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Question Number 75099    Answers: 1   Comments: 4

Given the matrix A = ((1,(−1),1),(0,2,( −1)),(2,3,0) ) and B= ((3,3,(−1)),((−2),(−2),1),((−4),(−5),2) ) find the matrix product AB and BA state the relationship between A and B find also the matrix product BM, where M= ((8),((−7)),(1) ) Hence solve the system of equations: x−y + z = 8, 2y −z =−7, 2x + 3y = 1.

$${Given}\:{the}\:{matrix}\: \\ $$$${A}\:=\:\begin{pmatrix}{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{2}}&{\:−\mathrm{1}}\\{\mathrm{2}}&{\mathrm{3}}&{\mathrm{0}}\end{pmatrix}\:\:{and}\:{B}=\:\begin{pmatrix}{\mathrm{3}}&{\mathrm{3}}&{−\mathrm{1}}\\{−\mathrm{2}}&{−\mathrm{2}}&{\mathrm{1}}\\{−\mathrm{4}}&{−\mathrm{5}}&{\mathrm{2}}\end{pmatrix} \\ $$$${find}\:{the}\:{matrix}\:{product}\:{AB}\:{and}\:{BA} \\ $$$${state}\:{the}\:{relationship}\:{between}\:{A}\:{and}\:{B} \\ $$$${find}\:{also}\:{the}\:{matrix}\:{product}\:{BM},\:{where}\:{M}=\begin{pmatrix}{\mathrm{8}}\\{−\mathrm{7}}\\{\mathrm{1}}\end{pmatrix} \\ $$$${Hence}\:{solve}\:{the}\:{system}\:{of}\:{equations}: \\ $$$$\:\:{x}−{y}\:+\:{z}\:=\:\mathrm{8}, \\ $$$$\:\:\:\:\:\:\:\mathrm{2}{y}\:−{z}\:=−\mathrm{7}, \\ $$$$\:\:\mathrm{2}{x}\:+\:\mathrm{3}{y}\:=\:\mathrm{1}. \\ $$

Question Number 75098    Answers: 1   Comments: 0

the function f is defined by f(x) = (2/(x^2 −1)) a) Express f into partial fraction b.show that ∫_3 ^5 f(x) dx = ln((4/3))

$$\mathrm{the}\:\mathrm{function}\:\mathrm{f}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.{a}\right)\:\mathrm{Express}\:\mathrm{f}\:\mathrm{into}\:\mathrm{partial}\:\mathrm{fraction} \\ $$$$\mathrm{b}.\mathrm{show}\:\mathrm{that}\:\int_{\mathrm{3}} ^{\mathrm{5}} {f}\left({x}\right)\:{dx}\:=\:{ln}\left(\frac{\mathrm{4}}{\mathrm{3}}\right) \\ $$

Question Number 75093    Answers: 1   Comments: 1

∫cos^3 xsin^3 xdx

$$\int{cos}^{\mathrm{3}} {xsin}^{\mathrm{3}} {xdx} \\ $$

Question Number 75083    Answers: 1   Comments: 0

A function f is given by f(x) = { ((x^2 −3, 0≤x<2)),((4x−7, 2≤x<4)) :} is such that f(x) = f(x + 4) find f(27) and f(−106).

$${A}\:{function}\:{f}\:{is}\:{given}\:{by}\: \\ $$$$\:{f}\left({x}\right)\:=\:\begin{cases}{{x}^{\mathrm{2}} −\mathrm{3},\:\:\mathrm{0}\leqslant{x}<\mathrm{2}}\\{\mathrm{4}{x}−\mathrm{7},\:\mathrm{2}\leqslant{x}<\mathrm{4}}\end{cases} \\ $$$${is}\:{such}\:{that}\:{f}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{4}\right)\: \\ $$$${find}\:\:{f}\left(\mathrm{27}\right)\:{and}\:{f}\left(−\mathrm{106}\right). \\ $$

Question Number 75082    Answers: 1   Comments: 0

find ∫_0 ^π e^(cosx) sinx dx

$${find} \\ $$$$\int_{\mathrm{0}} ^{\pi} {e}^{{cosx}} {sinx}\:{dx} \\ $$

Question Number 75081    Answers: 1   Comments: 0

Evaluate ∫_1 ^(3 ) (x^2 /(1+x)) dx

$${Evaluate}\: \\ $$$$\:\int_{\mathrm{1}} ^{\mathrm{3}\:} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}}\:{dx} \\ $$

Question Number 75080    Answers: 0   Comments: 1

Find out A=Σ_(n=0) ^∞ ∫_0 ^(π/2) (1−(√(sinx)))^n cosxdx

$$\mathrm{Find}\:\mathrm{out}\: \\ $$$$\mathrm{A}=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\sqrt{\mathrm{sinx}}\right)^{\mathrm{n}} \mathrm{cosxdx} \\ $$

