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Question Number 224989    Answers: 1   Comments: 0

Calculate D_n = determinant (((x_1 +a_1 ^2 ),(a_1 a_2 ),…,(a_1 a_n )),((a_2 a_1 ),(x_2 +a_2 ^2 ),…,(a_2 a_n )),(⋮,⋮,⋱,⋮),((a_n a_1 ),(a_n a_2 ),…,(x_n +a_n ^2 )))

$$\mathrm{Calculate} \\ $$$${D}_{{n}} =\begin{vmatrix}{{x}_{\mathrm{1}} +{a}_{\mathrm{1}} ^{\mathrm{2}} }&{{a}_{\mathrm{1}} {a}_{\mathrm{2}} }&{\ldots}&{{a}_{\mathrm{1}} {a}_{{n}} }\\{{a}_{\mathrm{2}} {a}_{\mathrm{1}} }&{{x}_{\mathrm{2}} +{a}_{\mathrm{2}} ^{\mathrm{2}} }&{\ldots}&{{a}_{\mathrm{2}} {a}_{{n}} }\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{{a}_{{n}} {a}_{\mathrm{1}} }&{{a}_{{n}} {a}_{\mathrm{2}} }&{\ldots}&{{x}_{{n}} +{a}_{{n}} ^{\mathrm{2}} }\end{vmatrix} \\ $$

Question Number 224985    Answers: 0   Comments: 0

A dairy man gives ada and amaka eight litre of milk to share and ada carry five litre container and amaka Carry three litre container how can ada and amaka share the eight litre of milk equally.

$$\mathrm{A}\:\mathrm{dairy}\:\mathrm{man}\:\mathrm{gives}\:\mathrm{ada}\:\mathrm{and}\: \\ $$$$\mathrm{amaka}\:\mathrm{eight}\:\mathrm{litre}\:\mathrm{of}\:\mathrm{milk}\:\mathrm{to}\:\mathrm{share}\: \\ $$$$\mathrm{and}\:\mathrm{ada}\:\mathrm{carry}\:\mathrm{five}\:\mathrm{litre}\:\mathrm{container} \\ $$$$\mathrm{and}\:\mathrm{amaka}\:\mathrm{Carry}\:\mathrm{three}\:\mathrm{litre} \\ $$$$\mathrm{container}\:\mathrm{how}\:\mathrm{can}\:\mathrm{ada}\:\mathrm{and}\:\mathrm{amaka} \\ $$$$\mathrm{share}\:\mathrm{the}\:\mathrm{eight}\:\mathrm{litre}\:\mathrm{of}\:\mathrm{milk}\:\mathrm{equally}. \\ $$

Question Number 224966    Answers: 2   Comments: 2

cos(π/7) − cos((2π)/7) + cos((3π)/7) = ? Help me please

$$ \\ $$$$\:\:\:{cos}\frac{\pi}{\mathrm{7}}\:−\:{cos}\frac{\mathrm{2}\pi}{\mathrm{7}}\:+\:{cos}\frac{\mathrm{3}\pi}{\mathrm{7}}\:=\:? \\ $$$$\:\:\:{Help}\:{me}\:{please} \\ $$$$ \\ $$

Question Number 224948    Answers: 2   Comments: 1

∫ (√x)sin x dx

$$\int\:\sqrt{{x}}\mathrm{sin}\:{x}\:{dx} \\ $$

Question Number 225262    Answers: 1   Comments: 4

Question Number 224927    Answers: 1   Comments: 2

Question Number 224926    Answers: 1   Comments: 0

∫ ((sin x)/x) dx

$$\int\:\frac{\mathrm{sin}\:{x}}{{x}}\:{dx} \\ $$

Question Number 224924    Answers: 1   Comments: 1

∫_0 ^(2π) ((xsin^(2n) x)/(sin^(2n) x+cos^(2n) x)) dx=π^2 (prove)

$$\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\frac{{x}\mathrm{sin}^{\mathrm{2}{n}} {x}}{\mathrm{sin}^{\mathrm{2}{n}} {x}+\mathrm{cos}^{\mathrm{2}{n}} {x}}\:{dx}=\pi^{\mathrm{2}} \:\left({prove}\right) \\ $$

