Question and Answers Forum

AllQuestion and Answers: Page 1125

Question Number 105039 Answers: 7 Comments: 0

(1) ∫ (dx/(x^6 −64)) (2) y′ + y tan x = sin 2x y(0) = 1

$$\left(\mathrm{1}\right)\:\int\:\frac{{dx}}{{x}^{\mathrm{6}} −\mathrm{64}} \\ $$$$\left(\mathrm{2}\right)\:{y}'\:+\:{y}\:\mathrm{tan}\:{x}\:=\:\mathrm{sin}\:\mathrm{2}{x} \\ $$$${y}\left(\mathrm{0}\right)\:=\:\mathrm{1} \\ $$

Question Number 105037 Answers: 1 Comments: 0

A heavy man , slips from a roof of a high building of height ′H′. When he was at a height of ′h′ he realised that the ground beneath him was pretty hard. To avoid injury, he thrown away his suitcase of mass ′m′. After doing this,he prevented his falling on the hard ground and fell on a pond. The mass of the man was ′M′. The distance of the pond from the hard ground was ′x′.And he just avoided from falling on the hard ground. Find the velocity required to throw away the suitcase as the man don′t fall on the hard ground.(There was no horizontal velocity of the suitcase−man system)

$${A}\:{heavy}\:{man}\:,\:{slips}\:\:{from}\:{a}\:{roof}\:{of}\:{a}\:{high}\:{building}\:{of} \\ $$$${height}\:'{H}'.\:{When}\:{he}\:{was}\:{at}\:{a}\:{height}\:{of}\:'{h}'\:\:{he}\:{realised}\:{that}\:{the} \\ $$$${ground}\:{beneath}\:{him}\:{was}\:{pretty}\:{hard}.\:{To}\:{avoid}\:{injury},\:{he}\:{thrown} \\ $$$${away}\:{his}\:{suitcase}\:{of}\:{mass}\:'{m}'.\:{After}\:{doing}\:{this},{he}\:{prevented}\:{his} \\ $$$${falling}\:{on}\:{the}\:{hard}\:{ground}\:{and}\:{fell}\:{on}\:{a}\:{pond}.\:{The}\:{mass}\:{of}\:{the} \\ $$$${man}\:{was}\:'{M}'.\:{The}\:{distance}\:{of}\:{the}\:{pond}\:{from}\:{the}\:{hard}\:{ground}\:{was}\: \\ $$$$'{x}'.{And}\:{he}\:{just}\:{avoided}\:{from}\:{falling}\:{on}\:{the}\:{hard}\:{ground}. \\ $$$$ \\ $$$${Find}\:{the}\:{velocity}\:{required}\:{to}\:{throw}\:{away}\:{the}\:{suitcase}\:{as}\:{the}\:{man} \\ $$$${don}'{t}\:{fall}\:{on}\:{the}\:{hard}\:{ground}.\left({There}\:{was}\:{no}\:{horizontal}\:{velocity}\:{of}\:{the}\right. \\ $$$$\left.{suitcase}−{man}\:{system}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 105036 Answers: 2 Comments: 0

lim_(x→0) x.[ (1/x) ] ? note [ ] = greatest integer function

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}.\left[\:\frac{\mathrm{1}}{{x}}\:\right]\:? \\ $$$${note}\:\left[\:\right]\:=\:{greatest}\:{integer}\:{function} \\ $$

Question Number 105026 Answers: 5 Comments: 0

Question Number 105023 Answers: 1 Comments: 0

(((x^2 −y^2 )^2 )/((1−x^2 )(y^2 −1)))+(((1−x^2 )^2 )/((x^2 −y^2 )(y^2 −1)))+(((y^2 −1)^2 )/((1−x^2 )(x^2 −y^2 )))=

$$\frac{\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left({y}^{\mathrm{2}} −\mathrm{1}\right)}+\frac{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\left({y}^{\mathrm{2}} −\mathrm{1}\right)}+\frac{\left({y}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{2}} }{\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)}= \\ $$$$ \\ $$

Question Number 105018 Answers: 3 Comments: 1

(((√((√5)+2)) + (√((√5)−2)))/(√((√5)+1))) ?

$$\frac{\sqrt{\sqrt{\mathrm{5}}+\mathrm{2}}\:+\:\sqrt{\sqrt{\mathrm{5}}−\mathrm{2}}}{\sqrt{\sqrt{\mathrm{5}}+\mathrm{1}}}\:? \\ $$

Question Number 105014 Answers: 0 Comments: 4

1)Determinate α then show that I belong to the circle 2) show that (BF)⊥(DI).

