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Question Number 51959 Answers: 0 Comments: 4

Question Number 51950 Answers: 2 Comments: 0

a^2 + b^2 + c^2 = 2019 a, b, c are prime numbers . how many possible triples of (a, b, c) which that suitable for equation above .

$${a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:+\:{c}^{\mathrm{2}} \:\:=\:\:\mathrm{2019} \\ $$$${a},\:{b},\:{c}\:\:\:{are}\:\:{prime}\:\:{numbers}\:. \\ $$$${how}\:\:{many}\:\:{possible}\:\:{triples}\:\:{of}\:\:\left({a},\:{b},\:{c}\right)\:\:{which}\:\:{that}\:\:{suitable}\:\:{for}\:\:{equation}\:\:{above}\:. \\ $$

Question Number 51942 Answers: 1 Comments: 5

Question Number 51938 Answers: 0 Comments: 2

To which interaction among the four fundamental interactions is linked the combustion ? the fission ? Thanks.

$$\mathrm{To}\:\mathrm{which}\:\mathrm{interaction}\:\mathrm{among}\:\mathrm{the}\:\mathrm{four} \\ $$$$\mathrm{fundamental}\:\mathrm{interactions}\:\mathrm{is}\:\mathrm{linked} \\ $$$$\mathrm{the}\:\mathrm{combustion}\:?\:\mathrm{the}\:\mathrm{fission}\:? \\ $$$$ \\ $$$$\mathrm{Thanks}. \\ $$

Question Number 51933 Answers: 2 Comments: 1

If p = cos θ + i sin θ and q = cos φ + i sin φ Show that (((p + q)(pq − 1))/((p − q)(pq + 1))) = ((sin θ + sin φ)/(sin θ − sin φ))

$$\mathrm{If}\:\:\mathrm{p}\:=\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\:\theta\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$

Question Number 51922 Answers: 1 Comments: 0

find the value of... 1−(1/(1+(1/(i/(1+(i/(1+i))))))) pls help.

$${find}\:{the}\:{value}\:{of}... \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{1}−\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\frac{{i}}{\mathrm{1}+\frac{{i}}{\mathrm{1}+{i}}}}} \\ $$$${pls}\:{help}. \\ $$

Question Number 51921 Answers: 1 Comments: 0

A normal chord to an ellipse (x^2 /a^2 )+(y^2 /b^2 )=1 make an angle of 45^° with the axis.prove that the square of its length is equal to ((32a^4 b^4 )/((a^2 +b^2 )^3 ))

$${A}\:{normal}\:{chord}\:{to}\:{an}\: \\ $$$${ellipse}\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1} \\ $$$${make}\:{an}\:{angle}\:{of}\:\mathrm{45}^{°} \\ $$$${with}\:{the}\:{axis}.{prove} \\ $$$${that}\:{the}\:{square}\:{of}\:{its}\: \\ $$$${length}\:{is}\:{equal}\:{to} \\ $$$$\frac{\mathrm{32}{a}^{\mathrm{4}} {b}^{\mathrm{4}} }{\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right)^{\mathrm{3}} }\: \\ $$

Question Number 51911 Answers: 0 Comments: 2

Question Number 51910 Answers: 0 Comments: 4

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Question Number 51905 Answers: 2 Comments: 0

If p = cos θ + i sinθ and q = cos φ + i sin φ Show that: (i) ((p − q)/(p + q)) = i tan (((θ − φ)/2)) (ii) (((p + q)(pq − 1))/((p − q)(pq + 1))) = ((sin θ + sin φ)/(sin θ − sin φ))

$$\mathrm{If}\:\:\:\mathrm{p}\:\:=\:\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\theta\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}: \\ $$$$\left(\mathrm{i}\right)\:\:\:\:\:\:\frac{\mathrm{p}\:−\:\mathrm{q}}{\mathrm{p}\:+\:\mathrm{q}}\:\:=\:\:\mathrm{i}\:\mathrm{tan}\:\left(\frac{\theta\:−\:\phi}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{ii}\right)\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$

