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AllQuestion and Answers: Page 1210

Question Number 93414 Answers: 0 Comments: 0

let p(x)=((x^n (4−2x)^n )/(n!)) 1) prove that p^((k)) (0)=p^((k)) (2)=0 for all k∈[1,n−1] 2) prove that ∀m∈N p^((m)) (0) and p^((m)) (2) are integrs

$${let}\:{p}\left({x}\right)=\frac{{x}^{{n}} \left(\mathrm{4}−\mathrm{2}{x}\right)^{{n}} }{{n}!} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:{p}^{\left({k}\right)} \left(\mathrm{0}\right)={p}^{\left({k}\right)} \left(\mathrm{2}\right)=\mathrm{0}\:{for}\:{all}\:{k}\in\left[\mathrm{1},{n}−\mathrm{1}\right] \\ $$$$\left.\mathrm{2}\right)\:\:{prove}\:{that}\:\:\forall{m}\in{N}\:\:\:\:{p}^{\left({m}\right)} \left(\mathrm{0}\right)\:{and}\:{p}^{\left({m}\right)} \left(\mathrm{2}\right)\:{are}\:{integrs} \\ $$

Question Number 93410 Answers: 0 Comments: 7

∫((1+x^6 )/(1+x^8 ))dx

$$\int\frac{\mathrm{1}+{x}^{\mathrm{6}} }{\mathrm{1}+{x}^{\mathrm{8}} }{dx} \\ $$

Question Number 93397 Answers: 0 Comments: 2

prove that ((1−tan^3 θ)/(1 + tan^3 θ)) = 1−2sin^3 θ

$$\mathrm{prove}\:\mathrm{that}\:\frac{\mathrm{1}−\mathrm{tan}^{\mathrm{3}} \theta}{\mathrm{1}\:+\:\mathrm{tan}^{\mathrm{3}} \theta}\:=\:\mathrm{1}−\mathrm{2sin}^{\mathrm{3}} \theta\: \\ $$

Question Number 93388 Answers: 0 Comments: 2

Question Number 93376 Answers: 0 Comments: 1

x^x^x =((1/2))^(1/2) x=?

$$ \\ $$$${x}^{{x}^{{x}} } =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$${x}=? \\ $$

Question Number 93371 Answers: 0 Comments: 21

New version of the app with usability enhacements is now available on playstore. Pleasd update app. Comment on this post for problem/feedback suggestions.

$$\mathrm{New}\:\mathrm{version}\:\mathrm{of}\:\mathrm{the}\:\mathrm{app}\:\mathrm{with}\:\mathrm{usability} \\ $$$$\mathrm{enhacements}\:\mathrm{is}\:\mathrm{now}\:\mathrm{available}\:\mathrm{on} \\ $$$$\mathrm{playstore}. \\ $$$$\mathrm{Pleasd}\:\mathrm{update}\:\mathrm{app}.\: \\ $$$$\mathrm{Comment}\:\mathrm{on}\:\mathrm{this}\:\mathrm{post}\:\mathrm{for} \\ $$$$\mathrm{problem}/\mathrm{feedback}\:\mathrm{suggestions}. \\ $$

Question Number 93368 Answers: 1 Comments: 0

given that f(r)= sin (1 + 2r)θ show that f(r)−f(r−1) = 2 cos 2r θ sin θ hence show that Σ_(r=1) ^n cos 2r θ sin θ = cos (n +1)θ sin nθ

$$\mathrm{given}\:\mathrm{that}\:{f}\left({r}\right)=\:\:\mathrm{sin}\:\left(\mathrm{1}\:+\:\mathrm{2}{r}\right)\theta \\ $$$$\mathrm{show}\:\mathrm{that}\:{f}\left({r}\right)−{f}\left({r}−\mathrm{1}\right)\:=\:\mathrm{2}\:\mathrm{cos}\:\mathrm{2}{r}\:\theta\:\mathrm{sin}\:\theta \\ $$$$\mathrm{hence}\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\mathrm{cos}\:\mathrm{2}{r}\:\theta\:\mathrm{sin}\:\theta\:\:=\:\mathrm{cos}\:\left({n}\:+\mathrm{1}\right)\theta\:\mathrm{sin}\:{n}\theta \\ $$

