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AllQuestion and Answers: Page 686

Question Number 147609 Answers: 0 Comments: 0

Question Number 147606 Answers: 0 Comments: 0

(R/r)=(5/2) ⇒ a:b:c=3:4:5 prove R,r− radius h_a :h_b :h_c =(1/a):(1/b):(1/c)=bc:ac:ab prove

$$\frac{\boldsymbol{\mathrm{R}}}{\boldsymbol{{r}}}=\frac{\mathrm{5}}{\mathrm{2}}\:\:\Rightarrow\:\boldsymbol{{a}}:\boldsymbol{{b}}:\boldsymbol{{c}}=\mathrm{3}:\mathrm{4}:\mathrm{5} \\ $$$$\boldsymbol{\mathrm{prove}}\:\: \\ $$$$\boldsymbol{\mathrm{R}},\boldsymbol{{r}}−\:\boldsymbol{\mathrm{radius}} \\ $$$$\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{a}}} :\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{b}}} :\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{c}}} =\frac{\mathrm{1}}{\boldsymbol{{a}}}:\frac{\mathrm{1}}{\boldsymbol{{b}}}:\frac{\mathrm{1}}{\boldsymbol{{c}}}=\boldsymbol{{bc}}:\boldsymbol{{ac}}:\boldsymbol{{ab}} \\ $$$$\boldsymbol{{prove}} \\ $$

Question Number 147603 Answers: 1 Comments: 0

Question Number 147602 Answers: 1 Comments: 0

Question Number 147593 Answers: 2 Comments: 0

Question Number 147587 Answers: 1 Comments: 1

Question Number 147585 Answers: 1 Comments: 0

Question Number 147582 Answers: 1 Comments: 0

if x;y;z>0 prove that ((x^3 +y^3 +z^3 )/(xyz)) ≥ 2((x/(y+z)) + (y/(z+x)) + (z/(x+y)))

$${if}\:\:{x};{y};{z}>\mathrm{0}\:\:{prove}\:{that} \\ $$$$\frac{{x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} }{{xyz}}\:\geqslant\:\mathrm{2}\left(\frac{{x}}{{y}+{z}}\:+\:\frac{{y}}{{z}+{x}}\:+\:\frac{{z}}{{x}+{y}}\right) \\ $$

Question Number 147581 Answers: 1 Comments: 0

2sin 2x −4sin^2 x = 7cos 2x (π/2)<x<π ⇒ sin 2x =?

$$\:\:\mathrm{2sin}\:\mathrm{2x}\:−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}\:=\:\mathrm{7cos}\:\mathrm{2x}\: \\ $$$$\:\frac{\pi}{\mathrm{2}}<\mathrm{x}<\pi\:\Rightarrow\:\mathrm{sin}\:\mathrm{2x}\:=? \\ $$

Question Number 147576 Answers: 1 Comments: 0

(1):: Σ_(i=1) ^n Σ_(j=1) ^n ∣i−j∣=? (2):: Σ_(i=1) ^n Σ_(j=i) ^n (1/j)=? (3):: Σ_(i=1) ^n^2 [(√i)]=?

$$\left(\mathrm{1}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mid\mathrm{i}−\mathrm{j}\mid=? \\ $$$$\left(\mathrm{2}\right)::\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\underset{\mathrm{j}=\mathrm{i}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{j}}=? \\ $$$$\left(\mathrm{3}\right)::\:\:\:\:\:\:\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}^{\mathrm{2}} } {\sum}}\left[\sqrt{\mathrm{i}}\right]=? \\ $$

Question Number 147573 Answers: 0 Comments: 1

Question Number 147572 Answers: 2 Comments: 0

Σ_(n≥1) ((4n−3)/((n^2 +2n)(n+3))) =?

$$\:\:\:\underset{{n}\geqslant\mathrm{1}} {\sum}\:\frac{\mathrm{4}{n}−\mathrm{3}}{\left({n}^{\mathrm{2}} +\mathrm{2}{n}\right)\left({n}+\mathrm{3}\right)}\:=? \\ $$

Question Number 147569 Answers: 1 Comments: 0

find the taylor series of f(z)=sinz ,z=(π/4) in complex number

$${find}\:{the}\:{taylor}\:{series}\:{of}\:{f}\left({z}\right)={sinz}\:,{z}=\frac{\pi}{\mathrm{4}}\:{in}\:{complex}\:{number} \\ $$

Question Number 147566 Answers: 1 Comments: 0

x^3 + 3367 = 2^n ⇒ x ; n = ?

$${x}^{\mathrm{3}} \:+\:\mathrm{3367}\:=\:\mathrm{2}^{\boldsymbol{{n}}} \:\:\Rightarrow\:{x}\:;\:{n}\:=\:? \\ $$

