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AlgebraQuestion and Answers: Page 252

Question Number 109611 Answers: 4 Comments: 1

Question Number 109595 Answers: 1 Comments: 1

For any Real numbers x,y and z, if (x+y+z)=2, then prove that xyz≥8(1−x)(1−y)(1−z)

$${For}\:{any}\:{Real}\:{numbers}\:{x},{y}\:{and}\:{z}, \\ $$$$\:{if}\:\:\left({x}+{y}+{z}\right)=\mathrm{2},\:{then}\:{prove}\:{that} \\ $$$$\:\:\:\:\:\:\:\:{xyz}\geqslant\mathrm{8}\left(\mathrm{1}−{x}\right)\left(\mathrm{1}−{y}\right)\left(\mathrm{1}−{z}\right) \\ $$

Question Number 109577 Answers: 4 Comments: 1

Given x^4 +x^2 y^2 +y^4 =133 and x^2 −xy+y^2 =7 then what is the value of xy ?

$$\:\:\:\mathrm{G}{iven}\:{x}^{\mathrm{4}} +{x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{4}} =\mathrm{133} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{and}\:{x}^{\mathrm{2}} −{xy}+{y}^{\mathrm{2}} =\mathrm{7} \\ $$$$\:\:{then}\:{what}\:{is}\:{the}\:{value}\:{of}\:{xy}\:? \\ $$

Question Number 109569 Answers: 1 Comments: 0

Question Number 109544 Answers: 2 Comments: 1

Exclude m and n from the equalities: a=m+n,b^3 =m^3 +n^3 ,c^5 =m^5 +n^5

$$\mathrm{Exclude}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n}\:\mathrm{from}\:\mathrm{the}\:\mathrm{equalities}: \\ $$$$\mathrm{a}=\mathrm{m}+\mathrm{n},\mathrm{b}^{\mathrm{3}} =\mathrm{m}^{\mathrm{3}} +\mathrm{n}^{\mathrm{3}} ,\mathrm{c}^{\mathrm{5}} =\mathrm{m}^{\mathrm{5}} +\mathrm{n}^{\mathrm{5}} \\ $$

Question Number 109516 Answers: 3 Comments: 0

cos (1−i)=a+ib Find a, b.

$$\mathrm{cos}\:\left(\mathrm{1}−{i}\right)={a}+{ib} \\ $$$${Find}\:\:{a},\:{b}. \\ $$

Question Number 109500 Answers: 3 Comments: 0

Given { ((a^2 +ab+bc+ac=a+c)),((b^2 +ab+bc+ac=b+a)),((c^2 +ab+bc+ac=c+b)) :} find the value of a+b+c

$${Given}\:\begin{cases}{{a}^{\mathrm{2}} +{ab}+{bc}+{ac}={a}+{c}}\\{{b}^{\mathrm{2}} +{ab}+{bc}+{ac}={b}+{a}}\\{{c}^{\mathrm{2}} +{ab}+{bc}+{ac}={c}+{b}}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:{a}+{b}+{c}\: \\ $$

Question Number 109495 Answers: 4 Comments: 0

1) ∫_0 ^(Π/2) sin x∙sin 2x∙sin 3x∙dx = ? 2) ∫_0 ^(1/2) arcsin x∙dx= ?

$$\left.\mathrm{1}\right)\:\:\:\:\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \mathrm{sin}\:{x}\centerdot\mathrm{sin}\:\mathrm{2}{x}\centerdot\mathrm{sin}\:\mathrm{3}{x}\centerdot{dx}\:=\:? \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \mathrm{arcsin}\:{x}\centerdot{dx}=\:? \\ $$

Question Number 109464 Answers: 0 Comments: 0

(√(1+(√(2+(√(3+(√(4+...))))))))

$$\sqrt{\mathrm{1}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}+\sqrt{\mathrm{4}+...}}}} \\ $$

