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AlgebraQuestion and Answers: Page 228

Question Number 127743 Answers: 2 Comments: 1

dx+ydy=x^2 ydy

$${dx}+{ydy}={x}^{\mathrm{2}} {ydy} \\ $$

Question Number 127742 Answers: 1 Comments: 0

∫xdx

$$\int{xdx} \\ $$

Question Number 127716 Answers: 2 Comments: 0

Let p and q be two positive real number such that { ((p(√p) +q(√q) = 32)),((p(√q) + q(√p) = 31)) :} find the value of ((5(p+q)?)/7)

$$\:\mathrm{Let}\:\mathrm{p}\:\mathrm{and}\:\mathrm{q}\:\mathrm{be}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{such}\:\mathrm{that}\:\begin{cases}{\mathrm{p}\sqrt{\mathrm{p}}\:+\mathrm{q}\sqrt{\mathrm{q}}\:=\:\mathrm{32}}\\{\mathrm{p}\sqrt{\mathrm{q}}\:+\:\mathrm{q}\sqrt{\mathrm{p}}\:=\:\mathrm{31}}\end{cases} \\ $$$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{\mathrm{5}\left(\mathrm{p}+\mathrm{q}\right)?}{\mathrm{7}} \\ $$

Question Number 127706 Answers: 2 Comments: 0

dx+ydy=x^2 ydy

$${dx}+{ydy}={x}^{\mathrm{2}} {ydy} \\ $$

Question Number 127641 Answers: 1 Comments: 0

Question Number 127607 Answers: 1 Comments: 0

Question Number 127547 Answers: 1 Comments: 0

Question Number 127525 Answers: 1 Comments: 2

Question Number 127519 Answers: 1 Comments: 0

Question Number 127500 Answers: 1 Comments: 0

if sin x_1 +sin x_2 +...+sin x_(100) =0, find the maximum value of sin^5 x_1 +sin^5 x_2 +...+sin^5 x_(100) . (x_1 ,x_2 ,...,x_(100) ∈R)

$${if}\:\mathrm{sin}\:{x}_{\mathrm{1}} +\mathrm{sin}\:{x}_{\mathrm{2}} +...+\mathrm{sin}\:{x}_{\mathrm{100}} =\mathrm{0}, \\ $$$${find}\:{the}\:{maximum}\:{value}\:{of} \\ $$$$\mathrm{sin}^{\mathrm{5}} \:{x}_{\mathrm{1}} +\mathrm{sin}^{\mathrm{5}} \:{x}_{\mathrm{2}} +...+\mathrm{sin}^{\mathrm{5}} \:{x}_{\mathrm{100}} . \\ $$$$\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,...,{x}_{\mathrm{100}} \in\mathbb{R}\right) \\ $$

Question Number 127481 Answers: 2 Comments: 1

Question Number 127429 Answers: 0 Comments: 1

6÷3(2)=???

$$\mathrm{6}\boldsymbol{\div}\mathrm{3}\left(\mathrm{2}\right)=??? \\ $$

Question Number 127306 Answers: 0 Comments: 6

Question Number 127301 Answers: 0 Comments: 2

Question Number 127344 Answers: 2 Comments: 0

S = Σ_(n=4) ^∞ ℓn (1−(1/n^2 )) =?

$${S}\:=\:\underset{{n}=\mathrm{4}} {\overset{\infty} {\sum}}\ell{n}\:\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right)\:=?\: \\ $$

Question Number 171745 Answers: 1 Comments: 0

Prove that in any triangle ABC , with area F holds: 7(m_a ^2 + m_b ^2 + m_c ^2 ) ≥ 36F + 3 Σ_(cyc) (a−m_a ^2 )

