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AlgebraQuestion and Answers: Page 93

Question Number 181099 Answers: 2 Comments: 0

prove that for every positivenumber p e q wee have: p+q≥(√(4pq))

$$ \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{every}\: \\ $$$$\mathrm{positivenumber}\:\mathrm{p}\:\mathrm{e}\:\mathrm{q}\:\mathrm{wee} \\ $$$$\mathrm{hav}{e}: \\ $$$${p}+{q}\geqslant\sqrt{\mathrm{4}{pq}} \\ $$

Question Number 181089 Answers: 1 Comments: 0

Question Number 181079 Answers: 6 Comments: 2

Solve for x : (((x + a)/(x + b)))^3 = ((x + 2a − b)/(x − a + 2b))

$$\mathrm{Solve}\:\mathrm{for}\:{x}\:: \\ $$$$\left(\frac{{x}\:+\:{a}}{{x}\:+\:{b}}\right)^{\mathrm{3}} =\:\frac{{x}\:+\:\mathrm{2}{a}\:−\:{b}}{{x}\:−\:{a}\:+\:\mathrm{2}{b}} \\ $$

Question Number 181041 Answers: 0 Comments: 0

If f(x)= x − ⌊ ((x−1)/3) ⌋ , g( x)= 2^( x) then , R_( gof) = ?

$$ \\ $$$$\:\:\:\mathrm{I}{f}\:\:\:\:{f}\left({x}\right)=\:{x}\:−\:\lfloor\:\frac{{x}−\mathrm{1}}{\mathrm{3}}\:\rfloor\:,\:\:{g}\left(\:{x}\right)=\:\mathrm{2}^{\:{x}} \\ $$$$\:\:\:\:\:\:\:{then}\:\:,\:\:\:\:\:\:{R}_{\:{gof}} \:=\:? \\ $$

Question Number 181020 Answers: 1 Comments: 0

solve for x>0 ∫_0 ^x ⌊t⌋^2 dt=2(x−1) (Q180780 reposted)

$${solve}\:{for}\:{x}>\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{{x}} \lfloor{t}\rfloor^{\mathrm{2}} {dt}=\mathrm{2}\left({x}−\mathrm{1}\right) \\ $$$$ \\ $$$$\left({Q}\mathrm{180780}\:{reposted}\right) \\ $$

Question Number 181007 Answers: 1 Comments: 0

[Question from Frix sir] ... I once played the game Tetris. You got points for each filled line, depending on how many lines you filled at once: 1 line = 1 point 2 lines = 3 points 3 lines = 6 points 4 lines = 10 points If you made 100 points, which possible combinations of lines did you fill when you filled 31 lines?

$$\left[{Question}\:{from}\:{Frix}\:{sir}\right] \\ $$$$...\:\mathrm{I}\:\mathrm{once}\:\mathrm{played}\:\mathrm{the}\:\mathrm{game}\:\mathrm{Tetris}.\: \\ $$$$\mathrm{You}\:\mathrm{got}\:\mathrm{points}\:\mathrm{for}\:\mathrm{each}\:\mathrm{filled}\:\mathrm{line},\: \\ $$$$\mathrm{depending}\:\mathrm{on}\:\mathrm{how}\:\mathrm{many}\:\mathrm{lines}\:\mathrm{you}\: \\ $$$$\mathrm{filled}\:\mathrm{at}\:\mathrm{once}: \\ $$$$\mathrm{1}\:\mathrm{line}\:=\:\mathrm{1}\:\mathrm{point} \\ $$$$\mathrm{2}\:\mathrm{lines}\:=\:\mathrm{3}\:\mathrm{points} \\ $$$$\mathrm{3}\:\mathrm{lines}\:=\:\mathrm{6}\:\mathrm{points} \\ $$$$\mathrm{4}\:\mathrm{lines}\:=\:\mathrm{10}\:\mathrm{points} \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{made}\:\mathrm{100}\:\mathrm{points},\:\mathrm{which}\:\mathrm{possible} \\ $$$$\mathrm{combinations}\:\mathrm{of}\:\mathrm{lines}\:\mathrm{did}\:\mathrm{you}\:\mathrm{fill}\:\mathrm{when} \\ $$$$\mathrm{you}\:\mathrm{filled}\:\mathrm{31}\:\mathrm{lines}? \\ $$

Question Number 180977 Answers: 3 Comments: 0

what is larger, 3^(100) +4^(100) or 5^(100) ?

$${what}\:{is}\:{larger},\:\mathrm{3}^{\mathrm{100}} +\mathrm{4}^{\mathrm{100}} \:{or}\:\:\mathrm{5}^{\mathrm{100}} ? \\ $$

Question Number 180953 Answers: 2 Comments: 0

What′s the smallest value of a^2 +b^2 +(1/(ab)) for a, b>0?

