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Question Number 30704    Answers: 0   Comments: 0

Question Number 30665    Answers: 0   Comments: 0

find ∫_0 ^π (dx/(1+cos(2x) +sin(2x))) .

$${find}\:\:\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}\left(\mathrm{2}{x}\right)\:+{sin}\left(\mathrm{2}{x}\right)}\:. \\ $$

Question Number 30657    Answers: 0   Comments: 2

lim_(x→∞) e^(−(x^2 /2))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{e}^{−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}} \\ $$

Question Number 30646    Answers: 1   Comments: 2

Question Number 30631    Answers: 0   Comments: 0

Question Number 30628    Answers: 1   Comments: 0

Question Number 30627    Answers: 1   Comments: 0

Question Number 30687    Answers: 1   Comments: 0

Given that LCM(A,B,C)=252 LCM(A,B)=36 & LCM(A,C)=63; then: LCM(B,C)=? Pl determine all possible answers.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{LCM}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right)=\mathrm{252} \\ $$$$\mathrm{LCM}\left(\mathrm{A},\mathrm{B}\right)=\mathrm{36}\:\&\:\mathrm{LCM}\left(\mathrm{A},\mathrm{C}\right)=\mathrm{63}; \\ $$$$\mathrm{then}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{LCM}\left(\mathrm{B},\mathrm{C}\right)=? \\ $$$$\mathrm{Pl}\:\mathrm{determine}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{answers}. \\ $$

Question Number 30615    Answers: 1   Comments: 0

Question Number 30614    Answers: 1   Comments: 0

Consider that two cars are accelerating along the same road and if the distance between them was observed to be increasing,what deduction can you make as regards the acceleration? a)it implies that the trailing car has the smaller acceleration b)it implies that the two cars are accelerating at the same rate c)it implies nothing about the acceleration d)it implies that the leading car has the greater acceleration.

$${Consider}\:{that}\:{two}\:{cars}\:{are}\: \\ $$$${accelerating}\:{along}\:{the}\:{same}\:{road} \\ $$$${and}\:{if}\:{the}\:{distance}\:{between}\:{them} \\ $$$${was}\:{observed}\:{to}\:{be}\:{increasing},{what} \\ $$$${deduction}\:{can}\:{you}\:{make}\:{as}\:{regards} \\ $$$${the}\:{acceleration}? \\ $$$$\left.{a}\right){it}\:{implies}\:{that}\:{the}\:{trailing}\:{car} \\ $$$${has}\:{the}\:{smaller}\:{acceleration} \\ $$$$\left.{b}\right){it}\:{implies}\:{that}\:{the}\:{two}\:{cars}\:{are} \\ $$$${accelerating}\:{at}\:{the}\:{same}\:{rate} \\ $$$$\left.{c}\right){it}\:{implies}\:{nothing}\:{about}\:{the} \\ $$$${acceleration} \\ $$$$\left.{d}\right){it}\:{implies}\:{that}\:{the}\:{leading}\:{car}\: \\ $$$${has}\:{the}\:{greater}\:{acceleration}. \\ $$

Question Number 30613    Answers: 1   Comments: 1

A car negotiates a bend of radius 20m with an acceleration of 12m/s^2 .What is the maximum speed the car can attain without skidding?

$${A}\:{car}\:{negotiates}\:{a}\:{bend}\:{of}\:{radius} \\ $$$$\mathrm{20}{m}\:{with}\:{an}\:{acceleration}\:{of}\: \\ $$$$\mathrm{12}{m}/{s}^{\mathrm{2}} .{What}\:{is}\:{the}\:{maximum} \\ $$$${speed}\:{the}\:{car}\:{can}\:{attain}\:{without} \\ $$$${skidding}? \\ $$

Question Number 30612    Answers: 0   Comments: 0

A car negotiates a bend of radius 20m with an acceleration of 12m/s^2 .What is the maximum speed the car can attain without skidding?

$${A}\:{car}\:{negotiates}\:{a}\:{bend}\:{of}\:{radius} \\ $$$$\mathrm{20}{m}\:{with}\:{an}\:{acceleration}\:{of}\: \\ $$$$\mathrm{12}{m}/{s}^{\mathrm{2}} .{What}\:{is}\:{the}\:{maximum} \\ $$$${speed}\:{the}\:{car}\:{can}\:{attain}\:{without} \\ $$$${skidding}? \\ $$

Question Number 30601    Answers: 0   Comments: 1

for 0<r≤1 and (θ,x)∈R^2 find S=Σ_(n=0) ^∞ r^n cos(nθ).

$${for}\:\mathrm{0}<{r}\leqslant\mathrm{1}\:{and}\:\left(\theta,{x}\right)\in{R}^{\mathrm{2}} \:\:{find} \\ $$$${S}=\sum_{{n}=\mathrm{0}} ^{\infty} \:{r}^{{n}} {cos}\left({n}\theta\right). \\ $$

Question Number 30600    Answers: 0   Comments: 0

let w_k =e^(i((2kπ)/n)) find A= Π_(k=0) ^(n−1) (a +bw_k ).

$${let}\:{w}_{{k}} ={e}^{{i}\frac{\mathrm{2}{k}\pi}{{n}}} \:\:\:\:{find}\:{A}=\:\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({a}\:+{bw}_{{k}} \:\right). \\ $$

Question Number 30599    Answers: 0   Comments: 1

decompose inside C(x) F= (1/((x−1)(x^n −1))) .

$${decompose}\:{inside}\:{C}\left({x}\right)\:\:{F}=\:\frac{\mathrm{1}}{\left({x}−\mathrm{1}\right)\left({x}^{{n}} \:−\mathrm{1}\right)}\:. \\ $$

