Question and Answers Forum

AllQuestion and Answers: Page 369

Question Number 181152 Answers: 0 Comments: 0

solve the integral ∫_0 ^( 1) ((ln(1+x)ln(1−x))/(1+x))dx=???

$$\boldsymbol{{solve}}\:\boldsymbol{{the}}\:\boldsymbol{{integral}} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\boldsymbol{{ln}}\left(\mathrm{1}+\boldsymbol{{x}}\right)\boldsymbol{{ln}}\left(\mathrm{1}−\boldsymbol{{x}}\right)}{\mathrm{1}+\boldsymbol{{x}}}\boldsymbol{{dx}}=??? \\ $$

Question Number 181151 Answers: 4 Comments: 0

Question Number 181196 Answers: 0 Comments: 0

Let a_1 , a_2 , a_3 , ...a_(2022) be numbers ranging from (0, +∞) \ {1}, for which the function f : R→R is defined as f(x)=a_1 ^x +a_2 ^x +a_3 ^x +...a_(2022) ^x . If f(2022)=f(−2022)=2022 prove that this function is constant.

$${Let}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:...{a}_{\mathrm{2022}} \:{be}\:{numbers} \\ $$$${ranging}\:{from}\:\left(\mathrm{0},\:+\infty\right)\:\backslash\:\left\{\mathrm{1}\right\},\:{for}\:{which} \\ $$$${the}\:{function}\:{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:{is}\:{defined}\:{as} \\ $$$${f}\left({x}\right)={a}_{\mathrm{1}} ^{{x}} +{a}_{\mathrm{2}} ^{{x}} +{a}_{\mathrm{3}} ^{{x}} +...{a}_{\mathrm{2022}} ^{{x}} . \\ $$$${If}\:{f}\left(\mathrm{2022}\right)={f}\left(−\mathrm{2022}\right)=\mathrm{2022}\:{prove} \\ $$$${that}\:{this}\:{function}\:{is}\:{constant}. \\ $$

Question Number 181140 Answers: 0 Comments: 6

I watch a favorite TV program daily for 30 min. If there were ads every 3 hours, what′s the probability that i will see ads once again the next day while watching that program?

$${I}\:{watch}\:{a}\:{favorite}\:{TV}\:{program}\:{daily}\:{for}\:\mathrm{30}\:{min}. \\ $$$$\:{If}\:{there}\:{were}\:{ads}\:{every}\:\mathrm{3}\:{hours},\:{what}'{s}\:{the} \\ $$$$\:{probability}\:{that}\:{i}\:{will}\:{see}\:{ads}\:{once}\:{again}\:{the}\:{next} \\ $$$$\:{day}\:{while}\:{watching}\:{that}\:{program}? \\ $$$$ \\ $$

Question Number 181138 Answers: 1 Comments: 0

If a + b + c = 0 then, (1/(2a^2 + bc)) + (1/(2b^2 + ca)) + (1/(2c^2 + ab)) = ?

$$\mathrm{If}\:{a}\:+\:{b}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{then}, \\ $$$$\frac{\mathrm{1}}{\mathrm{2}{a}^{\mathrm{2}} \:+\:{bc}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{b}^{\mathrm{2}} \:+\:{ca}}\:+\:\frac{\mathrm{1}}{\mathrm{2}{c}^{\mathrm{2}} \:+\:{ab}}\:=\:? \\ $$

Question Number 181153 Answers: 2 Comments: 0

Question Number 181130 Answers: 0 Comments: 0

Question Number 181128 Answers: 0 Comments: 2

Question Number 181154 Answers: 1 Comments: 0

(log_(15) 5)^2 +(log_(15) 3)(log_(15) 75)=?

$$\left({log}_{\mathrm{15}} \mathrm{5}\right)^{\mathrm{2}} +\left({log}_{\mathrm{15}} \mathrm{3}\right)\left({log}_{\mathrm{15}} \mathrm{75}\right)=? \\ $$$$ \\ $$

