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AllQuestion and Answers: Page 1323

Question Number 80503 Answers: 1 Comments: 0

prove that they are infinitely many primes

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{they}\:\mathrm{are}\:\mathrm{infinitely}\:\mathrm{many} \\ $$$$\mathrm{primes} \\ $$

Question Number 80504 Answers: 0 Comments: 2

Solve the system of congruences x ≡ 2 (mod 3) x ≡ 5( mod 7)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{congruences} \\ $$$${x}\:\equiv\:\mathrm{2}\:\left(\mathrm{mod}\:\mathrm{3}\right) \\ $$$${x}\:\equiv\:\mathrm{5}\left(\:\mathrm{mod}\:\mathrm{7}\right) \\ $$$$\: \\ $$

Question Number 80501 Answers: 0 Comments: 0

(5/7) = (a_2 /(2!)) + (a_3 /(3!)) + (a_4 /(4!)) + (a_5 /(5!)) + (a_6 /(6!)) + (a_7 /(7!)) 0 ≤ a_i < i , a_i ∈ N Find possible value of a_2 + a_3 + a_4 + a_5 + a_6 + a_7 .

$$\frac{\mathrm{5}}{\mathrm{7}}\:\:=\:\:\frac{{a}_{\mathrm{2}} }{\mathrm{2}!}\:+\:\frac{{a}_{\mathrm{3}} }{\mathrm{3}!}\:+\:\frac{{a}_{\mathrm{4}} }{\mathrm{4}!}\:+\:\frac{{a}_{\mathrm{5}} }{\mathrm{5}!}\:+\:\frac{{a}_{\mathrm{6}} }{\mathrm{6}!}\:+\:\frac{{a}_{\mathrm{7}} }{\mathrm{7}!} \\ $$$$\mathrm{0}\:\:\leqslant\:{a}_{{i}} \:<\:{i}\:\:,\:\:{a}_{{i}} \:\:\in\:\mathbb{N} \\ $$$${Find}\:\:{possible}\:\:{value}\:\:{of}\:\:\:{a}_{\mathrm{2}} \:+\:{a}_{\mathrm{3}} \:+\:{a}_{\mathrm{4}} \:+\:{a}_{\mathrm{5}} \:+\:{a}_{\mathrm{6}} \:+\:{a}_{\mathrm{7}} \:\:. \\ $$

Question Number 80455 Answers: 1 Comments: 2

lim_(x→0) (((1+mx)/(1−nx)))^((mn)/x)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}+{mx}}{\mathrm{1}−{nx}}\right)^{\frac{{mn}}{{x}}} \\ $$

Question Number 80452 Answers: 0 Comments: 1

find ∫_(−∞) ^(+∞) ((cos(2x^2 +1))/(x^4 −x^2 +3))dx

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}\right)}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

Question Number 80451 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(πx))/((x^2 +3)^2 ))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{cos}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 80448 Answers: 1 Comments: 3

Hello All of You verry Nice Day, God bless You love peace and happiness Solve for (x,y)∈R^2 { ((x^2 +y^2 =2x+3y+1)),((x^4 +y^4 =4x^2 +9y^2 +12xy+2x^2 y^2 +18)) :}

$${Hello}\:{All}\:{of}\:{You}\:{verry}\:{Nice}\:{Day},\:{God}\:{bless}\:{You}\:{love}\:{peace}\:{and}\: \\ $$$${happiness}\: \\ $$$${Solve}\:{for}\:\left({x},{y}\right)\in\mathbb{R}^{\mathrm{2}} \: \\ $$$$\begin{cases}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{2}{x}+\mathrm{3}{y}+\mathrm{1}}\\{{x}^{\mathrm{4}} +{y}^{\mathrm{4}} =\mathrm{4}{x}^{\mathrm{2}} +\mathrm{9}{y}^{\mathrm{2}} +\mathrm{12}{xy}+\mathrm{2}{x}^{\mathrm{2}} {y}^{\mathrm{2}} +\mathrm{18}}\end{cases} \\ $$$$ \\ $$

Question Number 80442 Answers: 1 Comments: 4

Question Number 80433 Answers: 1 Comments: 6

find the solution of (√(4−x))−2≤x∣x−3∣+4x

$${find}\:{the}\:{solution}\:{of} \\ $$$$\sqrt{\mathrm{4}−{x}}−\mathrm{2}\leqslant{x}\mid{x}−\mathrm{3}\mid+\mathrm{4}{x} \\ $$

Question Number 80432 Answers: 0 Comments: 7

Question Number 80417 Answers: 0 Comments: 5

lim_(x→0) (((1+x)^(1/x) −e)/x) = ?

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

Question Number 80416 Answers: 1 Comments: 0

∫_0 ^(π/2) ((xcos x)/((1+sin x)^2 )) dx ?

