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AlgebraQuestion and Answers: Page 336

Question Number 31336 Answers: 0 Comments: 1

Find the principal value of z=(1−i)^(1+i) .Hence find the modulus of the result.

$${Find}\:{the}\:{principal}\:{value}\:{of} \\ $$$${z}=\left(\mathrm{1}−{i}\right)^{\mathrm{1}+{i}} .{Hence}\:{find}\:{the} \\ $$$${modulus}\:{of}\:{the}\:{result}. \\ $$

Question Number 31335 Answers: 0 Comments: 3

Find the pricipal value of z=(1−i)^i

$${Find}\:{the}\:{pricipal}\:{value}\:{of}\: \\ $$$${z}=\left(\mathrm{1}−{i}\right)^{{i}} \\ $$

Question Number 31324 Answers: 1 Comments: 0

Question Number 31323 Answers: 1 Comments: 0

Question Number 31320 Answers: 1 Comments: 0

Let p and q are the roots of x^2 − 2mx − 5n = 0 and m and n are the roots of x^2 − 2px − 5q = 0 If p ≠ q ≠ m ≠ n, then the value of p + q + m + n is ...

$$\mathrm{Let}\:{p}\:\mathrm{and}\:{q}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{2}} \:−\:\mathrm{2}{mx}\:−\:\mathrm{5}{n}\:=\:\mathrm{0} \\ $$$$\mathrm{and}\:{m}\:\mathrm{and}\:{n}\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of} \\ $$$${x}^{\mathrm{2}} \:−\:\mathrm{2}{px}\:−\:\mathrm{5}{q}\:=\:\mathrm{0} \\ $$$$\mathrm{If}\:{p}\:\neq\:{q}\:\neq\:{m}\:\neq\:{n},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$${p}\:+\:{q}\:+\:{m}\:+\:{n}\:\mathrm{is}\:... \\ $$

Question Number 31317 Answers: 0 Comments: 1

Question Number 31286 Answers: 1 Comments: 1

Find all set of ordered triple/s (x,y,z), x,y,z∈ℜ, such that x−y=1−z 3(x^2 −y^2 )=5(1−z^2 ) 7(x^3 −y^3 )=19(1−z^3 ). Please show your solution.

$${Find}\:{all}\:{set}\:{of}\:{ordered}\:{triple}/{s}\:\left({x},{y},{z}\right),\:\:{x},{y},{z}\in\Re,\:{such}\:{that} \\ $$$${x}−{y}=\mathrm{1}−{z} \\ $$$$\mathrm{3}\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)=\mathrm{5}\left(\mathrm{1}−{z}^{\mathrm{2}} \right) \\ $$$$\mathrm{7}\left({x}^{\mathrm{3}} −{y}^{\mathrm{3}} \right)=\mathrm{19}\left(\mathrm{1}−{z}^{\mathrm{3}} \right). \\ $$$${Please}\:{show}\:{your}\:{solution}. \\ $$

Question Number 31290 Answers: 1 Comments: 1

Question Number 31259 Answers: 0 Comments: 4

Question Number 31214 Answers: 2 Comments: 2

Question Number 31194 Answers: 2 Comments: 1

Find the remainder when x^(203) −1 is divided by x^4 −1.

$${Find}\:{the}\:{remainder}\:{when}\:{x}^{\mathrm{203}} −\mathrm{1} \\ $$$${is}\:{divided}\:{by}\:{x}^{\mathrm{4}} −\mathrm{1}. \\ $$

Question Number 31047 Answers: 0 Comments: 0

let Δ={(x,y)∈N^2 /x+y=n , n∈N} find cardΔ 2) let A= {(x,y)∈N^2 / x+2y=n} find card A.

$${let}\:\Delta=\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} \:/{x}+{y}={n}\:,\:{n}\in{N}\right\}\:{find}\:{card}\Delta \\ $$$$\left.\mathrm{2}\right)\:{let}\:{A}=\:\left\{\left({x},{y}\right)\in{N}^{\mathrm{2}} /\:{x}+\mathrm{2}{y}={n}\right\}\:{find}\:{card}\:{A}. \\ $$

Question Number 31046 Answers: 0 Comments: 0

prove that C_n ^o C_n ^p +C_n ^1 C_(n−1) ^(p−1) +...C_n ^p C_(n−p) ^0 =2^p C_n ^p .

$${prove}\:{that}\:{C}_{{n}} ^{{o}} \:{C}_{{n}} ^{{p}} \:+{C}_{{n}} ^{\mathrm{1}} \:{C}_{{n}−\mathrm{1}} ^{{p}−\mathrm{1}} \:+...{C}_{{n}} ^{{p}} \:{C}_{{n}−{p}} ^{\mathrm{0}} =\mathrm{2}^{{p}} \:{C}_{{n}} ^{{p}} \:\:\:. \\ $$

Question Number 31042 Answers: 0 Comments: 0

solve in N^2 9y^2 −(x+1)^2 =32 .

