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AllQuestion and Answers: Page 65

Question Number 217402 Answers: 3 Comments: 0

Question Number 217389 Answers: 2 Comments: 0

Resolver ((∣x − 1∣ + ∣x + 2∣)/(∣x∣ − 1)) ≤ 3

$$\mathrm{Resolver} \\ $$$$\frac{\mid{x}\:−\:\mathrm{1}\mid\:+\:\mid{x}\:+\:\mathrm{2}\mid}{\mid{x}\mid\:−\:\mathrm{1}}\:\leq\:\mathrm{3} \\ $$

Question Number 217385 Answers: 0 Comments: 3

a,b,c∈R a^2 +b^2 +c^2 =9 a^3 +b^3 +c^3 =21 ab+bc+ca=?

$${a},{b},{c}\in\mathbb{R} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} =\mathrm{9} \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{21} \\ $$$${ab}+{bc}+{ca}=? \\ $$

Question Number 217384 Answers: 1 Comments: 0

Prove that sin351° = − (√(((4 − (√(10 + 2(√5))))/8) .))

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{sin351}°\:=\:−\:\sqrt{\frac{\mathrm{4}\:−\:\sqrt{\mathrm{10}\:+\:\mathrm{2}\sqrt{\mathrm{5}}}}{\mathrm{8}}\:.} \\ $$

Question Number 217383 Answers: 1 Comments: 0

show that the intergral ∫_1 ^∞ (1/x^p )dx converges for p>1 find it value

$$\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{intergral} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\mathrm{x}^{\mathrm{p}} }\mathrm{dx}\:\:\mathrm{converges}\:\mathrm{for}\:\mathrm{p}>\mathrm{1} \\ $$$$\:\mathrm{find}\:\mathrm{it}\:\mathrm{value} \\ $$

Question Number 217381 Answers: 1 Comments: 4

Let a,b,c be real numbers satisfying the equations (1) a +b + c= 4 (2) a^3 + b^3 + c^3 = 34 Find ab+bc+ca

$$\mathrm{Let}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{be}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{satisfying} \\ $$$$\:\mathrm{the}\:\mathrm{equations} \\ $$$$\left(\mathrm{1}\right)\:\:\mathrm{a}\:+\mathrm{b}\:+\:\mathrm{c}=\:\mathrm{4} \\ $$$$\left(\mathrm{2}\right)\:\:\mathrm{a}^{\mathrm{3}} \:+\:\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} =\:\mathrm{34} \\ $$$$\mathrm{Find}\:\mathrm{ab}+\mathrm{bc}+\mathrm{ca} \\ $$

Question Number 217378 Answers: 0 Comments: 0

Whether the following seriesconverge: determinant (((Σ_(n=1) ^∞ (−1)^n [1+n ln(1−(2/(2n+1)))])))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Whether}\:\mathrm{the}\:\mathrm{following}\: \\ $$$$\mathrm{seriesconverge}: \\ $$$$\begin{array}{|c|}{\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{{n}} \left[\mathrm{1}+{n}\:\mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{2}}{\mathrm{2}{n}+\mathrm{1}}\right)\right]}\\\hline\end{array} \\ $$

Question Number 217377 Answers: 2 Comments: 0

Let x,y,z be real numbers satisfying the equations x + y + z= 7 xy + yz + zx=10 xyz=6 Find the value of x^3 + y^3 + z^3

$$\mathrm{Let}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\mathrm{be}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{satisfying}\:\mathrm{the}\:\mathrm{equations} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}=\:\mathrm{7} \\ $$$$\mathrm{xy}\:+\:\mathrm{yz}\:+\:\mathrm{zx}=\mathrm{10} \\ $$$$\mathrm{xyz}=\mathrm{6} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{y}^{\mathrm{3}} \:+\:\mathrm{z}^{\mathrm{3}} \\ $$

Question Number 217359 Answers: 2 Comments: 0

Find all three-digit numbers such that when the number is divided by the sum of its digits the quotient is 7 and the remainder is 5.

$$\mathrm{Find}\:\mathrm{all}\:\mathrm{three}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{its}\:\mathrm{digits}\:\mathrm{the}\:\mathrm{quotient}\:\mathrm{is}\:\mathrm{7}\:\mathrm{and}\:\mathrm{the}\: \\ $$$$\mathrm{remainder}\:\mathrm{is}\:\mathrm{5}. \\ $$

