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Question Number 55418    Answers: 3   Comments: 1

Question Number 55375    Answers: 1   Comments: 1

Question Number 55368    Answers: 1   Comments: 0

Question Number 55335    Answers: 1   Comments: 0

Question Number 55301    Answers: 1   Comments: 2

find the fourth term in the expansion of (((√x)/y^2 )−(y/(√x)))^6

$${find}\:{the}\:{fourth}\:{term}\:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\frac{\sqrt{{x}}}{{y}^{\mathrm{2}} }−\frac{{y}}{\sqrt{{x}}}\right)^{\mathrm{6}} \\ $$

Question Number 55308    Answers: 1   Comments: 0

Question Number 55245    Answers: 1   Comments: 0

Question Number 55191    Answers: 2   Comments: 0

Question Number 55174    Answers: 0   Comments: 3

center and radius convergence of series Σ_(n=0) ^(∝) (((4−2i)/(1+5i)))^n z^n is...

$$\mathrm{center}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{convergence} \\ $$$$\:\mathrm{of}\:\mathrm{series}\:\underset{{n}=\mathrm{0}} {\overset{\propto} {\Sigma}}\:\left(\frac{\mathrm{4}−\mathrm{2}{i}}{\mathrm{1}+\mathrm{5}{i}}\right)^{{n}} {z}^{{n}} \:\mathrm{is}... \\ $$$$ \\ $$

Question Number 55172    Answers: 1   Comments: 0

Question Number 55171    Answers: 0   Comments: 3

Question Number 55190    Answers: 1   Comments: 0

Question Number 55087    Answers: 0   Comments: 1

Please any web site or ebook to learn LATEX ? Thank you.

$$\mathrm{Please}\:\mathrm{any}\:\mathrm{web}\:\mathrm{site}\:\mathrm{or}\:\mathrm{ebook}\:\mathrm{to}\:\mathrm{learn} \\ $$$${LATEX}\:? \\ $$$$\mathrm{Thank}\:\mathrm{you}. \\ $$

Question Number 55052    Answers: 0   Comments: 3

Question Number 54966    Answers: 1   Comments: 3

By differentiating x (√(1 + x)) with respect to x , Evaluate, ∫_( 0) ^( 2) (x/(√(1 + x))) dx

$$\mathrm{By}\:\mathrm{differentiating}\:\:\:\mathrm{x}\:\sqrt{\mathrm{1}\:+\:\mathrm{x}}\:\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{x}\:, \\ $$$$\mathrm{Evaluate},\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{2}} \:\frac{\mathrm{x}}{\sqrt{\mathrm{1}\:+\:\mathrm{x}}}\:\:\mathrm{dx} \\ $$

Question Number 54825    Answers: 1   Comments: 0

Question Number 54790    Answers: 3   Comments: 2

differenciatethefollowing i)x^x +(sinx)^(lnx) = ii)sin^(−1) (tanhx)= iii)(√(1+x^2 /1−x^2 =))

$${differenciatethefollowing} \\ $$$$\left.{i}\right){x}^{{x}} +\left({sinx}\right)^{{lnx}} = \\ $$$$\left.{ii}\right){sin}^{−\mathrm{1}} \left({tanhx}\right)= \\ $$$$\left.{iii}\right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} /\mathrm{1}−{x}^{\mathrm{2}} =} \\ $$

Question Number 54770    Answers: 0   Comments: 0

A glass bottle full of mercury has mass 500g. On being heated through 35°C, 2.43g of mercury are expelled. Calculate the mass of mercury remaining in the bottle. (Cubic expansivity of mercury is 1.8 × 10^(−4) K^(−1) , linear expansivity of glass is 8.0 × 10^(−6) K^(−1) .

