Question and Answers Forum

AllQuestion and Answers: Page 279

Question Number 193494 Answers: 3 Comments: 1

Let n be a fixed positive integer such that sin((π/(2n)))+cos((𝛑/(2n)))=((√n)/2) Then find n

$$\boldsymbol{\mathrm{Let}}\:\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{fixed}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integer}}\:\boldsymbol{\mathrm{such}}\:\boldsymbol{\mathrm{that}} \\ $$$$\mathrm{sin}\left(\frac{\pi}{\mathrm{2n}}\right)+\mathrm{cos}\left(\frac{\boldsymbol{\pi}}{\mathrm{2n}}\right)=\frac{\sqrt{\mathrm{n}}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{Then}}\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{n}} \\ $$

Question Number 193490 Answers: 0 Comments: 0

Question Number 193486 Answers: 1 Comments: 0

Question Number 193485 Answers: 2 Comments: 0

Show that 2^n −(−1)^n is divisible by 3 for all positive integers n.

$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{Show}}\:\boldsymbol{\mathrm{that}}\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} −\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} \:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{divisible}}\:\boldsymbol{\mathrm{by}} \\ $$$$\:\:\:\:\:\:\:\mathrm{3}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}}\:\boldsymbol{\mathrm{n}}. \\ $$

Question Number 193484 Answers: 0 Comments: 0

Nice problem: Find 8 distinctive numbers ∈N\{0} such that these are simultaniously true: (1) a+b+c+d = e+f+g+h (2) a^2 +b^2 +c^2 +d^2 = e^2 +f^2 +g^2 +h^2 (3) a^3 +b^3 +c^3 +d^3 = e^3 +f^3 +g^3 +h^3 [Find a method to generate such numbers]

$$\mathrm{Nice}\:\mathrm{problem}: \\ $$$$\mathrm{Find}\:\mathrm{8}\:\mathrm{distinctive}\:\mathrm{numbers}\:\in\mathbb{N}\backslash\left\{\mathrm{0}\right\}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{these}\:\mathrm{are}\:\mathrm{simultaniously}\:\mathrm{true}: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:{a}+{b}+{c}+{d}\:=\:{e}+{f}+{g}+{h} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \:=\:{e}^{\mathrm{2}} +{f}^{\mathrm{2}} +{g}^{\mathrm{2}} +{h}^{\mathrm{2}} \\ $$$$\left(\mathrm{3}\right)\:\:\:\:\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} \:=\:{e}^{\mathrm{3}} +{f}^{\mathrm{3}} +{g}^{\mathrm{3}} +{h}^{\mathrm{3}} \\ $$$$\left[\mathrm{Find}\:\mathrm{a}\:\mathrm{method}\:\mathrm{to}\:\mathrm{generate}\:\mathrm{such}\:\mathrm{numbers}\right] \\ $$

Question Number 193469 Answers: 0 Comments: 0

Question Number 193467 Answers: 3 Comments: 3

Proof : cot^(−1) ((((√(1+sint))+(√(1−sint)))/( (√(1+sint))−(√(1−sint)))))=(t/2)

$$\mathrm{Proof}\:: \\ $$$$\mathrm{cot}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}+\mathrm{sint}}+\sqrt{\mathrm{1}−\mathrm{sint}}}{\:\sqrt{\mathrm{1}+\mathrm{sint}}−\sqrt{\mathrm{1}−\mathrm{sint}}}\right)=\frac{\mathrm{t}}{\mathrm{2}}\: \\ $$

Question Number 193464 Answers: 1 Comments: 3

Question Number 193461 Answers: 2 Comments: 0

lim_(x→0) ((((√(1+sin x))−1)/(sin 2x))) = ??

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}−\mathrm{1}}{\mathrm{sin}\:\mathrm{2x}}\right)\:=\:?? \\ $$

Question Number 193458 Answers: 1 Comments: 1

lim_(x→0) (((1−cos(x))/(x sin(x)))) = ???

