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Question Number 203544    Answers: 1   Comments: 0

∫_0 ^1_ (dx/((1−x^6 )^(1/6) )) =(π/3)

$$\int_{\mathrm{0}} ^{\mathrm{1}_{} } \frac{{dx}}{\left(\mathrm{1}−{x}^{\mathrm{6}} \right)^{\frac{\mathrm{1}}{\mathrm{6}}} }\:=\frac{\pi}{\mathrm{3}} \\ $$

Question Number 203533    Answers: 2   Comments: 0

Solve for x : 3^(x+2) =15^(x−1)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:: \\ $$$$\mathrm{3}^{\mathrm{x}+\mathrm{2}} =\mathrm{15}^{\mathrm{x}−\mathrm{1}} \\ $$

Question Number 203529    Answers: 0   Comments: 0

Question Number 203527    Answers: 2   Comments: 0

Find the maximum value of the function f(x,y)=x^2 y^2 z^2 subject to the condition that x^2 +y^2 +z^2 =c^2 , where c is the constant. Thank you in advance

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function} \\ $$$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} \mathrm{z}^{\mathrm{2}} \:\mathrm{subject}\:\mathrm{to}\:\mathrm{the}\:\mathrm{condition}\:\mathrm{that} \\ $$$$\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{c}^{\mathrm{2}} ,\:\mathrm{where}\:\mathrm{c}\:\mathrm{is}\:\mathrm{the}\:\mathrm{constant}. \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{in}\:\mathrm{advance} \\ $$

Question Number 203526    Answers: 2   Comments: 0

Determine the maximum and minimum of the function: f(x,y)=x^4 +4x^2 y^2 −2x^2 +2y^2 −1 Thank you

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{and}\:\mathrm{minimum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{function}: \\ $$$$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{x}^{\mathrm{4}} +\mathrm{4x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} −\mathrm{2x}^{\mathrm{2}} +\mathrm{2y}^{\mathrm{2}} −\mathrm{1} \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$

Question Number 203525    Answers: 1   Comments: 0

Solve: 4x^3 +8xy^2 −4x=0 -----(1) 8x^2 y−4y =0 -----(2) simultaneosly i.e find the stationary point Thank you

$$\mathrm{Solve}:\: \\ $$$$\mathrm{4x}^{\mathrm{3}} +\mathrm{8xy}^{\mathrm{2}} −\mathrm{4x}=\mathrm{0}\:-----\left(\mathrm{1}\right) \\ $$$$\mathrm{8x}^{\mathrm{2}} \mathrm{y}−\mathrm{4y}\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{0}\:-----\left(\mathrm{2}\right) \\ $$$$\mathrm{simultaneosly}\:\mathrm{i}.\mathrm{e}\:\mathrm{find}\:\mathrm{the}\:\mathrm{stationary}\:\mathrm{point} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$

Question Number 203509    Answers: 0   Comments: 0

Find the value of: Π_(n=1) ^∞ ((2^n +1)/(2^n −1))

$${Find}\:{the}\:{value}\:{of}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\frac{\mathrm{2}^{{n}} +\mathrm{1}}{\mathrm{2}^{{n}} −\mathrm{1}} \\ $$

Question Number 203508    Answers: 1   Comments: 0

Evaluate the given limit: lim_(n→∞) (((8)^(1/n) −1)/( ((16))^(1/n) −1))

$${Evaluate}\:{the}\:{given}\:{limit}: \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\frac{\sqrt[{{n}}]{\mathrm{8}}−\mathrm{1}}{\:\sqrt[{{n}}]{\mathrm{16}}−\mathrm{1}} \\ $$

Question Number 203502    Answers: 1   Comments: 2

ze^z =e Obviously z=1 Now find at least one solution for z∈C

$${z}\mathrm{e}^{{z}} =\mathrm{e} \\ $$$$\mathrm{Obviously}\:{z}=\mathrm{1} \\ $$$$\mathrm{Now}\:\mathrm{find}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{solution}\:\mathrm{for}\:{z}\in\mathbb{C} \\ $$

Question Number 203498    Answers: 0   Comments: 0

(d^(3 ) y/dx^3 )=4(x+(1/4))^2 −4y

$$\frac{{d}^{\mathrm{3}\:} {y}}{{dx}^{\mathrm{3}} }=\mathrm{4}\left({x}+\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{2}} −\mathrm{4}{y} \\ $$

Question Number 203497    Answers: 0   Comments: 0

x^4 +cx+d=0 then find p from p^6 −4((d^( 3) /c^4 ))^(1/3) p^2 −1=0 x=(c^(1/3) /2)(p±(√(−p^2 −(8/p))) )

$${x}^{\mathrm{4}} +{cx}+{d}=\mathrm{0} \\ $$$${then}\:\:{find}\:{p}\:\:{from} \\ $$$${p}^{\mathrm{6}} −\mathrm{4}\left(\frac{{d}^{\:\mathrm{3}} }{{c}^{\mathrm{4}} }\right)^{\mathrm{1}/\mathrm{3}} {p}^{\mathrm{2}} −\mathrm{1}=\mathrm{0} \\ $$$${x}=\frac{{c}^{\mathrm{1}/\mathrm{3}} }{\mathrm{2}}\left({p}\pm\sqrt{−{p}^{\mathrm{2}} −\frac{\mathrm{8}}{{p}}}\:\right) \\ $$

Question Number 203490    Answers: 1   Comments: 0

1×3×5×7×9×...×2005 = ... (mod 1000)

$$\:\:\:\:\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×...×\mathrm{2005}\:=\:...\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$

