Question and Answers Forum

AllQuestion and Answers: Page 1817

Question Number 21294 Answers: 1 Comments: 0

Let z_1 , z_2 , z_3 be complex numbers, not all real, such that ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 and 2(z_1 + z_2 + z_3 ) − 3z_1 z_2 z_3 ∈ R. Prove that max(arg z_1 , arg z_2 , arg z_3 ) ≥ (π/6) . Where 0 < arg(z_1 ), arg(z_2 ), arg(z_3 ) < 2π.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers},\:\mathrm{not} \\ $$$$\mathrm{all}\:\mathrm{real},\:\mathrm{such}\:\mathrm{that}\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\mathrm{and}\:\mathrm{2}\left({z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \right)\:−\:\mathrm{3}{z}_{\mathrm{1}} {z}_{\mathrm{2}} {z}_{\mathrm{3}} \:\in\:{R}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{max}\left(\mathrm{arg}\:{z}_{\mathrm{1}} ,\:\mathrm{arg}\:{z}_{\mathrm{2}} ,\:\mathrm{arg}\:{z}_{\mathrm{3}} \right)\:\geqslant \\ $$$$\frac{\pi}{\mathrm{6}}\:.\:\mathrm{Where}\:\mathrm{0}\:<\:\mathrm{arg}\left({z}_{\mathrm{1}} \right),\:\mathrm{arg}\left({z}_{\mathrm{2}} \right),\:\mathrm{arg}\left({z}_{\mathrm{3}} \right) \\ $$$$<\:\mathrm{2}\pi. \\ $$

Question Number 21293 Answers: 1 Comments: 0

Let n be an even positive integer such that (n/2) is odd and let α_0 , α_1 , ...., α_(n−1) be the complex roots of unity of order n. Prove that Π_(k=0) ^(n−1) (a + bα_k ^2 ) = (a^(n/2) + b^(n/2) )^2 for any complex numbers a and b.

$$\mathrm{Let}\:{n}\:\mathrm{be}\:\mathrm{an}\:\mathrm{even}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such} \\ $$$$\mathrm{that}\:\frac{{n}}{\mathrm{2}}\:\mathrm{is}\:\mathrm{odd}\:\mathrm{and}\:\mathrm{let}\:\alpha_{\mathrm{0}} ,\:\alpha_{\mathrm{1}} ,\:....,\:\alpha_{{n}−\mathrm{1}} \:\mathrm{be} \\ $$$$\mathrm{the}\:\mathrm{complex}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{unity}\:\mathrm{of}\:\mathrm{order}\:{n}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({a}\:+\:{b}\alpha_{{k}} ^{\mathrm{2}} \right)\:=\:\left({a}^{\frac{{n}}{\mathrm{2}}} \:+\:{b}^{\frac{{n}}{\mathrm{2}}} \right)^{\mathrm{2}} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{complex}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}. \\ $$

Question Number 21292 Answers: 0 Comments: 0

Let a,b∈Z 0<a<b How would you find the maximum/ largest prime gap in (a, b)? Note: Prime gaps are the distance between consecutive primes. e.g. 7 and 11 has a prime gap 4 p_k ∈P ∴∀p_x ∀p_(x+1) ∈(a,b):p_(x+1) >p_x p_(x+1) and p_x are consecutive primes Lets denote δ_x =p_(x+1) −p_x as prime gap for (1, 20), the primes are 2,3,5,7,11,13,17 The prime gaps are: 1,2,2,4,2,4 Therefore the largest δ = 4 Is there a more general method?

$$\mathrm{Let}\:\:{a},{b}\in\mathbb{Z} \\ $$$$\mathrm{0}<{a}<{b} \\ $$$$\: \\ $$$$\mathrm{How}\:\mathrm{would}\:\mathrm{you}\:\mathrm{find}\:\mathrm{the}\:\mathrm{maximum}/ \\ $$$$\mathrm{largest}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{in}\:\left({a},\:{b}\right)? \\ $$$$ \\ $$$$\mathrm{Note}: \\ $$$$\mathrm{Prime}\:\mathrm{gaps}\:\mathrm{are}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{consecutive}\:\mathrm{primes}. \\ $$$$\mathrm{e}.\mathrm{g}.\:\:\:\mathrm{7}\:\mathrm{and}\:\mathrm{11}\:\mathrm{has}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{gap}\:\mathrm{4} \\ $$$$\: \\ $$$${p}_{{k}} \in\mathbb{P} \\ $$$$\therefore\forall{p}_{{x}} \forall{p}_{{x}+\mathrm{1}} \in\left({a},{b}\right):{p}_{{x}+\mathrm{1}} >{p}_{{x}} \\ $$$${p}_{{x}+\mathrm{1}} \:\mathrm{and}\:{p}_{{x}} \:\mathrm{are}\:\mathrm{consecutive}\:\mathrm{primes} \\ $$$$\mathrm{Lets}\:\mathrm{denote}\:\delta_{{x}} ={p}_{{x}+\mathrm{1}} −{p}_{{x}} \:\mathrm{as}\:\mathrm{prime}\:\mathrm{gap} \\ $$$$\: \\ $$$$\mathrm{for}\:\left(\mathrm{1},\:\mathrm{20}\right),\:\mathrm{the}\:\mathrm{primes}\:\mathrm{are}\:\mathrm{2},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{11},\mathrm{13},\mathrm{17} \\ $$$$\mathrm{The}\:\mathrm{prime}\:\mathrm{gaps}\:\mathrm{are}: \\ $$$$\mathrm{1},\mathrm{2},\mathrm{2},\mathrm{4},\mathrm{2},\mathrm{4} \\ $$$$\mathrm{Therefore}\:\mathrm{the}\:\mathrm{largest}\:\delta\:=\:\mathrm{4} \\ $$$$\: \\ $$$$\mathrm{Is}\:\mathrm{there}\:\mathrm{a}\:\mathrm{more}\:\mathrm{general}\:\mathrm{method}? \\ $$

