Question and Answers Forum

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) .

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

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