$$\mathrm{A}\:\mathrm{glass}\:\mathrm{bottle}\:\mathrm{full}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{has}\:\mathrm{mass}\:\mathrm{500g}.\:\mathrm{On}\:\mathrm{being}\:\mathrm{heated}\:\mathrm{through} \\ $$$$\mathrm{35}°\mathrm{C},\:\:\mathrm{2}.\mathrm{43g}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{are}\:\mathrm{expelled}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{mass}\:\mathrm{of}\:\mathrm{mercury} \\ $$$$\mathrm{remaining}\:\mathrm{in}\:\mathrm{the}\:\mathrm{bottle}.\:\left(\mathrm{Cubic}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{mercury}\:\mathrm{is}\:\:\mathrm{1}.\mathrm{8}\:×\:\mathrm{10}^{−\mathrm{4}} \:\mathrm{K}^{−\mathrm{1}} \:,\right. \\ $$$$\mathrm{linear}\:\mathrm{expansivity}\:\mathrm{of}\:\mathrm{glass}\:\mathrm{is}\:\:\mathrm{8}.\mathrm{0}\:×\:\mathrm{10}^{−\mathrm{6}} \mathrm{K}^{−\mathrm{1}} \:. \\ $$

Question Number 54767    Answers: 1   Comments: 1

A stone is thrown vertically upwards from a cliff 20m high. After a time of 3 s it passes the edge of the cliff on its way down. Calculate a) the speed of projection b) the speed when it hits the ground c) the times when it is 10m above the top of the cliff d) the time it is 15m above the ground (take g=10ms^(−2) ).

$$\mathrm{A}\:\mathrm{stone}\:\mathrm{is}\:\mathrm{thrown}\:\mathrm{vertically}\:\mathrm{upwards} \\ $$$$\mathrm{from}\:\mathrm{a}\:\mathrm{cliff}\:\mathrm{20m}\:\mathrm{high}.\:\mathrm{After}\:\mathrm{a}\:\mathrm{time}\:\mathrm{of}\:\mathrm{3}\:\mathrm{s} \\ $$$$\mathrm{it}\:\mathrm{passes}\:\mathrm{the}\:\mathrm{edge}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cliff}\:\mathrm{on}\:\mathrm{its}\:\mathrm{way} \\ $$$$\mathrm{down}.\:\mathrm{Calculate} \\ $$$$\left.\mathrm{a}\right)\:\mathrm{the}\:\mathrm{speed}\:\mathrm{of}\:\mathrm{projection} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{the}\:\mathrm{speed}\:\mathrm{when}\:\mathrm{it}\:\mathrm{hits}\:\mathrm{the}\:\mathrm{ground} \\ $$$$\left.\mathrm{c}\right)\:\mathrm{the}\:\mathrm{times}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is}\:\mathrm{10m}\:\mathrm{above}\:\mathrm{the} \\ $$$$\mathrm{top}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cliff} \\ $$$$\left.\mathrm{d}\right)\:\mathrm{the}\:\mathrm{time}\:\mathrm{it}\:\mathrm{is}\:\mathrm{15m}\:\mathrm{above}\:\mathrm{the}\:\mathrm{ground} \\ $$$$\left(\mathrm{take}\:\mathrm{g}=\mathrm{10ms}^{−\mathrm{2}} \right). \\ $$

Question Number 54745    Answers: 0   Comments: 0

(√(1−x^2 +))(√(1−y^2 ))=a(x−y) (dy/dx)=(√((1−y^2 )/(1−x^2 ))) without solve put x=sinθand y=sinφ direct solve

$$\sqrt{\mathrm{1}−{x}^{\mathrm{2}} +}\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }={a}\left({x}−{y}\right) \\ $$$$\frac{{dy}}{{dx}}=\sqrt{\frac{\mathrm{1}−{y}^{\mathrm{2}} }{\mathrm{1}−{x}^{\mathrm{2}} }} \\ $$$${without}\:{solve}\:{put}\:{x}={sin}\theta{and} \\ $$$${y}={sin}\phi \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$${direct}\:{solve} \\ $$

Question Number 54716    Answers: 0   Comments: 6

A light horizontal meter rule PQR has the end P fixed to vertical wall, while a weight of 5N is suspended from the end R. A light string QS of length 50cm fixed to the wall at S is used to maintain the meter rule in equilibrium. If PQ = 40 cm, the tension in the string is ?

