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Question Number 50369 Answers: 1 Comments: 0

find x ,y from Z wich verify y^2 =x(x+1)(x+7)(x+8)

$${find}\:{x}\:,{y}\:{from}\:{Z}\:\:{wich}\:{verify} \\ $$$${y}^{\mathrm{2}} ={x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{7}\right)\left({x}+\mathrm{8}\right) \\ $$

Question Number 50368 Answers: 0 Comments: 1

find all (x,y)∈Q^(+★^2 ) and x^y =y^x and x<y

$${find}\:{all}\:\left({x},{y}\right)\in{Q}^{+\bigstar^{\mathrm{2}} } \:\:{and}\:\:{x}^{{y}} ={y}^{{x}} \:\:{and}\:{x}<{y} \\ $$

Question Number 50367 Answers: 0 Comments: 0

calculate Σ_(k=p) ^(2p) (C_k ^p /2^k )

$${calculate}\:\sum_{{k}={p}} ^{\mathrm{2}{p}} \:\:\frac{{C}_{{k}} ^{{p}} }{\mathrm{2}^{{k}} } \\ $$

Question Number 50366 Answers: 0 Comments: 0

let A = (((0 m m^2 )),(((1/m) 0 m)) ) ((1/m^2 ) (1/m) 0 ) A ∈ M_3 (R) and m not 0 1) find relation betwen I_3 , A and A^2 2) is A inversible .determine A^(−1) in case of exist 3) find the propers values of A.

$${let}\:\:{A}\:=\begin{pmatrix}{\mathrm{0}\:\:\:\:\:{m}\:\:\:\:\:\:{m}^{\mathrm{2}} }\\{\frac{\mathrm{1}}{{m}}\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:{m}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{\mathrm{1}}{{m}^{\mathrm{2}} }\:\:\:\:\frac{\mathrm{1}}{{m}}\:\:\:\:\:\:\mathrm{0}\:\:\:\right) \\ $$$${A}\:\in\:{M}_{\mathrm{3}} \left({R}\right)\:\:{and}\:{m}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{relation}\:{betwen}\:{I}_{\mathrm{3}} ,\:{A}\:{and}\:{A}^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{is}\:{A}\:{inversible}\:\:\:.{determine}\:{A}^{−\mathrm{1}} \:{in}\:{case}\:{of}\:{exist} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{propers}\:{values}\:{of}\:{A}. \\ $$

Question Number 50365 Answers: 0 Comments: 0

let J = [((1 1 1)),((1 1 1)) ] [((1 1 1)),() ] element of M_3 (R) find J^n

$${let}\:{J}\:=\begin{bmatrix}{\mathrm{1}\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{1}}\\{}\end{bmatrix} \\ $$$${element}\:{of}\:{M}_{\mathrm{3}} \left({R}\right) \\ $$$${find}\:{J}^{{n}} \\ $$$$ \\ $$

Question Number 50364 Answers: 0 Comments: 0

let A = (((2 1)),((1 2)) ) calculate A^n

$${let}\:{A}\:=\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\mathrm{1}}\\{\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{2}}\end{pmatrix} \\ $$$${calculate}\:{A}^{{n}} \\ $$

Question Number 50363 Answers: 0 Comments: 0

let p ∈K_n [x] snd A and H two rlements of K[x] 1) prove that p(A(x)+H(x))=Σ_(k=0) ^n ((p^((k)) (A(x)))/(k!)).(H(x))^k 2)find the condition that p(A(x)+H(x))is divided by H(x)≠0 3) if p(x)≠c prove that p(p(x))−x is divided by p(x)−x.

$${let}\:{p}\:\in{K}_{{n}} \left[{x}\right]\:{snd}\:{A}\:{and}\:{H}\:{two}\:{rlements}\:{of}\:{K}\left[{x}\right] \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{p}\left({A}\left({x}\right)+{H}\left({x}\right)\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \frac{{p}^{\left({k}\right)} \left({A}\left({x}\right)\right)}{{k}!}.\left({H}\left({x}\right)\right)^{{k}} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{condition}\:{that}\:{p}\left({A}\left({x}\right)+{H}\left({x}\right)\right){is} \\ $$$${divided}\:{by}\:{H}\left({x}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{if}\:{p}\left({x}\right)\neq{c}\:{prove}\:\:{that}\:{p}\left({p}\left({x}\right)\right)−{x}\:{is}\:{divided} \\ $$$${by}\:{p}\left({x}\right)−{x}. \\ $$

Question Number 50362 Answers: 0 Comments: 0

calculate S_1 =Σ_(k=0) ^n C_n ^k S_2 =Σ_(k=0) ^([(n/2)]) C_n ^(2k) S_3 = Σ_(k=0) ^([(n/3)]) C_n ^(3k)

$${calculate}\:{S}_{\mathrm{1}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \\ $$$${S}_{\mathrm{2}} =\sum_{{k}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:{C}_{{n}} ^{\mathrm{2}{k}} \\ $$$${S}_{\mathrm{3}} =\:\sum_{{k}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{3}}\right]} \:{C}_{{n}} ^{\mathrm{3}{k}} \\ $$