Question Number 93367 Answers: 0 Comments: 2

show using demoives theorem that sin^2 θ cos^2 4θ =(1/8)(1−cos4θ)

$$\mathrm{show}\:\mathrm{using}\:\mathrm{demoives}\:\mathrm{theorem}\:\mathrm{that}\: \\ $$$$\mathrm{sin}^{\mathrm{2}} \theta\:\mathrm{cos}^{\mathrm{2}} \mathrm{4}\theta\:=\frac{\mathrm{1}}{\mathrm{8}}\left(\mathrm{1}−\mathrm{cos4}\theta\right) \\ $$

Question Number 93366 Answers: 1 Comments: 0

solve the equation (z−2)^3 = (1/2)−i((√3)/2)

$$\:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\left({z}−\mathrm{2}\right)^{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}−{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}} \\ $$

Question Number 93365 Answers: 0 Comments: 3

given that α is a real number, use mathematical induction or otherwise to show that cos ((α/2))cos((α/2^2 ))cos((α/2^3 )) ...cos((α/2^n )) = ((sin α)/(2^n sin((α/2^n )))) hence find the lim_(n→∞) cos((α/2))cos((α/2^2 ))cos((α/2^3 )) ... cos((α/2^n ))

Question Number 93350 Answers: 1 Comments: 0

f(x)=x^3 +2x Find: f^(−1) (x)

$${f}\left({x}\right)={x}^{\mathrm{3}} +\mathrm{2}{x} \\ $$$${Find}:\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$

Question Number 93340 Answers: 1 Comments: 7

Σ_(k = 1) ^(2011) (1/(k(k+1)(k+2))) ?

$$\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\mathrm{2011}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{k}\left(\mathrm{k}+\mathrm{1}\right)\left(\mathrm{k}+\mathrm{2}\right)}\:? \\ $$

Question Number 93331 Answers: 1 Comments: 2

If f(x+y) = f(x)+f(y) and f(1)=5 then what is the value of f(2020) ?

$$\mathrm{If}\:\mathrm{f}\left(\mathrm{x}+\mathrm{y}\right)\:=\:\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{y}\right)\:\mathrm{and} \\ $$$$\mathrm{f}\left(\mathrm{1}\right)=\mathrm{5}\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\mathrm{f}\left(\mathrm{2020}\right)\:?\: \\ $$

Question Number 93330 Answers: 1 Comments: 0

A committee of five is to be chosen from a group of 9 people. The probability that a certain married couple will either serve together or not at all is

$$\mathrm{A}\:\mathrm{committee}\:\mathrm{of}\:\mathrm{five}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{chosen}\:\mathrm{from} \\ $$$$\mathrm{a}\:\mathrm{group}\:\mathrm{of}\:\:\mathrm{9}\:\mathrm{people}.\:\mathrm{The}\:\mathrm{probability} \\ $$$$\mathrm{that}\:\mathrm{a}\:\mathrm{certain}\:\mathrm{married}\:\mathrm{couple}\:\mathrm{will}\:\mathrm{either} \\ $$$$\mathrm{serve}\:\mathrm{together}\:\mathrm{or}\:\mathrm{not}\:\mathrm{at}\:\mathrm{all}\:\mathrm{is} \\ $$

Question Number 93315 Answers: 0 Comments: 0

If n>=2 and n⊂N . prove (1/2^2 )+(1/3^2 )+...(1/n^2 )<=n(1−(1/2)(1+(1/n)))^(1/n)

$${If}\:\:{n}>=\mathrm{2}\:\:{and}\:{n}\subset\mathbb{N}\:. \\ $$$${prove}\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }+...\frac{\mathrm{1}}{{n}^{\mathrm{2}} }<={n}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\right)^{\frac{\mathrm{1}}{{n}}} \\ $$

Question Number 93312 Answers: 1 Comments: 0

(y−xy^2 )dx+(x+x^2 y^2 )dy = 0

$$\left(\mathrm{y}−\mathrm{xy}^{\mathrm{2}} \right)\mathrm{dx}+\left(\mathrm{x}+\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} \right)\mathrm{dy}\:=\:\mathrm{0}\: \\ $$

Question Number 93304 Answers: 5 Comments: 5

∫(√(tan (x)))dx Who can solve this problem?

$$\int\sqrt{\mathrm{tan}\:\left({x}\right)}{dx}\:{Who}\:{can}\:{solve}\:{this}\:{problem}? \\ $$

Question Number 93299 Answers: 0 Comments: 2

find a reduction formulae for I_n = ∫_(−1) ^0 x^n (1 + x)^2 dx