Question Number 109460 Answers: 1 Comments: 0

Question Number 109414 Answers: 2 Comments: 0

Question Number 109377 Answers: 1 Comments: 0

Question Number 109369 Answers: 2 Comments: 0

Question Number 109307 Answers: 1 Comments: 0

Question Number 109222 Answers: 1 Comments: 0

((…♭emATH…)/(≅≅≅≅≅≅)) (√(5+(√(10)))) = x.((√(5+(√(15)) )) +(√(5−(√(15)))) ) x =?

$$\:\:\:\:\frac{\ldots\flat{em}\mathcal{ATH}\ldots}{\cong\cong\cong\cong\cong\cong} \\ $$$$\sqrt{\mathrm{5}+\sqrt{\mathrm{10}}}\:=\:{x}.\left(\sqrt{\mathrm{5}+\sqrt{\mathrm{15}}\:}\:+\sqrt{\mathrm{5}−\sqrt{\mathrm{15}}}\:\right) \\ $$$${x}\:=? \\ $$

Question Number 109193 Answers: 3 Comments: 8

solve x^x^3 =5

$${solve}\: \\ $$$${x}^{{x}^{\mathrm{3}} } =\mathrm{5} \\ $$

Question Number 109160 Answers: 4 Comments: 0

Find the equation of line through the point of intersection of the line x+3y−11=0 and 5x−4y+2=0 and perpendicular to 4x+2y+9=0.

$${Find}\:{the}\:{equation}\:{of}\:{line}\:{through} \\ $$$${the}\:{point}\:{of}\:{intersection}\:{of}\:{the} \\ $$$${line}\:{x}+\mathrm{3}{y}−\mathrm{11}=\mathrm{0}\:{and}\:\mathrm{5}{x}−\mathrm{4}{y}+\mathrm{2}=\mathrm{0} \\ $$$${and}\:{perpendicular}\:{to}\:\mathrm{4}{x}+\mathrm{2}{y}+\mathrm{9}=\mathrm{0}. \\ $$

Question Number 109147 Answers: 3 Comments: 0

The principal argument of z=1+cos (((6π)/5))+isin (((6π)/5)) is = ?

$${The}\:{principal}\:{argument}\:{of} \\ $$$${z}=\mathrm{1}+\mathrm{cos}\:\left(\frac{\mathrm{6}\pi}{\mathrm{5}}\right)+{i}\mathrm{sin}\:\left(\frac{\mathrm{6}\pi}{\mathrm{5}}\right)\:\:\:{is}\:=\:? \\ $$

Question Number 109237 Answers: 4 Comments: 0

((♭o♭hans)/(∠∠∠∠∠)) { ((x^3 +x^2 y = 9)),((y^3 +y^2 x = 25)) :}. find x and y.

$$\:\:\frac{\flat{o}\flat{hans}}{\angle\angle\angle\angle\angle} \\ $$$$\begin{cases}{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} {y}\:=\:\mathrm{9}}\\{{y}^{\mathrm{3}} +{y}^{\mathrm{2}} {x}\:=\:\mathrm{25}}\end{cases}.\:{find}\:{x}\:{and}\:{y}. \\ $$

Question Number 109090 Answers: 1 Comments: 0

Find all those roots of the equation z^(12) −56z^6 −512=0 whose imaginary part is positive.

$${Find}\:{all}\:{those}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\boldsymbol{{z}}^{\mathrm{12}} −\mathrm{56}\boldsymbol{{z}}^{\mathrm{6}} −\mathrm{512}=\mathrm{0}\:\:{whose}\:{imaginary} \\ $$$${part}\:{is}\:{positive}. \\ $$

Question Number 109029 Answers: 1 Comments: 0

Find the value(s) of a,b and c if: (x+a)(x+2020)+1=(x+b)(x+c) where a,b and c are natural numbers.

$${Find}\:{the}\:{value}\left({s}\right)\:{of}\:{a},{b}\:{and}\:{c}\:\:{if}: \\ $$$$\left({x}+{a}\right)\left({x}+\mathrm{2020}\right)+\mathrm{1}=\left({x}+{b}\right)\left({x}+{c}\right) \\ $$$${where}\:{a},{b}\:{and}\:{c}\:{are}\:{natural}\:{numbers}. \\ $$