$${solve}\:{in}\:{N}^{\mathrm{2}} \:\:\mathrm{9}{y}^{\mathrm{2}} \:−\left({x}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{32}\:. \\ $$

Question Number 31041 Answers: 0 Comments: 0

solve in Z^2 11x −5y =14

$${solve}\:{in}\:{Z}^{\mathrm{2}} \:\mathrm{11}{x}\:−\mathrm{5}{y}\:=\mathrm{14} \\ $$

Question Number 31040 Answers: 0 Comments: 0

solve in Z^2 x^2 −y^2 =1969

$${solve}\:{in}\:{Z}^{\mathrm{2}} \:{x}^{\mathrm{2}} \:−{y}^{\mathrm{2}} =\mathrm{1969} \\ $$

Question Number 31039 Answers: 0 Comments: 0

find the sum s_0 = C_n ^o +C_n ^4 +C_n ^8 +... s_1 = C_n ^1 +C_n ^5 +C_n ^9 +.... s_3 = C_n ^2 +C_n ^6 + C_n ^(10) +....

$${find}\:{the}\:{sum} \\ $$$${s}_{\mathrm{0}} =\:{C}_{{n}} ^{{o}} \:+{C}_{{n}} ^{\mathrm{4}} \:+{C}_{{n}} ^{\mathrm{8}} \:+... \\ $$$${s}_{\mathrm{1}} =\:{C}_{{n}} ^{\mathrm{1}} \:+{C}_{{n}} ^{\mathrm{5}} \:+{C}_{{n}} ^{\mathrm{9}} \:+.... \\ $$$${s}_{\mathrm{3}} =\:{C}_{{n}} ^{\mathrm{2}} \:\:+{C}_{{n}} ^{\mathrm{6}} \:+\:{C}_{{n}} ^{\mathrm{10}} \:+.... \\ $$

Question Number 31038 Answers: 0 Comments: 0

let give α ∈]−π ,π[ 1)prove that sin^2 α −2(1+cosα) =−4cos^4 ((α/2)) 2)solve inside C z^2 −2z sinα +2(1+cosα)=0 find the module and arg of the roots.

$$\left.{let}\:{give}\:\alpha\:\in\right]−\pi\:,\pi\left[\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:{sin}^{\mathrm{2}} \alpha\:−\mathrm{2}\left(\mathrm{1}+{cos}\alpha\right)\:=−\mathrm{4}{cos}^{\mathrm{4}} \left(\frac{\alpha}{\mathrm{2}}\right) \\ $$$$\left.\mathrm{2}\right){solve}\:{inside}\:{C}\:\:{z}^{\mathrm{2}} \:−\mathrm{2}{z}\:{sin}\alpha\:+\mathrm{2}\left(\mathrm{1}+{cos}\alpha\right)=\mathrm{0}\:{find} \\ $$$${the}\:{module}\:{and}\:{arg}\:{of}\:{the}\:{roots}. \\ $$

Question Number 31037 Answers: 0 Comments: 0

let θ∈[0,π] and Z=1+cosθ +isinθ Z^′ =cosθ +(1+sinθ)i find ∣ZZ^′ ∣ arg(ZZ^′ ) ∣(Z/Z^′ )∣ , arg((Z/Z^, )) , ∣Z−Z^, ∣ and arg(Z−Z^, ).

$${let}\:\:\theta\in\left[\mathrm{0},\pi\right]\:\:{and}\:{Z}=\mathrm{1}+{cos}\theta\:+{isin}\theta \\ $$$${Z}^{'} ={cos}\theta\:+\left(\mathrm{1}+{sin}\theta\right){i}\:\:{find}\:\mid{ZZ}^{'} \mid\:\:{arg}\left({ZZ}^{'} \right) \\ $$$$\mid\frac{{Z}}{{Z}^{'} }\mid\:,\:{arg}\left(\frac{{Z}}{{Z}^{,} }\right)\:\:,\:\:\mid{Z}−{Z}^{,} \mid\:{and}\:{arg}\left({Z}−{Z}^{,} \right). \\ $$

Question Number 31036 Answers: 0 Comments: 0

solve inside C z^6 = (z^− )^2 .

$${solve}\:{inside}\:{C}\:\:{z}^{\mathrm{6}} =\:\left({z}^{−} \right)^{\mathrm{2}} \:\:. \\ $$

Question Number 31035 Answers: 0 Comments: 0

1) solve x^5 =1 2) if x_i are roots of this equation prove that Σ_i x_i =0 3)prove that cos(((2π)/5)) +cos(((4π)/5)) =−(1/2) 4)calculate cos(((4π)/5)) interms cos(((2π)/5)) then find its values.

$$\left.\mathrm{1}\right)\:{solve}\:{x}^{\mathrm{5}} =\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{if}\:{x}_{{i}} \:{are}\:{roots}\:{of}\:{this}\:{equation}\:{prove}\:{that}\:\sum_{{i}} \:{x}_{{i}} =\mathrm{0} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:+{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\left.\mathrm{4}\right){calculate}\:{cos}\left(\frac{\mathrm{4}\pi}{\mathrm{5}}\right)\:{interms}\:{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{5}}\right)\:{then}\:{find}\:{its} \\ $$$${values}. \\ $$