Question Number 51901 Answers: 0 Comments: 0

Prove that the two parabola y^2 =4ax and y^2 =4c(x−b) cannot have a common normal other than the axis unless (b/(a−c))>2

$${Prove}\:{that}\:{the}\:{two}\: \\ $$$${parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:{and} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{c}\left({x}−{b}\right)\:{cannot} \\ $$$${have}\:{a}\:{common}\:{normal} \\ $$$${other}\:{than}\:{the}\:{axis}\:{unless} \\ $$$$\frac{{b}}{{a}−{c}}>\mathrm{2} \\ $$

Question Number 51897 Answers: 1 Comments: 0

If x + (1/x) = 2cosθ , y + (1/y) = 2cosφ , z + (1/z) = 2cosψ Show that xyz + (1/(xyz)) = 2cos(θ + φ + ψ)

$$\mathrm{If}\:\:\:\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\:\:=\:\:\mathrm{2cos}\theta\:,\:\:\:\:\:\:\mathrm{y}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:\:=\:\:\mathrm{2cos}\phi\:,\:\:\:\:\:\:\:\:\:\mathrm{z}\:+\:\frac{\mathrm{1}}{\mathrm{z}}\:\:=\:\:\mathrm{2cos}\psi \\ $$$$\mathrm{Show}\:\mathrm{that}\:\:\:\:\:\:\:\:\:\:\mathrm{xyz}\:+\:\frac{\mathrm{1}}{\mathrm{xyz}}\:\:=\:\:\mathrm{2cos}\left(\theta\:+\:\phi\:+\:\psi\right) \\ $$

Question Number 51887 Answers: 1 Comments: 0

Prove that; tanh(log (√3)) = (1/2)

$$\mathrm{Prove}\:\mathrm{that};\:\:\:\:\mathrm{tanh}\left(\mathrm{log}\:\sqrt{\mathrm{3}}\right)\:\:=\:\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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Question Number 51876 Answers: 1 Comments: 2

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Question Number 51849 Answers: 3 Comments: 3

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Question Number 51843 Answers: 1 Comments: 3

If ax^2 +bx+c+i=0 has purely imaginary roots where a,b,c are non−zero real. answer given: a=b^2 c I think question is wrong since if z_1 and z_2 are roots than z_1 +z_2 =−(b/a) purely imaginary=purely real not possible Can some point a mistake.

$${If}\:{ax}^{\mathrm{2}} +{bx}+{c}+{i}=\mathrm{0}\:\mathrm{has}\:\mathrm{purely} \\ $$$$\mathrm{imaginary}\:\mathrm{roots}\:\mathrm{where}\: \\ $$$${a},{b},{c}\:{are}\:{non}−{zero}\:{real}. \\ $$$${answer}\:{given}:\:{a}={b}^{\mathrm{2}} {c} \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{question}\:\mathrm{is}\:\mathrm{wrong} \\ $$$$\mathrm{since}\:\mathrm{if}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{are}\:\mathrm{roots}\:\mathrm{than} \\ $$$${z}_{\mathrm{1}} +{z}_{\mathrm{2}} =−\frac{{b}}{{a}} \\ $$$${purely}\:{imaginary}={purely}\:{real} \\ $$$${not}\:{possible} \\ $$$$\mathrm{Can}\:\mathrm{some}\:\mathrm{point}\:\mathrm{a}\:\mathrm{mistake}. \\ $$

Question Number 51841 Answers: 0 Comments: 2

Question Number 51840 Answers: 2 Comments: 0

solve (dy/(dx )) + ((2y)/(3x )) = (x/(√y))

$${solve} \\ $$$$\frac{{dy}}{{dx}\:}\:+\:\frac{\mathrm{2}{y}}{\mathrm{3}{x}\:}\:=\:\frac{{x}}{\sqrt{{y}}} \\ $$

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