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Question Number 21341 Answers: 2 Comments: 0

the angle between the straight lines x^2 +4xy+3y^2 =0 is

$${the}\:{angle}\:{between}\:{the}\:{straight}\:{lines}\:{x}^{\mathrm{2}} +\mathrm{4}{xy}+\mathrm{3}{y}^{\mathrm{2}} =\mathrm{0}\:{is} \\ $$

Question Number 21321 Answers: 1 Comments: 0

The number of real solutions of the equation 4x^(99) + 5x^(98) + 4x^(97) + 5x^(96) + ..... + 4x + 5 = 0 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{4}{x}^{\mathrm{99}} \:+\:\mathrm{5}{x}^{\mathrm{98}} \:+\:\mathrm{4}{x}^{\mathrm{97}} \:+\:\mathrm{5}{x}^{\mathrm{96}} \:+ \\ $$$$.....\:+\:\mathrm{4}{x}\:+\:\mathrm{5}\:=\:\mathrm{0}\:\mathrm{is} \\ $$

Question Number 21319 Answers: 0 Comments: 0

If x, y, z are three real numbers such that x + y + z = 4 and x^2 + y^2 + z^2 = 6, then (1) (2/3) ≤ x, y, z ≤ 2 (2) 0 ≤ x, y, z ≤ 2 (3) 1 ≤ x, y, z ≤ 3 (4) 2 ≤ x, y, z ≤ 3

$$\mathrm{If}\:{x},\:{y},\:{z}\:\mathrm{are}\:\mathrm{three}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that}\:{x}\:+\:{y}\:+\:{z}\:=\:\mathrm{4}\:\mathrm{and}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \:=\:\mathrm{6}, \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{2}}{\mathrm{3}}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{0}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{2} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{1}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{3} \\ $$$$\left(\mathrm{4}\right)\:\mathrm{2}\:\leqslant\:{x},\:{y},\:{z}\:\leqslant\:\mathrm{3} \\ $$

Question Number 21316 Answers: 0 Comments: 0

Let p = (x_1 − x_2 )^2 + (x_1 − x_3 )^2 + .... + (x_1 − x_6 )^2 + (x_2 − x_3 )^2 + (x_2 − x_4 )^2 + .... + (x_2 − x_6 )^2 + .... + (x_5 − x_6 )^2 = Σ_(1≤i<j≤6) ^6 (x_i − x_j )^2 . Then the maximum value of p if each x_i (i = 1, 2, ....., 6) has the value 0 and 1 is

Question Number 21315 Answers: 0 Comments: 0

The number of real solutions of the equation ((97 − x))^(1/4) + (x)^(1/4) = 5

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\sqrt[{\mathrm{4}}]{\mathrm{97}\:−\:{x}}\:+\:\sqrt[{\mathrm{4}}]{{x}}\:=\:\mathrm{5} \\ $$

Question Number 21314 Answers: 1 Comments: 0

Let α and β be the root of x^2 + px − (1/(2p^2 )) = 0, p ∈ R. The minimum value of α^4 + β^4 is

$$\mathrm{Let}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{be}\:\mathrm{the}\:\mathrm{root}\:\mathrm{of}\:{x}^{\mathrm{2}} \:+\:{px}\:−\:\frac{\mathrm{1}}{\mathrm{2}{p}^{\mathrm{2}} }\:=\:\mathrm{0}, \\ $$$${p}\:\in\:{R}.\:\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\alpha^{\mathrm{4}} \:+\:\beta^{\mathrm{4}} \:\mathrm{is} \\ $$

Question Number 21313 Answers: 0 Comments: 4

Let k be a real number such that the inequality (√(x − 3)) + (√(6 − x)) ≥ k has a solution then the maximum value of k is

$$\mathrm{Let}\:{k}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}\:\mathrm{number}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{inequality}\:\sqrt{{x}\:−\:\mathrm{3}}\:+\:\sqrt{\mathrm{6}\:−\:{x}}\:\geqslant\:{k}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{solution}\:\mathrm{then}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:{k} \\ $$$$\mathrm{is} \\ $$

Question Number 21311 Answers: 0 Comments: 0

Let a and b be positive real numbers with a^3 + b^3 = a − b, and k = a^2 + 4b^2 , then (1) k < 1 (2) k >1 (3) k = 1 (4) k > 2

$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\mathrm{with}\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:=\:{a}\:−\:{b},\:\mathrm{and}\:{k}\:=\:{a}^{\mathrm{2}} \:+\:\mathrm{4}{b}^{\mathrm{2}} , \\ $$$$\mathrm{then} \\ $$$$\left(\mathrm{1}\right)\:{k}\:<\:\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:{k}\:>\mathrm{1} \\ $$$$\left(\mathrm{3}\right)\:{k}\:=\:\mathrm{1} \\ $$$$\left(\mathrm{4}\right)\:{k}\:>\:\mathrm{2} \\ $$