Question Number 224922    Answers: 1   Comments: 0

Domain f(x)=(√(log _e (x^2 −6x+6)))

$${Domain} \\ $$$${f}\left({x}\right)=\sqrt{\mathrm{log}\:_{{e}} \left({x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{6}\right)} \\ $$

Question Number 224920    Answers: 0   Comments: 0

∫_( 0) ^( 1) ((x ln(1 + x) Li_2 (x))/(1 + x^2 )) dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{ln}\left(\mathrm{1}\:\:\:+\:\:\:\mathrm{x}\right)\:\mathrm{Li}_{\mathrm{2}} \left(\mathrm{x}\right)}{\mathrm{1}\:\:\:\:+\:\:\:\:\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 224908    Answers: 1   Comments: 2

Hello Everyone We recently did an update that cause view older button to be partially hidden behind the navigation button. We will release an update soon. workaround Tap + sign at top of screen to open editor and then press back. Then the button will be shown correctly Part of the button is still visible and you can press at the edge. Fix We will upload another update today amd it will be available to download in two to three days.

$$\mathrm{Hello}\:\mathrm{Everyone} \\ $$$$\mathrm{We}\:\mathrm{recently}\:\mathrm{did}\:\mathrm{an}\:\mathrm{update}\:\mathrm{that} \\ $$$$\mathrm{cause}\:\mathrm{view}\:\mathrm{older}\:\mathrm{button}\:\mathrm{to}\:\mathrm{be}\:\mathrm{partially} \\ $$$$\mathrm{hidden}\:\mathrm{behind}\:\mathrm{the}\:\mathrm{navigation}\:\mathrm{button}. \\ $$$$\mathrm{We}\:\mathrm{will}\:\mathrm{release}\:\mathrm{an}\:\mathrm{update}\:\mathrm{soon}. \\ $$$$ \\ $$$$\boldsymbol{\mathrm{workaround}} \\ $$$$\mathrm{Tap}\:+\:\mathrm{sign}\:\mathrm{at}\:\mathrm{top}\:\mathrm{of}\:\mathrm{screen}\:\mathrm{to}\:\mathrm{open} \\ $$$$\mathrm{editor}\:\mathrm{and}\:\mathrm{then}\:\mathrm{press}\:\mathrm{back}.\:\mathrm{Then}\:\mathrm{the} \\ $$$$\mathrm{button}\:\mathrm{will}\:\mathrm{be}\:\mathrm{shown}\:\mathrm{correctly} \\ $$$$\mathrm{Part}\:\mathrm{of}\:\mathrm{the}\:\mathrm{button}\:\mathrm{is}\:\mathrm{still}\:\mathrm{visible}\:\mathrm{and} \\ $$$$\mathrm{you}\:\mathrm{can}\:\mathrm{press}\:\mathrm{at}\:\mathrm{the}\:\mathrm{edge}. \\ $$$$\boldsymbol{\mathrm{Fix}} \\ $$$$\mathrm{We}\:\mathrm{will}\:\mathrm{upload}\:\mathrm{another}\:\mathrm{update}\:\mathrm{today} \\ $$$$\mathrm{amd}\:\mathrm{it}\:\mathrm{will}\:\mathrm{be}\:\mathrm{available}\:\mathrm{to}\:\mathrm{download} \\ $$$$\mathrm{in}\:\mathrm{two}\:\mathrm{to}\:\mathrm{three}\:\mathrm{days}. \\ $$

Question Number 224907    Answers: 1   Comments: 0

Let B={(x,y,z)∈R^3 :x^2 +y^2 +z^2 ≤1} and define u(x,y,z)=sin ((1−x^2 −y^2 −z^2 )^2 ) for (x,y,z)∈B. Then the value of ∫∫∫_(B) ((∂^2 u/∂x^2 )+(∂^2 u/∂x^2 )+(∂^2 u/∂z^2 ))dxdydz is ..........

$${Let}\:{B}=\left\{\left({x},{y},{z}\right)\in\mathbb{R}^{\mathrm{3}} :{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \leqslant\mathrm{1}\right\} \\ $$$${and}\:{define}\:{u}\left({x},{y},{z}\right)=\mathrm{sin}\:\left(\left(\mathrm{1}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{z}^{\mathrm{2}} \right)^{\mathrm{2}} \right) \\ $$$${for}\:\left({x},{y},{z}\right)\in{B}. \\ $$$${Then}\:{the}\:{value}\:{of} \\ $$$$\underset{{B}} {\int\int\int}\left(\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {u}}{\partial{z}^{\mathrm{2}} }\right){dxdydz}\:{is}\:.......... \\ $$

