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Question Number 90727 Answers: 0 Comments: 0

Σ_(a,b≥1) (1/((a+b^2 )(a+b^2 +1))) nature and sum value

$$\underset{{a},{b}\geqslant\mathrm{1}} {\sum}\:\frac{\mathrm{1}}{\left({a}+{b}^{\mathrm{2}} \right)\left({a}+{b}^{\mathrm{2}} +\mathrm{1}\right)}\:\:\:\:\:\:{nature}\:{and}\:\:{sum}\:{value}\:\: \\ $$$$ \\ $$

Question Number 90722 Answers: 1 Comments: 3

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

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

Question Number 90717 Answers: 0 Comments: 0

Prove that for all integer r≥2 HCF(r^n −1;r^m −1)=r^(HCF(n;m)) −1

$${Prove}\:{that}\:{for}\:{all}\:{integer}\:{r}\geqslant\mathrm{2}\: \\ $$$$\:{HCF}\left({r}^{{n}} −\mathrm{1};{r}^{{m}} −\mathrm{1}\right)={r}^{{HCF}\left({n};{m}\right)} −\mathrm{1} \\ $$

Question Number 90716 Answers: 0 Comments: 1

If x + (1/x) = 4 , what the value of ((x^6 −1)/x^3 )

$${If}\:{x}\:+\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{4}\:,\:{what}\:{the}\: \\ $$$${value}\:{of}\:\frac{{x}^{\mathrm{6}} −\mathrm{1}}{{x}^{\mathrm{3}} } \\ $$

Question Number 90709 Answers: 0 Comments: 2

α,β and γ are the roots of x^3 −9x+9=0 find the value of (1) α^(−3) +β^(−3) +γ^(−3) (2) α^(−5) +β^(−5) +γ^(−5)

$$\alpha,\beta\:{and}\:\gamma\:{are}\:{the}\:{roots}\:{of}\:\:{x}^{\mathrm{3}} −\mathrm{9}{x}+\mathrm{9}=\mathrm{0} \\ $$$${find}\:{the}\:{value}\:{of}\:\left(\mathrm{1}\right)\:\alpha^{−\mathrm{3}} +\beta^{−\mathrm{3}} +\gamma^{−\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)\:\alpha^{−\mathrm{5}} +\beta^{−\mathrm{5}} +\gamma^{−\mathrm{5}} \\ $$

Question Number 90706 Answers: 2 Comments: 1

Question Number 90700 Answers: 0 Comments: 3

sin^2 (((7π)/8))+sin^2 (((3π)/8))+sin^2 (((5π)/8))+sin^2 ((π/8)) ?

$$\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)+\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\pi}{\mathrm{8}}\right)\:? \\ $$

Question Number 90698 Answers: 1 Comments: 3

how many solution the equation ⌊ x ⌋ +2016. {x} = 38?

$${how}\:{many}\:{solution}\:{the}\:{equation} \\ $$$$\lfloor\:{x}\:\rfloor\:+\mathrm{2016}.\:\left\{{x}\right\}\:=\:\mathrm{38}? \\ $$

Question Number 90692 Answers: 1 Comments: 2

Question Number 90679 Answers: 0 Comments: 1

if y=sin(msin^(−1) x), prove that (1−x^2 )y_(n+2) −(2n+1)xy_(n+1) +(m^2 −n^2 )y_n =0

$${if}\:{y}={sin}\left({m}\mathrm{sin}^{−\mathrm{1}} {x}\right),\:{prove}\:{that}\:\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}_{{n}+\mathrm{2}} −\left(\mathrm{2}{n}+\mathrm{1}\right){xy}_{{n}+\mathrm{1}} +\left({m}^{\mathrm{2}} −{n}^{\mathrm{2}} \right){y}_{{n}} =\mathrm{0} \\ $$

Question Number 92777 Answers: 1 Comments: 2

a_(n+1) =(2n+1)a_n a_1 =1 a_n =?

$$\mathrm{a}_{\mathrm{n}+\mathrm{1}} =\left(\mathrm{2n}+\mathrm{1}\right)\mathrm{a}_{\mathrm{n}} \\ $$$$\mathrm{a}_{\mathrm{1}} =\mathrm{1} \\ $$$$\mathrm{a}_{\mathrm{n}} =? \\ $$$$ \\ $$