Question Number 21310 Answers: 1 Comments: 8

What do you guys think of creating a Telegram group to discuss theory and more descriptive questions?

$$\mathrm{What}\:\mathrm{do}\:\mathrm{you}\:\mathrm{guys}\:\mathrm{think}\:\mathrm{of} \\ $$$$\mathrm{creating}\:\mathrm{a}\:\mathrm{Telegram}\:\mathrm{group}\:\mathrm{to} \\ $$$$\mathrm{discuss}\:\mathrm{theory}\:\mathrm{and}\:\mathrm{more}\:\mathrm{descriptive} \\ $$$$\mathrm{questions}? \\ $$

Question Number 21309 Answers: 0 Comments: 0

Suppose p is a polynomial with complex coefficients and an even degree. If all the roots of p are complex non-real numbers with modulus 1, prove that p(1) ∈ R iff p(−1) ∈ R.

$$\mathrm{Suppose}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{with}\:\mathrm{complex} \\ $$$$\mathrm{coefficients}\:\mathrm{and}\:\mathrm{an}\:\mathrm{even}\:\mathrm{degree}.\:\mathrm{If}\:\mathrm{all} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{p}\:\mathrm{are}\:\mathrm{complex}\:\mathrm{non}-\mathrm{real} \\ $$$$\mathrm{numbers}\:\mathrm{with}\:\mathrm{modulus}\:\mathrm{1},\:\mathrm{prove}\:\mathrm{that} \\ $$$${p}\left(\mathrm{1}\right)\:\in\:{R}\:\mathrm{iff}\:{p}\left(−\mathrm{1}\right)\:\in\:{R}. \\ $$

Question Number 21308 Answers: 0 Comments: 0

Find all complex numbers z such that ∣z − ∣z + 1∣∣ = ∣z + ∣z − 1∣∣

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{complex}\:\mathrm{numbers}\:{z}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mid{z}\:−\:\mid{z}\:+\:\mathrm{1}\mid\mid\:=\:\mid{z}\:+\:\mid{z}\:−\:\mathrm{1}\mid\mid \\ $$

Question Number 21307 Answers: 0 Comments: 5

Let z_1 , z_2 , z_3 be complex numbers such that (i) ∣z_1 ∣ = ∣z_2 ∣ = ∣z_3 ∣ = 1 (ii) z_1 + z_2 + z_3 ≠ 0 (iii) z_1 ^2 + z_2 ^2 + z_3 ^2 = 0 Prove that for all n ≥ 2, ∣z_1 ^n + z_2 ^n + z_3 ^n ∣ ∈ {0, 1, 2, 3}.

$$\mathrm{Let}\:{z}_{\mathrm{1}} ,\:{z}_{\mathrm{2}} ,\:{z}_{\mathrm{3}} \:\mathrm{be}\:\mathrm{complex}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that} \\ $$$$\left(\mathrm{i}\right)\:\mid{z}_{\mathrm{1}} \mid\:=\:\mid{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{3}} \mid\:=\:\mathrm{1} \\ $$$$\left(\mathrm{ii}\right)\:{z}_{\mathrm{1}} \:+\:{z}_{\mathrm{2}} \:+\:{z}_{\mathrm{3}} \:\neq\:\mathrm{0} \\ $$$$\left(\mathrm{iii}\right)\:{z}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{z}_{\mathrm{3}} ^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:{n}\:\geqslant\:\mathrm{2}, \\ $$$$\mid{z}_{\mathrm{1}} ^{{n}} \:+\:{z}_{\mathrm{2}} ^{{n}} \:+\:{z}_{\mathrm{3}} ^{{n}} \mid\:\in\:\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right\}. \\ $$

Question Number 21295 Answers: 1 Comments: 0

For a particle performing uniform circular motion, angular momentum is constant in magnitude but direction keeps changing. Am I right or wrong?

$$\mathrm{For}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{performing}\:\mathrm{uniform} \\ $$$$\mathrm{circular}\:\mathrm{motion},\:\mathrm{angular}\:\mathrm{momentum}\:\mathrm{is} \\ $$$$\mathrm{constant}\:\mathrm{in}\:\mathrm{magnitude}\:\mathrm{but}\:\mathrm{direction} \\ $$$$\mathrm{keeps}\:\mathrm{changing}. \\ $$$$\mathrm{Am}\:\mathrm{I}\:\mathrm{right}\:\mathrm{or}\:\mathrm{wrong}? \\ $$

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)

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

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