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AlgebraQuestion and Answers: Page 234

Question Number 122580 Answers: 0 Comments: 3

Solve x^x =2x

$${Solve} \\ $$$${x}^{{x}} =\mathrm{2}{x} \\ $$

Question Number 122551 Answers: 1 Comments: 0

Question Number 122523 Answers: 2 Comments: 0

Question Number 122445 Answers: 2 Comments: 0

Question Number 122360 Answers: 3 Comments: 0

Question Number 122214 Answers: 1 Comments: 0

Question Number 122166 Answers: 3 Comments: 0

Given that I_n = ∫_0 ^1 x(1−x)^n dx obtain a reduction formulae for I_(n ) in terms of I_(n−2) Hence evaluate ∫_0 ^1 x(1−x)^5 dx.

$$\mathrm{Given}\:\mathrm{that}\:{I}_{{n}} \:=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}\left(\mathrm{1}−{x}\right)^{{n}} {dx} \\ $$$$\mathrm{obtain}\:\mathrm{a}\:\mathrm{reduction}\:\mathrm{formulae}\:\mathrm{for}\:{I}_{{n}\:} \:\mathrm{in}\:\mathrm{terms} \\ $$$$\mathrm{of}\:{I}_{{n}−\mathrm{2}} \:\mathrm{Hence}\:\mathrm{evaluate}\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}\left(\mathrm{1}−{x}\right)^{\mathrm{5}} {dx}. \\ $$

Question Number 122095 Answers: 0 Comments: 0

Find the number of triplets(x/a/b) where x is a real number and (a/b) belongs to the set {1/2/3/4/5/6/7/8/9} such that x^2 −a{x}+b =0 where {x} denotes the fractional part of real number x

$${Find}\:{the}\:\:{number}\:{of}\:{triplets}\left({x}/{a}/{b}\right)\:{where}\:{x} \\ $$$${is}\:{a}\:{real}\:{number}\:{and}\:\left({a}/{b}\right)\:{belongs}\:{to}\:{the}\:{set} \\ $$$$\left\{\mathrm{1}/\mathrm{2}/\mathrm{3}/\mathrm{4}/\mathrm{5}/\mathrm{6}/\mathrm{7}/\mathrm{8}/\mathrm{9}\right\}\:{such}\:{that}\: \\ $$$$ \\ $$$${x}^{\mathrm{2}} −{a}\left\{{x}\right\}+{b}\:=\mathrm{0} \\ $$$${where}\:\left\{{x}\right\}\:{denotes}\:{the}\:{fractional}\:{part}\:{of}\:{real}\:{number}\:{x} \\ $$$$ \\ $$$$ \\ $$

Question Number 122081 Answers: 2 Comments: 0

solve { ((∣x−1∣+∣y−1∣=1)),((∣x−1∣−y=−5)) :}

$$\:{solve}\:\begin{cases}{\mid{x}−\mathrm{1}\mid+\mid{y}−\mathrm{1}\mid=\mathrm{1}}\\{\mid{x}−\mathrm{1}\mid−{y}=−\mathrm{5}}\end{cases} \\ $$

Question Number 122075 Answers: 1 Comments: 0

Question Number 122012 Answers: 0 Comments: 1

r4width10calculatesurfacearea

$${r}\mathrm{4}{width}\mathrm{10}{calculatesurfacearea} \\ $$

Question Number 122010 Answers: 1 Comments: 0

Question Number 122002 Answers: 2 Comments: 0

Solve the system of equations { ((x^2 +y^2 +((2xy)/(x+y)) = 1)),(((√(x+y)) = x^2 −y)) :} in real numbers x,y.

$$\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}\: \\ $$$$\begin{cases}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\frac{\mathrm{2xy}}{\mathrm{x}+\mathrm{y}}\:=\:\mathrm{1}}\\{\sqrt{\mathrm{x}+\mathrm{y}}\:=\:\mathrm{x}^{\mathrm{2}} −\mathrm{y}}\end{cases}\:\mathrm{in}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x},\mathrm{y}. \\ $$

Question Number 121996 Answers: 1 Comments: 0

⌊x⌋ +⌊2x⌋+⌊4x⌋+⌊8x⌋+⌊16x⌋+⌊32x⌋=123456 find all values of x for which this relation holds?