Question Number 50360 Answers: 0 Comments: 0

find radius of convergence for the serie Σ_(n=0) ^∞ {(1+i)^n −(1−i)^n }x^n

$${find}\:{radius}\:{of}\:{convergence}\:{for}\:{the}\:{serie} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \left\{\left(\mathrm{1}+{i}\right)^{{n}} −\left(\mathrm{1}−{i}\right)^{{n}} \right\}{x}^{{n}} \\ $$

Question Number 50359 Answers: 0 Comments: 0

1) calculate Σ_(n=0) ^∞ (−1)^k C_n ^k (1/(k+1)) 2)calculate S_n (p)=Σ_(k=0) ^n (−1)^k (C_n ^k /(p+k+1)) p integr natural.

$$\left.\mathrm{1}\right)\:{calculate}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:\frac{\mathrm{1}}{{k}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{S}_{{n}} \left({p}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:\:\:\frac{{C}_{{n}} ^{{k}} }{{p}+{k}+\mathrm{1}} \\ $$$${p}\:{integr}\:{natural}. \\ $$

Question Number 50357 Answers: 1 Comments: 1

Question Number 50356 Answers: 0 Comments: 0

devompose inside C[x] and R[x] the polynom 1)x^4 +1 2)x^6 −1 3)x^8 +x^4 +1

$${devompose}\:{inside}\:{C}\left[{x}\right]\:{and}\:{R}\left[{x}\right]\:{the}\:{polynom} \\ $$$$\left.\mathrm{1}\right){x}^{\mathrm{4}} +\mathrm{1}\:\:\:\: \\ $$$$\left.\mathrm{2}\right){x}^{\mathrm{6}} −\mathrm{1} \\ $$$$\left.\mathrm{3}\right){x}^{\mathrm{8}} \:+{x}^{\mathrm{4}} \:+\mathrm{1} \\ $$

Question Number 50355 Answers: 0 Comments: 0

let (α_k ) (k∈[[0,n−1]] the n^(eme) roots of 1 calculate p(x,y)=(x+α_0 y)(x+α_1 y)....(x+α_(n−1) y)

$${let}\:\left(\alpha_{{k}} \right)\:\:\left({k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]\:{the}\:{n}^{{eme}} \:{roots}\:{of}\:\mathrm{1}\right. \\ $$$${calculate}\:{p}\left({x},{y}\right)=\left({x}+\alpha_{\mathrm{0}} {y}\right)\left({x}+\alpha_{\mathrm{1}} {y}\right)....\left({x}+\alpha_{{n}−\mathrm{1}} {y}\right) \\ $$

Question Number 50354 Answers: 0 Comments: 0

((x)=a_k ) _(1≤k≤n) is a sequence of reals let p(x) =Π_(k=1) ^n (cos(a_k )+xsin(a_k )) if p(x)=(x^2 +1)q +r find q and r

$$\left(\left({x}\right)={a}_{{k}} \right)\:_{\mathrm{1}\leqslant{k}\leqslant{n}} \:{is}\:{a}\:{sequence}\:{of}\:{reals}\:{let} \\ $$$${p}\left({x}\right)\:=\prod_{{k}=\mathrm{1}} ^{{n}} \left({cos}\left({a}_{{k}} \right)+{xsin}\left({a}_{{k}} \right)\right) \\ $$$${if}\:{p}\left({x}\right)=\left({x}^{\mathrm{2}} +\mathrm{1}\right){q}\:+{r}\:\:\:{find}\:{q}\:{and}\:{r} \\ $$$$ \\ $$

Question Number 50353 Answers: 0 Comments: 0

let p(x)=x^(4n) −x^(3n) +x^(2n) −x^n +1 and q(x)=x^4 −x^3 +x^2 −x+1 determine the integr n to have q divide p.

$${let}\:{p}\left({x}\right)={x}^{\mathrm{4}{n}} −{x}^{\mathrm{3}{n}} +{x}^{\mathrm{2}{n}} −{x}^{{n}} +\mathrm{1}\:{and} \\ $$$${q}\left({x}\right)={x}^{\mathrm{4}} −{x}^{\mathrm{3}} +{x}^{\mathrm{2}} −{x}+\mathrm{1}\:\:{determine}\:{the}\:{integr}\:{n} \\ $$$${to}\:{have}\:{q}\:{divide}\:{p}. \\ $$

Question Number 50352 Answers: 1 Comments: 5

The distances from a point to the sides of a triangle are p,q,r. Find the maximum (or minimum) area of the triangle, if it exists. Assume r≤q≤p.