Question Number 21297 Answers: 1 Comments: 0

Question Number 21282 Answers: 0 Comments: 0

soit (u_n )_(n∈N^∗ ) une suite a termes positifs telle que: ∀n∈N^∗ ,Σ_(k=1) ^n u_k ^3 =(Σ_(k=1) ^n u_k )^2 montrer que ∀n∈N^∗ , u_n =n

$${soit}\:\left({u}_{{n}} \right)_{{n}\in\mathbb{N}^{\ast} } {une}\:{suite}\:{a}\:{termes}\:{positifs}\:{telle}\:{que}: \\ $$$$\forall{n}\in\mathbb{N}^{\ast} ,\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{u}_{{k}} ^{\mathrm{3}} =\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{u}_{{k}} \right)^{\mathrm{2}} \\ $$$${montrer}\:{que}\:\forall{n}\in\mathbb{N}^{\ast} ,\:{u}_{{n}} ={n} \\ $$

Question Number 21280 Answers: 1 Comments: 0

cosec^2 67^0 −tan^2 23^0 =

$${cosec}^{\mathrm{2}} \mathrm{67}^{\mathrm{0}} −{tan}^{\mathrm{2}} \mathrm{23}^{\mathrm{0}} = \\ $$

Question Number 21279 Answers: 1 Comments: 0

1+sinθ/cosθ=

$$\mathrm{1}+{sin}\theta/{cos}\theta= \\ $$

Question Number 21296 Answers: 1 Comments: 3

Question Number 21276 Answers: 1 Comments: 0

Question Number 21273 Answers: 0 Comments: 0

cos 70+4cos 70

$$\mathrm{cos}\:\mathrm{70}+\mathrm{4cos}\:\mathrm{70} \\ $$

Question Number 21272 Answers: 1 Comments: 0

Let α, β ∈ (−π, π) be such that cos(α − β) = 1 and cos(α + β) = (1/e). The number of pairs of α, β satisfying the above system of equation is

$$\mathrm{Let}\:\alpha,\:\beta\:\in\:\left(−\pi,\:\pi\right)\:\mathrm{be}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{cos}\left(\alpha\:−\:\beta\right)\:=\:\mathrm{1}\:\mathrm{and}\:\mathrm{cos}\left(\alpha\:+\:\beta\right)\:=\:\frac{\mathrm{1}}{{e}}. \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{pairs}\:\mathrm{of}\:\alpha,\:\beta\:\mathrm{satisfying} \\ $$$$\mathrm{the}\:\mathrm{above}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equation}\:\mathrm{is} \\ $$

Question Number 21270 Answers: 0 Comments: 2

why any infinitely differentiable function is a power series? mathematically, if f(x) is A infinitely differentiable function then why f(x)=ax^0 +bx^1 +cx^2 +dx^3 +ex^4 +..... for example sin(x)=x−(x^3 /(3!))+(x^5 /(5!))−(x^7 /(7!))+.....up to∞

$$\mathrm{why}\:\mathrm{any}\:\mathrm{infinitely}\:\mathrm{differentiable}\: \\ $$$$\mathrm{function}\:\mathrm{is}\:\mathrm{a}\:\mathrm{power}\:\mathrm{series}? \\ $$$${mathematically}, \\ $$$${if}\:{f}\left({x}\right)\:{is}\:{A}\:{infinitely}\:{differentiable} \\ $$$${function}\:{then}\:{why} \\ $$$${f}\left({x}\right)={ax}^{\mathrm{0}} +{bx}^{\mathrm{1}} +{cx}^{\mathrm{2}} +{dx}^{\mathrm{3}} +{ex}^{\mathrm{4}} +..... \\ $$$${for}\:{example} \\ $$$${sin}\left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}!}+.....{up}\:{to}\infty \\ $$$$ \\ $$

