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AlgebraQuestion and Answers: Page 327

Question Number 44691 Answers: 1 Comments: 1

Question Number 44570 Answers: 1 Comments: 0

Question Number 44543 Answers: 1 Comments: 3

If y =f(x) = ax^2 +bx+c and at some x, say x= p ∫_0 ^( p) ydx = y(p)= y ′(p) = y ′′(p)= p , then find p .

$${If}\:\:{y}\:={f}\left({x}\right)\:=\:{ax}^{\mathrm{2}} +{bx}+{c} \\ $$$${and}\:\:{at}\:{some}\:{x},\:{say}\:\:{x}=\:{p} \\ $$$$\int_{\mathrm{0}} ^{\:\:{p}} {ydx}\:=\:{y}\left({p}\right)=\:{y}\:'\left({p}\right)\:=\:{y}\:''\left({p}\right)=\:{p}\:, \\ $$$${then}\:{find}\:\boldsymbol{{p}}\:. \\ $$

Question Number 44502 Answers: 1 Comments: 0

If a>b,and c>d,prove that a−c may be greater than, equal to or less than b−d.

$$\mathrm{If}\:\mathrm{a}>\mathrm{b},\mathrm{and}\:\mathrm{c}>\mathrm{d},\mathrm{prove}\:\mathrm{that}\:\mathrm{a}−\mathrm{c}\:\mathrm{may}\:\mathrm{be}\:\mathrm{greater}\:\mathrm{than}, \\ $$$$\mathrm{equal}\:\mathrm{to}\:\mathrm{or}\:\mathrm{less}\:\mathrm{than}\:\mathrm{b}−\mathrm{d}. \\ $$$$ \\ $$

Question Number 44444 Answers: 1 Comments: 0

simplify (√((4x^2 y)^(2/3) +(8x^2 y^2 )^4 ))

$${simplify}\:\:\:\:\sqrt{\left(\mathrm{4}{x}^{\mathrm{2}} {y}\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\left(\mathrm{8}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \right)^{\mathrm{4}} } \\ $$

Question Number 44397 Answers: 2 Comments: 4

If x is nearly equal to 1 then ((mx^m −nx^n )/(m−n))=

$${If}\:{x}\:{is}\:{nearly}\:{equal}\:{to}\:\mathrm{1}\:{then} \\ $$$$\frac{{mx}^{{m}} −{nx}^{{n}} }{{m}−{n}}= \\ $$

Question Number 44384 Answers: 2 Comments: 6

Let a and b are real numbers such that a > b > 0 Find the minimum value of (√2)a^3 + (3/(ab − b^2 ))

$$\mathrm{Let}\:{a}\:\mathrm{and}\:{b}\:\mathrm{are}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such}\:\mathrm{that} \\ $$$${a}\:>\:{b}\:>\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\sqrt{\mathrm{2}}{a}^{\mathrm{3}} \:+\:\frac{\mathrm{3}}{{ab}\:−\:{b}^{\mathrm{2}} } \\ $$

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Question Number 44200 Answers: 0 Comments: 4

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Question Number 44062 Answers: 1 Comments: 0

Question Number 44038 Answers: 2 Comments: 0

How many times does the digit 6 appear when writing from 6 to 400 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{does}\:\mathrm{the}\:\mathrm{digit}\:\mathrm{6}\:\mathrm{appear}\:\mathrm{when}\:\mathrm{writing}\:\mathrm{from}\:\:\mathrm{6}\:\mathrm{to}\:\mathrm{400}\:? \\ $$

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Question Number 43971 Answers: 1 Comments: 0

Question Number 43902 Answers: 2 Comments: 4

(√(a−b)) + (√(a+b)) = c (√(a−c)) + (√(a+c)) = b Solve for real b, and c ; in terms of real a.

$$\sqrt{{a}−{b}}\:+\:\sqrt{{a}+{b}}\:=\:{c} \\ $$$$\sqrt{{a}−{c}}\:+\:\sqrt{{a}+{c}}\:=\:{b} \\ $$$${Solve}\:{for}\:{real}\:{b},\:{and}\:{c}\:;\:{in}\:{terms}\: \\ $$$${of}\:{real}\:{a}. \\ $$

Question Number 43894 Answers: 0 Comments: 0

∣z∣=∣Arg ((a/b)π)∣=1∧k, n∈Z∧b≠0≤k<n: x^n =z⇒x=e^(((2k+a)/(bm))πi) To prove that, please.

$$\mid{z}\mid=\mid{Arg}\:\left(\frac{{a}}{{b}}\pi\right)\mid=\mathrm{1}\wedge{k},\:{n}\in\mathbb{Z}\wedge{b}\neq\mathrm{0}\leqslant{k}<{n}: \\ $$$${x}^{{n}} ={z}\Rightarrow{x}={e}^{\frac{\mathrm{2}{k}+{a}}{{bm}}\pi{i}} \\ $$$$\mathrm{To}\:\mathrm{prove}\:\mathrm{that},\:\mathrm{please}. \\ $$

Question Number 43804 Answers: 1 Comments: 0

solve for ε s(1−α)=(1−ε)σT^4

$${solve}\:{for}\:\epsilon \\ $$$$ \\ $$$${s}\left(\mathrm{1}−\alpha\right)=\left(\mathrm{1}−\epsilon\right)\sigma{T}^{\mathrm{4}} \\ $$

