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Question Number 24622 Answers: 0 Comments: 4

Question Number 24612 Answers: 1 Comments: 0

If f(x) = [x] then is fof(x) = f(x)?

$$\mathrm{If}\:{f}\left({x}\right)\:=\:\left[{x}\right]\:{then}\:{is}\:{fof}\left({x}\right)\:=\:{f}\left({x}\right)? \\ $$$$ \\ $$

Question Number 24628 Answers: 0 Comments: 1

Question Number 24605 Answers: 3 Comments: 1

Question Number 24604 Answers: 2 Comments: 0

x^2 −xsin x−cos x=0

$${x}^{\mathrm{2}} −{x}\mathrm{sin}\:{x}−\mathrm{cos}\:{x}=\mathrm{0} \\ $$

Question Number 24598 Answers: 2 Comments: 0

if y=x^3 +x^2 +3x.... find its turning point

$${if}\:{y}={x}^{\mathrm{3}} +{x}^{\mathrm{2}} +\mathrm{3}{x}.... \\ $$$${find}\:{its}\:{turning}\:{point} \\ $$

Question Number 24641 Answers: 1 Comments: 1

Question Number 24576 Answers: 1 Comments: 1

Question Number 24569 Answers: 1 Comments: 3

Question Number 24565 Answers: 1 Comments: 0

y=ax^3 +bx^2 +cx+d , then prove that the equation y=0 has only one real root if a[(9ad−bc)^2 −4(b^2 −3ac)(c^2 −3bd)] > 0 provided b^2 > 3ac .

$$\:\:\boldsymbol{{y}}=\boldsymbol{{ax}}^{\mathrm{3}} +\boldsymbol{{bx}}^{\mathrm{2}} +\boldsymbol{{cx}}+\boldsymbol{{d}}\:,\:{then} \\ $$$${prove}\:{that}\:{the}\:{equation}\:{y}=\mathrm{0} \\ $$$${has}\:{only}\:{one}\:{real}\:{root}\:{if} \\ $$$$\:\boldsymbol{{a}}\left[\left(\mathrm{9}\boldsymbol{{ad}}−\boldsymbol{{bc}}\right)^{\mathrm{2}} −\mathrm{4}\left(\boldsymbol{{b}}^{\mathrm{2}} −\mathrm{3}\boldsymbol{{ac}}\right)\left(\boldsymbol{{c}}^{\mathrm{2}} −\mathrm{3}\boldsymbol{{bd}}\right)\right] \\ $$$$\:\:\:\:>\:\mathrm{0}\:\:\:\:\:{provided}\:\:\:\boldsymbol{{b}}^{\mathrm{2}} \:>\:\mathrm{3}\boldsymbol{{ac}}\:. \\ $$

Question Number 24555 Answers: 1 Comments: 3

Question Number 24554 Answers: 1 Comments: 4

Show that: tan^(−1) ((p/(p + 2q))) + tan^(−1) ((p/(p + q))) = (π/2)

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{p}}{\mathrm{p}\:+\:\mathrm{2q}}\right)\:+\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{p}}{\mathrm{p}\:+\:\mathrm{q}}\right)\:=\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 24549 Answers: 0 Comments: 5

Question Number 24548 Answers: 1 Comments: 0

prove that Σ_(n=1) ^r {n(n−(r/2))^2 }= r∙Σ_(n=1) ^(r/2) n^2 where r = 2k ; k ∈ N

$${prove}\:{that}\: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{{r}} {\sum}}\left\{{n}\left({n}−\frac{{r}}{\mathrm{2}}\right)^{\mathrm{2}} \right\}=\:{r}\centerdot\underset{{n}=\mathrm{1}} {\overset{{r}/\mathrm{2}} {\sum}}{n}^{\mathrm{2}} \\ $$$$\:{where}\:\:\:{r}\:=\:\mathrm{2}{k}\:;\:{k}\:\in\:\mathbb{N} \\ $$

Question Number 24542 Answers: 0 Comments: 2

Prove that coefficient of x^n in ((a+bx+cx^2 )/e^x ) is (((−1)^n )/(n!))[cn^2 −(b+c)n+a]

$${Prove}\:{that}\:{coefficient}\:{of}\:{x}^{{n}} \:{in} \\ $$$$\frac{{a}+{bx}+{cx}^{\mathrm{2}} }{{e}^{{x}} }\:{is}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\left[{cn}^{\mathrm{2}} −\left({b}+{c}\right){n}+{a}\right] \\ $$

