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Question Number 37230    Answers: 0   Comments: 0

let A = (((1 −2)),((1 4)) ) calculate A^n 2) find e^A , e^(−A) 3) find e^(iA) , e^(−iA) and e^(iA) +e^(−iA) .

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{2}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}}\end{pmatrix} \\ $$$${calculate}\:\:{A}^{{n}} \: \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{e}^{{A}} \:\:,\:{e}^{−{A}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:\:{e}^{{iA}} ,\:{e}^{−{iA}} \:\:\:{and}\:{e}^{{iA}} \:+{e}^{−{iA}} \:\:. \\ $$

Question Number 37228    Answers: 0   Comments: 0

E id k vectorial space and f∈L(E) 1)prove that if f is nilpotent with indice p≥1 ,I −f is bijective and (I−f)^(−1) =Σ_(i=0) ^(p−1) f^i 2)let E=R_n [x] and f∈L(E) / f(p) =p−p^′ prove that f is inversible and find f^(−1) .

$${E}\:{id}\:{k}\:{vectorial}\:{space}\:{and}\:{f}\in{L}\left({E}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{if}\:{f}\:{is}\:{nilpotent}\:{with}\:{indice} \\ $$$${p}\geqslant\mathrm{1}\:,{I}\:−{f}\:{is}\:{bijective}\:{and} \\ $$$$\left({I}−{f}\right)^{−\mathrm{1}} =\sum_{{i}=\mathrm{0}} ^{{p}−\mathrm{1}} {f}^{{i}} \\ $$$$\left.\mathrm{2}\right){let}\:{E}={R}_{{n}} \left[{x}\right]\:{and}\:{f}\in{L}\left({E}\right)\:/ \\ $$$${f}\left({p}\right)\:={p}−{p}^{'} \:\:{prove}\:{that}\:{f}\:{is}\:{inversible} \\ $$$${and}\:{find}\:{f}^{−\mathrm{1}} \:. \\ $$

Question Number 37225    Answers: 0   Comments: 0

let n≥2 and f : R_n [x]→R_2 [x] / f(p) =xp(1) +(x^2 −4)p(0) 1) prove that f is linear 2) find dim Kerf and dimIm(f)

$${let}\:{n}\geqslant\mathrm{2}\:{and}\:{f}\:\::\:{R}_{{n}} \left[{x}\right]\rightarrow{R}_{\mathrm{2}} \left[{x}\right]\:/ \\ $$$${f}\left({p}\right)\:={xp}\left(\mathrm{1}\right)\:+\left({x}^{\mathrm{2}} \:−\mathrm{4}\right){p}\left(\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{f}\:{is}\:{linear} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{dim}\:{Kerf}\:{and}\:{dimIm}\left({f}\right) \\ $$

Question Number 37224    Answers: 0   Comments: 0

let p(x)=(1+jx)^n −(1−jx)^n with j=e^(i((2π)/3)) find p at form r(x)e^(iθ(x)) 2) calculate ∫_0 ^1 r(x) e^(iθ(x)) dx .

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{jx}\right)^{{n}} \:−\left(\mathrm{1}−{jx}\right)^{{n}} \:{with} \\ $$$${j}={e}^{{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\:\:{find}\:{p}\:{at}\:\:{form}\:{r}\left({x}\right){e}^{{i}\theta\left({x}\right)} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {r}\left({x}\right)\:{e}^{{i}\theta\left({x}\right)} {dx}\:. \\ $$

Question Number 37221    Answers: 0   Comments: 8

Hi! MrW is back! I′m sorry for having been absent for months without giving you a message. It′s alright with me. The last few months I was very very busy with our new house and I had no time for other things. I changed my smartphone and lost my old ID. Sorry and thanks to all those who have cared about me! Sir Rasheed, your post #35656 has moved me especially, thank you! I have viewed some new posts and I′m happy to see that a lot of old friends are still active in the forum. I′ll try to find time to read more new posts.

