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Question Number 97380    Answers: 0   Comments: 0

E is a vectorial plane. his base is B=(i^→ ;j^→ ). f is an endomorphism defined by f(i^→ )=−((√2)/2)i^→ +((√2)/2)j^→ and f(j^→ )=((√2)/2)i^→ −((√2)/2)j^→ 1)Show that ker f is a vectorial straigh line and his base is e_1 ^→ =(√2)i^→ +(√2)j^→ 2)show that G, the set of vectors u^→ ∈ E such as f(u^→ )=(√2)u^→ is a vectorial straigh line and his Base is e_(2 ) ^→ =i^→ +j^→ 3) Determine the matrix A′ of f in B′ if B′=(e_1 ^→ ;e_2 ^→ ).

$${E}\:{is}\:{a}\:{vectorial}\:{plane}.\:{his}\:{base}\:{is}\: \\ $$$${B}=\left(\overset{\rightarrow} {{i}};\overset{\rightarrow} {{j}}\right).\:{f}\:{is}\:{an}\:{endomorphism}\:{defined} \\ $$$${by}\:{f}\left(\overset{\rightarrow} {{i}}\right)=−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{i}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{j}}\:{and}\:{f}\left(\overset{\rightarrow} {{j}}\right)=\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{i}}−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{1}\right){Show}\:{that}\:{ker}\:{f}\:{is}\:{a}\:{vectorial}\:{straigh} \\ $$$${line}\:{and}\:{his}\:{base}\:{is}\:\overset{\rightarrow} {{e}}_{\mathrm{1}} =\sqrt{\mathrm{2}}\overset{\rightarrow} {{i}}+\sqrt{\mathrm{2}}\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{2}\right){show}\:{that}\:{G},\:{the}\:{set}\:{of}\:{vectors}\:\overset{\rightarrow} {{u}} \\ $$$$\:\in\:{E}\:{such}\:{as}\:{f}\left(\overset{\rightarrow} {{u}}\right)=\sqrt{\mathrm{2}}\overset{\rightarrow} {{u}}\:{is}\:{a}\:{vectorial}\:{straigh} \\ $$$${line}\:{and}\:{his}\:{Base}\:{is}\:\overset{\rightarrow} {{e}}_{\mathrm{2}\:\:} =\overset{\rightarrow} {{i}}+\overset{\rightarrow} {{j}} \\ $$$$\left.\mathrm{3}\right)\:{Determine}\:{the}\:{matrix}\:{A}'\:{of}\:{f}\:{in} \\ $$$${B}'\:{if}\:{B}'=\left(\overset{\rightarrow} {{e}}_{\mathrm{1}} ;\overset{\rightarrow} {{e}}_{\mathrm{2}} \right). \\ $$

Question Number 97371    Answers: 0   Comments: 1

Question Number 97369    Answers: 1   Comments: 0

∫_0 ^1 (((ln(x))^2 )/(1+x^2 ))dx=(π^3 /(16))

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left({ln}\left({x}\right)\right)^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=\frac{\pi^{\mathrm{3}} }{\mathrm{16}} \\ $$

Question Number 97368    Answers: 3   Comments: 11

Question Number 97361    Answers: 1   Comments: 6

∫((xdx)/(sin^2 x−3))=?

$$\int\frac{\mathrm{xdx}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{3}}=? \\ $$

Question Number 97355    Answers: 0   Comments: 1

Question Number 97353    Answers: 5   Comments: 0

Question Number 97346    Answers: 1   Comments: 0

Question Number 97335    Answers: 0   Comments: 2

Question Number 97323    Answers: 1   Comments: 0

Mr Peter has 4 children. x are in class C and y are in class D. x≥1 and y≥1. Show that the number of possibility to choose at random and simultaneous 2 children in same class verify this equation p(x)=x^2 −4x+6

$${Mr}\:{Peter}\:{has}\:\mathrm{4}\:{children}.\:{x}\:{are}\:{in}\: \\ $$$${class}\:{C}\:{and}\:{y}\:{are}\:{in}\:{class}\:{D}.\:{x}\geqslant\mathrm{1}\:{and} \\ $$$${y}\geqslant\mathrm{1}.\:{Show}\:{that}\:{the}\:{number}\:{of}\:{possibility}\: \\ $$$${to}\:{choose}\:{at}\:{random}\:{and}\:{simultaneous} \\ $$$$\mathrm{2}\:{children}\:{in}\:{same}\:{class}\:{verify}\:{this} \\ $$$${equation}\:{p}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{6} \\ $$

Question Number 97322    Answers: 1   Comments: 0

∫sin^4 x∙cos^5 xdx=?

$$\int\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\centerdot\mathrm{cos}\:^{\mathrm{5}} \mathrm{xdx}=? \\ $$

Question Number 97319    Answers: 1   Comments: 0

If 5,6,11,17,28,45,73,x,y,z then the value of x,y,z?__ (a) 118,192,309 (b) 117,191,310 (c) 117,191,308 (d) 118,192,310 (e) 118,191,309

$$\mathrm{If}\:\mathrm{5},\mathrm{6},\mathrm{11},\mathrm{17},\mathrm{28},\mathrm{45},\mathrm{73},\mathrm{x},\mathrm{y},\mathrm{z} \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x},\mathrm{y},\mathrm{z}?\_\_ \\ $$$$\left(\mathrm{a}\right)\:\mathrm{118},\mathrm{192},\mathrm{309} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{117},\mathrm{191},\mathrm{310} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{117},\mathrm{191},\mathrm{308} \\ $$$$\left(\mathrm{d}\right)\:\mathrm{118},\mathrm{192},\mathrm{310} \\ $$$$\left(\mathrm{e}\right)\:\mathrm{118},\mathrm{191},\mathrm{309} \\ $$