Question Number 24540 Answers: 2 Comments: 1

Prove that (i) Σ_(n=0) ^∞ (n^2 /(n!))=2e. (ii) Σ_(n=0) ^∞ (n^3 /(n!))=5e. (iii) Σ_(n=0) ^∞ (n^4 /(n!))=15e.

$${Prove}\:{that} \\ $$$$\left({i}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{2}} }{{n}!}=\mathrm{2}{e}. \\ $$$$\left({ii}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{3}} }{{n}!}=\mathrm{5}{e}. \\ $$$$\left({iii}\right)\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{4}} }{{n}!}=\mathrm{15}{e}. \\ $$

Question Number 24539 Answers: 0 Comments: 1

Question Number 24526 Answers: 2 Comments: 1

Question Number 24524 Answers: 1 Comments: 0

Question Number 24520 Answers: 2 Comments: 0

A particle moves in a straight line along x-axis. At t = 0 it passes origin with some velocity towards positive x-axis and with an acceleration a which is given as, a = − Kx, where x is in metre and K is a positive constant. The time at which its velocity becomes half of its value at t = 0 for the first time, is

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{along} \\ $$$${x}-\mathrm{axis}.\:\mathrm{At}\:{t}\:=\:\mathrm{0}\:\mathrm{it}\:\mathrm{passes}\:\mathrm{origin}\:\mathrm{with} \\ $$$$\mathrm{some}\:\mathrm{velocity}\:\mathrm{towards}\:\mathrm{positive}\:{x}-\mathrm{axis} \\ $$$$\mathrm{and}\:\mathrm{with}\:\mathrm{an}\:\mathrm{acceleration}\:{a}\:\mathrm{which}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{as},\:{a}\:=\:−\:{Kx},\:\mathrm{where}\:{x}\:\mathrm{is}\:\mathrm{in}\:\mathrm{metre} \\ $$$$\mathrm{and}\:{K}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{constant}.\:\mathrm{The}\:\mathrm{time} \\ $$$$\mathrm{at}\:\mathrm{which}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{becomes}\:\mathrm{half}\:\mathrm{of}\:\mathrm{its} \\ $$$$\mathrm{value}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{for}\:\mathrm{the}\:\mathrm{first}\:\mathrm{time},\:\mathrm{is} \\ $$

Question Number 24490 Answers: 0 Comments: 2

I hold my hand behind my back, and secretly hold up n fingers (0≤n≤5). I then as you, am I holding up k fingers? Where k is also a random number 0≤k≤5. You randomly guess Yes or No. What is the probability you guess correctly?

$$\mathrm{I}\:\mathrm{hold}\:\mathrm{my}\:\mathrm{hand}\:\mathrm{behind}\:\mathrm{my}\:\mathrm{back},\:\mathrm{and}\:\mathrm{secretly} \\ $$$$\mathrm{hold}\:\mathrm{up}\:{n}\:\mathrm{fingers}\:\left(\mathrm{0}\leqslant{n}\leqslant\mathrm{5}\right). \\ $$$$\: \\ $$$$\mathrm{I}\:\mathrm{then}\:\mathrm{as}\:\mathrm{you},\:\mathrm{am}\:\mathrm{I}\:\mathrm{holding}\:\mathrm{up}\:{k}\:\mathrm{fingers}? \\ $$$$\mathrm{Where}\:{k}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{random}\:\mathrm{number}\:\mathrm{0}\leqslant{k}\leqslant\mathrm{5}. \\ $$$$\: \\ $$$$\mathrm{You}\:\mathrm{randomly}\:\mathrm{guess}\:\mathrm{Yes}\:\mathrm{or}\:\mathrm{No}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{you}\:\mathrm{guess}\:\mathrm{correctly}? \\ $$

Question Number 24482 Answers: 0 Comments: 4

Neglecting friction and mass of pulleys, what is the acceleration of mass B?

$$\mathrm{Neglecting}\:\mathrm{friction}\:\mathrm{and}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{pulleys}, \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{mass}\:{B}? \\ $$

Question Number 24479 Answers: 0 Comments: 0

Describe the energy transformations that take place when a skier starts sking down a hill, but after a time is brought to rest by striking a snowdrift.