$$\left.\mathrm{1}\right){Determinate}\:\alpha\:{then}\:{show}\:{that}\:{I}\:{belong}\:{to}\:{the}\:{circle} \\ $$$$\left.\mathrm{2}\right)\:{show}\:{that}\:\left({BF}\right)\bot\left({DI}\right).\: \\ $$$$ \\ $$$$ \\ $$

Question Number 104975 Answers: 0 Comments: 2

Dear Forum-Friends There′s a trend to make answers short in the forum. To follow this trend some friends are omitting key-steps and for this reason the answers become difficult to understand and many readers like me are compelled to leave out such answers at the price of no-understanding!!! So please omit only calculation steps in order to make your answers short-&-understandable.

$$\mathrm{Dear}\:\mathrm{Forum}-\mathrm{Friends} \\ $$$$\:\:\:\:\:\:\mathcal{T}{here}'{s}\:{a}\:{trend}\:{to}\:{make} \\ $$$${answers}\:{short}\:{in}\:{the}\:{forum}. \\ $$$$\:\:\:\:\:\:\:\mathcal{T}{o}\:{follow}\:{this}\:{trend}\:{some} \\ $$$${friends}\:{are}\:{omitting}\:\boldsymbol{{key}}-\boldsymbol{{steps}} \\ $$$${and}\:{for}\:{this}\:{reason}\:{the}\:{answers} \\ $$$${become}\:\boldsymbol{{difficult}}\:\boldsymbol{{to}} \\ $$$$\boldsymbol{{understand}}\:{and}\: \\ $$$${many}\:{readers}\:{like}\:{me} \\ $$$$\:{are}\:{compelled}\:{to}\:{leave}\:{out} \\ $$$${such}\:{answers}\:{at}\:{the}\:{price}\:{of} \\ $$$$\boldsymbol{{no}}-\boldsymbol{{understanding}}!!! \\ $$$$\mathcal{S}{o}\:{please}\:{omit}\:{only} \\ $$$$\boldsymbol{{calculation}}\:\boldsymbol{{steps}}\:{in}\:{order}\:{to} \\ $$$${make}\:{your}\:{answers} \\ $$$${short}-\&-{understandable}. \\ $$

Question Number 104972 Answers: 1 Comments: 0

In the a sport camp, 65% children know playing the football,70%−in voleyball,75%−in basketball.What is least number of children who know playing all above three sport games? (Answer 10%)

$$ \\ $$$$\mathrm{In}\:\mathrm{the}\:\mathrm{a}\:\:\mathrm{sport}\:\mathrm{camp},\:\mathrm{65\%}\:\mathrm{children}\:\mathrm{know} \\ $$$$\mathrm{playing}\:\mathrm{the}\:\mathrm{football},\mathrm{70\%}−\mathrm{in}\:\mathrm{voleyball},\mathrm{75\%}−\mathrm{in} \\ $$$$\mathrm{basketball}.\mathrm{What}\:\mathrm{is}\:\mathrm{least}\:\mathrm{number}\:\mathrm{of}\:\mathrm{children}\:\mathrm{who} \\ $$$$\mathrm{know}\:\mathrm{playing}\:\mathrm{all}\:\mathrm{above}\:\mathrm{three}\:\mathrm{sport}\:\mathrm{games}? \\ $$$$\left(\mathrm{Answer}\:\mathrm{10\%}\right) \\ $$

Question Number 104948 Answers: 0 Comments: 0

Question Number 104993 Answers: 0 Comments: 0

Question Number 104928 Answers: 0 Comments: 5

∫ (e^x −(2x+3)^4 )^3 dx

$$\int\:\left({e}^{{x}} −\left(\mathrm{2}{x}+\mathrm{3}\right)^{\mathrm{4}} \right)^{\mathrm{3}} \:{dx} \\ $$

Question Number 104927 Answers: 1 Comments: 0

∫(dx/(1+x+x^2 ))

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

Question Number 104919 Answers: 0 Comments: 3

let f(x) =(1/(√(1−x^2 ))) finf f^((n)) (x)

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{1}}{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }} \\ $$$$\mathrm{finf}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right) \\ $$

Question Number 104940 Answers: 1 Comments: 1

Question Number 104939 Answers: 0 Comments: 0

Σ_(n=0) ^∞ (((−1)^(n+1) )/((n+1)!(2n+1))) = ((√π)/2)

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\left({n}+\mathrm{1}\right)!\left(\mathrm{2}{n}+\mathrm{1}\right)}\:=\:\frac{\sqrt{\pi}}{\mathrm{2}}\: \\ $$

