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Category: Geometry

proof-or-given-a-counter-example-if-p-q-are-prines-with-p-gt-q-and-s-prime-such-s-q-p-then-p-q-r-q-p-r-is-prime-r-

Question Number 510 by 123456 last updated on 21/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${if}\:{p},{q}\:{are}\:{prines}\:{with}\:{p}>{q},\:{and}\:\exists{s}\:{prime} \\ $$$${such}\:{s}\in\left({q},{p}\right)\:{then} \\ $$$${p}−{q}\leqslant\underset{{r}\in\left({q},{p}\right),{r}\:{is}\:{prime}} {\sum}{r} \\ $$$$ \\ $$ Commented by prakash jain…

e-x-sin-2x-dx-

Question Number 497 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int{e}^{{x}} \mathrm{sin}\:\mathrm{2}{x}\:{dx} \\ $$ Answered by prakash jain last updated on 16/Jan/15 $$\mathrm{Integrate}\:\mathrm{by}\:\mathrm{parts} \\ $$$${I}=\int{e}^{{x}} \mathrm{sin}\:\mathrm{2}{x}\:{dx}…

1-1-2x-dx-

Question Number 487 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:\:{dx} \\ $$ Answered by prakash jain last updated on 14/Jan/15 $${x}=\mathrm{2}{t}^{\mathrm{2}} \\ $$$${dx}=\mathrm{4}{t}\:{dt} \\ $$$$\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}\:{dx}=\int\:\frac{\mathrm{4}{tdt}}{\mathrm{1}+\mathrm{2}{t}}…

1-x-1-dx-

Question Number 483 by 13/NaSaNa(N)056565 last updated on 25/Jan/15 $$\int\frac{\mathrm{1}}{\:\sqrt{{x}−\mathrm{1}}}{dx} \\ $$$$ \\ $$ Answered by 123456 last updated on 12/Jan/15 $$\int\frac{{dx}}{\:\sqrt{{x}−\mathrm{1}}} \\ $$$${u}={x}−\mathrm{1} \\…

proof-or-given-a-counter-example-for-s-2-3-4-5-i-1-n-1-s-i-1-i-i-s-i-1-n-s-1-si-s-

Question Number 480 by 123456 last updated on 11/Jan/15 $${proof}\:{or}\:{given}\:{a}\:{counter}\:{example}: \\ $$$${for}\:{s}\in\left\{\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5}\right\} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left[\frac{\mathrm{1}}{{s}^{{i}} }−\frac{\left(−\mathrm{1}\right)^{{i}} }{{i}^{{s}} }\right]\leqslant\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{s}+\mathrm{1}}{{si}^{{s}} } \\ $$ Commented…

proof-or-give-a-counter-example-if-nm-is-prime-then-mdc-n-2-m-2-1-

Question Number 472 by 123456 last updated on 25/Jan/15 $${proof}\:{or}\:{give}\:{a}\:{counter}−{example}: \\ $$$${if}\:{nm}\:{is}\:{prime}\:{then}\:\mathrm{mdc}\left({n}^{\mathrm{2}} ,{m}^{\mathrm{2}} \right)=\mathrm{1} \\ $$ Answered by prakash jain last updated on 10/Jan/15 $$\mathrm{If}\:{n},{m}\:\in\mathbb{Z}\:\mathrm{then}\:{n}=\mathrm{1}\:\mathrm{or}\:{m}=\mathrm{1}…

A-particle-moves-with-a-central-acceration-varies-as-the-cube-of-the-distance-if-it-be-projected-from-an-apse-at-distance-a-from-the-origin-with-a-velocity-which-is-2-times-the-velocity-for-a-c

Question Number 465 by shubham last updated on 09/Jan/15 $${A}\:{particle}\:{moves}\:{with}\:{a}\:{central}\:{acceration}\:{varies}\:{as}\:{the}\:{cube}\:{of}\:{the}\:{distance}.\:{if}\:{it}\:{be}\:{projected}\:{from}\:{an}\:{apse}\:{at}\:{distance}\:{a}\:{from}\:{the}\:{origin}\:{with}\:{a}\:{velocity}\:{which}\:{is}\:\sqrt{\mathrm{2}\:}\:{times}\:{the}\:{velocity}\:{for}\:{a}\:{circle}\:{of}\:{radius}\:{a}.\:{show}\:{that}\:{the}\:{equation}\:{of}\:{path}\:{is}\:{its}\:{r}\mathrm{cos}\theta/\sqrt{\mathrm{2}}\:=\:{a} \\ $$ Commented by prakash jain last updated on 09/Jan/15 $$\mathrm{Reformatted} \\ $$$${A}\:{particle}\:{moves}\:{with}\:{a}\:{central} \\ $$$${acceration}\:{varies}\:{as}\:{the}\:{cube}\:…

Question-131534

Question Number 131534 by mr W last updated on 06/Feb/21 Commented by mr W last updated on 05/Feb/21 $${find}\:{the}\:{radii}\:{of}\:{the}\:{inscribing}\:{and} \\ $$$${the}\:{circumscribing}\:{sphere}\:{of}\:{a} \\ $$$${tetrahedron}\:{with}\:{edge}\:{lengthes}\:{a},{b},{c}, \\ $$$${p},{q},{r}\:{as}\:{shown}.…