Question Number 217358 Answers: 1 Comments: 0

Question Number 217356 Answers: 1 Comments: 0

Question Number 217326 Answers: 3 Comments: 0

Find three consecutive integers such that the sum of their squares is 50.

$${Find}\:{three}\:{consecutive}\:{integers}\: \\ $$$${such}\:{that}\:{the}\:{sum}\:{of}\:{their}\:{squares} \\ $$$$\:{is}\:\mathrm{50}. \\ $$

Question Number 217321 Answers: 2 Comments: 0

Find three consecutive integers such that the product of the first two is 16 more than twice the third integer. Provide all possible solutions.

$$\mathrm{Find}\:\mathrm{three}\:\mathrm{consecutive}\:\mathrm{integers}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{first}\:\mathrm{two}\:\mathrm{is}\:\mathrm{16}\:\mathrm{more}\:\mathrm{than}\:\mathrm{twice}\:\mathrm{the}\:\mathrm{third}\:\mathrm{integer}.\:\mathrm{Provide}\:\mathrm{all}\: \\ $$$$\mathrm{possible}\:\mathrm{solutions}. \\ $$

Question Number 217314 Answers: 1 Comments: 9

How is ψ^2 (1) = − 2ζ(3) and ψ^2 ((1/2)) = − 14ζ(3) ???

$$\mathrm{How}\:\mathrm{is}\:\:\:\psi^{\mathrm{2}} \left(\mathrm{1}\right)\:\:=\:\:−\:\:\mathrm{2}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\mathrm{and}\:\:\:\:\psi^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right)\:\:\:=\:\:\:−\:\:\mathrm{14}\zeta\left(\mathrm{3}\right)\:\:\:\:\:\:\:??? \\ $$

Question Number 217300 Answers: 2 Comments: 0

Question Number 217299 Answers: 1 Comments: 0

Question Number 217291 Answers: 5 Comments: 0

Solve: ((x+2)/(x−3))−((x−1)/(x+4))=((10)/(x^2 +x−12))

$$\mathrm{Solve}: \\ $$$$\frac{{x}+\mathrm{2}}{{x}−\mathrm{3}}−\frac{{x}−\mathrm{1}}{{x}+\mathrm{4}}=\frac{\mathrm{10}}{{x}^{\mathrm{2}} +{x}−\mathrm{12}} \\ $$

Question Number 217290 Answers: 4 Comments: 0

∫_0 ^( 1) ((ln^2 (1−x))/x^2 ) dx= ?

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}^{\mathrm{2}} }\:{dx}=\:? \\ $$$$ \\ $$$$ \\ $$

Question Number 217289 Answers: 1 Comments: 1

∫_0 ^1 sin^(−1) (1/( (√(1+x−x^2 )))) dx=?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+{x}−{x}^{\mathrm{2}} }}\:{dx}=? \\ $$

Question Number 217278 Answers: 0 Comments: 0

Question Number 217256 Answers: 1 Comments: 0

Question Number 217255 Answers: 1 Comments: 0

∫_(−∞) ^(+∞) e^(−(x^2 /2)) dx=(√(2π)),∫_(−∞) ^(+∞) e^(−(x^2 /2)+x) dx.

$$\int_{−\infty} ^{+\infty} {e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} {dx}=\sqrt{\mathrm{2}\pi},\int_{−\infty} ^{+\infty} {e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}+{x}} {dx}. \\ $$

Question Number 217252 Answers: 3 Comments: 5

Question Number 217245 Answers: 1 Comments: 0

find the following differential equation by eliminating the arbritrary constant (1)y=Ae^x +Bcosx (2) xy=Ae^x +Be^(−x) +x^2

$${find}\:{the}\:{following}\:{differential}\:{equation}\: \\ $$$${by}\:{eliminating}\:{the}\:{arbritrary}\:{constant} \\ $$$$\left(\mathrm{1}\right){y}={Ae}^{{x}} +{Bcosx} \\ $$$$\left(\mathrm{2}\right)\:{xy}={Ae}^{{x}} +{Be}^{−{x}} +{x}^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 217238 Answers: 2 Comments: 0

Question Number 217235 Answers: 3 Comments: 0

∫_( 0) ^( 1) ((x ln^2 (x))/(1 + x^2 )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{ln}^{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{1}\:\:+\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

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