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Question Number 50936    Answers: 1   Comments: 1

Question Number 50925    Answers: 0   Comments: 1

factor x^3 +x^2 +x−(1/3) and x^3 +x^2 −x+(1/3) and x^3 +x^2 −x−(1/3)

$$\mathrm{factor}\:\:\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\frac{\mathrm{1}}{\mathrm{3}}\:\:\:\mathrm{and} \\ $$$$\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\frac{\mathrm{1}}{\mathrm{3}}\:\:\mathrm{and} \\ $$$$\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 50932    Answers: 1   Comments: 0

Question Number 50915    Answers: 1   Comments: 0

The range of riffle bullet is 1000m when θ is the angle of projection.if the bullet is fired with the same angle from a car travelling at 36km/h towards the target show that the range will be increased by ((1000)/7).(√(tanθ)) m.

$${The}\:{range}\:{of}\:{riffle}\:{bullet} \\ $$$${is}\:\mathrm{1000}{m}\:{when}\:\theta\:{is}\:{the}\:{angle} \\ $$$${of}\:{projection}.{if}\:{the}\:{bullet} \\ $$$${is}\:{fired}\:\:{with}\:{the}\:{same}\: \\ $$$${angle}\:\:{from}\:{a}\:{car}\:{travelling} \\ $$$${at}\:\mathrm{36}{km}/{h}\:{towards}\:{the}\:{target} \\ $$$${show}\:{that}\:{the}\:{range}\:{will} \\ $$$${be}\:{increased}\:{by} \\ $$$$\frac{\mathrm{1000}}{\mathrm{7}}.\sqrt{{tan}\theta}\:{m}. \\ $$

Question Number 50914    Answers: 2   Comments: 0

prove that relative velocity is reversed by a head on collision

$${prove}\:{that}\:{relative}\:{velocity} \\ $$$${is}\:{reversed}\:{by}\:{a}\:{head}\:{on} \\ $$$${collision} \\ $$

Question Number 50913    Answers: 1   Comments: 0

A ball is dropped from height h.it strikes the ground and rises,the coefficiate of restitution being e what is the total distance it moves and the time before it comes to rest?

$${A}\:{ball}\:{is}\:{dropped}\:{from} \\ $$$${height}\:{h}.{it}\:{strikes}\:{the}\:{ground} \\ $$$${and}\:{rises},{the}\:{coefficiate} \\ $$$${of}\:{restitution}\:{being}\:\:{e} \\ $$$${what}\:{is}\:{the}\:{total}\:{distance} \\ $$$${it}\:{moves}\:\:{and}\:\:{the}\:{time}\: \\ $$$${before}\:{it}\:{comes}\:{to}\:{rest}? \\ $$$$ \\ $$$$ \\ $$

Question Number 50912    Answers: 2   Comments: 0

Show that in collision where kinetic energy is conserved linear momemtum is also conserved

$${Show}\:{that}\:{in}\:{collision} \\ $$$${where}\:{kinetic}\:{energy}\:{is} \\ $$$${conserved}\:{linear}\:{momemtum} \\ $$$${is}\:{also}\:{conserved} \\ $$

Question Number 50908    Answers: 1   Comments: 0

Given f(x)=Σ_(k=0) ^n ^n C_k sin(kx)cos((n−k)x) Find a simple form for f(x) (Your answer should be written like c(n).g(nx))

$${Given}\:{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right) \\ $$$${Find}\:{a}\:{simple}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left({Your}\:{answer}\:{should}\:{be}\:{written}\:{like}\:{c}\left({n}\right).{g}\left({nx}\right)\right)\: \\ $$

Question Number 50898    Answers: 3   Comments: 1

Question Number 50888    Answers: 0   Comments: 2

Question Number 50890    Answers: 1   Comments: 0

Determine whether the following is true for all value of x 0≤(((x+1)^2 )/(x^2 +x+1))≤(4/3)

$${Determine}\:{whether} \\ $$$${the}\:{following}\:\:{is}\:{true}\:{for}\:{all} \\ $$$${value}\:{of}\:{x} \\ $$$$\mathrm{0}\leqslant\frac{\left({x}+\mathrm{1}\right)^{\mathrm{2}} }{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\leqslant\frac{\mathrm{4}}{\mathrm{3}} \\ $$

