Question and Answers Forum

AllQuestion and Answers: Page 114

Question Number 212347 Answers: 2 Comments: 1

Question Number 212342 Answers: 1 Comments: 0

The number abc^(−) is divisible by 37. Prove that bca^(−) + cab^(−) is divisible by 37.

$$\mathrm{The}\:\mathrm{number}\:\:\overline {\mathrm{abc}}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{37}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\overline {\mathrm{bca}}\:+\:\overline {\mathrm{cab}}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{37}.\: \\ $$

Question Number 212341 Answers: 2 Comments: 0

Question Number 212339 Answers: 3 Comments: 0

Question Number 212332 Answers: 0 Comments: 4

Please help 1.1.Let XUY=X for all sets X. Prove that Y=0(empty set). From Singler book "Excercises in set theory". I think this task is totaly wrong and cannot be proved. I would ask someone to provide me valid proof of that. I have sets X and Y such as Y is subset of X. For example. If Y={1} and X={1,2} then XUY=X is correct but that doesn't imply Y is empty. Another example when X=Y since X is any set. I can choose X=Y. Why not? Then YUY=Y is always true, but again, that doesnt imply Y is empty set Proof in book claim that is correct if we suppose Y is not empty and if we choose for instance X is empty set. Then 0UY=0 but this is wrong since 0UY=Y. Therefore, Y must be empty?

$$ \\ $$Please help 1.1.Let XUY=X for all sets X. Prove that Y=0(empty set). From Singler book "Excercises in set theory". I think this task is totaly wrong and cannot be proved. I would ask someone to provide me valid proof of that. I have sets X and Y such as Y is subset of X. For example. If Y={1} and X={1,2} then XUY=X is correct but that doesn't imply Y is empty. Another example when X=Y since X is any set. I can choose X=Y. Why not? Then YUY=Y is always true, but again, that doesnt imply Y is empty set Proof in book claim that is correct if we suppose Y is not empty and if we choose for instance X is empty set. Then 0UY=0 but this is wrong since 0UY=Y. Therefore, Y must be empty?

Question Number 212327 Answers: 3 Comments: 0

if (a+(1/a))=15 find the value of (a^2 +(1/a^2 ))

$${if}\:\left({a}+\frac{\mathrm{1}}{{a}}\right)=\mathrm{15}\:{find}\:{the}\:{value}\:{of}\:\left({a}^{\mathrm{2}} \:+\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$

Question Number 212326 Answers: 0 Comments: 0

Σ_(n=1) ^∞ (((−3)^n n!)/(1.4...(3n+1))) (1) check its a absolute conergent series (2) show that its a convergent series

$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{3}\right)^{{n}} \:{n}!}{\mathrm{1}.\mathrm{4}...\left(\mathrm{3}{n}+\mathrm{1}\right)} \\ $$$$\:\left(\mathrm{1}\right)\:{check}\:\:{its}\:{a}\:{absolute}\:{conergent}\:{series} \\ $$$$\:\:\left(\mathrm{2}\right)\:{show}\:{that}\:{its}\:{a}\:{convergent}\:{series} \\ $$

Question Number 212325 Answers: 1 Comments: 0

show that 1+(√(2 ))+2+..........+32(√(2 )) is 63(√2) +63

$${show}\:{that}\:\mathrm{1}+\sqrt{\mathrm{2}\:}+\mathrm{2}+..........+\mathrm{32}\sqrt{\mathrm{2}\:}\:{is}\:\mathrm{63}\sqrt{\mathrm{2}}\:+\mathrm{63} \\ $$

Question Number 212320 Answers: 1 Comments: 1

a+b+c+d=2, a^2 +b^2 +c^2 +d^2 =2 a^3 +b^3 +c^3 +d^3 =−4, a^4 +b^4 +c^4 +d^4 =−6 find real value of a^(2023) +b^(2023) +c^(2023) +d^(2023) .

