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Question Number 143011 Answers: 0 Comments: 0

x^4 +bx^2 +cx=s let x^2 =px+t ⇒ p^2 x^2 +2ptx+t^2 +bx^2 +cx=s ⇒ (p^2 +b)x^2 +(2pt+c)x =s−t^2 ⇒ (p^2 +b)(px+t)+(2pt+c)x =s−t^2 ⇒ p(p^2 +b)+2pt+c=0 and (p^2 +b)t=s−t^2 ((s/t)−t−b)((s/t)+t)^2 =c^2 ⇒ (A−b)(A^2 +4s)=c^2 ⇒ A^3 −bA^2 +4sA−4bs−c^2 =0 let A=z+(b/3) ⇒ z^3 +(4s−(b^2 /3))z−(((2b^3 )/(27))+((8bs)/3)+c^2 )=0 D=((b^3 /(27))+((4bs)/3)+(c^2 /2))^2 −((b^2 /9)−((4s)/3))^3 If s=0, b=−1, c→−c then D=(−(1/(27))+(c^2 /2))^2 −((1/9))^3 ...

$$\:\:{x}^{\mathrm{4}} +{bx}^{\mathrm{2}} +{cx}={s} \\ $$$${let}\:\:{x}^{\mathrm{2}} ={px}+{t} \\ $$$$\Rightarrow\:{p}^{\mathrm{2}} {x}^{\mathrm{2}} +\mathrm{2}{ptx}+{t}^{\mathrm{2}} +{bx}^{\mathrm{2}} +{cx}={s} \\ $$$$\Rightarrow\:\left({p}^{\mathrm{2}} +{b}\right){x}^{\mathrm{2}} +\left(\mathrm{2}{pt}+{c}\right){x} \\ $$$$\:\:\:\:\:\:\:\:\:\:={s}−{t}^{\mathrm{2}} \\ $$$$\Rightarrow\:\left({p}^{\mathrm{2}} +{b}\right)\left({px}+{t}\right)+\left(\mathrm{2}{pt}+{c}\right){x} \\ $$$$\:\:\:\:\:\:\:={s}−{t}^{\mathrm{2}} \\ $$$$\Rightarrow\:\:{p}\left({p}^{\mathrm{2}} +{b}\right)+\mathrm{2}{pt}+{c}=\mathrm{0} \\ $$$${and}\:\:\left({p}^{\mathrm{2}} +{b}\right){t}={s}−{t}^{\mathrm{2}} \\ $$$$\left(\frac{{s}}{{t}}−{t}−{b}\right)\left(\frac{{s}}{{t}}+{t}\right)^{\mathrm{2}} ={c}^{\mathrm{2}} \\ $$$$\Rightarrow\:\left({A}−{b}\right)\left({A}^{\mathrm{2}} +\mathrm{4}{s}\right)={c}^{\mathrm{2}} \\ $$$$\Rightarrow\:{A}^{\mathrm{3}} −{bA}^{\mathrm{2}} +\mathrm{4}{sA}−\mathrm{4}{bs}−{c}^{\mathrm{2}} =\mathrm{0} \\ $$$${let}\:\:{A}={z}+\frac{{b}}{\mathrm{3}}\:\Rightarrow \\ $$$${z}^{\mathrm{3}} +\left(\mathrm{4}{s}−\frac{{b}^{\mathrm{2}} }{\mathrm{3}}\right){z}−\left(\frac{\mathrm{2}{b}^{\mathrm{3}} }{\mathrm{27}}+\frac{\mathrm{8}{bs}}{\mathrm{3}}+{c}^{\mathrm{2}} \right)=\mathrm{0} \\ $$$${D}=\left(\frac{{b}^{\mathrm{3}} }{\mathrm{27}}+\frac{\mathrm{4}{bs}}{\mathrm{3}}+\frac{{c}^{\mathrm{2}} }{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{{b}^{\mathrm{2}} }{\mathrm{9}}−\frac{\mathrm{4}{s}}{\mathrm{3}}\right)^{\mathrm{3}} \\ $$$${If}\:\:{s}=\mathrm{0},\:{b}=−\mathrm{1},\:{c}\rightarrow−{c}\:\: \\ $$$${then}\:{D}=\left(−\frac{\mathrm{1}}{\mathrm{27}}+\frac{{c}^{\mathrm{2}} }{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{1}}{\mathrm{9}}\right)^{\mathrm{3}} \\ $$$$... \\ $$

