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Question Number 51279    Answers: 1   Comments: 1

Question Number 51271    Answers: 1   Comments: 0

Show that the locus of a point which moves so that its distance from the point (ae,0) is e times its distance from the line x=(a/e) is given by the equation (x^2 /a^2 )+(y^2 /(a^2 (1−e^2 )))=1

$${Show}\:{that}\:{the}\:{locus}\:{of}\:{a} \\ $$$${point}\:{which}\:{moves}\:{so} \\ $$$${that}\:{its}\:{distance}\:{from} \\ $$$${the}\:{point}\:\left({ae},\mathrm{0}\right)\:{is}\:{e}\:{times} \\ $$$${its}\:{distance}\:{from}\:{the}\: \\ $$$${line}\:{x}=\frac{{a}}{{e}}\:{is}\:{given}\:{by}\:{the} \\ $$$${equation} \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{a}^{\mathrm{2}} \left(\mathrm{1}−{e}^{\mathrm{2}} \right)}=\mathrm{1} \\ $$$$ \\ $$

Question Number 51269    Answers: 1   Comments: 0

Find the ecentricity If (1)lactus rectum is half major axis (2)lactus rectum is half minor axis

$${Find}\:{the}\:{ecentricity}\:{If} \\ $$$$\left(\mathrm{1}\right){lactus}\:{rectum}\:{is}\:{half} \\ $$$${major}\:{axis} \\ $$$$\left(\mathrm{2}\right){lactus}\:{rectum}\:{is}\:{half} \\ $$$${minor}\:{axis} \\ $$

Question Number 51263    Answers: 1   Comments: 0

If P = 2 + j3 and Q = 2 − j3 and R = j1 Show that angle PRQ is right angle

$$\mathrm{If}\:\:\mathrm{P}\:=\:\mathrm{2}\:+\:\mathrm{j3}\:\mathrm{and}\:\mathrm{Q}\:=\:\mathrm{2}\:−\:\mathrm{j3}\:\mathrm{and}\:\mathrm{R}\:=\:\mathrm{j1} \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\mathrm{angle}\:\:\mathrm{PRQ}\:\mathrm{is}\:\mathrm{right}\:\mathrm{angle} \\ $$

Question Number 51250    Answers: 1   Comments: 0

Given that z_1 = R_1 + R + jωL ; z_2 = R_2 ; z_3 = (1/(jωC_3 )) and z_4 = R_4 + (1/(jωC_4 )) and also that z_1 z_3 = z_2 z_4 , express R and L in terms of the real constants R_1 , R_2 , R_4 , C_3 and C_4 Answer: R = ((R_2 C_3 − R_1 C_4 )/C_4 ) , L = R_2 R_4 C_3

$$\mathrm{Given}\:\mathrm{that}\:\:\:\mathrm{z}_{\mathrm{1}} \:=\:\mathrm{R}_{\mathrm{1}} \:+\:\mathrm{R}\:+\:\mathrm{j}\omega\mathrm{L}\:;\:\:\:\mathrm{z}_{\mathrm{2}} \:=\:\mathrm{R}_{\mathrm{2}} \:;\:\:\mathrm{z}_{\mathrm{3}} \:=\:\frac{\mathrm{1}}{\mathrm{j}\omega\mathrm{C}_{\mathrm{3}} } \\ $$$$\mathrm{and}\:\:\mathrm{z}_{\mathrm{4}} \:=\:\mathrm{R}_{\mathrm{4}} \:+\:\frac{\mathrm{1}}{\mathrm{j}\omega\mathrm{C}_{\mathrm{4}} }\:\:\mathrm{and}\:\mathrm{also}\:\mathrm{that}\:\:\:\mathrm{z}_{\mathrm{1}} \mathrm{z}_{\mathrm{3}} \:\:=\:\:\mathrm{z}_{\mathrm{2}} \mathrm{z}_{\mathrm{4}} \:,\:\:\:\mathrm{express}\: \\ $$$$\mathrm{R}\:\mathrm{and}\:\mathrm{L}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{the}\:\mathrm{real}\:\mathrm{constants}\:\:\mathrm{R}_{\mathrm{1}} ,\:\mathrm{R}_{\mathrm{2}} ,\:\mathrm{R}_{\mathrm{4}} ,\:\mathrm{C}_{\mathrm{3}} \:\mathrm{and}\:\mathrm{C}_{\mathrm{4}} \\ $$$$ \\ $$$$\mathrm{Answer}:\:\:\:\:\:\:\mathrm{R}\:=\:\frac{\mathrm{R}_{\mathrm{2}} \mathrm{C}_{\mathrm{3}} \:−\:\mathrm{R}_{\mathrm{1}} \mathrm{C}_{\mathrm{4}} }{\mathrm{C}_{\mathrm{4}} }\:,\:\:\:\:\:\:\:\:\mathrm{L}\:=\:\mathrm{R}_{\mathrm{2}} \mathrm{R}_{\mathrm{4}} \mathrm{C}_{\mathrm{3}} \\ $$