Question Number 75079    Answers: 0   Comments: 0

Let f∈C([0,1],[0,1]) Prove that lim_(n→∞) ∫_([0,1]^n ) f((1/n)Σ_(i=1) ^n x_i )dx_1 ....dx_n =f((1/2))

$$\mathrm{Let}\:\mathrm{f}\in\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\left[\mathrm{0},\mathrm{1}\right]\right)\:\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{n}} } \mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{x}_{\mathrm{i}} \:\right)\mathrm{dx}_{\mathrm{1}} ....\mathrm{dx}_{\mathrm{n}} \:=\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 75078    Answers: 1   Comments: 0

Question Number 75066    Answers: 1   Comments: 0

Prove that if f is a function R→R and there exist x_0 >0 , such as L(f)(x_0 ) exist then lim_(t→∞) f(t)e^(−x_0 t) =0 and ∀ x>x_0 L(f)(x) exist. L(f) is the Laplace transformed function

$$\mathrm{Prove}\:\:\mathrm{that}\:\mathrm{if}\:\:\mathrm{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathbb{R}\rightarrow\mathbb{R}\: \\ $$$$\mathrm{and}\:\:\mathrm{there}\:\mathrm{exist}\:\mathrm{x}_{\mathrm{0}} >\mathrm{0}\:\:,\:\mathrm{such}\:\mathrm{as}\:\:\mathrm{L}\left(\mathrm{f}\right)\left(\mathrm{x}_{\mathrm{0}} \right)\:\mathrm{exist}\: \\ $$$$\mathrm{then}\:\underset{\mathrm{t}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{x}_{\mathrm{0}} \mathrm{t}} =\mathrm{0}\:\mathrm{and}\:\forall\:\mathrm{x}>\mathrm{x}_{\mathrm{0}} \:\:\mathrm{L}\left(\mathrm{f}\right)\left(\mathrm{x}\right)\:\mathrm{exist}. \\ $$$$\mathrm{L}\left(\mathrm{f}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{Laplace}\:\mathrm{transformed}\:\mathrm{function} \\ $$

Question Number 75063    Answers: 1   Comments: 1

calculate Σ_(n=1) ^(17) (((−1)^n )/n^3 )

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\mathrm{17}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} } \\ $$

Question Number 75059    Answers: 0   Comments: 0

Question Number 75058    Answers: 1   Comments: 2

in triangle: ABC: a=(√(2 )),b−c=(((√2)+1)/2),B^ −C^ =(𝛑/2) find: h_a , S_(ABC ) ,d_(a ) , R ,A^ .

$$\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{triangle}}:\:\:\boldsymbol{\mathrm{ABC}}: \\ $$$$\boldsymbol{\mathrm{a}}=\sqrt{\mathrm{2}\:},\boldsymbol{\mathrm{b}}−\boldsymbol{\mathrm{c}}=\frac{\sqrt{\mathrm{2}}+\mathrm{1}}{\mathrm{2}},\overset{} {\boldsymbol{\mathrm{B}}}−\overset{} {\boldsymbol{\mathrm{C}}}=\frac{\boldsymbol{\pi}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{find}}:\:\:\boldsymbol{\mathrm{h}}_{\boldsymbol{\mathrm{a}}} ,\:\:\boldsymbol{\mathrm{S}}_{\boldsymbol{\mathrm{ABC}}\:\:} ,\boldsymbol{\mathrm{d}}_{\boldsymbol{\mathrm{a}}\:\:\:} ,\:\boldsymbol{\mathrm{R}}\:\:\:\:,\overset{} {\boldsymbol{\mathrm{A}}}. \\ $$

Question Number 75048    Answers: 1   Comments: 1

I would like that you help me to show this equality: 16cos (Π/(24))cos((5Π)/(24))cos((7Π)/(24))cos((11Π)/(24))=1

$$\mathrm{I}\:\mathrm{would}\:\mathrm{like}\:\mathrm{that}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\: \\ $$$$\mathrm{show}\:\mathrm{this}\:\mathrm{equality}: \\ $$$$\mathrm{16cos}\:\frac{\Pi}{\mathrm{24}}\mathrm{cos}\frac{\mathrm{5}\Pi}{\mathrm{24}}\mathrm{cos}\frac{\mathrm{7}\Pi}{\mathrm{24}}\mathrm{cos}\frac{\mathrm{11}\Pi}{\mathrm{24}}=\mathrm{1} \\ $$

Question Number 75046    Answers: 0   Comments: 3

Question Number 75041    Answers: 1   Comments: 1

1) Show that for a∈]01]the function f_a :R_+ →R defined by f_a (x)=x^a is a−holder function in other way there exist K>0 such as ∀ x,y>0 ∣f_a (x)−f_a (y)∣≤K∣x−y∣^a

$$\left.\mathrm{1}\left.\right)\left.\:\mathrm{Show}\:\mathrm{that}\:\:\mathrm{for}\:\mathrm{a}\in\right]\mathrm{01}\right]\mathrm{the}\:\mathrm{function}\:\:\mathrm{f}_{\mathrm{a}} \::\mathbb{R}_{+} \rightarrow\mathbb{R}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{f}_{\mathrm{a}} \left(\mathrm{x}\right)=\mathrm{x}^{\mathrm{a}} \: \\ $$$$\mathrm{is}\:\:\:\mathrm{a}−\mathrm{holder}\:\mathrm{function}\:\:\mathrm{in}\:\mathrm{other}\:\mathrm{way}\:\:\mathrm{there}\:\mathrm{exist}\:\:\mathrm{K}>\mathrm{0}\:\mathrm{such}\:\mathrm{as}\:\forall\:\mathrm{x},\mathrm{y}>\mathrm{0}\: \\ $$$$\mid\mathrm{f}_{\mathrm{a}} \left(\mathrm{x}\right)−\mathrm{f}_{\mathrm{a}} \left(\mathrm{y}\right)\mid\leqslant\mathrm{K}\mid\mathrm{x}−\mathrm{y}\mid^{\mathrm{a}} \:\: \\ $$$$ \\ $$

