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AllQuestion and Answers: Page 1707

Question Number 37249 Answers: 0 Comments: 0

a triangle with vertices A(2,1),B(6,1) and (3,3) is transformed by (((2 0)),((0 2)) ) a)find the image A′B′C′ after this transformstion b) state the type of transformation

$${a}\:{triangle}\:{with}\:{vertices}\: \\ $$$$\:{A}\left(\mathrm{2},\mathrm{1}\right),{B}\left(\mathrm{6},\mathrm{1}\right)\:{and}\:\left(\mathrm{3},\mathrm{3}\right)\:{is}\:{transformed} \\ $$$${by}\:\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{0}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\end{pmatrix}\: \\ $$$$\left.{a}\right){find}\:{the}\:{image}\:{A}'{B}'{C}'\:{after}\: \\ $$$${this}\:{transformstion} \\ $$$$\left.{b}\right)\:{state}\:{the}\:{type}\:{of}\:{transformation} \\ $$

Question Number 37244 Answers: 1 Comments: 0

sin^2 (π/11)+sin^2 (2π/11)+...+sin^2 (5π/11)=?

$$\mathrm{sin}^{\mathrm{2}} \left(\pi/\mathrm{11}\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}\pi/\mathrm{11}\right)+...+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{5}\pi/\mathrm{11}\right)=? \\ $$

Question Number 37243 Answers: 0 Comments: 0

find f (t) =∫_0 ^∞ e^x ln(1+e^(−tx) )dx with t >0 . 2) let u_n = ∫_0 ^∞ e^x ln(1+e^(−nx) ) dx find lim_(n→+∞) u_n .

$${find}\:{f}\:\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{e}^{{x}} {ln}\left(\mathrm{1}+{e}^{−{tx}} \right){dx}\:{with}\:{t}\:>\mathrm{0}\:. \\ $$$$\left.\mathrm{2}\right)\:{let}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{{x}} {ln}\left(\mathrm{1}+{e}^{−{nx}} \right)\:{dx} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {u}_{{n}} \:. \\ $$

Question Number 37237 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) ((cosθ.sinθ)/(cosθ +sinθ)) dθ .

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{cos}\theta.{sin}\theta}{{cos}\theta\:+{sin}\theta}\:{d}\theta\:. \\ $$

Question Number 37236 Answers: 0 Comments: 0

calculate ∫_0 ^(2π) ∣sin(((kt)/2))∣ dt with k integr and k≥3

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \mid{sin}\left(\frac{{kt}}{\mathrm{2}}\right)\mid\:{dt}\:\:{with}\:{k}\:{integr} \\ $$$${and}\:{k}\geqslant\mathrm{3} \\ $$

Question Number 37235 Answers: 1 Comments: 0

calculate ∫_0 ^(π/2) (√(4sin^2 t +cos^2 (t))) dt

$${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}\:+{cos}^{\mathrm{2}} \left({t}\right)}\:\:{dt} \\ $$

Question Number 37233 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) (((1+ix)/(1−ix)))^(m−n) (dx/(1+x^2 )) dx with m and n integrs

$${calculate} \\ $$$$\int_{−\infty} ^{+\infty} \:\:\left(\frac{\mathrm{1}+{ix}}{\mathrm{1}−{ix}}\right)^{{m}−{n}} \:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{with}\:{m}\:{and}\:{n} \\ $$$${integrs}\: \\ $$

Question Number 37232 Answers: 0 Comments: 0

let A = (((1 −1 0)),((−1 1 1)) ) (0 0 3 ) calculate A^n .

$${let}\:{A}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}}\\{−\mathrm{1}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{3}\:\right) \\ $$$${calculate}\:{A}^{{n}} \:. \\ $$

Question Number 37231 Answers: 0 Comments: 0

let A = (((0 1 1)),((1 0 1)) ) (1 1 0 ) 1) calculate p_c (A) the caracteristic polunom of A 2) calculate A^n with n integr natural 3) calcypulate e^(tA) t∈ R

$${let}\:{A}\:=\:\begin{pmatrix}{\mathrm{0}\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{1}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{0}\:\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{p}_{{c}} \left({A}\right)\:{the}\:{caracteristic}\: \\ $$$${polunom}\:{of}\:{A} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{A}^{{n}} \:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{3}\right)\:{calcypulate}\:{e}^{{tA}} \:\:\:\:\:{t}\in\:{R}\:\: \\ $$

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

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

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