Question Number 224915    Answers: 1   Comments: 0

evaluate ∫_C (3y^2 +2z^2 )dx+(6x−10z)y dy +(4xz−5y^2 )dz along the portion from (1,0,1) to (3,4,5) of the curve C, which is the intersection of the two surfaces z^2 =x^2 +y^2 and z=y+1

$${evaluate}\: \\ $$$$\int_{{C}} \left(\mathrm{3}{y}^{\mathrm{2}} +\mathrm{2}{z}^{\mathrm{2}} \right){dx}+\left(\mathrm{6}{x}−\mathrm{10}{z}\right){y}\:{dy}\:+\left(\mathrm{4}{xz}−\mathrm{5}{y}^{\mathrm{2}} \right){dz} \\ $$$${along}\:{the}\:{portion}\:{from}\:\left(\mathrm{1},\mathrm{0},\mathrm{1}\right)\:{to}\:\left(\mathrm{3},\mathrm{4},\mathrm{5}\right)\:{of} \\ $$$${the}\:{curve}\:{C}, \\ $$$${which}\:{is}\:{the}\:{intersection}\:{of}\:{the} \\ $$$${two}\:{surfaces}\:{z}^{\mathrm{2}} ={x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{and}\:{z}={y}+\mathrm{1} \\ $$

Question Number 224892    Answers: 1   Comments: 1

lim_(n→0) ∫_0 ^1 e^x^2 sin (nx)dx=?

$$\underset{{n}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:{e}^{{x}^{\mathrm{2}} } \mathrm{sin}\:\left({nx}\right){dx}=? \\ $$

Question Number 224888    Answers: 0   Comments: 2

Question Number 224886    Answers: 1   Comments: 0

Find for acute θ, sin θ and cos θ in terms of 0<k<1, if ((sin θ(1−cos θ))/(cos θ(1−sin θ)))=k.

$${Find}\:{for}\:{acute}\:\theta,\:\mathrm{sin}\:\theta\:{and}\:\mathrm{cos}\:\theta \\ $$$${in}\:{terms}\:{of}\:\mathrm{0}<{k}<\mathrm{1}, \\ $$$${if}\:\:\:\:\frac{\mathrm{sin}\:\theta\left(\mathrm{1}−\mathrm{cos}\:\theta\right)}{\mathrm{cos}\:\theta\left(\mathrm{1}−\mathrm{sin}\:\theta\right)}={k}. \\ $$

Question Number 224883    Answers: 0   Comments: 0

∫ vol(g^ )=∫_( V) (√(det g_(μν) )) dx^1 ∧dx^2 ∧dx^3 parametric Surface S^→ (u,v,w);R^3 →R^3 S^→ (r,θ,ρ) { ((rsin(θ)cos(ρ))),((rsin(θ)sin(ρ))),((rcos(θ))) :} find metric tensor g_(μν) = ((g_(11) ,g_(12) ,g_(13) ),(g_(21) ,g_(22) ,g_(23) ),(g_(31) ,g_(32) ,g_(33) ) ) Describe it in the same as ds^2 =g_(μν) dx^μ dx^ν ds^2 =(dr dθ dρ) ((g_(11) ,g_(12) ,g_(13) ),(g_(21) ,g_(22) ,g_(23) ),(g_(31) ,g_(32) ,g_(33) ) ) ((dr),(dθ),(dρ) ) and find volume V=∫ vol(g)