Question Number 104910 Answers: 0 Comments: 0

∫_0 ^1 (1/(√(1+x^3 )))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\sqrt{\mathrm{1}+{x}^{\mathrm{3}} }}{dx} \\ $$

Question Number 104905 Answers: 3 Comments: 0

Question Number 104904 Answers: 3 Comments: 0

4cos^2 x sin x −2sin^2 x = 3sin x where −(π/2)≤x≤(π/2)

$$\mathrm{4cos}\:^{\mathrm{2}} {x}\:\mathrm{sin}\:{x}\:−\mathrm{2sin}\:^{\mathrm{2}} {x}\:=\:\mathrm{3sin}\:{x} \\ $$$${where}\:−\frac{\pi}{\mathrm{2}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}} \\ $$

Question Number 104899 Answers: 2 Comments: 0

lim_(x→0) ((cos (sin x)−cos (x))/x^4 ) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)−\mathrm{cos}\:\left({x}\right)}{{x}^{\mathrm{4}} }\:?\: \\ $$

Question Number 104895 Answers: 2 Comments: 0

1) decompose the fraction F(x) =(1/(x^3 (x+1)^4 )) 2) find the sumA = Σ_(n=1) ^∞ (1/(n^3 (n+1)^4 )) and B =Σ_(n=1) ^(∞ ) (((−1)^n )/(n^3 (n+1)^4 )) 3) what is the value of Σ_(n=0) ^∞ (1/((n+1)^4 (2n+1)^3 )) ?

$$\left.\mathrm{1}\right)\:\mathrm{decompose}\:\mathrm{the}\:\mathrm{fraction}\:\:\mathrm{F}\left(\mathrm{x}\right)\:=\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} \left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{sumA}\:=\:\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{4}} }\:\:\mathrm{and}\:\mathrm{B}\:=\sum_{\mathrm{n}=\mathrm{1}} ^{\infty\:} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right)\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{4}} \left(\mathrm{2n}+\mathrm{1}\right)^{\mathrm{3}} }\:? \\ $$

Question Number 104894 Answers: 1 Comments: 0

prove that cos^6 a −sin^6 a = cos 2a (1−(1/4)sin^2 2a)

$${prove}\:{that}\:\mathrm{cos}\:^{\mathrm{6}} {a}\:−\mathrm{sin}\:^{\mathrm{6}} {a}\:=\: \\ $$$$\mathrm{cos}\:\mathrm{2}{a}\:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{a}\right)\: \\ $$

Question Number 104893 Answers: 1 Comments: 0

(d^2 y/dx^2 ) + tan x (dy/dx) = sec x + cot x

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{tan}\:{x}\:\frac{{dy}}{{dx}}\:=\:\mathrm{sec}\:{x}\:+\:\mathrm{cot}\:{x} \\ $$

Question Number 104891 Answers: 1 Comments: 0

1) decompose the fraction F(x) =(1/(x^3 (x+1)^3 )) 2) find the sum Σ_(n=1) ^∞ (((−1)^n )/(n^3 (n+1)^3 ))

$$\left.\mathrm{1}\right)\:\mathrm{decompose}\:\mathrm{the}\:\mathrm{fraction}\:\mathrm{F}\left(\mathrm{x}\right)\:=\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} \left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 104890 Answers: 1 Comments: 0

lim_(n→∞) (1/n)Σ_(k=1) ^n [1+(k^3 /n^3 )]^(−(1/2))

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left[\mathrm{1}+\frac{\mathrm{k}^{\mathrm{3}} }{\mathrm{n}^{\mathrm{3}} }\right]^{−\frac{\mathrm{1}}{\mathrm{2}}} \\ $$

Question Number 104973 Answers: 1 Comments: 1

2(√3) + i is a cubic root for 18(√3) + 35i find the other 2 cubic roots

$$\mathrm{2}\sqrt{\mathrm{3}}\:+\:\boldsymbol{{i}}\:\:\:{is}\:{a}\:{cubic}\:{root}\:{for} \\ $$$$\mathrm{18}\sqrt{\mathrm{3}}\:+\:\mathrm{35}\boldsymbol{{i}}\: \\ $$$$\boldsymbol{{find}}\:\:\boldsymbol{{the}}\:\boldsymbol{{other}}\:\mathrm{2}\:\boldsymbol{{cubic}}\:\boldsymbol{{roots}} \\ $$

Pg 1120 Pg 1121 Pg 1122 Pg 1123 Pg 1124 Pg 1125 Pg 1126 Pg 1127 Pg 1128 Pg 1129

Contact: info@tinkutara.com