$$\lfloor{x}\rfloor\:+\lfloor\mathrm{2}{x}\rfloor+\lfloor\mathrm{4}{x}\rfloor+\lfloor\mathrm{8}{x}\rfloor+\lfloor\mathrm{16}{x}\rfloor+\lfloor\mathrm{32}{x}\rfloor=\mathrm{123456} \\ $$$${find}\:{all}\:{values}\:{of}\:{x}\:{for}\:{which}\:{this}\:{relation}\:{holds}? \\ $$

Question Number 122174 Answers: 1 Comments: 0

If x = (√(5 )) + (√3),then x^3 + (1/x^3 ) = ? or, is it possible at all?

$${If}\:\:\boldsymbol{{x}}\:=\:\sqrt{\mathrm{5}\:}\:+\:\sqrt{\mathrm{3}},{then}\:\boldsymbol{{x}}^{\mathrm{3}} \:+\:\frac{\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\:? \\ $$$$\boldsymbol{{or}},\:\boldsymbol{{is}}\:\boldsymbol{{it}}\:\boldsymbol{{possible}}\:\boldsymbol{{at}}\:\boldsymbol{{all}}? \\ $$

Question Number 121973 Answers: 1 Comments: 4

solve this equation { ((xy+x+y=19)),((yz + y+z = 11)),((z+x+zx = 14)) :}

$$\:{solve}\:{this}\:{equation}\: \\ $$$$\:\begin{cases}{{xy}+{x}+{y}=\mathrm{19}}\\{{yz}\:+\:{y}+{z}\:=\:\mathrm{11}}\\{{z}+{x}+{zx}\:=\:\mathrm{14}}\end{cases} \\ $$

Question Number 121957 Answers: 0 Comments: 4

resoudre equation∣−2x−(√(3/))4∣=5−2(√7)

$${resoudre}\:{equation}\mid−\mathrm{2}{x}−\sqrt{\mathrm{3}/}\mathrm{4}\mid=\mathrm{5}−\mathrm{2}\sqrt{\mathrm{7}} \\ $$

Question Number 121950 Answers: 1 Comments: 0

Calculate the sum Σ_(k = 1) ^n (1/( (√(k+(√(k^2 +1)))))) .

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{sum}\:\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\:\sqrt{\mathrm{k}+\sqrt{\mathrm{k}^{\mathrm{2}} +\mathrm{1}}}}\:. \\ $$

Question Number 121948 Answers: 0 Comments: 0

Question Number 121919 Answers: 2 Comments: 0

Σ_(k=p) ^∞ 4∙3^(2−k) = (2/9) p = ?

$$\: \\ $$$$\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\underset{{k}={p}} {\overset{\infty} {\sum}}\:\mathrm{4}\centerdot\mathrm{3}^{\mathrm{2}−{k}} \:=\:\frac{\mathrm{2}}{\mathrm{9}} \\ $$$$\:\:\:\:\:\:\:\:\:\:{p}\:=\:? \\ $$$$\: \\ $$$$\: \\ $$

Question Number 121918 Answers: 1 Comments: 6

a, b and c are solutions of x^3 −4x^2 −5x+8=0. Without determinating a, b and c ; calculate a+b+c.

$${a},\:{b}\:{and}\:{c}\:{are}\:{solutions}\:{of}\:{x}^{\mathrm{3}} −\mathrm{4}{x}^{\mathrm{2}} −\mathrm{5}{x}+\mathrm{8}=\mathrm{0}. \\ $$$${Without}\:{determinating}\:{a},\:{b}\:{and}\:{c}\:;\: \\ $$$${calculate}\:{a}+{b}+{c}. \\ $$

Question Number 121913 Answers: 1 Comments: 0

{ ((x^2 +y=36)),((x+y^2 =25 x=? y=?)) :}

$$\begin{cases}{{x}^{\mathrm{2}} +{y}=\mathrm{36}}\\{{x}+{y}^{\mathrm{2}} =\mathrm{25}\:\:{x}=?\:{y}=?}\end{cases} \\ $$

Question Number 121890 Answers: 2 Comments: 1

Question Number 121889 Answers: 2 Comments: 0

Question Number 121847 Answers: 0 Comments: 0

Question Number 121754 Answers: 2 Comments: 0

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