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Question Number 65227 Answers: 0 Comments: 3

Question Number 65219 Answers: 1 Comments: 1

Question Number 65217 Answers: 0 Comments: 1

Question Number 65214 Answers: 1 Comments: 3

Prove or disprove that 2^(101) ∣ n^n − 101 .

$${Prove}\:\:{or}\:\:{disprove}\:\:\:{that}\:\:\:\mathrm{2}^{\mathrm{101}} \:\mid\:{n}^{{n}} \:−\:\mathrm{101}\:. \\ $$

Question Number 65212 Answers: 1 Comments: 1

Question Number 65203 Answers: 1 Comments: 1

∫_0 ^∞ (x^2 /(x^4 +x^2 +1))dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 65202 Answers: 1 Comments: 0

∫_0 ^π ((cos 2θ)/(1−2acos θ+a^2 ))dθ, a^2 <1 answer?

$$\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{cos}\:\mathrm{2}\theta}{\mathrm{1}−\mathrm{2}{a}\mathrm{cos}\:\theta+{a}^{\mathrm{2}} }{d}\theta,\:{a}^{\mathrm{2}} <\mathrm{1} \\ $$$$\mathrm{answer}? \\ $$

Question Number 65200 Answers: 2 Comments: 0

Question Number 65199 Answers: 1 Comments: 0

Question Number 65198 Answers: 0 Comments: 5

let a∈R^+ , and x>0 x^4 +(1−2a)x^2 −2ax+1=0 find x

$${let}\:{a}\in\mathbb{R}^{+} \:,\:{and}\:{x}>\mathrm{0} \\ $$$${x}^{\mathrm{4}} +\left(\mathrm{1}−\mathrm{2}{a}\right){x}^{\mathrm{2}} −\mathrm{2}{ax}+\mathrm{1}=\mathrm{0} \\ $$$${find}\:{x} \\ $$

Question Number 65196 Answers: 0 Comments: 0

find ∫_0 ^1 ((1/(1−x)) +(1/(lnx)))dx

$$\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{1}−{x}}\:+\frac{\mathrm{1}}{{lnx}}\right){dx} \\ $$

Question Number 65195 Answers: 0 Comments: 0

find ∫_(π/4) ^(π/2) ln(ln(tanx)dx

$${find}\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left({ln}\left({tanx}\right){dx}\right. \\ $$

Question Number 65194 Answers: 0 Comments: 0

calculate ∫_0 ^1 ln(Γ(x))dx with Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt and x>0

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\Gamma\left({x}\right)\right){dx}\:{with}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:\:{and}\:{x}>\mathrm{0} \\ $$

Question Number 65193 Answers: 0 Comments: 1

U_n is a sequence wich verify U_n +U_(n+1) =n for all integr n 1) calculate U_n intrem of n 2) find nature of the serie Σ (U_n /n^2 )

$${U}_{{n}} \:{is}\:{a}\:{sequence}\:{wich}\:{verify}\:{U}_{{n}} +{U}_{{n}+\mathrm{1}} ={n}\:{for}\:{all}\:{integr}\:{n} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} \:{intrem}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:{the}\:{serie}\:\Sigma\:\frac{{U}_{{n}} }{{n}^{\mathrm{2}} } \\ $$

Question Number 65192 Answers: 1 Comments: 0

Minimum value of ∣ sin x + cos x + tan x + cot x + sec x + cosec x ∣ is ...

$${Minimum}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mid\:\mathrm{sin}\:{x}\:+\:\mathrm{cos}\:{x}\:+\:\mathrm{tan}\:{x}\:+\:\mathrm{cot}\:{x}\:+\:\mathrm{sec}\:{x}\:+\:\mathrm{cosec}\:{x}\:\mid\:\:\:{is}\:\:... \\ $$

Question Number 65189 Answers: 1 Comments: 0

all square is a rhombus. why?

$${all}\:{square}\:{is}\:{a}\:{rhombus}.\:{why}? \\ $$

Question Number 65170 Answers: 0 Comments: 0

three forces F_1 , F_2 and F_3 acts through the points with position vectors r_1 ,r_2 and r_3 respectively where F_1 =(3i −2j−4k)N, r_1 = (i +k)m F_2 =(−i+j)N, r_2 =(j+k)m F_3 =(−i+4k)N, r_3 =(i+j+k)m a. show that this system does not reduce to a single force. When a fourth force F is added, the system of forces is in equilibrium b. Show that F acts through the point with vectors (3k)m.

$${three}\:{forces}\:{F}_{\mathrm{1}} ,\:{F}_{\mathrm{2}} \:{and}\:{F}_{\mathrm{3}} \:{acts}\:{through}\:{the}\:{points}\:{with}\:{position}\:{vectors} \\ $$$$\boldsymbol{{r}}_{\mathrm{1}} ,{r}_{\mathrm{2}} \:{and}\:{r}_{\mathrm{3}} \:{respectively}\:{where} \\ $$$$\:{F}_{\mathrm{1}} \:=\left(\mathrm{3}{i}\:−\mathrm{2}{j}−\mathrm{4}{k}\right){N},\:\:\:{r}_{\mathrm{1}} =\:\left({i}\:+{k}\right){m} \\ $$$${F}_{\mathrm{2}} =\left(−{i}+{j}\right){N},\:\:{r}_{\mathrm{2}} =\left({j}+{k}\right){m} \\ $$$${F}_{\mathrm{3}} =\left(−{i}+\mathrm{4}{k}\right){N},\:{r}_{\mathrm{3}} =\left({i}+{j}+{k}\right){m} \\ $$$${a}.\:{show}\:{that}\:{this}\:{system}\:{does}\:{not}\:{reduce}\:{to}\:{a}\:{single}\:{force}. \\ $$$$\:{When}\:{a}\:{fourth}\:{force}\:{F}\:{is}\:{added},\:{the}\:{system}\:{of}\:{forces}\:{is}\:{in}\:{equilibrium} \\ $$$${b}.\:{Show}\:{that}\:{F}\:{acts}\:{through}\:\:{the}\:{point}\:{with}\:{vectors}\:\left(\mathrm{3}{k}\right){m}. \\ $$

Question Number 65168 Answers: 2 Comments: 0

z = 1− i(√3) express z in the form r(cosθ +isinθ) also express z^7 in the form re^(iθ) .

$${z}\:=\:\mathrm{1}−\:\mathrm{i}\sqrt{\mathrm{3}} \\ $$$${express}\:{z}\:{in}\:{the}\:{form}\:\:{r}\left({cos}\theta\:+\mathrm{i}{sin}\theta\right)\:{also}\:{express}\:{z}^{\mathrm{7}} \:{in}\:{the}\:{form} \\ $$$${re}^{\mathrm{i}\theta} . \\ $$

Question Number 65166 Answers: 2 Comments: 1

Given that f(x) = (2/(x^2 −1)) a) Express f(x) in partial fraction. b.Evaluate ∫_3 ^5 f (x) dx

$${Given}\:{that}\:\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.{a}\right)\:{Express}\:{f}\left({x}\right)\:{in}\:{partial}\:{fraction}. \\ $$$${b}.{Evaluate}\:\:\int_{\mathrm{3}} ^{\mathrm{5}} {f}\:\left({x}\right)\:{dx} \\ $$

Question Number 65162 Answers: 0 Comments: 0