Question Number 50861    Answers: 2   Comments: 0

((n!)/((n−5)!))=20((n!)/((n−3)!)) n=? ________ please give me simple solve. thanks

$$\frac{\mathrm{n}!}{\left(\mathrm{n}−\mathrm{5}\right)!}=\mathrm{20}\frac{\mathrm{n}!}{\left(\mathrm{n}−\mathrm{3}\right)!} \\ $$$$ \\ $$$$\mathrm{n}=? \\ $$$$\_\_\_\_\_\_\_\_ \\ $$$$\mathrm{please}\:\mathrm{give}\:\mathrm{me}\:\mathrm{simple}\:\mathrm{solve}. \\ $$$$\mathrm{thanks} \\ $$

Question Number 50892    Answers: 3   Comments: 0

solve the equation tan 3θcotθ+1=0 for 0≤θ≤180 b)show that if cos 2θ is not zero then cos 2θ+sec 2θ=2[((cos^4 θ+sin^4 θ)/(cos^4 θ−sin^4 θ))] c)find the limit of ((tan (θ/3))/(3θ)) as θ→0

$${solve}\:{the}\:{equation} \\ $$$$\mathrm{tan}\:\mathrm{3}\theta{cot}\theta+\mathrm{1}=\mathrm{0}\:{for} \\ $$$$\mathrm{0}\leqslant\theta\leqslant\mathrm{180} \\ $$$$\left.{b}\right){show}\:{that}\:{if}\:\mathrm{cos}\:\mathrm{2}\theta\:{is}\:{not}\:{zero} \\ $$$${then} \\ $$$$\mathrm{cos}\:\mathrm{2}\theta+\mathrm{sec}\:\mathrm{2}\theta=\mathrm{2}\left[\frac{\mathrm{cos}\:^{\mathrm{4}} \theta+\mathrm{sin}\:^{\mathrm{4}} \theta}{\mathrm{cos}\:^{\mathrm{4}} \theta−\mathrm{sin}\:^{\mathrm{4}} \theta}\right] \\ $$$$\left.{c}\right){find}\:{the}\:{limit}\:{of} \\ $$$$\frac{\mathrm{tan}\:\frac{\theta}{\mathrm{3}}}{\mathrm{3}\theta}\:{as}\:\theta\rightarrow\mathrm{0} \\ $$$$ \\ $$

Question Number 50856    Answers: 1   Comments: 0

show that Σ_(x=0) ^n xp(x)=np given that p(x)=^n C_x p^x q^(n−x)

$${show}\:{that} \\ $$$$\underset{\mathrm{x}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}{xp}\left({x}\right)={np}\:{given}\: \\ $$$${that}\:{p}\left({x}\right)=^{\mathrm{n}} {C}_{\mathrm{x}} {p}^{{x}} {q}^{{n}−{x}} \\ $$

Question Number 50855    Answers: 1   Comments: 0

For the random variable x show that a)Var(x)=E^2 (x)−[E(x)]^2 b)Var(ax+b)=a^2 Var(x)

$${F}\mathrm{or}\:{the}\:{random}\:{variable} \\ $$$${x}\:{show}\:{that} \\ $$$$\left.{a}\right){Var}\left({x}\right)={E}^{\mathrm{2}} \left({x}\right)−\left[{E}\left({x}\right)\right]^{\mathrm{2}} \\ $$$$\left.{b}\right){Var}\left({ax}+{b}\right)={a}^{\mathrm{2}} {Var}\left({x}\right) \\ $$$$ \\ $$

Question Number 50849    Answers: 3   Comments: 0

solve for z in the form x+iy if tanz=0.5

$$\mathrm{solve}\:{for}\:{z}\:\:{in}\:{the}\:{form}\:\:{x}+{iy}\: \\ $$$${if}\:{tanz}=\mathrm{0}.\mathrm{5}\: \\ $$