Question Number 90661 Answers: 2 Comments: 0

show that (n^4 −n^2 ) is divisible by 12

$${show}\:{that}\:\left({n}^{\mathrm{4}} −{n}^{\mathrm{2}} \right)\:{is}\:{divisible}\:{by}\:\mathrm{12} \\ $$

Question Number 90647 Answers: 0 Comments: 14

Question Number 90641 Answers: 0 Comments: 4

Solve x^2 y′′+xy′+x^2 y=0

$${Solve}\:{x}^{\mathrm{2}} {y}''+{xy}'+{x}^{\mathrm{2}} {y}=\mathrm{0} \\ $$

Question Number 90637 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (1/2^n )tan((1/2^n ))

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{{n}} }{tan}\left(\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\right) \\ $$

Question Number 90632 Answers: 1 Comments: 0

Question Number 90630 Answers: 1 Comments: 1

f(x) = xe^(−x) f^((2020)) (x) =

$${f}\left({x}\right)\:=\:{xe}^{−{x}} \\ $$$${f}^{\left(\mathrm{2020}\right)} \left({x}\right)\:=\: \\ $$

Question Number 90629 Answers: 1 Comments: 1

If sin^(−1) (1−x)−2 sin^(−1) x = (π/2), then x=

$$\mathrm{If}\:\:\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)−\mathrm{2}\:\mathrm{sin}^{−\mathrm{1}} {x}\:=\:\frac{\pi}{\mathrm{2}},\:\mathrm{then}\:{x}= \\ $$

Question Number 90628 Answers: 0 Comments: 1

The value of sin (π/(14)) sin ((3π)/(14)) sin ((5π)/(14)) is

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{sin}\:\frac{\pi}{\mathrm{14}}\:\mathrm{sin}\:\frac{\mathrm{3}\pi}{\mathrm{14}}\:\mathrm{sin}\:\frac{\mathrm{5}\pi}{\mathrm{14}}\:\:\mathrm{is} \\ $$

Question Number 90625 Answers: 0 Comments: 0

∫e^(arcsinx) dx

$$\int{e}^{{arcsinx}} {dx} \\ $$

Question Number 90609 Answers: 0 Comments: 3

find the infinite sumΣ_(n=0) ^∞ (F_n /2^n ) where F_n =(1/(√5))(((1+(√5))/2))^(n+1) −(1/(√5))(((1−(√5))/2))^(n+1)

$${find}\:{the}\:{infinite}\:{sum}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{F}_{{n}} }{\mathrm{2}^{{n}} }\: \\ $$$${where}\:{F}_{{n}} =\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}+\mathrm{1}} −\frac{\mathrm{1}}{\sqrt{\mathrm{5}}}\left(\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}+\mathrm{1}} \\ $$

Question Number 90596 Answers: 1 Comments: 0

If sin(28) = a and cos(32) = b Find (i) cos(28) (ii) cos(64) (iii) sin(4)

$$\mathrm{If}\:\:\:\mathrm{sin}\left(\mathrm{28}\right)\:\:=\:\:\mathrm{a}\:\:\:\:\mathrm{and}\:\:\:\mathrm{cos}\left(\mathrm{32}\right)\:\:=\:\:\mathrm{b} \\ $$$$\mathrm{Find}\:\:\left(\mathrm{i}\right)\:\:\mathrm{cos}\left(\mathrm{28}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{ii}\right)\:\mathrm{cos}\left(\mathrm{64}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{iii}\right)\:\mathrm{sin}\left(\mathrm{4}\right) \\ $$

Question Number 90594 Answers: 0 Comments: 3

Question Number 90592 Answers: 0 Comments: 3

∫_0 ^(√(arccos(((−2φ)/π)+1))) x sin(x^2 ) dx

$$\int_{\mathrm{0}} ^{\sqrt{{arccos}\left(\frac{−\mathrm{2}\phi}{\pi}+\mathrm{1}\right)}} {x}\:{sin}\left({x}^{\mathrm{2}} \right)\:{dx} \\ $$

Question Number 90590 Answers: 1 Comments: 2

Question Number 90589 Answers: 0 Comments: 3

∫(1/(x+(√(x^2 +x+1))))dx

$$\int\frac{\mathrm{1}}{{x}+\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$

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