$$\mathrm{Describe}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{transformations} \\ $$$$\mathrm{that}\:\mathrm{take}\:\mathrm{place}\:\mathrm{when}\:\mathrm{a}\:\mathrm{skier}\:\mathrm{starts} \\ $$$$\mathrm{sking}\:\mathrm{down}\:\mathrm{a}\:\mathrm{hill},\:\mathrm{but}\:\mathrm{after}\:\mathrm{a}\:\mathrm{time}\:\mathrm{is} \\ $$$$\mathrm{brought}\:\mathrm{to}\:\mathrm{rest}\:\mathrm{by}\:\mathrm{striking}\:\mathrm{a}\:\mathrm{snowdrift}. \\ $$

Question Number 24477 Answers: 1 Comments: 2

A particle moving horizonatally collides perpendiculrly at one end of a rod having equal mass and placed on a horizontal surface. The book says that particle will continue to move along the same direction regardless of value of e (coefficient of restitution). I did not understand the logic. please help.

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moving}\:\mathrm{horizonatally} \\ $$$$\mathrm{collides}\:\mathrm{perpendiculrly}\:\mathrm{at}\:\mathrm{one}\:\mathrm{end} \\ $$$$\mathrm{of}\:\:\mathrm{a}\:\mathrm{rod}\:\mathrm{having}\:\mathrm{equal}\:\mathrm{mass}\:\mathrm{and} \\ $$$$\mathrm{placed}\:\mathrm{on}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{surface}. \\ $$$$\mathrm{The}\:\mathrm{book}\:\mathrm{says}\:\mathrm{that}\:\mathrm{particle}\:\mathrm{will} \\ $$$$\mathrm{continue}\:\mathrm{to}\:\mathrm{move}\:\mathrm{along}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{direction}\:\mathrm{regardless}\:\mathrm{of}\:\mathrm{value}\:\mathrm{of} \\ $$$${e}\:\left(\mathrm{coefficient}\:\mathrm{of}\:\mathrm{restitution}\right). \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{did}\:\mathrm{not}\:\mathrm{understand}\:\mathrm{the}\:\mathrm{logic}. \\ $$$$\mathrm{please}\:\mathrm{help}. \\ $$

Question Number 24469 Answers: 0 Comments: 2

Let 2x + 3y + 4z = 9, x, y, z > 0 then the maximum value of (1 + x)^2 (2 + y)^3 (4 + z)^4 is

$$\mathrm{Let}\:\mathrm{2}{x}\:+\:\mathrm{3}{y}\:+\:\mathrm{4}{z}\:=\:\mathrm{9},\:{x},\:{y},\:{z}\:>\:\mathrm{0}\:\mathrm{then} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\left(\mathrm{1}\:+\:{x}\right)^{\mathrm{2}} \:\left(\mathrm{2}\:+\:{y}\right)^{\mathrm{3}} \\ $$$$\left(\mathrm{4}\:+\:{z}\right)^{\mathrm{4}} \:\mathrm{is} \\ $$

Question Number 24465 Answers: 0 Comments: 3

Two blocks of masses m_1 and m_2 are placed in contact with each other on a horizontal platform. The coefficient of friction between the platform and the two blocks is the same. The platform moves with an acceleration. The force of interaction between the blocks is

$$\mathrm{Two}\:\mathrm{blocks}\:\mathrm{of}\:\mathrm{masses}\:{m}_{\mathrm{1}} \:\mathrm{and}\:{m}_{\mathrm{2}} \:\mathrm{are} \\ $$$$\mathrm{placed}\:\mathrm{in}\:\mathrm{contact}\:\mathrm{with}\:\mathrm{each}\:\mathrm{other}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{platform}.\:\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of} \\ $$$$\mathrm{friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{platform}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{two}\:\mathrm{blocks}\:\mathrm{is}\:\mathrm{the}\:\mathrm{same}.\:\mathrm{The}\:\mathrm{platform} \\ $$$$\mathrm{moves}\:\mathrm{with}\:\mathrm{an}\:\mathrm{acceleration}.\:\mathrm{The}\:\mathrm{force} \\ $$$$\mathrm{of}\:\mathrm{interaction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{blocks}\:\mathrm{is} \\ $$

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