$$\int\:\mathrm{vol}\left({g}^{\:} \right)=\int_{\:{V}} \:\sqrt{\mathrm{det}\:\boldsymbol{\mathrm{g}}_{\mu\nu} }\:\mathrm{d}{x}^{\mathrm{1}} \wedge\mathrm{d}{x}^{\mathrm{2}} \wedge\mathrm{d}{x}^{\mathrm{3}} \\ $$$$\mathrm{parametric}\:\mathrm{Surface}\: \\ $$$$\overset{\rightarrow} {\mathcal{S}}\left({u},{v},{w}\right);\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}^{\mathrm{3}} \\ $$$$\overset{\rightarrow} {\mathcal{S}}\left({r},\theta,\rho\right)\begin{cases}{{r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)}\\{{r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)}\\{{r}\mathrm{cos}\left(\theta\right)}\end{cases}\: \\ $$$$\mathrm{find}\:\mathrm{metric}\:\mathrm{tensor}\:\boldsymbol{\mathrm{g}}_{\mu\nu} =\begin{pmatrix}{\mathrm{g}_{\mathrm{11}} }&{\mathrm{g}_{\mathrm{12}} }&{\mathrm{g}_{\mathrm{13}} }\\{\mathrm{g}_{\mathrm{21}} }&{\mathrm{g}_{\mathrm{22}} }&{\mathrm{g}_{\mathrm{23}} }\\{\mathrm{g}_{\mathrm{31}} }&{\mathrm{g}_{\mathrm{32}} }&{\mathrm{g}_{\mathrm{33}} }\end{pmatrix} \\ $$$$\: \\ $$$$\mathrm{Describe}\:\mathrm{it}\:\mathrm{in}\:\mathrm{the}\:\mathrm{same}\:\mathrm{as}\:\mathrm{d}{s}^{\mathrm{2}} =\boldsymbol{\mathrm{g}}_{\mu\nu} \mathrm{d}{x}^{\mu} \mathrm{d}{x}^{\nu} \\ $$$$\mathrm{d}{s}^{\mathrm{2}} =\left(\mathrm{d}{r}\:\:\mathrm{d}\theta\:\:\mathrm{d}\rho\right)\begin{pmatrix}{\mathrm{g}_{\mathrm{11}} }&{\mathrm{g}_{\mathrm{12}} }&{\mathrm{g}_{\mathrm{13}} }\\{\mathrm{g}_{\mathrm{21}} }&{\mathrm{g}_{\mathrm{22}} }&{\mathrm{g}_{\mathrm{23}} }\\{\mathrm{g}_{\mathrm{31}} }&{\mathrm{g}_{\mathrm{32}} }&{\mathrm{g}_{\mathrm{33}} }\end{pmatrix}\begin{pmatrix}{\mathrm{d}{r}}\\{\mathrm{d}\theta}\\{\mathrm{d}\rho}\end{pmatrix} \\ $$$$\mathrm{and}\:\mathrm{find}\:\mathrm{volume}\:{V}=\int\:\:\mathrm{vol}\left(\mathrm{g}\right) \\ $$

Question Number 224879    Answers: 1   Comments: 0

prove lim_(n→∞) (1/(ln(p_n ))) Π_k (1/(1−(1/p_k )))=e^Υ_0 , Υ_0 =0.57721566490153286060..

$$\mathrm{prove} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{ln}\left({p}_{{n}} \right)}\:\underset{{k}} {\prod}\:\frac{\mathrm{1}}{\mathrm{1}−\frac{\mathrm{1}}{{p}_{{k}} }}={e}^{\Upsilon_{\mathrm{0}} } \:,\: \\ $$$$\Upsilon_{\mathrm{0}} =\mathrm{0}.\mathrm{57721566490153286060}.. \\ $$

Question Number 224873    Answers: 0   Comments: 5

Question Number 224872    Answers: 0   Comments: 0

Question Number 224866    Answers: 1   Comments: 0

can you guys explan why metric tensor g_(μν) =0 → Riemann metric tensor R_(αγβ) ^δ =0

$$\mathrm{can}\:\mathrm{you}\:\mathrm{guys}\:\mathrm{explan}\:\mathrm{why} \\ $$$$\mathrm{metric}\:\mathrm{tensor}\:\mathrm{g}_{\mu\nu} =\mathrm{0}\:\:\rightarrow\:\mathrm{Riemann}\:\mathrm{metric}\:\mathrm{tensor}\:{R}_{\alpha\gamma\beta} ^{\delta} =\mathrm{0} \\ $$

Question Number 224861    Answers: 2   Comments: 0

∫((sin x)/( (√(1+sin x)))) dx

$$\int\frac{\mathrm{sin}\:{x}}{\:\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}}\:{dx} \\ $$