Question Number 21268 Answers: 1 Comments: 0

if A+B+C=π so proof sin A+sin B+sin C=4cos (A/2)cos (B/2)cos (C/2)

$${if}\:{A}+{B}+{C}=\pi \\ $$$${so}\:{proof}\: \\ $$$$\mathrm{sin}\:{A}+\mathrm{sin}\:{B}+\mathrm{sin}\:{C}=\mathrm{4cos}\:\frac{{A}}{\mathrm{2}}\mathrm{cos}\:\frac{{B}}{\mathrm{2}}\mathrm{cos}\:\frac{{C}}{\mathrm{2}} \\ $$

Question Number 21267 Answers: 0 Comments: 0

cos A+cos B+cos C=1+(r/R)

$$\mathrm{cos}\:{A}+\mathrm{cos}\:{B}+\mathrm{cos}\:{C}=\mathrm{1}+\frac{{r}}{{R}} \\ $$

Question Number 21266 Answers: 1 Comments: 0

((b^2 −c^2 )/(tan A))+((c^2 −a^2 )/(tan B))+((a^2 −b^2 )/(tan C))=0

$$\frac{{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{tan}\:{A}}+\frac{{c}^{\mathrm{2}} −{a}^{\mathrm{2}} }{\mathrm{tan}\:{B}}+\frac{{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{tan}\:{C}}=\mathrm{0} \\ $$

Question Number 21264 Answers: 1 Comments: 0

2x+3y+2z=2

$$\mathrm{2x}+\mathrm{3y}+\mathrm{2z}=\mathrm{2} \\ $$

Question Number 21262 Answers: 0 Comments: 0

Question Number 21260 Answers: 1 Comments: 0

if (θ−ϕ) subtle and sin θ+sin ϕ=(√(3(cos ϕ)) −cos θ) so proof sin 3θ+sin 3ϕ=0

$${if}\:\left(\theta−\varphi\right)\:{subtle}\:{and}\:\mathrm{sin}\:\theta+\mathrm{sin}\:\varphi=\sqrt{\mathrm{3}\left(\mathrm{cos}\:\varphi\right.} \\ $$$$\left.−\mathrm{cos}\:\theta\right) \\ $$$${so}\:{proof}\:\mathrm{sin}\:\mathrm{3}\theta+\mathrm{sin}\:\mathrm{3}\varphi=\mathrm{0} \\ $$

Question Number 21259 Answers: 1 Comments: 0

2cos (π/(13))cos ((9π)/(13))+cos ((3π)/(13))+cos ((5π)/(13))=0

$$\mathrm{2cos}\:\frac{\pi}{\mathrm{13}}\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}}=\mathrm{0} \\ $$

Question Number 21253 Answers: 1 Comments: 0

the hcf of two numbers is 75 and their lcm is 3375 if one of the numbers is 675 find the other number

$${the}\:{hcf}\:{of}\:{two}\:{numbers}\:{is}\:\mathrm{75}\:{and}\:{their}\:{lcm}\:{is}\:\mathrm{3375}\:{if}\:{one}\:{of}\:{the}\:{numbers}\:{is}\:\mathrm{675}\:{find}\:{the}\:{other}\:{number} \\ $$

Question Number 21252 Answers: 0 Comments: 0

prove,∀x_1 ,...,x_n y_1 ,...,y_n ∈R^+ (√(x_1 x_2 ...x_n ))+(√(y_1 y_2 ...y_n ))≤(√((x_1 +y_1 )(x_2 +y_2 )...(x_n +y_n )))

$${prove},\forall{x}_{\mathrm{1}} ,...,{x}_{{n}} {y}_{\mathrm{1}} ,...,{y}_{{n}} \in\mathbb{R}^{+} \\ $$$$\sqrt{{x}_{\mathrm{1}} {x}_{\mathrm{2}} ...{x}_{{n}} }+\sqrt{{y}_{\mathrm{1}} {y}_{\mathrm{2}} ...{y}_{{n}} }\leqslant\sqrt{\left({x}_{\mathrm{1}} +{y}_{\mathrm{1}} \right)\left({x}_{\mathrm{2}} +{y}_{\mathrm{2}} \right)...\left({x}_{{n}} +{y}_{{n}} \right)} \\ $$

Question Number 21251 Answers: 0 Comments: 0

resolve : ∀x∈R, (√(x+1))∣x−2∣=(√(x+2))∣x−1∣−2

$${resolve}\:: \\ $$$$\forall{x}\in\mathbb{R},\:\sqrt{{x}+\mathrm{1}}\mid{x}−\mathrm{2}\mid=\sqrt{{x}+\mathrm{2}}\mid{x}−\mathrm{1}\mid−\mathrm{2} \\ $$$$ \\ $$