Question Number 181125 Answers: 1 Comments: 3

Let the acute triangle ΔABC have an outer circumscribed circle, whose tangents at the points B and C intersect at point P. Let D and E be the projections of perpendicular lines from point P on AC and AB. Prove that the interdection point of the heights of ΔADE is the midpoint of BC

$${Let}\:{the}\:{acute}\:{triangle}\:\Delta{ABC}\:\:{have} \\ $$$${an}\:{outer}\:{circumscribed}\:{circle}, \\ $$$${whose}\:{tangents}\:{at}\:{the}\:{points}\:{B}\:{and}\:{C} \\ $$$${intersect}\:{at}\:{point}\:{P}.\:{Let}\:{D}\:{and}\:{E}\:{be} \\ $$$${the}\:{projections}\:{of}\:{perpendicular} \\ $$$${lines}\:{from}\:{point}\:{P}\:{on}\:{AC}\:{and}\:{AB}. \\ $$$${Prove}\:{that}\:{the}\:{interdection}\:{point}\:{of} \\ $$$${the}\:{heights}\:{of}\:\Delta{ADE}\:{is}\:{the}\:{midpoint} \\ $$$${of}\:{BC} \\ $$

Question Number 181144 Answers: 3 Comments: 0

Solve for x : (((3x − 28)/(3x − 26)))^3 = ((x − 10)/(x − 8))

$$\mathrm{Solve}\:\mathrm{for}\:{x}\:: \\ $$$$\left(\frac{\mathrm{3}{x}\:−\:\mathrm{28}}{\mathrm{3}{x}\:−\:\mathrm{26}}\right)^{\mathrm{3}} \:=\:\frac{{x}\:−\:\mathrm{10}}{{x}\:−\:\mathrm{8}} \\ $$

Question Number 181104 Answers: 2 Comments: 1

lim_(x→∞) ((ln (1+(4/x)))/(π−arctan (2x))) =?

$$\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{4}}{\mathrm{x}}\right)}{\pi−\mathrm{arctan}\:\left(\mathrm{2x}\right)}\:=?\: \\ $$

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 181085 Answers: 1 Comments: 3

Question Number 181084 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 181070 Answers: 2 Comments: 2

Question Number 181062 Answers: 0 Comments: 0

Q180886

$${Q}\mathrm{180886} \\ $$

Question Number 181055 Answers: 2 Comments: 14

Question Number 181052 Answers: 2 Comments: 3

Question Number 181051 Answers: 1 Comments: 0

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 181039 Answers: 0 Comments: 3

Prove that for all a^2 divisible by b, a is also divisible by b for all a, b ∈ N

$${Prove}\:{that}\:{for}\:{all}\:{a}^{\mathrm{2}} \:{divisible}\:{by}\:{b}, \\ $$$${a}\:{is}\:{also}\:{divisible}\:{by}\:{b} \\ $$$${for}\:{all}\:{a},\:{b}\:\in\:\mathbb{N} \\ $$

Question Number 181053 Answers: 1 Comments: 0

An open pipe of r_o = 10 Cm, ℓ= 3m has an outer layer of ice that is melting at the rate of 2π Cm^3 per minute with thickness of 20 mm. How many days untill all the ice melts? and how fast is the thickness of the ice decreasing per hour?

$${An}\:{open}\:{pipe}\:{of}\:{r}_{{o}} =\:\mathrm{10}\:{Cm},\:\ell=\:\mathrm{3}{m}\:{has}\:{an}\:{outer} \\ $$$$\:{layer}\:{of}\:{ice}\:{that}\:{is}\:{melting}\:{at}\:{the}\:{rate}\:{of}\:\mathrm{2}\pi\:{Cm}^{\mathrm{3}} \\ $$$$\:{per}\:{minute}\:{with}\:{thickness}\:{of}\:\mathrm{20}\:{mm}.\:{How}\:{many} \\ $$$$\:{days}\:{untill}\:{all}\:{the}\:{ice}\:{melts}?\:{and}\:{how}\:{fast}\:{is}\:{the} \\ $$$$\:{thickness}\:{of}\:{the}\:{ice}\:{decreasing}\:{per}\:{hour}? \\ $$

Question Number 181025 Answers: 0 Comments: 1

Send you solutions to kinmatics@gmail.com

$$\mathrm{Send}\:\mathrm{you}\:\mathrm{solutions}\:\mathrm{to}\:\boldsymbol{\mathrm{kinmatics}}@\boldsymbol{\mathrm{gmail}}.\boldsymbol{\mathrm{com}} \\ $$

Pg 364 Pg 365 Pg 366 Pg 367 Pg 368 Pg 369 Pg 370 Pg 371 Pg 372 Pg 373

Contact: info@tinkutara.com