Question Number 109005 Answers: 2 Comments: 0

i^i =?

$${i}^{{i}} =? \\ $$

Question Number 109004 Answers: 4 Comments: 0

^ Solve ∣3x+5∣ = ∣4x−3∣ where x ∈ R

$$\overset{} {\:}\:\:{Solve}\:\:\:\:\mid\mathrm{3}{x}+\mathrm{5}\mid\:=\:\mid\mathrm{4}{x}−\mathrm{3}\mid \\ $$$$\:\:\:{where}\:{x}\:\in\:\mathbb{R} \\ $$$$ \\ $$

Question Number 109000 Answers: 0 Comments: 0

Question Number 108984 Answers: 7 Comments: 1

B_≈ eM_≈ ath_≈ (1)Σ_(n = 1) ^(100) [ (2/(4n^2 −1)) ] ? (2) (2/(4.9)) + (2/(9.14)) + (2/(14.19)) + ... + (2/(49.54)) ? (3) Given a quadratic equation x^2 (Σ_(i = 1) ^2 2)+ x(Σ_(i = 1) ^5 i)−(2/3)(Σ_(i = 1) ^3 i) = 0 has the roots are x_1 and x_2 with x_1 < x_2 . Find the value of x_1 +8x_2 .

$$\:\:\:\underset{\approx} {\mathcal{B}}{e}\underset{\approx} {\mathcal{M}}{at}\underset{\approx} {{h}} \\ $$$$\left(\mathrm{1}\right)\underset{{n}\:=\:\mathrm{1}} {\overset{\mathrm{100}} {\sum}}\left[\:\frac{\mathrm{2}}{\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}}\:\right]\:? \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{2}}{\mathrm{4}.\mathrm{9}}\:+\:\frac{\mathrm{2}}{\mathrm{9}.\mathrm{14}}\:+\:\frac{\mathrm{2}}{\mathrm{14}.\mathrm{19}}\:+\:...\:+\:\frac{\mathrm{2}}{\mathrm{49}.\mathrm{54}}\:? \\ $$$$\left(\mathrm{3}\right)\:{Given}\:{a}\:{quadratic}\:{equation} \\ $$$${x}^{\mathrm{2}} \left(\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{2}} {\sum}}\mathrm{2}\right)+\:{x}\left(\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{5}} {\sum}}{i}\right)−\frac{\mathrm{2}}{\mathrm{3}}\left(\underset{{i}\:=\:\mathrm{1}} {\overset{\mathrm{3}} {\sum}}{i}\right)\:=\:\mathrm{0} \\ $$$${has}\:{the}\:{roots}\:{are}\:{x}_{\mathrm{1}} \:{and}\:{x}_{\mathrm{2}} \:{with}\: \\ $$$${x}_{\mathrm{1}} \:<\:{x}_{\mathrm{2}} .\:{Find}\:{the}\:{value}\:{of}\:{x}_{\mathrm{1}} +\mathrm{8}{x}_{\mathrm{2}} . \\ $$

Question Number 108967 Answers: 3 Comments: 0

b^★ o^★ b^★ h^∼ a^∼ n^∼ s^∼ 1+(4/(11))+(9/(121))+((16)/(1331))+((25)/(14641))+...

$$\:\:\overset{\bigstar} {{b}}\overset{\bigstar} {{o}}\overset{\bigstar} {{b}}\overset{\sim} {{h}}\overset{\sim} {{a}}\overset{\sim} {{n}}\overset{\sim} {{s}} \\ $$$$\mathrm{1}+\frac{\mathrm{4}}{\mathrm{11}}+\frac{\mathrm{9}}{\mathrm{121}}+\frac{\mathrm{16}}{\mathrm{1331}}+\frac{\mathrm{25}}{\mathrm{14641}}+... \\ $$

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