$$\mathrm{A}\:\mathrm{light}\:\mathrm{horizontal}\:\mathrm{meter}\:\mathrm{rule}\:\:\mathrm{PQR}\:\:\mathrm{has}\:\mathrm{the}\:\mathrm{end}\:\:\mathrm{P}\:\mathrm{fixed}\:\mathrm{to}\: \\ $$$$\mathrm{vertical}\:\mathrm{wall},\:\mathrm{while}\:\mathrm{a}\:\mathrm{weight}\:\mathrm{of}\:\:\mathrm{5N}\:\mathrm{is}\:\mathrm{suspended}\:\mathrm{from}\:\mathrm{the}\:\mathrm{end} \\ $$$$\mathrm{R}.\:\mathrm{A}\:\mathrm{light}\:\mathrm{string}\:\mathrm{QS}\:\:\mathrm{of}\:\mathrm{length}\:\:\mathrm{50cm}\:\mathrm{fixed}\:\mathrm{to}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{at}\:\mathrm{S}\:\mathrm{is}\: \\ $$$$\mathrm{used}\:\mathrm{to}\:\mathrm{maintain}\:\mathrm{the}\:\mathrm{meter}\:\mathrm{rule}\:\mathrm{in}\:\mathrm{equilibrium}. \\ $$$$\mathrm{If}\:\:\mathrm{PQ}\:=\:\mathrm{40}\:\mathrm{cm},\:\:\:\mathrm{the}\:\mathrm{tension}\:\mathrm{in}\:\mathrm{the}\:\mathrm{string}\:\mathrm{is}\:? \\ $$

Question Number 54658    Answers: 0   Comments: 0

Question Number 54647    Answers: 0   Comments: 3

show that a. Σ_(r=1) ^(n) r^3 ._n C_r =n^2 (n+3).2^(n−3) b. _n C_0 ._n C_1 +_n C_1 ._n C_2 +...+_n C_(n−1) ._n C_n =(((2n)!)/((n−1)!.(n+1)!))

$$\mathrm{show}\:\mathrm{that} \\ $$$${a}.\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\Sigma}}\:{r}^{\mathrm{3}} ._{{n}} {C}_{{r}} ={n}^{\mathrm{2}} \left({n}+\mathrm{3}\right).\mathrm{2}^{{n}−\mathrm{3}} \\ $$$${b}.\:_{{n}} {C}_{\mathrm{0}} ._{{n}} {C}_{\mathrm{1}} +_{{n}} {C}_{\mathrm{1}} ._{{n}} {C}_{\mathrm{2}} +...+_{{n}} {C}_{{n}−\mathrm{1}} ._{{n}} {C}_{{n}} =\frac{\left(\mathrm{2}{n}\right)!}{\left({n}−\mathrm{1}\right)!.\left({n}+\mathrm{1}\right)!} \\ $$

Question Number 54607    Answers: 1   Comments: 2

If ((u^5 +v^5 )/((u+v)^5 )) = −(1/5) , find ((u^3 +v^3 )/((u+v)^3 )) = ?

$${If}\:\:\frac{{u}^{\mathrm{5}} +{v}^{\mathrm{5}} }{\left({u}+{v}\right)^{\mathrm{5}} }\:=\:−\frac{\mathrm{1}}{\mathrm{5}}\:, \\ $$$${find}\:\:\:\frac{{u}^{\mathrm{3}} +{v}^{\mathrm{3}} }{\left({u}+{v}\right)^{\mathrm{3}} }\:\:=\:? \\ $$

Question Number 54502    Answers: 0   Comments: 5

In △ABC cos A+cos B+cos C=(3/2) prove that trianle is equilateral

$${In}\:\bigtriangleup{ABC}\:\mathrm{cos}\:{A}+\mathrm{cos}\:{B}+\mathrm{cos}\:{C}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${prove}\:{that}\:{trianle}\:{is}\:{equilateral} \\ $$

Question Number 54457    Answers: 0   Comments: 0

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