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{cos}\left(\mathrm{x}\right)}{\mathrm{x}\:\mathrm{sin}\left(\mathrm{x}\right)}\right)\:=\:\:\:??? \\ $$

Question Number 193449 Answers: 1 Comments: 4

Question Number 193448 Answers: 1 Comments: 0

Question Number 193439 Answers: 1 Comments: 2

∫^(π/2) _( 0) (((tanx))^(1/3) /((sinx+cosx)^2 ))dx

$$\underset{\:\:\mathrm{0}} {\int}^{\pi/\mathrm{2}} \frac{\sqrt[{\mathrm{3}}]{{tanx}}}{\left({sinx}+{cosx}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 193438 Answers: 1 Comments: 0

Question Number 193437 Answers: 0 Comments: 0

in the malicol of CaCO_3 how many σ and π bond has?

$$\mathrm{in}\:\mathrm{the}\:\mathrm{malicol}\:\mathrm{of}\:\mathrm{CaCO}_{\mathrm{3}} \:\mathrm{how}\:\mathrm{many} \\ $$$$\sigma\:\mathrm{and}\:\pi\:\mathrm{bond}\:\mathrm{has}? \\ $$

Question Number 193436 Answers: 1 Comments: 0

((sin(x+18^o ))/(sin(48^o )))=((sin(x))/(sin(18^o )))

$$ \\ $$$$\frac{{sin}\left({x}+\mathrm{18}^{{o}} \right)}{{sin}\left(\mathrm{48}^{{o}} \right)}=\frac{{sin}\left({x}\right)}{{sin}\left(\mathrm{18}^{{o}} \right)} \\ $$$$ \\ $$

Question Number 193435 Answers: 1 Comments: 1

Question Number 193426 Answers: 1 Comments: 0

∫((x^3 +1))^(1/3) dx=? solution?

$$\int\sqrt[{\mathrm{3}}]{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}\mathrm{dx}=? \\ $$$$\mathrm{solution}? \\ $$

Question Number 193423 Answers: 3 Comments: 0

when tan(θ/2)=(1/a) then find cosθ=? from the a

$$\mathrm{when}\:\:\:\mathrm{tan}\frac{\theta}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{a}} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{cos}\theta=?\:\mathrm{from}\:\mathrm{the}\:\mathrm{a} \\ $$

Question Number 193414 Answers: 2 Comments: 0

Question Number 193411 Answers: 2 Comments: 1

∫_0 ^1 (√((1−x)/(1+x))) dx =?

$$\:\: \\ $$$$ \underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\sqrt{\frac{\mathrm{1}−\mathrm{x}}{\mathrm{1}+\mathrm{x}}}\:\mathrm{dx}\:=? \\ $$

Question Number 193410 Answers: 2 Comments: 0

{ ((x=(√(3−(√(5+2(√3))))))),((y=(√(3+(√(5+2(√3))))))) :}

$$\:\:\begin{cases}{\mathrm{x}=\sqrt{\mathrm{3}−\sqrt{\mathrm{5}+\mathrm{2}\sqrt{\mathrm{3}}}}}\\{\mathrm{y}=\sqrt{\mathrm{3}+\sqrt{\mathrm{5}+\mathrm{2}\sqrt{\mathrm{3}}}}}\end{cases}\: \\ $$$$\:\:\: \\ $$

Question Number 193409 Answers: 0 Comments: 0

lim_(n→∞) ((∫_0 ^2 (1+6x−7x^2 +4x^3 −x^4 )^n dx))^(1/n)

$$\:\: \underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\mathrm{n}}]{\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\left(\mathrm{1}+\mathrm{6x}−\mathrm{7x}^{\mathrm{2}} +\mathrm{4x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{4}} \right)^{\mathrm{n}} \:\mathrm{dx}} \\ $$

Question Number 193408 Answers: 1 Comments: 0

lim_(n→∞) ((1/n). ((1−(e^(a/n) )^(n−1) )/(1−e^(a/n) )) )

$$ \\ $$$$ \underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{n}}.\:\frac{\mathrm{1}−\left(\mathrm{e}^{\frac{\mathrm{a}}{\mathrm{n}}} \right)^{\mathrm{n}−\mathrm{1}} }{\mathrm{1}−\mathrm{e}^{\frac{\mathrm{a}}{\mathrm{n}}} }\:\right)\: \\ $$

Question Number 193405 Answers: 0 Comments: 0

Question Number 193399 Answers: 1 Comments: 0

Pg 274 Pg 275 Pg 276 Pg 277 Pg 278 Pg 279 Pg 280 Pg 281 Pg 282 Pg 283

Contact: info@tinkutara.com