Question Number 43759 Answers: 2 Comments: 0

Question Number 43757 Answers: 0 Comments: 0

Probably if x^n =Am ((a/b)π), x=e^(((2k+a)/(bn))iπ) about 0<(k∈N∪{0})<(n∈N) and b≠0. p.s. Am (0°)=1, Am (90°)=i etc., and s°=(π/(180))s rad(ians)=(π/(180))s.

$$\mathrm{Probably}\:\mathrm{if}\:{x}^{{n}} ={Am}\:\left(\frac{{a}}{{b}}\pi\right),\:{x}={e}^{\frac{\mathrm{2}{k}+{a}}{{bn}}{i}\pi} \\ $$$$\mathrm{about}\:\mathrm{0}<\left({k}\in\mathbb{N}\cup\left\{\mathrm{0}\right\}\right)<\left({n}\in\mathbb{N}\right)\:\mathrm{and}\:{b}\neq\mathrm{0}. \\ $$$$\mathrm{p}.\mathrm{s}.\:{Am}\:\left(\mathrm{0}°\right)=\mathrm{1},\:{Am}\:\left(\mathrm{90}°\right)={i}\:\mathrm{etc}., \\ $$$$\mathrm{and}\:{s}°=\frac{\pi}{\mathrm{180}}{s}\:\mathrm{rad}\left(\mathrm{ians}\right)=\frac{\pi}{\mathrm{180}}{s}. \\ $$

Question Number 43756 Answers: 1 Comments: 0

x^3 +px+q = 0 If equation has all its roots real, find them.

$$\:\:\boldsymbol{{x}}^{\mathrm{3}} +\boldsymbol{{px}}+\boldsymbol{{q}}\:=\:\mathrm{0} \\ $$$$\boldsymbol{{If}}\:\boldsymbol{{equation}}\:\boldsymbol{{has}}\:\boldsymbol{{all}}\:\boldsymbol{{its}}\:\boldsymbol{{roots}} \\ $$$$\boldsymbol{{real}},\:\boldsymbol{{find}}\:\boldsymbol{{them}}. \\ $$

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Question Number 43716 Answers: 1 Comments: 3

Question Number 43707 Answers: 1 Comments: 3

Simplify: (x + y + z)(x^(−1) + y^(−1) + z^(−1) ) = (x^(−1) y^(−1) z^(−1) )(x + y)(y + z)(z + x)

$$\mathrm{Simplify}:\:\:\: \\ $$$$\left(\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\right)\left(\mathrm{x}^{−\mathrm{1}} \:+\:\mathrm{y}^{−\mathrm{1}} \:+\:\mathrm{z}^{−\mathrm{1}} \right)\:=\:\left(\mathrm{x}^{−\mathrm{1}} \:\mathrm{y}^{−\mathrm{1}} \:\mathrm{z}^{−\mathrm{1}} \right)\left(\mathrm{x}\:+\:\mathrm{y}\right)\left(\mathrm{y}\:+\:\mathrm{z}\right)\left(\mathrm{z}\:+\:\mathrm{x}\right) \\ $$

Question Number 43706 Answers: 1 Comments: 2

If pqr = 1 Hence evaluate: (1/(1 + e + f^(−1) )) + (1/(1 + f + g^(−1) )) + (1/(1 + g + e^(−1) ))

$$\mathrm{If}\:\:\mathrm{pqr}\:=\:\mathrm{1} \\ $$$$\mathrm{Hence}\:\mathrm{evaluate}:\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{e}\:+\:\mathrm{f}^{−\mathrm{1}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{f}\:+\:\mathrm{g}^{−\mathrm{1}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{g}\:+\:\mathrm{e}^{−\mathrm{1}} } \\ $$

Question Number 43705 Answers: 0 Comments: 0

Prove that to each quadratic factor in the denominator of the form ax^2 + bx + c which does not have linear factors, there corresponds to a partial fraction of the form ((Ax + B)/(ax^2 + bx + c)) where A and B are constant.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{to}\:\mathrm{each}\:\mathrm{quadratic}\:\mathrm{factor}\:\mathrm{in}\:\mathrm{the}\:\mathrm{denominator}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\: \\ $$$$\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{bx}\:+\:\mathrm{c}\:\:\:\mathrm{which}\:\mathrm{does}\:\mathrm{not}\:\mathrm{have}\:\mathrm{linear}\:\mathrm{factors},\:\mathrm{there}\:\mathrm{corresponds}\:\mathrm{to} \\ $$$$\mathrm{a}\:\mathrm{partial}\:\mathrm{fraction}\:\mathrm{of}\:\mathrm{the}\:\mathrm{form}\:\:\:\:\frac{\mathrm{Ax}\:+\:\mathrm{B}}{\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{bx}\:+\:\mathrm{c}}\:\:\:\mathrm{where}\:\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{constant}. \\ $$

Question Number 43702 Answers: 1 Comments: 0

simplify [((12^(1/5) )/(27^(1/5) ))]^(5/2)

$${simplify}\:\:\:\left[\frac{\mathrm{12}^{\mathrm{1}/\mathrm{5}} }{\mathrm{27}^{\mathrm{1}/\mathrm{5}} }\right]^{\mathrm{5}/\mathrm{2}} \\ $$

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