Question Number 75040    Answers: 1   Comments: 0

Please can you help me to to show that: cos ((47Π)/(13))=sin ((23Π)/(26))=sin((3Π)/(26))

$$\mathrm{Please}\:\mathrm{can}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\: \\ $$$$\mathrm{to}\:\mathrm{show}\:\mathrm{that}: \\ $$$$\mathrm{cos}\:\frac{\mathrm{47}\Pi}{\mathrm{13}}=\mathrm{sin}\:\frac{\mathrm{23}\Pi}{\mathrm{26}}=\mathrm{sin}\frac{\mathrm{3}\Pi}{\mathrm{26}} \\ $$

Question Number 75034    Answers: 3   Comments: 0

Question Number 75033    Answers: 1   Comments: 0

(4/(11)) < (x/y) < (3/8) x, y ∈ Z^+ min {x+y} = ?

$$\frac{\mathrm{4}}{\mathrm{11}}\:<\:\frac{{x}}{{y}}\:<\:\frac{\mathrm{3}}{\mathrm{8}} \\ $$$${x},\:{y}\:\:\in\:\:\mathbb{Z}^{+} \\ $$$${min}\:\left\{{x}+{y}\right\}\:\:=\:\:? \\ $$

Question Number 75027    Answers: 1   Comments: 1

Question Number 75013    Answers: 1   Comments: 0

The largest interval for which x^(12) −x^9 +x^4 −x+1>0 is (a)−4<x≤0 (b)0<x<1 (c)−100<x<100 (d)−∞<x<∞

$${The}\:{largest}\:{interval}\:{for}\:{which} \\ $$$${x}^{\mathrm{12}} −{x}^{\mathrm{9}} +{x}^{\mathrm{4}} −{x}+\mathrm{1}>\mathrm{0}\:{is} \\ $$$$\left({a}\right)−\mathrm{4}<{x}\leqslant\mathrm{0} \\ $$$$\left({b}\right)\mathrm{0}<{x}<\mathrm{1} \\ $$$$\left({c}\right)−\mathrm{100}<{x}<\mathrm{100} \\ $$$$\left({d}\right)−\infty<{x}<\infty \\ $$

Question Number 75001    Answers: 0   Comments: 2

A block of mass 0.2kg rests on an incline plane of 30° to the horizontal with a velocity of 12m/s.If the coefficient of sliding friction is 0.16, (i)determine how far up the plane the mass travels before stoping. (ii)if the block returns,what is the velocity of the block at the bottom of the plane. (g=9.8m/s)

$${A}\:{block}\:{of}\:{mass}\:\mathrm{0}.\mathrm{2}{kg}\:{rests}\:{on}\:{an}\:{incline} \\ $$$${plane}\:{of}\:\mathrm{30}°\:{to}\:{the}\:{horizontal}\:{with}\:{a} \\ $$$${velocity}\:{of}\:\mathrm{12}{m}/{s}.{If}\:{the}\:{coefficient}\:{of} \\ $$$${sliding}\:{friction}\:{is}\:\mathrm{0}.\mathrm{16}, \\ $$$$\left({i}\right){determine}\:{how}\:{far}\:{up}\:{the}\:{plane}\:{the} \\ $$$${mass}\:{travels}\:{before}\:{stoping}. \\ $$$$\left({ii}\right){if}\:{the}\:{block}\:{returns},{what}\:{is}\:{the} \\ $$$${velocity}\:{of}\:{the}\:{block}\:{at}\:{the}\:{bottom}\:{of}\:{the} \\ $$$${plane}. \\ $$$$ \\ $$$$\left({g}=\mathrm{9}.\mathrm{8}{m}/{s}\right) \\ $$

Question Number 74997    Answers: 1   Comments: 1

Question Number 74995    Answers: 1   Comments: 1

find ∫_0 ^(π/2) Log cosx dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{Log}\:\mathrm{cos}{x}\:{dx} \\ $$

Question Number 74994    Answers: 0   Comments: 6

Question Number 74980    Answers: 1   Comments: 0

Prove that for n∈N^∗ Σ_(p=0) ^(n−1) [x+(p/n)]=[nx]

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\:\underset{\mathrm{p}=\mathrm{0}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\left[\mathrm{x}+\frac{\mathrm{p}}{\mathrm{n}}\right]=\left[\mathrm{nx}\right] \\ $$

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