$${What}'{s}\:{the}\:{smallest}\:{value}\:{of} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\frac{\mathrm{1}}{{ab}}\:{for}\:{a},\:{b}>\mathrm{0}? \\ $$

Question Number 181066 Answers: 1 Comments: 1

Find the value of the expression: 2arcctg(tg3) − 3arctg(ctg2) a)2π−3 b)12 c)−6 d)π−4/2 e)3π/2

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}: \\ $$$$\mathrm{2arcctg}\left(\mathrm{tg3}\right)\:−\:\mathrm{3arctg}\left(\mathrm{ctg2}\right) \\ $$$$\left.\mathrm{a}\left.\right)\left.\mathrm{2}\left.\pi\left.−\mathrm{3}\:\:\mathrm{b}\right)\mathrm{12}\:\:\mathrm{c}\right)−\mathrm{6}\:\:\mathrm{d}\right)\pi−\mathrm{4}/\mathrm{2}\:\:\mathrm{e}\right)\mathrm{3}\pi/\mathrm{2} \\ $$

Question Number 180943 Answers: 2 Comments: 0

if: a , bc^(−) = 1 + (1/2) + (3/(50)) find: a+b+c=?

$$\mathrm{if}:\:\:\:\overline {\mathrm{a}\:,\:\mathrm{bc}}\:=\:\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{3}}{\mathrm{50}} \\ $$$$\mathrm{find}:\:\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=? \\ $$

Question Number 180938 Answers: 2 Comments: 0

ab^(−) + ba^(−) = c3^(−) find: a+b+2c=?

$$\overline {\mathrm{ab}}\:+\:\overline {\mathrm{ba}}\:=\:\overline {\mathrm{c3}} \\ $$$$\mathrm{find}:\:\:\:\mathrm{a}+\mathrm{b}+\mathrm{2c}=? \\ $$

Question Number 180923 Answers: 1 Comments: 0

Question Number 180889 Answers: 0 Comments: 1

the average of 10 numbers is 5. the sum of their squares is 5000. how large can the largest number among them at most be and how small can the smallest number among them at most be?

$${the}\:{average}\:{of}\:\mathrm{10}\:{numbers}\:{is}\:\mathrm{5}.\:{the} \\ $$$${sum}\:{of}\:{their}\:{squares}\:{is}\:\mathrm{5000}.\:{how}\: \\ $$$${large}\:{can}\:{the}\:{largest}\:{number}\:{among} \\ $$$${them}\:{at}\:{most}\:{be}\:{and}\:{how}\:{small}\:{can}\: \\ $$$${the}\:{smallest}\:{number}\:{among}\:{them} \\ $$$${at}\:{most}\:{be}? \\ $$

Question Number 180873 Answers: 1 Comments: 2

If a,b,c<0 and abc(a+b+c)=64 Then find min of P=2a+b+c

$$\mathrm{If}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c}<\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{abc}\left(\mathrm{a}+\mathrm{b}+\mathrm{c}\right)=\mathrm{64} \\ $$$$\mathrm{Then}\:\mathrm{find}\:\mathrm{min}\:\mathrm{of}\:\:\:\mathrm{P}=\mathrm{2a}+\mathrm{b}+\mathrm{c} \\ $$

Question Number 180896 Answers: 0 Comments: 1

find the maximum of Σ_(i=1) ^(100) sin^3 x_i if Σ_(i=1) ^(100) sin x_i =0.

$${find}\:{the}\:{maximum}\:{of}\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{100}} {\sum}}\mathrm{sin}^{\mathrm{3}} \:{x}_{{i}} \\ $$$${if}\:\underset{{i}=\mathrm{1}} {\overset{\mathrm{100}} {\sum}}\mathrm{sin}\:{x}_{{i}} =\mathrm{0}. \\ $$

Question Number 180856 Answers: 1 Comments: 0

find all values of m∈R such that the equation: ∫_0 ^( x) ((arctany)/y) dy = mx has two real roots: x_1 ∈(−∞;0) , x_2 ∈(0;∞)

$$\mathrm{find}\:\mathrm{all}\:\mathrm{values}\:\mathrm{of}\:\:\mathrm{m}\in\mathbb{R}\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\boldsymbol{\mathrm{x}}} \:\frac{\mathrm{arctan}\boldsymbol{\mathrm{y}}}{\mathrm{y}}\:\mathrm{dy}\:=\:\mathrm{mx} \\ $$$$\mathrm{has}\:\mathrm{two}\:\mathrm{real}\:\mathrm{roots}:\:\:\:\mathrm{x}_{\mathrm{1}} \in\left(−\infty;\mathrm{0}\right)\:,\:\mathrm{x}_{\mathrm{2}} \in\left(\mathrm{0};\infty\right) \\ $$

Question Number 180894 Answers: 3 Comments: 2

x^3 +x=1 x^8 +3x^3 =?