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Question Number 115645    Answers: 1   Comments: 2

Question Number 115643    Answers: 0   Comments: 0

Question Number 115642    Answers: 0   Comments: 0

A vertical post of height h m rises from a plane which slopes down towards the South at an angle Ξ± to the horizontal. Prove that the length of its shadow when the sun is S𝛉W at an elevation Ξ² is ((h(√((1+tan^2 Ξ± cos^2 ΞΈ) )))/(tanΞ² + tanΞ± cos ΞΈ))m

$${A}\:{vertical}\:{post}\:{of}\:{height}\:{h}\:{m}\:{rises}\:{from}\:{a}\:{plane}\:{which}\: \\ $$$${slopes}\:{down}\:{towards}\:{the}\:{South}\:{at}\:{an}\:{angle} \\ $$$$\alpha\:{to}\:{the}\:{horizontal}.\:{Prove}\:{that}\:{the}\:{length} \\ $$$${of}\:{its}\:{shadow}\:{when}\:{the}\:{sun}\:{is}\:\boldsymbol{{S}\theta{W}}\:\: \\ $$$${at}\:{an}\:{elevation}\:\beta\:{is} \\ $$$$ \\ $$$$\frac{{h}\sqrt{\left(\mathrm{1}+{tan}^{\mathrm{2}} \alpha\:{cos}^{\mathrm{2}} \theta\right)\:}}{{tan}\beta\:+\:{tan}\alpha\:\mathrm{cos}\:\theta}{m} \\ $$

Question Number 115632    Answers: 1   Comments: 6

Question Number 115629    Answers: 0   Comments: 2

Question Number 115628    Answers: 0   Comments: 0

Question Number 115627    Answers: 0   Comments: 1

Question Number 115621    Answers: 2   Comments: 0

... advanced calculus... evaluate :: show that lim_(nβ†’βˆž) (1/n)[cos^(2p) (Ο€/(2n))+cos^(2p) ((2Ο€)/(2n))+cos^(2p) ((3Ο€)/(2n))......cos^(2p) (Ο€/2)] =Ξ _(r=1) ^p ((p+r)/(4r))

$$ \\ $$$$\:\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}...\: \\ $$$$\:\:\:\:\:\:\:{evaluate}\::: \\ $$$${show}\:{that}\:{lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\left[{cos}^{\mathrm{2}{p}} \frac{\pi}{\mathrm{2}{n}}+{cos}^{\mathrm{2}{p}} \frac{\mathrm{2}\pi}{\mathrm{2}{n}}+{cos}^{\mathrm{2}{p}} \frac{\mathrm{3}\pi}{\mathrm{2}{n}}......{cos}^{\mathrm{2}{p}} \frac{\pi}{\mathrm{2}}\right]\:=\underset{{r}=\mathrm{1}} {\overset{{p}} {\prod}}\frac{{p}+{r}}{\mathrm{4}{r}} \\ $$

Question Number 115616    Answers: 2   Comments: 0

lim_(xβ†’0) ((((256+tan x))^(1/(8 )) βˆ’((1+sin x))^(1/(3 )) βˆ’1)/( ((1+tan x))^(1/(6 )) βˆ’1)) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{8}\:}]{\mathrm{256}+\mathrm{tan}\:\:{x}}\:βˆ’\sqrt[{\mathrm{3}\:}]{\mathrm{1}+\mathrm{sin}\:{x}}\:βˆ’\mathrm{1}}{\:\sqrt[{\mathrm{6}\:}]{\mathrm{1}+\mathrm{tan}\:{x}}\:βˆ’\mathrm{1}}\:? \\ $$

Question Number 115604    Answers: 6   Comments: 1

(1)lim_(xβ†’(Ο€/2)) ((cos 2x)/(tan x)) =? (2) lim_(xβ†’Ο€) (((√(2 +cos x)) βˆ’1)/((Ο€βˆ’x)^2 )) = ?

$$\left(\mathrm{1}\right)\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{tan}\:{x}}\:=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}\:+\mathrm{cos}\:{x}}\:βˆ’\mathrm{1}}{\left(\piβˆ’{x}\right)^{\mathrm{2}} }\:=\:? \\ $$

Question Number 115597    Answers: 4   Comments: 0

Question Number 115595    Answers: 1   Comments: 0

Prove that, for all primes p>3, 13∣10^(2p) βˆ’10^p +1

$$\mathrm{Prove}\:\mathrm{that},\:\mathrm{for}\:\mathrm{all}\:\mathrm{primes}\:\mathrm{p}>\mathrm{3}, \\ $$$$\mathrm{13}\mid\mathrm{10}^{\mathrm{2p}} βˆ’\mathrm{10}^{\mathrm{p}} +\mathrm{1} \\ $$

Question Number 115594    Answers: 1   Comments: 0

old question, I couldnβ€²t find it: ∫(√(xβˆ’(√x)))dx=?

$$\mathrm{old}\:\mathrm{question},\:\mathrm{I}\:\mathrm{couldn}'\mathrm{t}\:\mathrm{find}\:\mathrm{it}: \\ $$$$\int\sqrt{{x}βˆ’\sqrt{{x}}}{dx}=? \\ $$

Question Number 115592    Answers: 0   Comments: 1

Question Number 115625    Answers: 0   Comments: 1

Question Number 115575    Answers: 3   Comments: 2

lim_(nβ†’βˆž) Ξ _(k=1) ^n (1βˆ’(1/(k+1)))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\prod}}\left(\mathrm{1}βˆ’\frac{\mathrm{1}}{\mathrm{k}+\mathrm{1}}\right)=? \\ $$

