Question and Answers Forum

AllQuestion and Answers: Page 262

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

Question Number 193792 Answers: 1 Comments: 0

Question Number 193790 Answers: 2 Comments: 0

Question Number 193785 Answers: 1 Comments: 0

prove that (1/2)×(3/4)×(5/6)×(7/8)×(9/(10))×...×((99)/(100))<(1/(10))

$${prove}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{5}}{\mathrm{6}}×\frac{\mathrm{7}}{\mathrm{8}}×\frac{\mathrm{9}}{\mathrm{10}}×...×\frac{\mathrm{99}}{\mathrm{100}}<\frac{\mathrm{1}}{\mathrm{10}}\:\:\: \\ $$

Question Number 193769 Answers: 0 Comments: 0

Question Number 193768 Answers: 1 Comments: 0

f(x) = 2+∫_0 ^( x) (2t+f(t))^2 dt then ∫_(−1) ^2 f(x) dx =

$$\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{2}+\underset{\mathrm{0}} {\overset{\:\mathrm{x}} {\int}}\left(\mathrm{2t}+\mathrm{f}\left(\mathrm{t}\right)\right)^{\mathrm{2}} \mathrm{dt}\: \\ $$$$\:\:\mathrm{then}\:\underset{−\mathrm{1}} {\overset{\mathrm{2}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:= \\ $$

Question Number 193767 Answers: 1 Comments: 0

prove that c^(log_b a) =a^(log_b c)

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

Question Number 193761 Answers: 2 Comments: 0

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

Pg 257 Pg 258 Pg 259 Pg 260 Pg 261 Pg 262 Pg 263 Pg 264 Pg 265 Pg 266

Contact: info@tinkutara.com