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AlgebraQuestion and Answers: Page 224

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)))=???

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

Question Number 126950 Answers: 1 Comments: 0

Question Number 126935 Answers: 0 Comments: 0

Prove or give a counter example: (a+1)^(n−1) =Σ_(k=1) ^n ( _(k−1) ^(n−1) )a^(k−1) ( _r ^n )=((n!)/(r!(n−r)!))

$${Prove}\:{or}\:{give}\:{a}\:{counter}\:{example}: \\ $$$$\left({a}+\mathrm{1}\right)^{{n}−\mathrm{1}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\underset{{k}−\mathrm{1}} {\overset{{n}−\mathrm{1}} {\:}}\right){a}^{{k}−\mathrm{1}} \\ $$$$\left(\underset{{r}} {\overset{{n}} {\:}}\right)=\frac{{n}!}{{r}!\left({n}−{r}\right)!} \\ $$

Question Number 126931 Answers: 1 Comments: 0

y=(1−x)^(cosx)

$${y}=\left(\mathrm{1}−{x}\right)^{{cosx}} \\ $$

Question Number 126930 Answers: 1 Comments: 4

(√(1997×1996×1995×1994+1)) =?

$$\:\:\sqrt{\mathrm{1997}×\mathrm{1996}×\mathrm{1995}×\mathrm{1994}+\mathrm{1}}\:=? \\ $$

Question Number 126887 Answers: 2 Comments: 2

... calculus... please solve: ( with explanation) {_(x^2 +y=3) ^(x+y^2 =5)

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{calculus}... \\ $$$${please}\:\:{solve}:\:\left(\:{with}\:{explanation}\right) \\ $$$$\:\:\:\:\left\{_{{x}^{\mathrm{2}} +{y}=\mathrm{3}} ^{{x}+{y}^{\mathrm{2}} =\mathrm{5}} \right. \\ $$$$ \\ $$

Question Number 126882 Answers: 1 Comments: 0

x^x =3

$$\boldsymbol{{x}}^{\boldsymbol{{x}}} =\mathrm{3} \\ $$

Question Number 126845 Answers: 3 Comments: 0

... calculus (I)... prove :: i:: ⌊2x⌋=^? ⌊x⌋+⌊x+(1/2)⌋ ii:: ⌊3x⌋=⌊x⌋+⌊x+(1/3)⌋+⌊x+(2/3)⌋

$$\:\:\:\:\:\:\:\:\:\:\:...\:{calculus}\:\:\left(\mathrm{I}\right)... \\ $$$$\:\:\:\:{prove}\:::\: \\ $$$$\:\:\:\:\:\:\:{i}::\:\:\lfloor\mathrm{2}{x}\rfloor\overset{?} {=}\lfloor{x}\rfloor+\lfloor{x}+\frac{\mathrm{1}}{\mathrm{2}}\rfloor \\ $$$$\:\:\:\:\:\:\:{ii}::\:\lfloor\mathrm{3}{x}\rfloor=\lfloor{x}\rfloor+\lfloor{x}+\frac{\mathrm{1}}{\mathrm{3}}\rfloor+\lfloor{x}+\frac{\mathrm{2}}{\mathrm{3}}\rfloor \\ $$$$ \\ $$

Question Number 126723 Answers: 0 Comments: 2

Question Number 126702 Answers: 1 Comments: 0

if tanh(x/2)=t prove that cosh(x)=((1+t^2 )/(1−t^2 ))

$${if}\:\:\:\:{tanh}\frac{{x}}{\mathrm{2}}={t}\:\:{prove}\:{that}\:\:{cosh}\left({x}\right)=\frac{\mathrm{1}+{t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 126701 Answers: 2 Comments: 0

use right triangles to explain why cos^(−1) (x)+sin^(−1) (x)=π/2

$${use}\:{right}\:{triangles}\:{to}\:{explain} \\ $$$${why}\:{cos}^{−\mathrm{1}} \left({x}\right)+{sin}^{−\mathrm{1}} \left({x}\right)=\pi/\mathrm{2} \\ $$

Question Number 126700 Answers: 2 Comments: 0

θ=sin^(−1) ((2/5)) find cos(θ) and tan(θ)

$$\theta={sin}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{\mathrm{5}}\right)\:{find}\:{cos}\left(\theta\right)\:{and}\:{tan}\left(\theta\right) \\ $$$$ \\ $$

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