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Question Number 75257 Answers: 1 Comments: 0

Let A={2,4,6,7,8,9} B={1,3,5,6,10} and C={x:3x+6=0 or 2x+6=0}.Find a. A∪B. b. is(A∪B)∪C=A∪(B∪C)?

$${Let}\:{A}=\left\{\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9}\right\} \\ $$$${B}=\left\{\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{6},\mathrm{10}\right\}\:{and} \\ $$$${C}=\left\{{x}:\mathrm{3}{x}+\mathrm{6}=\mathrm{0}\:{or}\:\mathrm{2}{x}+\mathrm{6}=\mathrm{0}\right\}.{Find} \\ $$$${a}.\:{A}\cup{B}. \\ $$$${b}.\:{is}\left({A}\cup{B}\right)\cup{C}={A}\cup\left({B}\cup{C}\right)? \\ $$

Question Number 75248 Answers: 1 Comments: 1

Question Number 75242 Answers: 2 Comments: 0

Find th greatest coefficients in the expansion of (3a+5b)^(18) 2)If three consecutive coefficient of (1+x)^n are 28,56,70. find the value of n

$${Find}\:{th}\:{greatest}\:{coefficients} \\ $$$${in}\:{the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{3}{a}+\mathrm{5}{b}\right)^{\mathrm{18}} \\ $$$$ \\ $$$$\left.\mathrm{2}\right){If}\:{three}\:{consecutive}\: \\ $$$${coefficient}\:{of}\:\left(\mathrm{1}+{x}\right)^{{n}} \:{are}\:\mathrm{28},\mathrm{56},\mathrm{70}. \\ $$$${find}\:{the}\:{value}\:{of}\:{n} \\ $$$$ \\ $$

Question Number 75230 Answers: 1 Comments: 1

Question Number 75262 Answers: 0 Comments: 3

Question Number 75225 Answers: 0 Comments: 0

Prove that lim_(x→∞) (xln∣x∣−(x−m)ln∣x−m∣)=+∞

$$\mathrm{Prove}\:\mathrm{that}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\mathrm{xln}\mid\mathrm{x}\mid−\left(\mathrm{x}−\mathrm{m}\right)\mathrm{ln}\mid\mathrm{x}−\mathrm{m}\mid\right)=+\infty \\ $$

Question Number 75224 Answers: 0 Comments: 0

Prove that lim_(x→∞) (xln∣x∣−(x−m)ln∣x−m∣)=+∞

Question Number 75223 Answers: 0 Comments: 0

Question Number 75222 Answers: 0 Comments: 1

Question Number 75220 Answers: 2 Comments: 0

Let us consider the function F(x)=∫_0 ^1 e^(−x) ln(x−lnt)dt 1)Prove that for all x≥1 , F(x) exist 2)Prove that lim_(t→0) tln(x−lnt)=0 3)Prove that F ∈ C^1 ([1:∞[,[1:∞[) and for all x≥1 F(x)=F′(x)+lnx 4) Find out the value lim_(x→∞) F(x) and lim_(x→1) F(x) 5)Can you prove that at least one of the both result is irrational???

$$\mathrm{Let}\:\mathrm{us}\:\mathrm{consider}\:\mathrm{the}\:\mathrm{function}\: \\ $$$$\mathrm{F}\left(\mathrm{x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{e}^{−\mathrm{x}} \mathrm{ln}\left(\mathrm{x}−\mathrm{lnt}\right)\mathrm{dt}\: \\ $$$$\left.\mathrm{1}\right)\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{all}\:\mathrm{x}\geqslant\mathrm{1}\:,\:\:\mathrm{F}\left(\mathrm{x}\right)\:\mathrm{exist} \\ $$$$\left.\mathrm{2}\right)\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{tln}\left(\mathrm{x}−\mathrm{lnt}\right)=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\mathrm{Prove}\:\mathrm{that}\:\mathrm{F}\:\in\:\mathrm{C}^{\mathrm{1}} \left(\left[\mathrm{1}:\infty\left[,\left[\mathrm{1}:\infty\left[\right)\:\mathrm{and}\:\mathrm{for}\:\right.\right.\right.\right. \\ $$$$\mathrm{all}\:\mathrm{x}\geqslant\mathrm{1}\:\:\mathrm{F}\left(\mathrm{x}\right)=\mathrm{F}'\left(\mathrm{x}\right)+\mathrm{lnx} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{Find}\:\mathrm{out}\:\mathrm{the}\:\mathrm{value}\: \\ $$$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{F}\left(\mathrm{x}\right)\:\:\:\:\:\:\mathrm{and}\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}F}\left(\mathrm{x}\right) \\ $$$$\left.\mathrm{5}\right)\mathrm{Can}\:\mathrm{you}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{both}\:\mathrm{result}\:\mathrm{is}\:\mathrm{irrational}??? \\ $$

Question Number 75228 Answers: 0 Comments: 1

Let consider A=lim_(x→0) (∫_0 ^1 (Γ(t))^x dt)^(1/x) Prove that A=∫_0 ^1 ln(Γ(t))dt Deduce the value of A

$$\:\mathrm{Let}\:\mathrm{consider}\: \\ $$$$\mathrm{A}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} \left(\Gamma\left(\mathrm{t}\right)\right)^{\mathrm{x}} \mathrm{dt}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\:\mathrm{A}=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\Gamma\left(\mathrm{t}\right)\right)\mathrm{dt}\:\: \\ $$$$\mathrm{Deduce}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{A} \\ $$