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{in}\:\mathrm{any}\:\mathrm{triangle}\:\:\mathrm{ABC}\:, \\ $$$$\mathrm{with}\:\mathrm{area}\:\:\mathrm{F}\:\:\mathrm{holds}: \\ $$$$\mathrm{7}\left(\mathrm{m}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{2}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{b}}} ^{\mathrm{2}} \:+\:\mathrm{m}_{\boldsymbol{\mathrm{c}}} ^{\mathrm{2}} \right)\:\geqslant\:\mathrm{36F}\:+\:\mathrm{3}\:\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\left(\mathrm{a}−\mathrm{m}_{\boldsymbol{\mathrm{a}}} ^{\mathrm{2}} \right) \\ $$

Question Number 171743 Answers: 0 Comments: 2

solve: 12^(x−2) =4^x , find x

$${solve}:\:\mathrm{12}^{{x}−\mathrm{2}} =\mathrm{4}^{{x}} ,\:{find}\:{x} \\ $$

Question Number 127155 Answers: 0 Comments: 1

∫_0 ^X −f(x) dx=f(X)∙c∙(X∙c_1 +c_2 ) f(x)=?

$$\underset{\mathrm{0}} {\overset{{X}} {\int}}−{f}\left({x}\right)\:{dx}={f}\left({X}\right)\centerdot{c}\centerdot\left({X}\centerdot{c}_{\mathrm{1}} +{c}_{\mathrm{2}} \right) \\ $$$${f}\left({x}\right)=? \\ $$$$ \\ $$

Question Number 127144 Answers: 1 Comments: 0

Question Number 127094 Answers: 1 Comments: 1

Question Number 127090 Answers: 4 Comments: 0

Given that I_n = ∫_0 ^1 x(1−x)^n dx , n ∈ Z^+ show that (n+2)I_n = nI_(n−1) , n ≥ 1.

$$\mathrm{Given}\:\mathrm{that}\: \\ $$$$\:\:\mathcal{I}_{{n}} \:=\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}\left(\mathrm{1}−{x}\right)^{{n}} {dx}\:,\:{n}\:\in\:\mathbb{Z}^{+} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:\:\left({n}+\mathrm{2}\right)\mathcal{I}_{{n}} \:=\:{nI}_{{n}−\mathrm{1}} ,\:{n}\:\geqslant\:\mathrm{1}. \\ $$

Question Number 127006 Answers: 1 Comments: 0

Two pipes A and B together can fill a cistern in 5 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

$${Two}\:{pipes}\:{A}\:{and}\:{B}\:{together}\:{can}\:{fill}\: \\ $$$${a}\:{cistern}\:{in}\:\mathrm{5}\:{hours}.\:{Had}\:{they}\:{been}\:{opened} \\ $$$${separately},\:{then}\:{B}\:{would}\:{have}\:{taken}\: \\ $$$$\mathrm{6}\:{hours}\:{more}\:{than}\:{A}\:{to}\:{fill}\:{the}\:{cistern}. \\ $$$${How}\:{much}\:{time}\:{will}\:{be}\:{taken}\:{by}\:{A}\:{to}\:{fill} \\ $$$${the}\:{cistern}\:{separately}?\: \\ $$

Question Number 126990 Answers: 1 Comments: 0

Obtain a formula for I_n = ∫_0 ^n [x] dx in terms of n where [x] is the greatest integer function of x.

$$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{formula}\:\mathrm{for}\: \\ $$$$\:{I}_{{n}} \:=\:\underset{\mathrm{0}} {\overset{{n}} {\int}}\left[{x}\right]\:{dx}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n} \\ $$$$\:\mathrm{where}\:\left[{x}\right]\:\mathrm{is}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}\:\mathrm{of}\:{x}. \\ $$

Question Number 126989 Answers: 1 Comments: 0

∫e^(√x) dx

$$\int{e}^{\sqrt{{x}}} {dx} \\ $$

Question Number 126966 Answers: 0 Comments: 2

Question Number 126952 Answers: 3 Comments: 1

Σ_(n=1) ^(10000) (1/(n(n+1)))=???

$$\underset{{n}=\mathrm{1}} {\overset{\mathrm{10000}} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}=??? \\ $$

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