Question Number 203480    Answers: 3   Comments: 2

Question Number 203471    Answers: 1   Comments: 0

Question Number 203428    Answers: 0   Comments: 0

GKJC−VFIX+2 (B) (1/2)(GKJC+VFIX)−(3/2) (C) (1/3)(GKJC+VFIX)−2 (D) (1/2)(GKJC−VFIX)+(5/2) (E) (1/3)(GKJC+VFIX)+2

$$ \\ $$$$ \\ $$$$ \mathrm{GKJC}−\mathrm{VFIX}+\mathrm{2} \\ $$$$\left(\mathrm{B}\right)\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{GKJC}+\mathrm{VFIX}\right)−\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\:\left(\mathrm{C}\right)\:\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{GKJC}+\mathrm{VFIX}\right)−\mathrm{2} \\ $$$$\left(\mathrm{D}\right)\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{GKJC}−\mathrm{VFIX}\right)+\frac{\mathrm{5}}{\mathrm{2}} \\ $$$$\:\left(\mathrm{E}\right)\:\frac{\mathrm{1}}{\mathrm{3}}\left(\mathrm{GKJC}+\mathrm{VFIX}\right)+\mathrm{2}\: \\ $$

Question Number 203424    Answers: 3   Comments: 0

Question Number 203419    Answers: 0   Comments: 4

Question Number 203468    Answers: 0   Comments: 0

let A B and C be sets prove using venn diagrams and by first principle a)A×(B∪C)=(A×B)∪(A×C) b)A×(B∩C)=(A×B)∩(A×C) c)(A−B)∩(A∩B)∩(B−A)={}

$$\:\boldsymbol{\mathrm{let}}\:\boldsymbol{\mathrm{A}}\:{B}\:{and}\:{C}\:\:\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{sets}}\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{venn}}\:\boldsymbol{\mathrm{diagrams}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{first}}\:\boldsymbol{\mathrm{principle}} \\ $$$$\left.\:\boldsymbol{\mathrm{a}}\right)\boldsymbol{\mathrm{A}}×\left(\boldsymbol{\mathrm{B}}\cup\boldsymbol{\mathrm{C}}\right)=\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{B}}\right)\cup\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{C}}\right) \\ $$$$\left.\boldsymbol{\mathrm{b}}\right)\mathrm{A}×\left(\boldsymbol{\mathrm{B}}\cap\boldsymbol{\mathrm{C}}\right)=\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{B}}\right)\cap\left(\boldsymbol{\mathrm{A}}×\boldsymbol{\mathrm{C}}\right) \\ $$$$\left.\boldsymbol{\mathrm{c}}\right)\left(\boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{B}}\right)\cap\left(\boldsymbol{{A}}\cap\boldsymbol{\mathrm{B}}\right)\cap\left(\boldsymbol{\mathrm{B}}−\boldsymbol{\mathrm{A}}\right)=\left\{\right\} \\ $$

Question Number 203474    Answers: 5   Comments: 0

((z^4 +4z^3 −6z^2 −4z+1)/(z^4 −4z^3 −6z^2 +4z+1))=((z+1)/(z−1)) Find z∈R.

$$\frac{{z}^{\mathrm{4}} +\mathrm{4}{z}^{\mathrm{3}} −\mathrm{6}{z}^{\mathrm{2}} −\mathrm{4}{z}+\mathrm{1}}{{z}^{\mathrm{4}} −\mathrm{4}{z}^{\mathrm{3}} −\mathrm{6}{z}^{\mathrm{2}} +\mathrm{4}{z}+\mathrm{1}}=\frac{{z}+\mathrm{1}}{{z}−\mathrm{1}} \\ $$$${Find}\:{z}\in\mathbb{R}. \\ $$

Question Number 203465    Answers: 2   Comments: 0

Focus and vertex of a parabola are at (3, 4) and (0,0). Find the equation of the directrix.

$$\mathrm{Focus}\:\mathrm{and}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{a}\:\mathrm{parabola}\:\mathrm{are}\:\mathrm{at}\:\left(\mathrm{3},\:\mathrm{4}\right)\:\mathrm{and}\:\left(\mathrm{0},\mathrm{0}\right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{directrix}. \\ $$

Question Number 203457    Answers: 1   Comments: 0

Question Number 203454    Answers: 1   Comments: 0

(√5)+(√5)=x x×5π=? find all answers until 24.01.2024

$$\sqrt{\mathrm{5}}+\sqrt{\mathrm{5}}={x} \\ $$$${x}×\mathrm{5}\pi=?\:{find}\:{all}\:{answers}\:{until}\:\mathrm{24}.\mathrm{01}.\mathrm{2024} \\ $$

Question Number 203445    Answers: 0   Comments: 1

Question Number 203417    Answers: 1   Comments: 1

If ∣−z + iz^(−) ∣ = 2 (√(13)) and z = 1 + (x + 1)∙i Find: mun(x) = ?

$$\mathrm{If} \\ $$$$\mid\overline {−\mathrm{z}\:+\:\boldsymbol{\mathrm{i}}\mathrm{z}}\mid\:=\:\mathrm{2}\:\sqrt{\mathrm{13}}\:\:\:\mathrm{and}\:\:\:\mathrm{z}\:=\:\mathrm{1}\:+\:\left(\mathrm{x}\:+\:\mathrm{1}\right)\centerdot\boldsymbol{\mathrm{i}} \\ $$$$ \\ $$$$\mathrm{Find}:\:\:\:\mathrm{mun}\left(\mathrm{x}\right)\:=\:? \\ $$

Question Number 203416    Answers: 2   Comments: 0

Question Number 203434    Answers: 2   Comments: 0

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