 0

Let n & k be positive integers and let S be a set of n points in The plane such that : For any point P of S there are at least K points of S Equidistant from p Prove that k<(1/2)+(√(2n))

$$ \\ $$$$\boldsymbol{{Let}}\:\boldsymbol{{n}}\:\&\:\boldsymbol{{k}}\:\boldsymbol{{be}}\:\boldsymbol{{positive}}\:\boldsymbol{{integers}}\:\boldsymbol{{and}}\:\boldsymbol{{let}} \\ $$$$\boldsymbol{{S}}\:\boldsymbol{{be}}\:\boldsymbol{{a}}\:\boldsymbol{{set}}\:\boldsymbol{{of}}\:\boldsymbol{{n}}\:\boldsymbol{{points}}\:\boldsymbol{{in}}\:\boldsymbol{{The}}\:\boldsymbol{{plane}}\:\boldsymbol{{such}}\:\boldsymbol{{that}}\:: \\ $$$$\boldsymbol{{For}}\:\boldsymbol{{any}}\:\boldsymbol{{point}}\:\boldsymbol{{P}}\:\boldsymbol{{of}}\:\boldsymbol{{S}}\:\boldsymbol{{there}}\:\boldsymbol{{are}}\:\boldsymbol{{at}}\:\boldsymbol{{least}}\:\boldsymbol{{K}}\:\boldsymbol{{points}}\:\boldsymbol{{of}}\:\boldsymbol{{S}}\:\boldsymbol{{Equidistant}}\:\boldsymbol{{from}}\:\boldsymbol{{p}} \\ $$$$\boldsymbol{{Prove}}\:\boldsymbol{{that}}\:\boldsymbol{{k}}<\frac{\mathrm{1}}{\mathrm{2}}+\sqrt{\mathrm{2}\boldsymbol{{n}}} \\ $$

Question Number 193852 Answers: 3 Comments: 0

Question Number 193848 Answers: 2 Comments: 0

Question Number 193847 Answers: 1 Comments: 0

Question Number 193835 Answers: 2 Comments: 1

Question Number 193821 Answers: 0 Comments: 1

Question Number 193819 Answers: 1 Comments: 0

lim_(x→0) cos ((((Π/3)+x)/x))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}cos}\:\left(\frac{\frac{\Pi}{\mathrm{3}}+{x}}{{x}}\right) \\ $$

Question Number 193809 Answers: 0 Comments: 0

A(-1, 2), B(3, 5) and C(4, 8) are the vertices of triangle ABC. Forces whose magnitudes are 5N and 3√10N act along (AB) ⃗ and (CB) ⃗ respectively. Find the direction of the resultant of the forces.

Question Number 193804 Answers: 2 Comments: 0

Ques. 6 Let (G, ∗) be a group. and let C={c∈G : c∗a = a∗c ∀a∈G}. Prove that C is subgroup of G. hence or otherwise show that C is Abelian. [Note C is called the center of group G] Ques. 7 If (G, ∗) is a group such that (a∗b)^2 = a^2 ∗b^2 (multiplicatively) for all a,b∈G. Show that G must be Abelian

$$\mathrm{Ques}.\:\mathrm{6}\: \\ $$$$\:\:\:\:\:\mathrm{Let}\:\left(\mathrm{G},\:\ast\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{group}.\:\mathrm{and}\:\mathrm{let} \\ $$$$\mathrm{C}=\left\{\mathrm{c}\in\mathrm{G}\::\:\mathrm{c}\ast\mathrm{a}\:=\:\mathrm{a}\ast\mathrm{c}\:\forall\mathrm{a}\in\mathrm{G}\right\}.\:\mathrm{Prove} \\ $$$$\mathrm{that}\:\mathrm{C}\:\mathrm{is}\:\mathrm{subgroup}\:\mathrm{of}\:\mathrm{G}.\:\mathrm{hence}\:\mathrm{or}\: \\ $$$$\mathrm{otherwise}\:\mathrm{show}\:\mathrm{that}\:\mathrm{C}\:\mathrm{is}\:\mathrm{Abelian}. \\ $$$$ \\ $$$$\left[\mathrm{Note}\:\mathrm{C}\:\mathrm{is}\:\mathrm{called}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{group}\:\mathrm{G}\right] \\ $$$$ \\ $$$$\mathrm{Ques}.\:\mathrm{7} \\ $$$$\:\:\:\:\:\mathrm{If}\:\left(\mathrm{G},\:\ast\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{group}\:\mathrm{such}\:\mathrm{that}\:\left(\mathrm{a}\ast\mathrm{b}\right)^{\mathrm{2}} \: \\ $$$$=\:\mathrm{a}^{\mathrm{2}} \ast\mathrm{b}^{\mathrm{2}} \:\left(\mathrm{multiplicatively}\right)\:\mathrm{for}\:\mathrm{all}\: \\ $$$$\mathrm{a},\mathrm{b}\in\mathrm{G}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{G}\:\mathrm{must}\:\mathrm{be}\:\mathrm{Abelian} \\ $$

Question Number 193803 Answers: 1 Comments: 0

lim_(x→2) (((√(11−x)) cos ((π/(x−2))))/(cot (x−2)))=?

$$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{11}−\mathrm{x}}\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{x}−\mathrm{2}}\right)}{\mathrm{cot}\:\left(\mathrm{x}−\mathrm{2}\right)}=? \\ $$

Question Number 193796 Answers: 0 Comments: 0

Let G be a finite group,f be an automorphism of G such that f(x)=x ⇒x=e . Then prove that, (i)∀g∈G, ∃x∈G such that g=x^(−1) f(x). (ii)If ∀x∈G , f(f(x))=x ⇒ G is Abelian.

$${Let}\:{G}\:{be}\:{a}\:{finite}\:{group},{f}\:{be}\:{an}\:{automorphism}\:{of}\:{G} \\ $$$${such}\:{that}\:{f}\left({x}\right)={x}\:\Rightarrow{x}={e}\:. \\ $$$${Then}\:{prove}\:{that}, \\ $$$$\left(\boldsymbol{{i}}\right)\forall{g}\in{G},\:\exists{x}\in{G}\:{such}\:{that}\:{g}={x}^{−\mathrm{1}} {f}\left({x}\right). \\ $$$$\left(\boldsymbol{{ii}}\right){If}\:\forall{x}\in{G}\:,\:{f}\left({f}\left({x}\right)\right)={x}\:\Rightarrow\:{G}\:{is}\:{Abelian}. \\ $$$$ \\ $$

Question Number 193794 Answers: 1 Comments: 0

Let H be a subgroup of (R,+) such that H∩[−1,1] contains a non zero element. Prove that H is cyclic.

$${Let}\:{H}\:{be}\:{a}\:{subgroup}\:{of}\:\left(\mathbb{R},+\right)\:{such}\:{that}\:{H}\cap\left[−\mathrm{1},\mathrm{1}\right]\: \\ $$$${contains}\:{a}\:{non}\:{zero}\:{element}. \\ $$$${Prove}\:{that}\:{H}\:{is}\:{cyclic}. \\ $$

Question Number 193793 Answers: 1 Comments: 4