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Question Number 173321 Answers: 1 Comments: 0

The diagonal of a rectangualar field is 169m. If the ratio of the lengh to the width is 12:5, find its dimensions.

$$\mathrm{The}\:\mathrm{diagonal}\:\mathrm{of}\:\mathrm{a}\:\mathrm{rectangualar}\:\mathrm{field}\:\mathrm{is}\:\mathrm{169m}. \\ $$$$\mathrm{If}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{lengh}\:\mathrm{to}\:\mathrm{the}\:\mathrm{width}\:\mathrm{is} \\ $$$$\mathrm{12}:\mathrm{5},\:\mathrm{find}\:\mathrm{its}\:\mathrm{dimensions}. \\ $$

Question Number 173386 Answers: 1 Comments: 0

Question Number 173311 Answers: 4 Comments: 2

Question Number 173309 Answers: 2 Comments: 0

lim_(x→∞) ((1+(1/x))^x x−ex)=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{x}} {x}−{ex}\right)=? \\ $$

Question Number 173302 Answers: 2 Comments: 2

Question Number 173303 Answers: 1 Comments: 0

Question Number 173298 Answers: 1 Comments: 0

Q: f(x)= e^( x) + x −4 is given put : h(x)= ln(x−f^( −1) (x)) find : D_( h ) = (domain of h )

$$\mathrm{Q}: \\ $$$$\:\:\:\:\:{f}\left({x}\right)=\:{e}^{\:{x}} +\:{x}\:−\mathrm{4}\:{is}\:{given} \\ $$$$\:\:\:\:\:{put}\::\:\:\:\:\:{h}\left({x}\right)=\:{ln}\left({x}−{f}^{\:−\mathrm{1}} \left({x}\right)\right) \\ $$$$\:\:\:\:\:\:{find}\::\:\:\:\:\mathrm{D}_{\:{h}\:} \:=\:\left({domain}\:{of}\:\:\:\:{h}\:\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 173296 Answers: 1 Comments: 0

I=∫(((cosx−sinx)(1+sin2x))/((sinx+cosx)))dx=

$$\mathrm{I}=\int\frac{\left(\mathrm{cosx}−\mathrm{sinx}\right)\left(\mathrm{1}+\mathrm{sin2x}\right)}{\left(\mathrm{sinx}+\mathrm{cosx}\right)}\mathrm{dx}=\: \\ $$

Question Number 173293 Answers: 1 Comments: 0

A particle P moves in a plane such that at time t seconds, its velocity, v=(2ti−t^3 )ms^(−1) . (a) Find, when t=2, the magnitudeof the: (i) velocity of P. (ii) acceleration of P. (b) Given that P is at the point with position vector (3i+2j) when t=1, find the position vector of P when t=2.

$$\mathrm{A}\:\mathrm{particle}\:\boldsymbol{\mathrm{P}}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{at}\:\mathrm{time}\:\boldsymbol{{t}}\:\mathrm{seconds},\:\mathrm{its}\:\mathrm{velocity},\:\boldsymbol{\mathrm{v}}=\left(\mathrm{2t}\boldsymbol{{i}}−\boldsymbol{{t}}^{\mathrm{3}} \right)\mathrm{ms}^{−\mathrm{1}} . \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find},\:\mathrm{when}\:{t}=\mathrm{2},\:\mathrm{the}\:\mathrm{magnitudeof}\:\mathrm{the}: \\ $$$$\left(\mathrm{i}\right)\:\mathrm{velocity}\:\mathrm{of}\:\boldsymbol{\mathrm{P}}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{acceleration}\:\mathrm{of}\:\boldsymbol{\mathrm{P}}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Given}\:\mathrm{that}\:\boldsymbol{\mathrm{P}}\:\mathrm{is}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\mathrm{with}\:\mathrm{position}\: \\ $$$$\mathrm{vector}\:\left(\mathrm{3i}+\mathrm{2j}\right)\:\mathrm{when}\:\mathrm{t}=\mathrm{1},\:\mathrm{find}\:\mathrm{the}\:\mathrm{position} \\ $$$$\mathrm{vector}\:\mathrm{of}\:\boldsymbol{\mathrm{P}}\:\mathrm{when}\:\mathrm{t}=\mathrm{2}. \\ $$$$ \\ $$

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Question Number 173275 Answers: 2 Comments: 1

Question Number 173273 Answers: 1 Comments: 1

Question Number 173254 Answers: 3 Comments: 4

solve for x,y,z ∈R x+y+z=(√3) xy+yz+zx=1

$${solve}\:{for}\:{x},{y},{z}\:\in{R} \\ $$$${x}+{y}+{z}=\sqrt{\mathrm{3}} \\ $$$${xy}+{yz}+{zx}=\mathrm{1} \\ $$