Question Number 21249 Answers: 0 Comments: 1

A particle slides down a frictionless parabolic (y = x^2 ) track (A − B − C) starting from rest at point A. Point B is at the vertex of parabola and point C is at a height less than that of point A. After C, the particle moves freely in air as a projectile. If the particle reaches highest point at P, then (a) KE at P = KE at B (b) height at P = height at A (c) total energy at P = total energy at A (d) time of travel from A to B = time of travel from B to P.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{slides}\:\mathrm{down}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{parabolic}\:\left({y}\:=\:{x}^{\mathrm{2}} \right)\:\mathrm{track}\:\left({A}\:−\:{B}\:−\:{C}\right) \\ $$$$\mathrm{starting}\:\mathrm{from}\:\mathrm{rest}\:\mathrm{at}\:\mathrm{point}\:{A}.\:\mathrm{Point}\:{B} \\ $$$$\mathrm{is}\:\mathrm{at}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{parabola}\:\mathrm{and}\:\mathrm{point}\:{C} \\ $$$$\mathrm{is}\:\mathrm{at}\:\mathrm{a}\:\mathrm{height}\:\mathrm{less}\:\mathrm{than}\:\mathrm{that}\:\mathrm{of}\:\mathrm{point}\:{A}. \\ $$$$\mathrm{After}\:{C},\:\mathrm{the}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{freely}\:\mathrm{in}\:\mathrm{air} \\ $$$$\mathrm{as}\:\mathrm{a}\:\mathrm{projectile}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{reaches} \\ $$$$\mathrm{highest}\:\mathrm{point}\:\mathrm{at}\:{P},\:\mathrm{then} \\ $$$$\left({a}\right)\:\mathrm{KE}\:\mathrm{at}\:{P}\:=\:\mathrm{KE}\:\mathrm{at}\:{B} \\ $$$$\left({b}\right)\:\mathrm{height}\:\mathrm{at}\:{P}\:=\:\mathrm{height}\:\mathrm{at}\:{A} \\ $$$$\left({c}\right)\:\mathrm{total}\:\mathrm{energy}\:\mathrm{at}\:{P}\:=\:\mathrm{total}\:\mathrm{energy}\:\mathrm{at} \\ $$$${A} \\ $$$$\left({d}\right)\:\mathrm{time}\:\mathrm{of}\:\mathrm{travel}\:\mathrm{from}\:{A}\:\mathrm{to}\:{B}\:=\:\mathrm{time}\:\mathrm{of} \\ $$$$\mathrm{travel}\:\mathrm{from}\:{B}\:\mathrm{to}\:{P}. \\ $$

Question Number 21248 Answers: 0 Comments: 0

The locus of the centre of a circle which touches the given circles ∣z − z_1 ∣ = ∣3 + 4i∣ and ∣z − z_2 ∣ = ∣1 + i(√3)∣ is a hyperbola, then the length of its transverse axis is

$$\mathrm{The}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{which} \\ $$$$\mathrm{touches}\:\mathrm{the}\:\mathrm{given}\:\mathrm{circles}\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:= \\ $$$$\mid\mathrm{3}\:+\:\mathrm{4}{i}\mid\:\mathrm{and}\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid\mathrm{1}\:+\:{i}\sqrt{\mathrm{3}}\mid\:\mathrm{is}\:\mathrm{a} \\ $$$$\mathrm{hyperbola},\:\mathrm{then}\:\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{transverse}\:\mathrm{axis}\:\mathrm{is} \\ $$

Question Number 21247 Answers: 1 Comments: 0

If [ ] represents the greatest integer function and f(x) = x − [x] then number of real roots of the equation f(x) + f((1/x)) = 1 are infinite. True/False

$$\mathrm{If}\:\left[\:\right]\:\mathrm{represents}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer} \\ $$$$\mathrm{function}\:\mathrm{and}\:{f}\left({x}\right)\:=\:{x}\:−\:\left[{x}\right]\:\mathrm{then} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${f}\left({x}\right)\:+\:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}\:\mathrm{are}\:\mathrm{infinite}. \\ $$$$\boldsymbol{\mathrm{True}}/\boldsymbol{\mathrm{False}} \\ $$

Question Number 21353 Answers: 1 Comments: 0

A censusman on duty visited a house which the lady inmates declined to reveal their individual ages, but said − “we do not mind giving you the sum of the ages of any two ladies you may choose”. Thereupon the censusman said − “In that case please give me the sum of the ages of every possible pair of you”. The gave the sums as follows : 30, 33, 41, 58, 66, 69. The censusman took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?

Pg 1812 Pg 1813 Pg 1814 Pg 1815 Pg 1816 Pg 1817 Pg 1818 Pg 1819 Pg 1820 Pg 1821

Contact: info@tinkutara.com