Question Number 51248    Answers: 1   Comments: 0

If ((R_1 + jωL)/R_3 ) = (R_2 /(R_4 − j (1/(ωC)))) , where R_1 , R_2 , R_3 , R_4 , ω, L and C are real , show that L = ((C R_2 R_3 )/(ω^2 C^2 R_4 ^2 + 1))

$$\mathrm{If}\:\:\:\:\:\frac{\mathrm{R}_{\mathrm{1}} \:+\:\mathrm{j}\omega\mathrm{L}}{\mathrm{R}_{\mathrm{3}} }\:\:=\:\:\frac{\mathrm{R}_{\mathrm{2}} }{\mathrm{R}_{\mathrm{4}} \:−\:\mathrm{j}\:\frac{\mathrm{1}}{\omega\mathrm{C}}}\:\:,\:\:\:\mathrm{where}\:\:\mathrm{R}_{\mathrm{1}} ,\:\mathrm{R}_{\mathrm{2}} ,\:\mathrm{R}_{\mathrm{3}} ,\:\mathrm{R}_{\mathrm{4}} ,\:\omega,\:\mathrm{L}\:\mathrm{and}\:\mathrm{C} \\ $$$$\mathrm{are}\:\mathrm{real}\:,\:\:\mathrm{show}\:\mathrm{that}\:\:\:\:\mathrm{L}\:=\:\frac{\mathrm{C}\:\mathrm{R}_{\mathrm{2}} \mathrm{R}_{\mathrm{3}} }{\omega^{\mathrm{2}} \mathrm{C}^{\mathrm{2}} \mathrm{R}_{\mathrm{4}} ^{\mathrm{2}} \:+\:\mathrm{1}} \\ $$

Question Number 51245    Answers: 2   Comments: 0

Question Number 51228    Answers: 1   Comments: 1

Question Number 51227    Answers: 1   Comments: 4

How many odd numbers with different digits are there from 2019 to 9102?

$${How}\:{many}\:{odd}\:{numbers}\:{with}\:{different} \\ $$$${digits}\:{are}\:{there}\:{from}\:\mathrm{2019}\:{to}\:\mathrm{9102}? \\ $$

Question Number 51220    Answers: 1   Comments: 1

Question Number 51218    Answers: 0   Comments: 0

show each of the following functions a entire functions a. f(z)=e^(−y) sin x−i e^(−y) cos x b. f(z)=(z^2 −2)e^(−x) e^(−iy)

$$\mathrm{show}\:\mathrm{each}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{functions}\:\mathrm{a}\:\mathrm{entire}\:\mathrm{functions} \\ $$$$\mathrm{a}.\:{f}\left({z}\right)={e}^{−{y}} \mathrm{sin}\:{x}−{i}\:{e}^{−{y}} \mathrm{cos}\:{x} \\ $$$${b}.\:{f}\left({z}\right)=\left({z}^{\mathrm{2}} −\mathrm{2}\right){e}^{−{x}} {e}^{−{iy}} \\ $$

Question Number 51217    Answers: 0   Comments: 0

show that f(z)=z^2 continuous at z=z_0

$$\mathrm{show}\:\mathrm{that}\:{f}\left({z}\right)={z}^{\mathrm{2}} \:\mathrm{continuous}\:\mathrm{at}\:{z}={z}_{\mathrm{0}} \\ $$

Question Number 51216    Answers: 1   Comments: 0

show lim f(z) for z→0 along the line y=x where: f(z)=((2xy)/(x^2 +y^2 ))−i(y^2 /x^2 )

$$\mathrm{show}\:\mathrm{lim}\:{f}\left({z}\right)\:\mathrm{for}\:{z}\rightarrow\mathrm{0}\:\mathrm{along}\:\mathrm{the}\:\mathrm{line}\:{y}={x} \\ $$$$\mathrm{where}:\:{f}\left({z}\right)=\frac{\mathrm{2}{xy}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }−{i}\frac{{y}^{\mathrm{2}} }{{x}^{\mathrm{2}} } \\ $$

Question Number 51215    Answers: 1   Comments: 0

∫_0 ^π e^((1+i)x) dx=...

$$\int_{\mathrm{0}} ^{\pi} {e}^{\left(\mathrm{1}+{i}\right){x}} {dx}=... \\ $$

Question Number 51199    Answers: 0   Comments: 0

Question Number 51236    Answers: 2   Comments: 0

Prove that line y=mx+(3/(4 ))m+(1/m) touches the parabola y^2 =4x+3 whatever the value of m

$${Prove}\:{that}\:{line}\:{y}={mx}+\frac{\mathrm{3}}{\mathrm{4}\:\:}{m}+\frac{\mathrm{1}}{{m}} \\ $$$${touches}\:{the}\:{parabola} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{x}+\mathrm{3}\:{whatever}\:{the} \\ $$$${value}\:{of}\:{m} \\ $$