Question Number 173250 Answers: 1 Comments: 0

Solve for real numbers: ∫_0 ^( x) (t^2 /((t ∙ sinh t − cosh t)^2 )) dt = 0

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\int_{\mathrm{0}} ^{\:\boldsymbol{\mathrm{x}}} \:\frac{\mathrm{t}^{\mathrm{2}} }{\left(\mathrm{t}\:\centerdot\:\mathrm{sinh}\:\mathrm{t}\:−\:\mathrm{cosh}\:\mathrm{t}\right)^{\mathrm{2}} }\:\mathrm{dt}\:=\:\mathrm{0} \\ $$

Question Number 173245 Answers: 1 Comments: 0

The ends X and Y of an inextensible strings 27m long are fixed at two points on the same horizontal line which are 20 m apart. A particle of mass 7.5 kg is suspended from a point P on the string 12 m from X. (a) Illustrate this information in a diagram. (b) calculate, correct to two decimal places, <YXP and <XYP. (c) Find, correct to the nearest hundredth, the magnitudes of the tensions in the string. [take g=10 ms^(−2) ]

$$\mathrm{The}\:\mathrm{ends}\:\boldsymbol{\mathrm{X}}\:\mathrm{and}\:\boldsymbol{\mathrm{Y}}\:\mathrm{of}\:\mathrm{an}\:\mathrm{inextensible}\:\mathrm{strings}\:\mathrm{27m} \\ $$$$\mathrm{long}\:\mathrm{are}\:\mathrm{fixed}\:\mathrm{at}\:\mathrm{two}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{horizontal}\:\mathrm{line}\:\mathrm{which}\:\mathrm{are}\:\mathrm{20}\:\mathrm{m}\:\mathrm{apart}. \\ $$$$\mathrm{A}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{7}.\mathrm{5}\:\mathrm{kg}\:\mathrm{is}\:\mathrm{suspended} \\ $$$$\mathrm{from}\:\mathrm{a}\:\mathrm{point}\:\boldsymbol{\mathrm{P}}\:\mathrm{on}\:\mathrm{the}\:\mathrm{string}\:\mathrm{12}\:\mathrm{m}\:\mathrm{from}\:\boldsymbol{\mathrm{X}}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Illustrate}\:\mathrm{this}\:\mathrm{information}\:\mathrm{in}\:\mathrm{a}\:\mathrm{diagram}. \\ $$$$\left(\mathrm{b}\right)\:\mathrm{calculate},\:\mathrm{correct}\:\mathrm{to}\:\boldsymbol{\mathrm{two}}\:\mathrm{decimal} \\ $$$$\mathrm{places},\:<\mathrm{YXP}\:\mathrm{and}\:<\mathrm{XYP}. \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Find},\:\mathrm{correct}\:\mathrm{to}\:\mathrm{the}\:\boldsymbol{\mathrm{nearest}}\:\mathrm{hundredth}, \\ $$$$\mathrm{the}\:\mathrm{magnitudes}\:\mathrm{of}\:\mathrm{the}\:\mathrm{tensions}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{string}.\:\:\left[\mathrm{take}\:\boldsymbol{\mathrm{g}}=\mathrm{10}\:\mathrm{ms}^{−\mathrm{2}} \right] \\ $$

Question Number 173242 Answers: 2 Comments: 1

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Question Number 173236 Answers: 2 Comments: 0

lim_(x→∞) ((x/(1+x)))^x =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\frac{{x}}{\mathrm{1}+{x}}\right)^{{x}} =? \\ $$

Question Number 173216 Answers: 0 Comments: 0

O-circumcenter , I-incenter, R-circumradii, r-radii, a,b,c,d-sides in a bicenteric quadrilateral. Prove that: 20I^2 + r Σ_(cyc) (√(4R^2 − a^2 )) = 2(R^2 + 2r^2 )

$$\mathrm{O}-\mathrm{circumcenter}\:,\:\mathrm{I}-\mathrm{incenter},\:\mathrm{R}-\mathrm{circumradii}, \\ $$$$\mathrm{r}-\mathrm{radii},\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}-\mathrm{sides}\:\mathrm{in}\:\mathrm{a}\:\mathrm{bicenteric} \\ $$$$\mathrm{quadrilateral}.\:\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{20I}^{\mathrm{2}} \:+\:\mathrm{r}\:\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\sqrt{\mathrm{4R}^{\mathrm{2}} \:−\:\mathrm{a}^{\mathrm{2}} }\:=\:\mathrm{2}\left(\mathrm{R}^{\mathrm{2}} \:+\:\mathrm{2r}^{\mathrm{2}} \right) \\ $$

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