$${Hi}!\:{MrW}\:{is}\:{back}! \\ $$$${I}'{m}\:{sorry}\:{for}\:{having}\:{been}\:{absent}\:{for} \\ $$$${months}\:{without}\:{giving}\:{you}\:{a}\:{message}. \\ $$$${It}'{s}\:{alright}\:{with}\:{me}.\: \\ $$$${The}\:{last}\:{few}\:{months}\:{I}\:{was}\:{very}\:{very} \\ $$$${busy}\:{with}\:{our}\:{new}\:{house}\:{and}\:{I}\:{had}\:{no} \\ $$$${time}\:{for}\:{other}\:{things}.\:{I}\:{changed}\:{my} \\ $$$${smartphone}\:{and}\:{lost}\:{my}\:{old}\:{ID}. \\ $$$${Sorry}\:{and}\:{thanks}\:{to}\:{all}\:{those}\:{who}\:{have} \\ $$$${cared}\:{about}\:{me}!\: \\ $$$${Sir}\:{Rasheed},\:{your}\:{post}\:#\mathrm{35656}\:{has} \\ $$$${moved}\:{me}\:{especially},\:{thank}\:{you}! \\ $$$${I}\:{have}\:{viewed}\:{some}\:{new}\:{posts}\:{and} \\ $$$${I}'{m}\:{happy}\:{to}\:{see}\:{that}\:{a}\:{lot}\:{of}\:{old}\:{friends} \\ $$$${are}\:{still}\:{active}\:{in}\:{the}\:{forum}. \\ $$$${I}'{ll}\:{try}\:{to}\:{find}\:{time}\:{to}\:{read}\:{more}\:{new} \\ $$$${posts}. \\ $$

Question Number 37220    Answers: 1   Comments: 0

Question Number 37218    Answers: 1   Comments: 0

Proove that a) ((1−cosA + cos B − cos(A+B))/(1+cos A − cosB−cos(A+B)))= tan(A/2).cot (B/2) b) cosα cos(60−α)cos(60+α)= (1/4)cos3α

$$\:{Proove}\:{that} \\ $$$$\left.{a}\right)\:\:\frac{\mathrm{1}−{cosA}\:+\:{cos}\:{B}\:−\:{cos}\left({A}+{B}\right)}{\mathrm{1}+{cos}\:{A}\:−\:{cosB}−{cos}\left({A}+{B}\right)}=\:{tan}\frac{{A}}{\mathrm{2}}.{cot}\:\frac{{B}}{\mathrm{2}} \\ $$$$\left.{b}\right)\:{cos}\alpha\:{cos}\left(\mathrm{60}−\alpha\right){cos}\left(\mathrm{60}+\alpha\right)=\:\frac{\mathrm{1}}{\mathrm{4}}{cos}\mathrm{3}\alpha \\ $$$$ \\ $$

Question Number 37217    Answers: 0   Comments: 0

Question Number 37204    Answers: 0   Comments: 0

Question Number 37189    Answers: 1   Comments: 0

If ∫ ((cos 4x+1)/(cot x−tan x)) dx=A cos 4x+B, then find A,B?

$$\mathrm{If}\:\int\:\:\frac{\mathrm{cos}\:\mathrm{4}{x}+\mathrm{1}}{\mathrm{cot}\:{x}−\mathrm{tan}\:{x}}\:{dx}={A}\:\mathrm{cos}\:\mathrm{4}{x}+{B},\:\mathrm{then}\: \\ $$$$\mathrm{find}\:\mathrm{A},\mathrm{B}? \\ $$

Question Number 37188    Answers: 2   Comments: 0

∫ (√((cos x−cos^3 x)/(1−cos^3 x))) dx =

$$\int\:\sqrt{\frac{\mathrm{cos}\:{x}−\mathrm{cos}^{\mathrm{3}} {x}}{\mathrm{1}−\mathrm{cos}^{\mathrm{3}} {x}}}\:{dx}\:= \\ $$

Question Number 37209    Answers: 1   Comments: 3

Question Number 37179    Answers: 0   Comments: 4

∫ ((3x^2 +1)/((x^2 −1)^3 )) dx = ?

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

Question Number 37178    Answers: 0   Comments: 0

suppose T is a matrix of enlargement by factor k find the matrix S such that ST[(x,y)]=(x,y).

$${suppose}\:{T}\:{is}\:{a}\:{matrix}\:{of}\:{enlargement} \\ $$$$ \\ $$$${by}\:{factor}\:{k}\:{find}\:{the}\:{matrix}\:{S}\:{such}\:{that} \\ $$$${ST}\left[\left({x},{y}\right)\right]=\left({x},{y}\right). \\ $$

Question Number 37171    Answers: 0   Comments: 0

Question Number 37168    Answers: 0   Comments: 0

A stone is thrown into a circular pond of radius 1m.Suppose the stone falls uniformly at random on the area of the pond.What will be the expected distance od the stone from the centre of the pond. a)1/3 b)1/2 c)2/3 d)1/(√2)

$${A}\:{stone}\:{is}\:{thrown}\:{into}\:{a}\:{circular} \\ $$$${pond}\:{of}\:{radius}\:\mathrm{1}{m}.{Suppose}\:{the} \\ $$$${stone}\:{falls}\:{uniformly}\:{at}\:{random} \\ $$$${on}\:{the}\:{area}\:{of}\:{the}\:{pond}.{What} \\ $$$${will}\:{be}\:{the}\:{expected}\:{distance}\:{od} \\ $$$${the}\:{stone}\:{from}\:{the}\:{centre}\:{of}\:{the} \\ $$$${pond}. \\ $$$$ \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left./\mathrm{3}\:{b}\right)\mathrm{1}/\mathrm{2}\:{c}\right)\mathrm{2}/\mathrm{3}\:{d}\right)\mathrm{1}/\sqrt{\mathrm{2}} \\ $$