Question Number 147561 Answers: 1 Comments: 1

Question Number 147557 Answers: 2 Comments: 0

Simlify (((1+(√x))/( (√(1+x)))) − ((√(1+x))/(1+(√x))))^2 - (((1−(√x))/( (√(1+x)))) − ((√(1+x))/(1−(√x))))^2

$${Simlify} \\ $$$$\left(\frac{\mathrm{1}+\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}}}\:−\:\frac{\sqrt{\mathrm{1}+{x}}}{\mathrm{1}+\sqrt{{x}}}\right)^{\mathrm{2}} -\:\left(\frac{\mathrm{1}−\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}}}\:−\:\frac{\sqrt{\mathrm{1}+{x}}}{\mathrm{1}−\sqrt{{x}}}\right)^{\mathrm{2}} \\ $$

Question Number 147554 Answers: 1 Comments: 0

A toroid core has N=1200 turns, length L=80cm,cross-section area A=60cm^2 ,current I=1.5A. Compute B and H.Assume an empty core

$$\mathrm{A}\:\mathrm{toroid}\:\mathrm{core}\:\mathrm{has}\:\mathrm{N}=\mathrm{1200}\:\mathrm{turns}, \\ $$$$\mathrm{length}\:\mathrm{L}=\mathrm{80cm},\mathrm{cross}-\mathrm{section}\:\mathrm{area} \\ $$$$\mathrm{A}=\mathrm{60cm}^{\mathrm{2}} ,\mathrm{current}\:\mathrm{I}=\mathrm{1}.\mathrm{5A}. \\ $$$$\:\:\mathrm{Compute}\:\mathrm{B}\:\mathrm{and}\:\mathrm{H}.\mathrm{Assume}\:\mathrm{an} \\ $$$$\:\mathrm{empty}\:\mathrm{core} \\ $$

Question Number 147553 Answers: 1 Comments: 0

Question Number 147543 Answers: 2 Comments: 0

Π_(m=1) ^n ((1/2))^m

$$\underset{{m}=\mathrm{1}} {\overset{{n}} {\prod}}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{m}} \\ $$

Question Number 147539 Answers: 2 Comments: 0

show that (1/2)∫_1 ^( 2) (1/x^5 )dx ≤ ∫_1 ^( 2) (1/(x^4 +1))dx ≤∫_1 ^( 2) (1/x^4 )

$${show}\:{that} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{5}} }{dx}\:\leqslant\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{4}} +\mathrm{1}}{dx}\:\leqslant\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{1}}{{x}^{\mathrm{4}} } \\ $$

Question Number 147535 Answers: 1 Comments: 1

Question Number 147529 Answers: 3 Comments: 1

if a_(n+1) = (√(a_1 + a_n )) find lim_(x→∞) a_n = ?

$${if}\:\:\:{a}_{\boldsymbol{{n}}+\mathrm{1}} \:=\:\sqrt{{a}_{\mathrm{1}} \:+\:{a}_{\boldsymbol{{n}}} } \\ $$$${find}\:\:\underset{\boldsymbol{{x}}\rightarrow\infty} {{lim}a}_{\boldsymbol{{n}}} \:=\:? \\ $$

Question Number 147528 Answers: 0 Comments: 3

q_n =Π_(n=0) ^∞ cos((x/2^n )) i) study the variation of q_n ii) show that cosx=((sin2x)/(2sinx)) , ∀x∈[0,(π/2)] iii) deduce that q_n =(1/2^(n+1) )×((sin2x)/(sin((x/2^n )))) iv)lim_(n→∞) q_n =? v) solve cos((x/2))≥−(1/2)

$${q}_{{n}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}{cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right) \\ $$$$\left.{i}\right)\:{study}\:{the}\:{variation}\:{of}\:{q}_{{n}} \\ $$$$\left.{ii}\right) \\ $$$$\:{show}\:{that}\:{cosx}=\frac{{sin}\mathrm{2}{x}}{\mathrm{2}{sinx}}\:,\:\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right] \\ $$$$\left.{iii}\right) \\ $$$${deduce}\:{that}\:{q}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}^{{n}+\mathrm{1}} }×\frac{{sin}\mathrm{2}{x}}{{sin}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right)} \\ $$$$\left.{iv}\right)\underset{{n}\rightarrow\infty} {\mathrm{lim}}{q}_{{n}} =? \\ $$$$\left.{v}\right)\: \\ $$$${solve}\:{cos}\left(\frac{{x}}{\mathrm{2}}\right)\geqslant−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 147524 Answers: 0 Comments: 5

Question Number 147513 Answers: 1 Comments: 1

Question Number 147661 Answers: 1 Comments: 0

41^3 + 42^3 + 43^3 + ... + 59^3 Find the last three digits of the number

$$\mathrm{41}^{\mathrm{3}} \:+\:\mathrm{42}^{\mathrm{3}} \:+\:\mathrm{43}^{\mathrm{3}} \:+\:...\:+\:\mathrm{59}^{\mathrm{3}} \\ $$$${Find}\:{the}\:{last}\:{three}\:{digits}\:{of}\:{the} \\ $$$${number} \\ $$

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