Question Number 97314    Answers: 0   Comments: 0

Question Number 97307    Answers: 2   Comments: 2

Question Number 97306    Answers: 0   Comments: 2

if sin14=x then cos^2 22−cos^2 8=?

$$\mathrm{if}\:\:\:\:\:\:\:\:\:\mathrm{sin14}=\mathrm{x} \\ $$$$\mathrm{then} \\ $$$$\mathrm{cos}^{\mathrm{2}} \mathrm{22}−\mathrm{cos}^{\mathrm{2}} \mathrm{8}=? \\ $$

Question Number 97305    Answers: 1   Comments: 0

Question Number 97303    Answers: 0   Comments: 0

Given x_1 +x_2 +x_3 = 0 , y_1 + y_2 +y_3 = 0 and x_1 y_1 + x_2 y_2 + x_3 y_3 = 0 . The value of (x_1 ^2 /(x_1 ^2 +x_2 ^2 +x_3 ^2 )) + (y_1 ^2 /(y_1 ^2 +y_2 ^2 +y_3 ^2 )) = ?

$$\boldsymbol{\mathrm{G}}\mathrm{iven}\:\mathrm{x}_{\mathrm{1}} +\mathrm{x}_{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} \:=\:\mathrm{0}\:,\:\mathrm{y}_{\mathrm{1}} \:+\:\mathrm{y}_{\mathrm{2}} +\mathrm{y}_{\mathrm{3}} \:=\:\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{x}_{\mathrm{1}} \mathrm{y}_{\mathrm{1}} +\:\mathrm{x}_{\mathrm{2}} \mathrm{y}_{\mathrm{2}} \:+\:\mathrm{x}_{\mathrm{3}} \mathrm{y}_{\mathrm{3}} \:=\:\mathrm{0}\:.\:\mathrm{The}\:\mathrm{value} \\ $$$$\mathrm{of}\:\frac{\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} ^{\mathrm{2}} }\:+\:\frac{\mathrm{y}_{\mathrm{1}} ^{\mathrm{2}} }{\mathrm{y}_{\mathrm{1}} ^{\mathrm{2}} \:+\mathrm{y}_{\mathrm{2}} ^{\mathrm{2}} \:+\mathrm{y}_{\mathrm{3}} ^{\mathrm{2}} }\:=\:?\: \\ $$

Question Number 97283    Answers: 0   Comments: 0

Question Number 97275    Answers: 1   Comments: 14

Trial version with additional colors is now available.

$$\mathrm{Trial}\:\mathrm{version}\:\mathrm{with}\:\mathrm{additional} \\ $$$$\mathrm{colors}\:\mathrm{is}\:\mathrm{now}\:\mathrm{available}. \\ $$

Question Number 97272    Answers: 0   Comments: 0

Question Number 97271    Answers: 1   Comments: 2

hello every one why do planets of the solar system revolve around the sun in an eliptical not circular orbit

$${hello}\:{every}\:{one} \\ $$$${why}\:{do}\:{planets}\:{of}\:{the}\:{solar}\:{system} \\ $$$${revolve}\:{around}\:{the}\:{sun}\:{in}\:{an}\:{eliptical} \\ $$$${not}\:{circular}\:{orbit} \\ $$

Question Number 97270    Answers: 1   Comments: 3

solve for all real value of x,y and z giving answer the form (x,y,z) { ((x(x+y)+z(x−y)= 4)),((y(y+z)+x(y−z) = −4)),((z(z+x)+y(z−x) = 5)) :}

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{real}}\:\:\boldsymbol{\mathrm{value}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{z}}\:\boldsymbol{\mathrm{giving}} \\ $$$$\boldsymbol{\mathrm{answer}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{form}}\:\left(\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}},\boldsymbol{\mathrm{z}}\right)\:\begin{cases}{\boldsymbol{\mathrm{x}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)+\boldsymbol{\mathrm{z}}\left(\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{y}}\right)=\:\mathrm{4}}\\{\boldsymbol{\mathrm{y}}\left(\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}\right)+\boldsymbol{\mathrm{x}}\left(\boldsymbol{\mathrm{y}}−\boldsymbol{\mathrm{z}}\right)\:=\:−\mathrm{4}}\\{\boldsymbol{\mathrm{z}}\left(\boldsymbol{\mathrm{z}}+\boldsymbol{\mathrm{x}}\right)+\boldsymbol{\mathrm{y}}\left(\boldsymbol{\mathrm{z}}−\boldsymbol{\mathrm{x}}\right)\:=\:\mathrm{5}}\end{cases} \\ $$

Question Number 97250    Answers: 1   Comments: 0

Question Number 97239    Answers: 0   Comments: 3

∫ ((sec^3 x dx)/(√(tan x))) ?

$$\int\:\frac{\mathrm{sec}\:^{\mathrm{3}} {x}\:{dx}}{\sqrt{\mathrm{tan}\:{x}}}\:?\: \\ $$

Question Number 97238    Answers: 0   Comments: 1

Question Number 97236    Answers: 3   Comments: 3

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