Question Number 115564    Answers: 1   Comments: 0

Question Number 115558    Answers: 2   Comments: 2

... advanced calculus... evaluate :: ∫_0 ^( ∞) ln(1+ax^2 )ln(1+(b/x^2 ))dx m.n.july

$$\:\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}...\: \\ $$$$\:\:\:\:\:\:\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} {ln}\left(\mathrm{1}+{ax}^{\mathrm{2}} \right){ln}\left(\mathrm{1}+\frac{{b}}{{x}^{\mathrm{2}} }\right){dx} \\ $$$$\:\:\:\:\:\:\:{m}.{n}.{july} \\ $$$$ \\ $$

Question Number 115555    Answers: 3   Comments: 1

Given that x,y∈R βˆ€ x^2 βˆ’y^2 =32, (x+y)^4 +(xβˆ’y)^4 =4352, Find the value of x^2 +y^2 .

$$\mathrm{Given}\:\mathrm{that}\:{x},{y}\in\mathbb{R}\:\forall\:{x}^{\mathrm{2}} βˆ’{y}^{\mathrm{2}} =\mathrm{32}, \\ $$$$\left({x}+{y}\right)^{\mathrm{4}} +\left({x}βˆ’{y}\right)^{\mathrm{4}} =\mathrm{4352},\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} . \\ $$

Question Number 115551    Answers: 2   Comments: 0

lim_(xβ†’2) (((x^4 βˆ’4x^3 +5x^2 βˆ’4x+4))^(1/(4 )) /( (√(x^2 βˆ’3x+2)))) = ?

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{4}\:}]{{x}^{\mathrm{4}} βˆ’\mathrm{4}{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} βˆ’\mathrm{4}{x}+\mathrm{4}}}{\:\sqrt{{x}^{\mathrm{2}} βˆ’\mathrm{3}{x}+\mathrm{2}}}\:=\:? \\ $$

Question Number 115544    Answers: 1   Comments: 1

Given x^2 +12(√x) = 5 then x+2(√x) ?

$${Given}\:{x}^{\mathrm{2}} +\mathrm{12}\sqrt{{x}}\:=\:\mathrm{5} \\ $$$${then}\:{x}+\mathrm{2}\sqrt{{x}}\:? \\ $$

Question Number 115541    Answers: 3   Comments: 0

lim_(xβ†’a) (2βˆ’(x/a))^(tan (((Ο€x)/(2a)))) =?

$$\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\left(\mathrm{2}βˆ’\frac{{x}}{{a}}\right)^{\mathrm{tan}\:\left(\frac{\pi{x}}{\mathrm{2}{a}}\right)} =? \\ $$

Question Number 115539    Answers: 1   Comments: 0

(dy/dx) = ((e^(tan^(βˆ’1) (x)) βˆ’y)/(1+x^2 ))

$$\frac{{dy}}{{dx}}\:=\:\frac{{e}^{\mathrm{tan}^{βˆ’\mathrm{1}} \left({x}\right)} \:βˆ’{y}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$

Question Number 115534    Answers: 4   Comments: 1

Let say r^((n)) = Ξ _(k=0) ^(nβˆ’1) (rβˆ’k) and r^((0)) =1 With n∈N and r∈R... 1. Show that (nβˆ’1βˆ’r)^((n)) = (βˆ’1)^((n)) (r)^((n)) 2. If m≀n, show that (r^((n)) /r^((m)) )=(rβˆ’m)^((nβˆ’m)) 3. Espress r^((n+m)) as w^((n)) wβ€²^((m)) 4. Show that (2r)^((2n)) =2^(2n) r^((n)) (rβˆ’(1/2))^((n)) Can you help me... please...

$$\mathrm{Let}\:\mathrm{say}\:{r}^{\left({n}\right)} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}βˆ’\mathrm{1}} {\prod}}\left({r}βˆ’{k}\right)\:\mathrm{and}\:{r}^{\left(\mathrm{0}\right)} =\mathrm{1} \\ $$$$\mathrm{With}\:{n}\in\mathbb{N}\:\mathrm{and}\:{r}\in\mathbb{R}... \\ $$$$\mathrm{1}.\:\:\:\mathrm{Show}\:\mathrm{that}\:\left({n}βˆ’\mathrm{1}βˆ’{r}\right)^{\left({n}\right)} \:=\:\left(βˆ’\mathrm{1}\right)^{\left({n}\right)} \left({r}\right)^{\left({n}\right)} \\ $$$$\mathrm{2}.\:\mathrm{If}\:{m}\leqslant{n},\:\mathrm{show}\:\mathrm{that}\:\:\frac{{r}^{\left({n}\right)} }{{r}^{\left({m}\right)} }=\left({r}βˆ’{m}\right)^{\left({n}βˆ’{m}\right)} \\ $$$$\mathrm{3}.\:\mathrm{Espress}\:{r}^{\left({n}+{m}\right)} \:\mathrm{as}\:{w}^{\left({n}\right)} {w}'^{\left({m}\right)} \\ $$$$\mathrm{4}.\:\mathrm{Show}\:\mathrm{that}\:\left(\mathrm{2}{r}\right)^{\left(\mathrm{2}{n}\right)} =\mathrm{2}^{\mathrm{2}{n}} {r}^{\left({n}\right)} \left({r}βˆ’\frac{\mathrm{1}}{\mathrm{2}}\right)^{\left({n}\right)} \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}...\:\mathrm{please}... \\ $$

Question Number 115533    Answers: 0   Comments: 0

Question Number 115531    Answers: 0   Comments: 0

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