Question Number 37166    Answers: 0   Comments: 2

if 3x^2 +2αxy+2y^2 +2ax−4y+1 can be resolved into two linear factors, prove that ′α′ is a root of the equation x^2 +4ax+2a^2 +6=0

$${if}\:\:\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}\alpha{xy}+\mathrm{2}{y}^{\mathrm{2}} +\mathrm{2}{ax}−\mathrm{4}{y}+\mathrm{1} \\ $$$${can}\:{be}\:{resolved}\:\:{into}\:\:{two}\:\:{linear} \\ $$$${factors},\:\:{prove}\:\:{that}\:\:'\alpha'\:\:{is}\:{a}\:{root}\: \\ $$$${of}\:{the}\:{equation}\:{x}^{\mathrm{2}} +\mathrm{4}{ax}+\mathrm{2}{a}^{\mathrm{2}} +\mathrm{6}=\mathrm{0} \\ $$

Question Number 37155    Answers: 0   Comments: 4

3cos x−4sin x=tan^2 x locate x.

$$\mathrm{3cos}\:{x}−\mathrm{4sin}\:{x}=\mathrm{tan}\:^{\mathrm{2}} {x} \\ $$$${locate}\:{x}. \\ $$

Question Number 37146    Answers: 0   Comments: 0

Question Number 37143    Answers: 0   Comments: 0

Why are following statements wrong? a) There exists a function with domain R satisfying f(x)<0 ∀x , f′(x)>0∀x and f′′(x)>0∀x. b) If f′′(c)=0 then (c,f(c)) is an inflection point.

$$\mathrm{Why}\:\mathrm{are}\:\mathrm{following}\:\mathrm{statements}\:\mathrm{wrong}? \\ $$$$\left.\mathrm{a}\right)\:\mathrm{There}\:\mathrm{exists}\:\mathrm{a}\:\mathrm{function}\:\mathrm{with}\:\mathrm{domain}\: \\ $$$$\mathrm{R}\:\mathrm{satisfying}\:\mathrm{f}\left(\mathrm{x}\right)<\mathrm{0}\:\forall\mathrm{x}\:,\:\mathrm{f}'\left(\mathrm{x}\right)>\mathrm{0}\forall\mathrm{x}\:\mathrm{and} \\ $$$$\mathrm{f}''\left(\mathrm{x}\right)>\mathrm{0}\forall\mathrm{x}. \\ $$$$ \\ $$$$\left.\mathrm{b}\right)\:\mathrm{If}\:\mathrm{f}''\left(\mathrm{c}\right)=\mathrm{0}\:\mathrm{then}\:\left(\mathrm{c},\mathrm{f}\left(\mathrm{c}\right)\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{inflection} \\ $$$$\mathrm{point}. \\ $$

Question Number 37133    Answers: 0   Comments: 0

Question Number 37132    Answers: 0   Comments: 0

Question Number 37131    Answers: 0   Comments: 2

Question Number 37136    Answers: 0   Comments: 1

Question Number 37139    Answers: 2   Comments: 0

if α , β are the roots of the quadratic equation ax^2 +bx+c =0 then find the quadratic equation whose roots are α^(2 ) , β^2

$${if}\:\alpha\:,\:\beta\:\:{are}\:{the}\:{roots}\:{of}\:{the}\:{quadratic} \\ $$$${equation}\:{ax}^{\mathrm{2}} +{bx}+{c}\:=\mathrm{0}\:{then}\:\:{find} \\ $$$${the}\:{quadratic}\:{equation}\:{whose}\:{roots} \\ $$$${are}\:\:\alpha^{\mathrm{2}\:\:\:} ,\:\beta^{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$

Question Number 37137    Answers: 2   Comments: 4

Find minimum distance between y^2 =8x and x^2 +(y+6)^2 =1.

$$\mathrm{Find}\:\mathrm{minimum}\:\mathrm{distance}\:\mathrm{between} \\ $$$$\mathrm{y}^{\mathrm{2}} =\mathrm{8}{x}\:{and}\:{x}^{\mathrm{2}} +\left({y}+\mathrm{6}\right)^{\mathrm{2}} =\mathrm{1}. \\ $$

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