$${The}\:{distances}\:{from}\:{a}\:{point}\:{to}\:{the}\:{sides} \\ $$$${of}\:{a}\:{triangle}\:{are}\:{p},{q},{r}.\:{Find}\:{the}\: \\ $$$${maximum}\:\left({or}\:{minimum}\right)\:{area}\:{of}\:{the} \\ $$$${triangle},\:{if}\:{it}\:{exists}. \\ $$$${Assume}\:{r}\leqslant{q}\leqslant{p}. \\ $$

Question Number 50328 Answers: 2 Comments: 0

Question Number 50330 Answers: 1 Comments: 1

Question Number 50299 Answers: 1 Comments: 7

please help me solve these two questions 1. A magician cuts a rope into two parts at a point selected at random. what is the probability that the length of the longer rope is at least 8 times the length of the shorter rope. 2. find the sum of co−efficient of thebinomial expansion of (4x−1)^(16)

$${please}\:{help}\:{me}\:{solve}\:{these}\:{two} \\ $$$${questions}\: \\ $$$$\mathrm{1}.\:{A}\:{magician}\:{cuts}\:{a}\:{rope}\:{into}\:{two}\: \\ $$$$\:\:{parts}\:{at}\:{a}\:{point}\:{selected}\:{at}\: \\ $$$${random}.\:{what}\:{is}\:{the}\:{probability}\:{that} \\ $$$${the}\:{length}\:{of}\:\:{the}\:{longer}\:\:{rope}\:{is}\:{at}\:{least}\: \\ $$$$\mathrm{8}\:{times}\:{the}\:{length}\:{of}\:{the}\:{shorter}\: \\ $$$${rope}. \\ $$$$\mathrm{2}.\:{find}\:{the}\:{sum}\:{of}\:{co}−{efficient}\:{of}\: \\ $$$${thebinomial}\:\:{expansion}\:\:{of}\:\: \\ $$$$\left(\mathrm{4}{x}−\mathrm{1}\right)^{\mathrm{16}} \\ $$$$ \\ $$

Question Number 50293 Answers: 2 Comments: 4

lim_(x→0) Σ_(r=1) ^(2n) ((1/(r+n)))=? please help

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{2}{n}} {\sum}}\left(\frac{\mathrm{1}}{{r}+{n}}\right)=? \\ $$$${please}\:{help} \\ $$

Question Number 50279 Answers: 2 Comments: 0

{ ((x+y=6)),((y+z=10)) :} (x,y,z>0)

$$\begin{cases}{\mathrm{x}+\mathrm{y}=\mathrm{6}}\\{\mathrm{y}+\mathrm{z}=\mathrm{10}}\end{cases}\:\:\left(\mathrm{x},\mathrm{y},\mathrm{z}>\mathrm{0}\right) \\ $$

Question Number 50278 Answers: 1 Comments: 6

(√(a+(√(a−x)))) + (√(a−(√(a+x)))) = 2x please i beg u guys please solve this question

$$\sqrt{\mathrm{a}+\sqrt{\mathrm{a}−\mathrm{x}}}\:+\:\sqrt{\mathrm{a}−\sqrt{\mathrm{a}+\mathrm{x}}}\:=\:\mathrm{2x} \\ $$$$\mathrm{please}\:\mathrm{i}\:\mathrm{beg}\:\mathrm{u}\:\mathrm{guys}\: \\ $$$$\mathrm{please}\:\mathrm{solve}\:\mathrm{this}\:\mathrm{question} \\ $$

Question Number 50275 Answers: 1 Comments: 0

Question Number 50274 Answers: 2 Comments: 0

Question Number 50258 Answers: 1 Comments: 0

A delegation of 4 students is to be selected from a total of 12 students. In how many ways can the delegation be selected if: 1) 2 particular students wish to be included together only in the delegation? 2) 2 particular students refuse to be together and 2 other particular students wish to be together only in the delegation?

$${A}\:{delegation}\:{of}\:\mathrm{4}\:{students}\:{is}\:{to}\:{be} \\ $$$${selected}\:{from}\:{a}\:{total}\:{of}\:\mathrm{12}\:{students}. \\ $$$${In}\:{how}\:{many}\:{ways}\:{can}\:{the}\:{delegation} \\ $$$${be}\:{selected}\:{if}: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{2}\:{particular}\:{students}\:{wish}\:{to}\:{be}\: \\ $$$${included}\:{together}\:{only}\:{in}\:{the}\:{delegation}? \\ $$$$\left.\mathrm{2}\right)\:\mathrm{2}\:{particular}\:{students}\:{refuse}\:{to}\:{be}\: \\ $$$${together}\:{and}\:\mathrm{2}\:{other}\:{particular}\:{students} \\ $$$${wish}\:{to}\:{be}\:{together}\:{only}\:{in}\:{the}\:{delegation}? \\ $$

Question Number 50257 Answers: 1 Comments: 0

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