Question Number 50835    Answers: 1   Comments: 1

Question Number 50834    Answers: 0   Comments: 4

Question Number 50829    Answers: 2   Comments: 1

Question Number 50825    Answers: 1   Comments: 0

x^4 =ax^2 +by^2 y^4 =bx^2 +ay^2 solve for x, y. [a ,b∈ R; a, b≠0]

$$\boldsymbol{\mathrm{x}}^{\mathrm{4}} =\boldsymbol{\mathrm{ax}}^{\mathrm{2}} +\boldsymbol{\mathrm{by}}^{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{y}}^{\mathrm{4}} =\boldsymbol{\mathrm{bx}}^{\mathrm{2}} +\boldsymbol{\mathrm{ay}}^{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}},\:\boldsymbol{\mathrm{y}}.\:\left[\boldsymbol{\mathrm{a}}\:,\boldsymbol{\mathrm{b}}\in\:\boldsymbol{\mathrm{R}};\:\:\boldsymbol{\mathrm{a}},\:\boldsymbol{\mathrm{b}}\neq\mathrm{0}\right] \\ $$

Question Number 50820    Answers: 1   Comments: 1

Question Number 51822    Answers: 1   Comments: 0

If p and q are the length of perpendicular from the origin to the lines xcos θ−ysin θ=kcos2θ and xsec θ+ycosec θ=k respectively prove that p^2 +4q^2 =k^2

$${If}\:\:{p}\:{and}\:{q}\:\:{are}\:{the}\:{length} \\ $$$${of}\:{perpendicular}\:{from} \\ $$$${the}\:{origin}\:{to}\:{the}\:{lines} \\ $$$${x}\mathrm{cos}\:\theta−{y}\mathrm{sin}\:\:\theta={kcos}\mathrm{2}\theta \\ $$$${and}\:{x}\mathrm{sec}\:\theta+{y}\mathrm{cosec}\:\theta={k} \\ $$$${respectively} \\ $$$${prove}\:{that} \\ $$$${p}^{\mathrm{2}} +\mathrm{4}{q}^{\mathrm{2}} ={k}^{\mathrm{2}} \\ $$

Question Number 50818    Answers: 1   Comments: 0

If a right angled triangle has same area and double perimeter as that of a circle of unit radius, find the mutually perpendicular sides of the triangle.

$${If}\:{a}\:{right}\:{angled}\:{triangle}\:{has} \\ $$$${same}\:{area}\:{and}\:{double}\:{perimeter} \\ $$$${as}\:{that}\:{of}\:{a}\:{circle}\:{of}\:{unit}\:{radius}, \\ $$$${find}\:{the}\:{mutually}\:{perpendicular} \\ $$$${sides}\:{of}\:{the}\:{triangle}. \\ $$

Question Number 51823    Answers: 0   Comments: 0

Give a proof for : Σ_(n=1) ^k (Π_(n′=0) ^m ( n+n′)) = ((Π_(x=0) ^(m+1) (k+x))/(m+2)) In other terms : (it is the same) Σ_(n=1) ^k n(n+1)(n+2) ... (n+m) = (( k(k+1)(k+2) ... (k+m)(k+m+1) )/(m + 2)) Thank you !!!

$$ \\ $$$$\:\:\:\:\:\:\boldsymbol{\mathrm{Give}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{proof}}\:\boldsymbol{\mathrm{for}}\:: \\ $$$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}\:\left(\underset{{n}'=\mathrm{0}} {\overset{{m}} {\prod}}\left(\:{n}+\mathrm{n}'\right)\right)\:\:=\:\:\frac{\underset{{x}=\mathrm{0}} {\overset{{m}+\mathrm{1}} {\prod}}\left({k}+{x}\right)}{{m}+\mathrm{2}}\: \\ $$$$ \\ $$$$\:\:\:\:\:\mathrm{In}\:\mathrm{other}\:\mathrm{terms}\::\:\left({it}\:{is}\:{the}\:{same}\right) \\ $$$$ \\ $$$$\:\:\:\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\:...\:\left({n}+{m}\right) \\ $$$$\:=\:\frac{\:{k}\left({k}+\mathrm{1}\right)\left({k}+\mathrm{2}\right)\:...\:\left({k}+{m}\right)\left({k}+{m}+\mathrm{1}\right)\:}{{m}\:+\:\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}\:!!! \\ $$

Question Number 50811    Answers: 1   Comments: 2

Question Number 50808    Answers: 2   Comments: 0

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