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AlgebraQuestion and Answers: Page 237

Question Number 110182 Answers: 1 Comments: 2

Solve x^3 +15x−92=0

$$\mathrm{Solve}\:{x}^{\mathrm{3}} +\mathrm{15}{x}−\mathrm{92}=\mathrm{0} \\ $$

Question Number 110173 Answers: 2 Comments: 0

If Σ_(r=1) ^n t_r =((n(n+1)(n+2)(n+3))/8) then lim_(n→∞) Σ_(r=1) ^n (1/t_r ) = ?

$${If}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}{t}_{{r}} =\frac{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)}{\mathrm{8}} \\ $$$${then}\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{{t}_{{r}} }\:=\:? \\ $$$$ \\ $$

Question Number 110157 Answers: 2 Comments: 0

If we have 5 people, how many ways can they be seated on a round table, if there are, (a) 7 chairs (b) 3 chairs available

$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\:\:\mathrm{5}\:\:\mathrm{people},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{they}\:\mathrm{be}\:\mathrm{seated} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{round}\:\mathrm{table},\:\mathrm{if}\:\mathrm{there}\:\mathrm{are}, \\ $$$$\left(\mathrm{a}\right)\:\:\:\mathrm{7}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\mathrm{3}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\:\:\:\:\mathrm{available} \\ $$

Question Number 110156 Answers: 1 Comments: 0

If we have 5 people, how many ways can they be seated in a row on a chair, if their are, (a) 7 chairs (b) 3 chairs available

$$\mathrm{If}\:\mathrm{we}\:\mathrm{have}\:\:\mathrm{5}\:\mathrm{people},\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{can}\:\mathrm{they}\:\mathrm{be}\:\mathrm{seated}\:\mathrm{in}\:\mathrm{a}\:\mathrm{row} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{chair},\:\mathrm{if}\:\mathrm{their}\:\mathrm{are}, \\ $$$$\left(\mathrm{a}\right)\:\:\:\mathrm{7}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\:\:\:\mathrm{3}\:\:\mathrm{chairs}\:\:\:\:\:\:\:\:\mathrm{available} \\ $$

Question Number 110136 Answers: 4 Comments: 0

Question Number 110087 Answers: 0 Comments: 3

Solve the system following of equations { ((x+y+z=2)),((2x+3y+z=1)),((x^2 +(y+2)^2 +(z−1)^2 =9)) :}

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{following}\:\mathrm{of}\:\mathrm{equations} \\ $$$$\begin{cases}{\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{2}}\\{\mathrm{2x}+\mathrm{3y}+\mathrm{z}=\mathrm{1}}\\{\mathrm{x}^{\mathrm{2}} +\left(\mathrm{y}+\mathrm{2}\right)^{\mathrm{2}} +\left(\mathrm{z}−\mathrm{1}\right)^{\mathrm{2}} =\mathrm{9}}\end{cases} \\ $$

Question Number 109954 Answers: 1 Comments: 0

!3=????

$$!\mathrm{3}=???? \\ $$

Question Number 109914 Answers: 1 Comments: 0

Prove that tan142°30′+(√6)+(√3)−(√2) is an integer.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{tan142}°\mathrm{30}'+\sqrt{\mathrm{6}}+\sqrt{\mathrm{3}}−\sqrt{\mathrm{2}} \\ $$$$\mathrm{is}\:\mathrm{an}\:\mathrm{integer}. \\ $$

Question Number 109901 Answers: 1 Comments: 0

If first n terms of an A.P. is cn^2 , find sum of squares of its first n terms.

$${If}\:{first}\:{n}\:{terms}\:{of}\:{an}\:{A}.{P}.\:{is}\:{cn}^{\mathrm{2}} , \\ $$$${find}\:{sum}\:{of}\:{squares}\:{of}\:{its}\:{first} \\ $$$${n}\:{terms}. \\ $$

Question Number 109895 Answers: 1 Comments: 3

Question Number 109754 Answers: 1 Comments: 0

1.specify value absolute x if ? b.∣2x+3∣+x−3=0

$$\mathrm{1}.{specify}\:{value}\:{absolute}\:{x}\:{if}\:? \\ $$$$ \\ $$$${b}.\mid\mathrm{2}{x}+\mathrm{3}\mid+{x}−\mathrm{3}=\mathrm{0} \\ $$$$ \\ $$

Question Number 109721 Answers: 1 Comments: 0

Question Number 109699 Answers: 1 Comments: 0

x(x−1)^2 ≥ 12(x−1)

$$\:\:\:\:\:{x}\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:\geqslant\:\mathrm{12}\left({x}−\mathrm{1}\right) \\ $$

Question Number 109626 Answers: 1 Comments: 1

Question Number 109625 Answers: 0 Comments: 2

((sin 2𝛂+2sin 𝛂∙cos 2𝛂)/(1+cos 𝛂+cos2 𝛂+cos3 𝛂))

$$\frac{\mathrm{sin}\:\mathrm{2}\boldsymbol{\alpha}+\mathrm{2sin}\:\boldsymbol{\alpha}\centerdot\mathrm{cos}\:\mathrm{2}\boldsymbol{\alpha}}{\mathrm{1}+\mathrm{cos}\:\boldsymbol{\alpha}+\mathrm{cos2}\:\boldsymbol{\alpha}+\mathrm{cos3}\:\boldsymbol{\alpha}} \\ $$

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

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