Question Number 31034 Answers: 0 Comments: 0

1) solve inside C z^4 =6 2)sove inside C (((z+i)/(z−i)))^3 +(((z+i)/(z−i)))^2 +(((z+i)/(z−i))) +1=0

$$\left.\mathrm{1}\right)\:{solve}\:{inside}\:{C}\:\:{z}^{\mathrm{4}} =\mathrm{6} \\ $$$$\left.\mathrm{2}\right){sove}\:{inside}\:{C}\:\left(\frac{{z}+{i}}{{z}−{i}}\right)^{\mathrm{3}} \:+\left(\frac{{z}+{i}}{{z}−{i}}\right)^{\mathrm{2}} \:+\left(\frac{{z}+{i}}{{z}−{i}}\right)\:+\mathrm{1}=\mathrm{0} \\ $$

Question Number 31033 Answers: 0 Comments: 0

factorize p(x)=(1+ix)^n −e^(inθ) wth θ∈R.

$${factorize}\:{p}\left({x}\right)=\left(\mathrm{1}+{ix}\right)^{{n}} \:−{e}^{{in}\theta} \:\:\:{wth}\:\theta\in{R}. \\ $$

Question Number 31032 Answers: 0 Comments: 0

let give x_0 =0 ,y_0 =1 and {_(y_n =x_(n−1) +y_(n−1) ) ^(x_n =x_(n−1) −y_(n−1) ) for n≥1 let z_n =x_n +i y_n ∀n∈N 1)calculate z_0 ,z_1 and z_2 2)prove that ∀n∈N^ ,n≥1 z_n =(1+i)z_(n−1) find z_n then find the expression of x_n and y_n 3)let put S_n =z_0 +z_1 +....z_n s_n =x_0 +x_1 +...+x_n s_n ^′ =y_0 +y_1 +...+y_n find those sum interms of n.

$${let}\:{give}\:{x}_{\mathrm{0}} =\mathrm{0}\:,{y}_{\mathrm{0}} =\mathrm{1}\:{and}\:\left\{_{{y}_{{n}} ={x}_{{n}−\mathrm{1}} \:+{y}_{{n}−\mathrm{1}} } ^{{x}_{{n}} ={x}_{{n}−\mathrm{1}} \:−{y}_{{n}−\mathrm{1}} } \:\:\:\:\:\:{for}\:{n}\geqslant\mathrm{1}\:{let}\right. \\ $$$${z}_{{n}} ={x}_{{n}} \:+{i}\:{y}_{{n}} \:\:\:\:\:\forall{n}\in{N} \\ $$$$\left.\mathrm{1}\right){calculate}\:{z}_{\mathrm{0}} \:,{z}_{\mathrm{1}} \:{and}\:{z}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{n}\in{N}^{} ,{n}\geqslant\mathrm{1}\:\:{z}_{{n}} =\left(\mathrm{1}+{i}\right){z}_{{n}−\mathrm{1}} \:{find}\:{z}_{{n}} {then} \\ $$$${find}\:{the}\:{expression}\:{of}\:{x}_{{n}} \:{and}\:{y}_{{n}} \\ $$$$\left.\mathrm{3}\right){let}\:{put}\:{S}_{{n}} ={z}_{\mathrm{0}} \:+{z}_{\mathrm{1}} \:+....{z}_{{n}} \\ $$$${s}_{{n}} ={x}_{\mathrm{0}} \:+{x}_{\mathrm{1}} \:+...+{x}_{{n}} \\ $$$${s}_{{n}} ^{'} ={y}_{\mathrm{0}} \:+{y}_{\mathrm{1}} \:+...+{y}_{{n}} \:\:{find}\:{those}\:{sum}\:{interms}\:{of}\:{n}. \\ $$$$ \\ $$

Question Number 31031 Answers: 0 Comments: 0

1) solve inside C z^(12) =1 and give the solution at form r e^(iθ) 2)calculate 1+u +u^2 +... +u^n then find the solution of z∈C z^8 +z^4 +1=0

$$\left.\mathrm{1}\right)\:{solve}\:{inside}\:{C}\:\:{z}^{\mathrm{12}} =\mathrm{1}\:{and}\:{give}\:{the}\:{solution}\:{at}\:{form} \\ $$$${r}\:{e}^{{i}\theta} \\ $$$$\left.\mathrm{2}\right){calculate}\:\mathrm{1}+{u}\:+{u}^{\mathrm{2}} \:+...\:+{u}^{{n}} \:{then}\:{find}\:{the}\:{solution} \\ $$$${of}\:{z}\in{C}\:\:\:\:\:\:\:{z}^{\mathrm{8}} \:+{z}^{\mathrm{4}} \:+\mathrm{1}=\mathrm{0} \\ $$

Question Number 31023 Answers: 0 Comments: 0

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