Question Number 51235    Answers: 2   Comments: 0

Without using tables, find tha value of ((((√5) +2)^6 −((√5)−2)^6 )/(8(√5) ))

$${Without}\:{using}\:{tables}, \\ $$$${find}\:{tha}\:{value}\:{of} \\ $$$$\frac{\left(\sqrt{\mathrm{5}}\:+\mathrm{2}\right)^{\mathrm{6}} −\left(\sqrt{\mathrm{5}}−\mathrm{2}\right)^{\mathrm{6}} }{\mathrm{8}\sqrt{\mathrm{5}}\:}\:\: \\ $$

Question Number 51190    Answers: 0   Comments: 0

find lim_(ξ→0^+ ) ^ ∫_0 ^ξ (x/(sinx −(√(sin^2 x +ξ^2 ))))dx

$${find}\:{lim}_{\xi\rightarrow\mathrm{0}^{+} } ^{} \:\:\:\:\:\int_{\mathrm{0}} ^{\xi} \:\:\:\:\:\:\frac{{x}}{{sinx}\:−\sqrt{{sin}^{\mathrm{2}} {x}\:+\xi^{\mathrm{2}} }}{dx} \\ $$

Question Number 51188    Answers: 0   Comments: 0

find ∫ ((cos^2 x)/(cosx +2sinx))dx

$${find}\:\:\int\:\:\:\:\:\frac{{cos}^{\mathrm{2}} {x}}{{cosx}\:+\mathrm{2}{sinx}}{dx} \\ $$

Question Number 51186    Answers: 1   Comments: 1

calculate ∫_0 ^1 (([nx])/(2x+1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\left[{nx}\right]}{\mathrm{2}{x}+\mathrm{1}}{dx} \\ $$

Question Number 51185    Answers: 0   Comments: 1

calculate Σ_(n=0) ^∞ (n/((n+1)^4 (2n+1)^2 ))

$${calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{n}}{\left({n}+\mathrm{1}\right)^{\mathrm{4}} \left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 51181    Answers: 2   Comments: 1

Question Number 51174    Answers: 0   Comments: 7

please can you make arabic app like this ? or add arabic language to this great app ?

$$\mathrm{please} \\ $$$$\mathrm{can}\:\mathrm{you}\:\mathrm{make}\:\mathrm{arabic}\:\mathrm{app} \\ $$$$\mathrm{like}\:\mathrm{this}\:? \\ $$$$\mathrm{or}\:\mathrm{add}\:\mathrm{arabic}\:\mathrm{language} \\ $$$$\mathrm{to}\:\mathrm{this}\:\mathrm{great}\:\mathrm{app}\:? \\ $$

Question Number 51167    Answers: 3   Comments: 0

Prove that: (a) If ∣z_1 + z_2 ∣ = ∣z_1 − z_2 ∣, the difference of the arguements of z_1 and z_2 is (π/2) (b) If arg{((z_1 + z_2 )/(z_1 − z_2 ))} = (π/2) , then ∣z_1 ∣ = ∣z_2 ∣

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\left(\mathrm{a}\right)\:\:\mathrm{If}\:\:\mid\mathrm{z}_{\mathrm{1}} \:+\:\mathrm{z}_{\mathrm{2}} \mid\:=\:\mid\mathrm{z}_{\mathrm{1}} \:−\:\mathrm{z}_{\mathrm{2}} \mid,\:\:\mathrm{the}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{arguements}\:\mathrm{of}\:\mathrm{z}_{\mathrm{1}} \\ $$$$\mathrm{and}\:\mathrm{z}_{\mathrm{2}} \:\mathrm{is}\:\:\frac{\pi}{\mathrm{2}} \\ $$$$\left(\mathrm{b}\right)\:\:\mathrm{If}\:\:\mathrm{arg}\left\{\frac{\mathrm{z}_{\mathrm{1}} \:+\:\mathrm{z}_{\mathrm{2}} }{\mathrm{z}_{\mathrm{1}} \:−\:\mathrm{z}_{\mathrm{2}} }\right\}\:=\:\frac{\pi}{\mathrm{2}}\:,\:\:\:\mathrm{then}\:\:\:\:\mid\mathrm{z}_{\mathrm{1}} \mid\:=\:\mid\mathrm{z}_{\mathrm{2}} \mid \\ $$

Question Number 51163    Answers: 0   Comments: 0

Question Number 51156    Answers: 2   Comments: 0

Show that the equation of tangent to the ellipse (x^2 /a^2 )+(y^2 /b^2 )=1 at the end of lactus rectum which lie in the 1^(st) quadrant is xe+y−a=0 ∗merry X−mas and happy new year∗

$${Show}\:{that}\:{the}\:{equation} \\ $$$${of}\:{tangent}\:{to}\:{the}\:{ellipse} \\ $$$$\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}\:{at}\:{the}\:{end}\:{of} \\ $$$${lactus}\:{rectum}\:{which} \\ $$$${lie}\:{in}\:{the}\:\mathrm{1}^{{st}} {quadrant}\:{is} \\ $$$${xe}+{y}−{a}=\mathrm{0} \\ $$$$ \\ $$$$\ast{merry}\:{X}−{mas}\:{and}\:{happy}\:{new}\:{year}\ast \\ $$

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