Question Number 224859    Answers: 1   Comments: 1

A homogeneous rod AB of length L=1.8m and mass M is pivoted at the centre O in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal position.An insect S of the same mass M falls vertically with speed V on point C, midway between the points O and B. Immediaty after falling, the insect moves towards the end B such that the rod rotates with a constant angular velocity ω. a)determine the angular velocity ω in terms of V and L. b)If the imsect reaches the end B when the rod turned through an angle 90^0 , determine V

$${A}\:{homogeneous}\:{rod}\:{AB}\:{of}\:{length} \\ $$$${L}=\mathrm{1}.\mathrm{8}{m}\:{and}\:{mass}\:{M}\:{is}\:{pivoted} \\ $$$${at}\:{the}\:{centre}\:{O}\:{in}\:{such}\:{a}\:{way}\:{that} \\ $$$${it}\:{can}\:{rotate}\:{freely}\:{in}\:{the}\:{vertical}\:{plane}. \\ $$$$ \\ $$$${The}\:{rod}\:{is}\:{initially}\:{in}\:{the}\:{horizontal} \\ $$$${position}.{An}\:{insect}\:{S}\:\:{of}\:{the}\: \\ $$$${same}\:{mass}\:{M}\:{falls}\:{vertically} \\ $$$${with}\:{speed}\:{V}\:{on}\:{point}\:{C},\:{midway} \\ $$$${between}\:{the}\:{points}\:{O}\:{and}\:{B}. \\ $$$${Immediaty}\:{after}\:{falling},\:{the} \\ $$$${insect}\:{moves}\:{towards}\:{the}\:{end}\:{B} \\ $$$${such}\:{that}\:{the}\:{rod}\:{rotates} \\ $$$${with}\:{a}\:{constant}\:{angular}\:{velocity}\:\omega. \\ $$$$\left.{a}\right){determine}\:{the}\:{angular}\:{velocity}\:\omega \\ $$$${in}\:{terms}\:{of}\:{V}\:{and}\:{L}. \\ $$$$\left.{b}\right){If}\:{the}\:{imsect}\:{reaches}\:{the}\:{end}\:{B} \\ $$$${when}\:{the}\:{rod}\:{turned}\:{through} \\ $$$${an}\:{angle}\:\mathrm{90}^{\mathrm{0}} ,\:{determine}\:{V} \\ $$

Question Number 224853    Answers: 1   Comments: 15

a piece of chalk rests on a horizontal board with μ=0.1 Suddenly the board starts to move horizontally at a speed of 2m per second and after a time τ it stops abruptly. find the length of the line drawn by the chalk on the board for folowing cases τ=5sec τ=1sec g=10m/s^2

$$ \\ $$$$\mathrm{a}\:\mathrm{piece}\:\mathrm{of}\:\mathrm{chalk}\:\mathrm{rests}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{horizontal}\:\mathrm{board}\:\mathrm{with}\:\mu=\mathrm{0}.\mathrm{1} \\ $$$$\mathrm{Suddenly}\:\mathrm{the}\:\mathrm{board}\:\mathrm{starts}\:\mathrm{to} \\ $$$$\mathrm{move}\:\mathrm{horizontally}\:\mathrm{at}\:\mathrm{a}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{2m}\:\mathrm{per}\:\mathrm{second}\:\mathrm{and}\:\mathrm{after}\:\mathrm{a} \\ $$$$\mathrm{time}\:\tau\:\mathrm{it}\:\mathrm{stops}\:\mathrm{abruptly}.\:\mathrm{find}\: \\ $$$$\mathrm{the}\:\mathrm{length}\:\mathrm{of}\:\mathrm{the}\:\mathrm{line}\:\mathrm{drawn} \\ $$$$\mathrm{by}\:\mathrm{the}\:\mathrm{chalk}\:\mathrm{on}\:\mathrm{the}\:\mathrm{board}\:\mathrm{for} \\ $$$$\mathrm{folowing}\:\mathrm{cases} \\ $$$$\tau=\mathrm{5}{sec} \\ $$$$\tau=\mathrm{1}{sec} \\ $$$${g}=\mathrm{10}{m}/{s}^{\mathrm{2}} \\ $$

Question Number 224852    Answers: 2   Comments: 0

lim_(n→∞) ((1+(1/2)+(1/3)+(1/4)+...+(1/n))/(ln(n)))=?

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+...+\frac{\mathrm{1}}{{n}}}{{ln}\left({n}\right)}=? \\ $$

Question Number 224848    Answers: 0   Comments: 1

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