Question Number 30598    Answers: 0   Comments: 1

prove that it exist one polynomial p/ p(cosx)=cos(nx) find the roots of p(x) .

$${prove}\:{that}\:{it}\:{exist}\:{one}\:{polynomial}\:{p}/ \\ $$$${p}\left({cosx}\right)={cos}\left({nx}\right)\:{find}\:{the}\:{roots}\:{of}\:{p}\left({x}\right)\:. \\ $$

Question Number 30597    Answers: 0   Comments: 0

let p(x)=(1+x)^m −e^(2imx) (1−x)^m factorize p(x) inside C[x].

$${let}\:{p}\left({x}\right)=\left(\mathrm{1}+{x}\right)^{{m}} \:−{e}^{\mathrm{2}{imx}} \left(\mathrm{1}−{x}\right)^{{m}} \:{factorize}\:{p}\left({x}\right) \\ $$$${inside}\:{C}\left[{x}\right]. \\ $$

Question Number 30596    Answers: 0   Comments: 0

find all polynomial wich verify p(x^2 ) +p(x)p(x+1)=0.

$${find}\:{all}\:{polynomial}\:{wich}\:{verify}\: \\ $$$${p}\left({x}^{\mathrm{2}} \right)\:+{p}\left({x}\right){p}\left({x}+\mathrm{1}\right)=\mathrm{0}. \\ $$

Question Number 30595    Answers: 0   Comments: 1

let f(x)= (1/(x^2 −2cosαx+1)) find f^((n)) .

$${let}\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{{x}^{\mathrm{2}} \:−\mathrm{2}{cos}\alpha{x}+\mathrm{1}}\:\:{find}\:{f}^{\left({n}\right)} . \\ $$

Question Number 30594    Answers: 0   Comments: 0

let p(x)=x^3 +1 and q(x)=x^4 +1 prove that D(p,q)=1.

$${let}\:{p}\left({x}\right)={x}^{\mathrm{3}} \:+\mathrm{1}\:{and}\:{q}\left({x}\right)={x}^{\mathrm{4}} \:+\mathrm{1}\:{prove}\:{that} \\ $$$${D}\left({p},{q}\right)=\mathrm{1}. \\ $$

Question Number 30593    Answers: 1   Comments: 0

factorize inside C[x] p(x)=(1+i(x/n))^n −(1−i(x/n))^n .

$${factorize}\:{inside}\:{C}\left[{x}\right]\:{p}\left({x}\right)=\left(\mathrm{1}+{i}\frac{{x}}{{n}}\right)^{{n}} \:−\left(\mathrm{1}−{i}\frac{{x}}{{n}}\right)^{{n}} . \\ $$

Question Number 30592    Answers: 1   Comments: 0

let p(x)=x^(2n) −2cosα x^n +1 1) find roots lf p(x) 2)factorize p(x) inside C[x] 3)factorize p(x) inside R[x].

$${let}\:{p}\left({x}\right)={x}^{\mathrm{2}{n}} \:−\mathrm{2}{cos}\alpha\:{x}^{{n}} \:+\mathrm{1} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{roots}\:{lf}\:{p}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){factorize}\:{p}\left({x}\right)\:{inside}\:{C}\left[{x}\right] \\ $$$$\left.\mathrm{3}\right){factorize}\:{p}\left({x}\right)\:{inside}\:{R}\left[{x}\right]. \\ $$

Question Number 30590    Answers: 1   Comments: 0

decompose sur R[x] x^(2n+1) −1.

$${decompose}\:{sur}\:{R}\left[{x}\right]\:\:{x}^{\mathrm{2}{n}+\mathrm{1}} \:−\mathrm{1}. \\ $$

Question Number 30589    Answers: 0   Comments: 0

let U_n = {z∈C / z^n =1} find S= Σ_(z∈U_n ) (z/((x−z)^2 )) .

$${let}\:{U}_{{n}} =\:\left\{{z}\in{C}\:/\:{z}^{{n}} =\mathrm{1}\right\}\:\:{find} \\ $$$${S}=\:\sum_{{z}\in{U}_{{n}} } \:\:\frac{{z}}{\left({x}−{z}\right)^{\mathrm{2}} }\:\:. \\ $$

Question Number 30588    Answers: 0   Comments: 0

(n_k )_(1≤k≤n) is a family of integrs numbers let put p(x)=Σ_(k=1) ^n x^n_k and q(x)= Σ_(j=0) ^(n−1) x^j if n_k ≡k−1[n] prove that q divide p.

$$\left({n}_{{k}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} \:{is}\:{a}\:{family}\:{of}\:{integrs}\:{numbers}\:{let}\:{put} \\ $$$${p}\left({x}\right)=\sum_{{k}=\mathrm{1}} ^{{n}} \:{x}^{{n}_{{k}} } \:\:\:{and}\:{q}\left({x}\right)=\:\sum_{{j}=\mathrm{0}} ^{{n}−\mathrm{1}} \:{x}^{{j}} \: \\ $$$${if}\:{n}_{{k}} \equiv{k}−\mathrm{1}\left[{n}\right]\:{prove}\:{that}\:{q}\:{divide}\:{p}. \\ $$

Question Number 30587    Answers: 0   Comments: 0

find Σ_(k=0) ^(n−1) (−1)^k cos^n (((kπ)/n)).

$${find}\:\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \:\left(−\mathrm{1}\right)^{{k}} \:{cos}^{{n}} \left(\frac{{k}\pi}{{n}}\right). \\ $$

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