$$\:{a}+{b}+{c}+{d}=\mathrm{2},\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} =\mathrm{2} \\ $$$$\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} =−\mathrm{4},\:{a}^{\mathrm{4}} +{b}^{\mathrm{4}} +{c}^{\mathrm{4}} +{d}^{\mathrm{4}} =−\mathrm{6} \\ $$$$\:{find}\:{real}\:{value}\:{of}\:{a}^{\mathrm{2023}} +{b}^{\mathrm{2023}} +{c}^{\mathrm{2023}} +{d}^{\mathrm{2023}} . \\ $$

Question Number 212319 Answers: 2 Comments: 2

find G=(1/4)∫_0 ^(π/2) ln ((1+sin x)/(1−sin x)) dx

$$\mathrm{find} \\ $$$${G}=\frac{\mathrm{1}}{\mathrm{4}}\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\mathrm{ln}\:\frac{\mathrm{1}+\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{sin}\:{x}}\:{dx} \\ $$

Question Number 212314 Answers: 2 Comments: 0

Question Number 212457 Answers: 1 Comments: 0

Question Number 212456 Answers: 2 Comments: 0

∫ ((√(9x^2 −49))/x^3 ) dx = ?

$$\int\:\:\frac{\sqrt{\mathrm{9}{x}^{\mathrm{2}} −\mathrm{49}}}{{x}^{\mathrm{3}} }\:\:{dx}\:=\:? \\ $$

Question Number 212307 Answers: 1 Comments: 0

I=∫_0 ^∞ (((x−arctan x)^2 )/x^4 )dx.

$$ \\ $$$$\:\:{I}=\int_{\mathrm{0}} ^{\infty} \frac{\left({x}−\mathrm{arctan}\:{x}\right)^{\mathrm{2}} }{{x}^{\mathrm{4}} }{dx}. \\ $$

Question Number 212304 Answers: 0 Comments: 1

Hmmm...... does this Series Convergence?? Let′s define a_h as j_(ν,h) zero point of J_ν (z) ex. j_(1,1) is first zeros of J_1 (z) j_(2,2) is secondary zeros of J_2 (z)..... and that′s Sum S=Σ (1/(h!))a_(h ) , h=1,2,3.... div conv?? pls answer me...

$$\mathrm{Hmmm}...... \\ $$$$\mathrm{does}\:\mathrm{this}\:\mathrm{Series}\:\mathrm{Convergence}?? \\ $$$$\mathrm{Let}'\mathrm{s}\:\mathrm{define}\:{a}_{{h}} \:\mathrm{as}\:{j}_{\nu,{h}} \:\mathrm{zero}\:\mathrm{point}\:\mathrm{of}\:{J}_{\nu} \left({z}\right) \\ $$$$\mathrm{ex}.\:\:{j}_{\mathrm{1},\mathrm{1}} \:\mathrm{is}\:\mathrm{first}\:\mathrm{zeros}\:\mathrm{of}\:{J}_{\mathrm{1}} \left({z}\right) \\ $$$$\:\:\:\:\:{j}_{\mathrm{2},\mathrm{2}} \:\mathrm{is}\:\mathrm{secondary}\:\mathrm{zeros}\:\mathrm{of}\:\:{J}_{\mathrm{2}} \left({z}\right)..... \\ $$$$\mathrm{and}\:\mathrm{that}'\mathrm{s}\:\mathrm{Sum}\:{S}=\Sigma\:\frac{\mathrm{1}}{{h}!}{a}_{{h}\:} \:,\:{h}=\mathrm{1},\mathrm{2},\mathrm{3}.... \\ $$$$\mathrm{div}\:\mathrm{conv}??\:\mathrm{pls}\:\mathrm{answer}\:\mathrm{me}... \\ $$

Question Number 212301 Answers: 2 Comments: 0

3x+(2/( (√x)))=1, x−(√x) =?

$$\:\:\:\:\mathrm{3}{x}+\frac{\mathrm{2}}{\:\sqrt{{x}}}=\mathrm{1},\:{x}−\sqrt{{x}}\:=? \\ $$$$\:\: \\ $$

Question Number 212291 Answers: 1 Comments: 2