Question Number 75218 Answers: 1 Comments: 1

Simplify: (x^4 −3x^3 +4x^2 −12x) : (x^2 +4)

$$\mathrm{Simplify}:\:\left({x}^{\mathrm{4}} −\mathrm{3}{x}^{\mathrm{3}} +\mathrm{4}{x}^{\mathrm{2}} −\mathrm{12}{x}\right)\::\:\left({x}^{\mathrm{2}} +\mathrm{4}\right) \\ $$

Question Number 75212 Answers: 1 Comments: 0

Question Number 75208 Answers: 1 Comments: 0

Prove that : cos 18^0 −sin 18^0 = (√2) sin 2

$$\:\mathrm{Prove}\:\mathrm{that}\::\:\mathrm{cos}\:\mathrm{18}^{\mathrm{0}} −\mathrm{sin}\:\mathrm{18}^{\mathrm{0}} \:=\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{2} \\ $$

Question Number 75209 Answers: 2 Comments: 0

Question Number 75204 Answers: 0 Comments: 0

Question Number 75203 Answers: 1 Comments: 0

Question Number 75193 Answers: 2 Comments: 0

it is given that cos(π/5)=((1+(√5))/4) calculate the exact value of cos((2π)/5) and cos((3π)/5)

$${it}\:{is}\:{given}\:{that}\:{cos}\frac{\pi}{\mathrm{5}}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$${calculate}\:{the}\:{exact}\:{val}\mathrm{ue}\:{of}\: \\ $$$${cos}\frac{\mathrm{2}\pi}{\mathrm{5}}\:\:\:{and}\:\:{cos}\frac{\mathrm{3}\pi}{\mathrm{5}} \\ $$

Question Number 75186 Answers: 1 Comments: 7

7n ≡ 1 (mod 5) What is the general form of n ?

$$\mathrm{7}{n}\:\equiv\:\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{5}\right) \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{general}\:\mathrm{form}\:\mathrm{of}\:{n}\:? \\ $$

Question Number 75180 Answers: 0 Comments: 2

There are 6 girls and four boys in a class. 3 students are choosen at random so as to be awarded a scholaship.In how many ways can this be done if atlease 1 boy and 1 girl must in the selection

$${There}\:{are}\:\mathrm{6}\:{girls}\:{and}\:{four}\:{boys}\:{in}\:{a}\:{class}. \\ $$$$\mathrm{3}\:{students}\:{are}\:{choosen}\:{at}\:{random}\:{so}\:{as}\:{to}\:{be} \\ $$$${awarded}\:{a}\:{scholaship}.{In}\:{how}\:{many}\:{ways}\:{can} \\ $$$${this}\:{be}\:{done}\:{if}\:{atlease}\:\mathrm{1}\:{boy}\:{and}\:\mathrm{1}\:{girl}\:{must} \\ $$$${in}\:{the}\:{selection} \\ $$$$ \\ $$

Question Number 75178 Answers: 1 Comments: 0

givn that z = 1−i(√3) express z in the form z = r(cosθ + isinθ), hence express z^7 in the form re^(iθ)

$${givn}\:{that}\:{z}\:=\:\mathrm{1}−{i}\sqrt{\mathrm{3}}\:{express}\:{z}\:{in}\:{the}\:{form}\: \\ $$$$\:{z}\:=\:{r}\left({cos}\theta\:+\:{isin}\theta\right),\:{hence}\:{express} \\ $$$${z}^{\mathrm{7}} \:{in}\:{the}\:{form}\:{re}^{{i}\theta} \\ $$

Question Number 75177 Answers: 2 Comments: 2

Question Number 75176 Answers: 1 Comments: 0

P=(√(25x−50))−14(√((x−2)/4))+(√(9x−18)), x≥2 a) Simplify the equation b) Find x if P=3

$$\mathrm{P}=\sqrt{\mathrm{25}{x}−\mathrm{50}}−\mathrm{14}\sqrt{\frac{{x}−\mathrm{2}}{\mathrm{4}}}+\sqrt{\mathrm{9}{x}−\mathrm{18}},\:\:{x}\geqslant\mathrm{2} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{Simplify}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{Find}\:\mathrm{x}\:\mathrm{if}\:\mathrm{P}=\mathrm{3} \\ $$

Question Number 75175 Answers: 1 Comments: 0

solve the differential equation (x^2 −1)(dy/dx) + 2y = 0 when y=3 and x= 2,expressing your answer in the form y=f(x)

$${solve}\:{the}\:{differential}\:{equation} \\ $$$$\:\left({x}^{\mathrm{2}} −\mathrm{1}\right)\frac{{dy}}{{dx}}\:+\:\mathrm{2}{y}\:=\:\mathrm{0}\:{when}\:{y}=\mathrm{3}\:{and}\:{x}=\:\mathrm{2},{expressing} \\ $$$${your}\:{answer}\:{in}\:{the}\:{form}\:{y}={f}\left({x}\right) \\ $$

Question Number 75173 Answers: 1 Comments: 1

Question Number 75166 Answers: 1 Comments: 0

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