Question Number 143008 Answers: 1 Comments: 1

Solve : x=p^3 −p+2 , y^′ =p

$${Solve}\::\:\:{x}={p}^{\mathrm{3}} −{p}+\mathrm{2}\:\:,\:{y}^{'} ={p} \\ $$

Question Number 143006 Answers: 0 Comments: 0

1. y(∂z/∂x) + z(∂z/∂y) = (y/x) 2. x^2 (∂z/∂x) − xy(∂z/∂y) + y^2 = 0 3. { (((∂z/∂x) = (z/x))),(((∂z/∂y) = ((2z)/y))) :}

$$\mathrm{1}.\:{y}\frac{\partial{z}}{\partial{x}}\:+\:{z}\frac{\partial{z}}{\partial{y}}\:=\:\frac{{y}}{{x}} \\ $$$$\mathrm{2}.\:{x}^{\mathrm{2}} \frac{\partial{z}}{\partial{x}}\:−\:{xy}\frac{\partial{z}}{\partial{y}}\:+\:{y}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\mathrm{3}.\:\begin{cases}{\frac{\partial{z}}{\partial{x}}\:=\:\frac{{z}}{{x}}}\\{\frac{\partial{z}}{\partial{y}}\:=\:\frac{\mathrm{2}{z}}{{y}}}\end{cases} \\ $$

Question Number 143003 Answers: 1 Comments: 0

Question Number 142992 Answers: 0 Comments: 3

Question Number 142990 Answers: 2 Comments: 0

find ∫_0 ^∞ (e^(−x^2 ) /((3+x^2 )^2 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } }{\left(\mathrm{3}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 142989 Answers: 2 Comments: 0

calculate ∫_0 ^∞ (e^(−3x^2 ) /(1+x^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{e}^{−\mathrm{3x}^{\mathrm{2}} } }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 142988 Answers: 0 Comments: 0

find the sequence u_n wich verify u_n +u_(n+1) =(2/( (√n))) give a equivalent of u_n (n→∞)

$$\mathrm{find}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{u}_{\mathrm{n}} \mathrm{wich}\:\mathrm{verify}\:\mathrm{u}_{\mathrm{n}} +\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\frac{\mathrm{2}}{\:\sqrt{\mathrm{n}}} \\ $$$$\mathrm{give}\:\mathrm{a}\:\mathrm{equivalent}\:\mathrm{of}\:\mathrm{u}_{\mathrm{n}} \:\:\left(\mathrm{n}\rightarrow\infty\right) \\ $$

Question Number 142987 Answers: 1 Comments: 0

find the sequence u_n wich verify u_(n+1) =u_n −λu_(n−1) λ real

$$\mathrm{find}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{u}_{\mathrm{n}} \mathrm{wich}\:\mathrm{verify}\:\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{u}_{\mathrm{n}} −\lambda\mathrm{u}_{\mathrm{n}−\mathrm{1}} \\ $$$$\lambda\:\mathrm{real} \\ $$

Question Number 142986 Answers: 2 Comments: 2

Question Number 142983 Answers: 0 Comments: 0

Question Number 142980 Answers: 0 Comments: 0

find U_n =∫_0 ^∞ e^(−nx^2 ) log(2+e^x )dx (n≥1) determine nature of Σ U_n and Σ nU_n

$$\mathrm{find}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{nx}^{\mathrm{2}} } \mathrm{log}\left(\mathrm{2}+\mathrm{e}^{\mathrm{x}} \right)\mathrm{dx}\:\:\:\left(\mathrm{n}\geqslant\mathrm{1}\right) \\ $$$$\mathrm{determine}\:\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} \:\mathrm{and}\:\Sigma\:\mathrm{nU}_{\mathrm{n}} \\ $$

Question Number 142971 Answers: 2 Comments: 1

Question Number 142970 Answers: 1 Comments: 0

..........CALCULUS........... prove that:: 𝛗:=Σ_(n=1) ^∞ (((−1)^(n−1) ((n−1)!)^2 )/((2n)!))=2log^2 (ϕ) ϕ=golden ratio.... .............