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{{x}\mathrm{cos}\:{x}}{\left(\mathrm{1}+\mathrm{sin}\:{x}\right)^{\mathrm{2}} }\:{dx}\:? \\ $$

Question Number 80405 Answers: 1 Comments: 11

Solve: (a) (x − 3)^2 > − 5 (b) 3x^2 > − 12

$$\mathrm{Solve}: \\ $$$$\left(\mathrm{a}\right)\:\:\:\:\:\:\:\:\:\left(\mathrm{x}\:−\:\mathrm{3}\right)^{\mathrm{2}} \:\:>\:\:−\:\mathrm{5} \\ $$$$\left(\mathrm{b}\right)\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3x}^{\mathrm{2}} \:\:>\:\:−\:\mathrm{12} \\ $$

Question Number 80404 Answers: 1 Comments: 0

Question Number 80402 Answers: 0 Comments: 0

Question Number 80397 Answers: 0 Comments: 3

show that ∫_0 ^(π/2) ∫_0 ^∞ (1/((x^π )^(1/y) +1)) dx dy =2c whrre c denote tha catalan^, s constant

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\sqrt[{{y}}]{{x}^{\pi} }\:+\mathrm{1}}\:{dx}\:{dy}\:=\mathrm{2}{c}\: \\ $$$${whrre}\:{c}\:{denote}\:{tha}\:{catalan}^{,} {s}\:{constant} \\ $$

Question Number 80386 Answers: 1 Comments: 2

Question Number 80376 Answers: 1 Comments: 2

Question Number 80374 Answers: 0 Comments: 5

Question Number 80369 Answers: 0 Comments: 1

Question Number 80365 Answers: 1 Comments: 6

Question Number 80362 Answers: 1 Comments: 0

Question Number 80341 Answers: 1 Comments: 0

A particle moves round the polar curve r = a(1 + cos θ) with constant angular velocity ω . Find the transverse component of the velocity.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{round}\:\mathrm{the}\:\mathrm{polar}\:\mathrm{curve} \\ $$$${r}\:=\:{a}\left(\mathrm{1}\:+\:\mathrm{cos}\:\theta\right)\:\mathrm{with}\:\mathrm{constant}\:\mathrm{angular}\: \\ $$$$\mathrm{velocity}\:\omega\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{transverse}\:\mathrm{component} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{velocity}. \\ $$

Question Number 80340 Answers: 1 Comments: 0

If P = ((a,b,c,d),(c,d,a,b) ) , Q = ((a,b,c,d),(b,a,d,c) ) are permutations of the elements (a,b,c,d), then QP ≡

$$\mathrm{If}\:{P}\:=\:\begin{pmatrix}{{a}}&{{b}}&{{c}}&{{d}}\\{{c}}&{{d}}&{{a}}&{{b}}\end{pmatrix}\:\:,\:{Q}\:=\:\begin{pmatrix}{{a}}&{{b}}&{{c}}&{{d}}\\{{b}}&{{a}}&{{d}}&{{c}}\end{pmatrix}\:\mathrm{are} \\ $$$$\mathrm{permutations}\:\mathrm{of}\:\mathrm{the}\:\mathrm{elements}\:\left({a},{b},{c},{d}\right),\:\mathrm{then}\: \\ $$$${QP}\:\equiv \\ $$$$\: \\ $$

Question Number 80334 Answers: 0 Comments: 1

let f∈L^1 (R) let u_n = ∫_a ^b f(t)sin(nt)dt , v_n =∫_a ^b ((f(t))/t)sin(nt) 1)Prove that lim_(n→∞) u_n =0 2)Deduce in term of a,b,f(0) the value of lim_(n→∞) v_n

$$\:{let}\:\:\:{f}\in{L}^{\mathrm{1}} \left(\mathbb{R}\right)\:\:\: \\ $$$${let}\:\:{u}_{{n}} =\:\int_{{a}} ^{{b}} {f}\left({t}\right){sin}\left({nt}\right){dt}\:,\:{v}_{{n}} =\int_{{a}} ^{{b}} \frac{{f}\left({t}\right)}{{t}}{sin}\left({nt}\right)\: \\ $$$$\left.\mathrm{1}\right){Prove}\:{that}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{u}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right){Deduce}\:\:{in}\:{term}\:{of}\:{a},{b},{f}\left(\mathrm{0}\right)\:{the}\:{value}\:{of}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{v}_{{n}} \:\: \\ $$

Question Number 80332 Answers: 0 Comments: 1

let α ∈R and a_n =Σ_(k=1) ^n ((sin(kα))/(n+k)) Find lim_(n→∞) a_n

$$\:\:{let}\:\alpha\:\in\mathbb{R}\:\:{and}\:\:\:\:{a}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{sin}\left({k}\alpha\right)}{{n}+{k}} \\ $$$${Find}\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:{a}_{{n}} \: \\ $$

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