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:..........{CALCULUS}........... \\ $$$$\:\:\:\:\:\:\:{prove}\:{that}::\:\: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}:=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left(\left({n}−\mathrm{1}\right)!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}\right)!}=\mathrm{2}{log}^{\mathrm{2}} \left(\varphi\right) \\ $$$$\:\:\:\:\varphi={golden}\:{ratio}.... \\ $$$$\:\:\:\:............. \\ $$

Question Number 142968 Answers: 1 Comments: 0

Given p<x<q is solution set inequality 1+2^x +2^(2x) +2^(3x) +...>2 for x≠1. find the value of 5p−3q .

$${Given}\:{p}<{x}<{q}\:{is}\:{solution}\:{set} \\ $$$${inequality}\:\mathrm{1}+\mathrm{2}^{{x}} +\mathrm{2}^{\mathrm{2}{x}} +\mathrm{2}^{\mathrm{3}{x}} +...>\mathrm{2} \\ $$$${for}\:{x}\neq\mathrm{1}.\:{find}\:{the}\:{value}\:{of}\: \\ $$$$\mathrm{5}{p}−\mathrm{3}{q}\:. \\ $$

Question Number 142966 Answers: 0 Comments: 0

Question Number 142977 Answers: 0 Comments: 2

Question Number 142947 Answers: 0 Comments: 5

What is the probability that 3 points will fall twice when a dice is thrown 4 times?

$${What}\:{is}\:{the}\:{probability}\:{that}\:\mathrm{3}\:{points} \\ $$$${will}\:{fall}\:{twice}\:{when}\:{a}\:{dice}\:{is}\:{thrown} \\ $$$$\mathrm{4}\:{times}? \\ $$

Question Number 142945 Answers: 1 Comments: 0

∫_0 ^(0.5) ((1+x^3 ))^(1/3) dx

$$\int_{\mathrm{0}} ^{\mathrm{0}.\mathrm{5}} \sqrt[{\mathrm{3}}]{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{3}} }\boldsymbol{{dx}} \\ $$$$ \\ $$

Question Number 142939 Answers: 2 Comments: 0

Q#141663 by ajfour sir reposted. x^2 (x−12)(x−15)=k(x−16) ;k>0 Find x in terms of k.

$${Q}#\mathrm{141663}\:{by}\:\:{ajfour}\:{sir}\:{reposted}. \\ $$$$\:\:\:\:\:{x}^{\mathrm{2}} \left({x}−\mathrm{12}\right)\left({x}−\mathrm{15}\right)={k}\left({x}−\mathrm{16}\right)\:;{k}>\mathrm{0} \\ $$$$\:\:\:\:\:{Find}\:{x}\:{in}\:{terms}\:{of}\:{k}. \\ $$

Question Number 142936 Answers: 0 Comments: 0

Find all real values of a such that f(x)=((x^2 +ax+1)/(x^2 +x+1)) is surjective f :ℜ⇒ℜ

$${Find}\:{all}\:{real}\:{values}\:{of}\:{a}\:\:\:{such}\:{that}\:\:{f}\left({x}\right)=\frac{{x}^{\mathrm{2}} +{ax}+\mathrm{1}}{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:\: \\ $$$${is}\:{surjective}\:\:{f}\::\boldsymbol{\Re}\Rightarrow\Re \\ $$

Question Number 142935 Answers: 2 Comments: 0

Question Number 142931 Answers: 0 Comments: 0

Question Number 142922 Answers: 2 Comments: 0

lim_(x→1) ((sin(x+1))/(2x−(√(x^2 +3))))=?

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}+\mathrm{1}\right)}{\mathrm{2}{x}−\sqrt{{x}^{\mathrm{2}} +\mathrm{3}}}=? \\ $$

Question Number 142927 Answers: 2 Comments: 0

Question Number 142920 Answers: 1 Comments: 2

A student did not notice that the multiplication sign between two 7−digits numbers amd wrote one 14−digits number which turned out to be 3 times the would be product. What are the initial numbers ?

$$\mathrm{A}\:\mathrm{student}\:\mathrm{did}\:\mathrm{not}\:\mathrm{notice}\:\mathrm{that}\:\mathrm{the}\:\mathrm{multiplication} \\ $$$$\mathrm{sign}\:\mathrm{between}\:\mathrm{two}\:\mathrm{7}−\mathrm{digits}\:\mathrm{numbers}\:\mathrm{amd}\:\mathrm{wrote} \\ $$$$\mathrm{one}\:\mathrm{14}−\mathrm{digits}\:\mathrm{number}\:\mathrm{which}\:\mathrm{turned}\:\mathrm{out}\:\mathrm{to}\:\mathrm{be} \\ $$$$\mathrm{3}\:\mathrm{times}\:\mathrm{the}\:\mathrm{would}\:\mathrm{be}\:\mathrm{product}.\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{initial} \\ $$$$\mathrm{numbers}\:? \\ $$

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