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Question Number 173908    Answers: 0   Comments: 6

find the value of b so that the line y=b divides the region bound by the graphs of the two functinos , into two regions of equal area. f(x)=9−x^2 and g(x)=0

$${find}\:{the}\:{value}\:{of}\:{b}\:{so}\:{that}\:{the}\:{line}\:{y}={b} \\ $$$${divides}\:{the}\:{region}\:{bound}\:{by}\:{the}\:{graphs}\:{of} \\ $$$${the}\:{two}\:{functinos}\:,\:{into}\:{two}\:{regions}\:{of}\:{equal} \\ $$$${area}. \\ $$$${f}\left({x}\right)=\mathrm{9}−{x}^{\mathrm{2}} \:{and}\:{g}\left({x}\right)=\mathrm{0} \\ $$

Question Number 173956    Answers: 0   Comments: 1

Question Number 173909    Answers: 0   Comments: 2

Ω=∫_0 ^( 1) (( ln^( 2) (1−x))/((1+x )^( 2) )) = Li_2 ((1/2) ) −−− Solution −−− Ω =^(i.b.p) {[−(1/(1+x))ln^( 2) (1−x)]_0 ^1 −2∫_0 ^( 1) ((ln(1−x))/((1−x)(1+x)))dx =lim_( x→1^− ) −(1/(1+x)) ln^( 2) (1−x)−∫_0 ^( 1) ((ln(1−x))/(1−x)) +((ln(1−x))/(1+x))dx = lim_( x→1^− ) (1/2) ln^( 2) (1−x)−(1/(1+x))ln^( 2) (1−x)−Φ =lim_( x→1^( −) ) (((x−1)/(2(1+x))))ln^( 2) (1−x)−Φ = −Φ =_(derived) ^(earlier) −(−(π^( 2) /(12)) +(1/2)ln^( 2) (2)) ⇒ Ω = (π^( 2) /(12)) −(1/2) ln^( 2) (2 )=Li_2 ((1/2))

$$ \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}\right)}{\left(\mathrm{1}+{x}\:\right)^{\:\mathrm{2}} }\:=\:{Li}_{\mathrm{2}} \:\left(\frac{\mathrm{1}}{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:−−−\:\:\:{Solution}\:−−− \\ $$$$\:\:\:\:\Omega\:\overset{{i}.{b}.{p}} {=}\left\{\left[−\frac{\mathrm{1}}{\mathrm{1}+{x}}{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} −\mathrm{2}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right)}{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+{x}\right)}{dx}\right. \\ $$$$\:\:\:\:\:={lim}_{\:{x}\rightarrow\mathrm{1}^{−} } −\frac{\mathrm{1}}{\mathrm{1}+{x}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}\right)−\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}−{x}}\:+\frac{{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}+{x}}{dx} \\ $$$$\:\:\:\:\:=\:{lim}_{\:{x}\rightarrow\mathrm{1}^{−} } \frac{\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}\right)−\frac{\mathrm{1}}{\mathrm{1}+{x}}{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}\right)−\Phi \\ $$$$\:\:\:\:={lim}_{\:{x}\rightarrow\mathrm{1}^{\:−} } \:\left(\frac{{x}−\mathrm{1}}{\mathrm{2}\left(\mathrm{1}+{x}\right)}\right){ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}\right)−\Phi \\ $$$$\:\:\:\:\:=\:−\Phi\:\underset{{derived}} {\overset{{earlier}} {=}}\:−\left(−\frac{\pi^{\:\mathrm{2}} }{\mathrm{12}}\:+\frac{\mathrm{1}}{\mathrm{2}}{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\right) \\ $$$$\Rightarrow\:\:\Omega\:=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{12}}\:−\frac{\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{2}\:\right)={Li}_{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$$$ \\ $$

Question Number 173902    Answers: 1   Comments: 0

Q: How many common three−digit numbers are there in the following two sequences? { (( a_n = 1 , 5 , 9 ,13 , ...)),(( b_( m) = 4 , 7 , 10 , 13 ,...)) :}

$$ \\ $$$$\:\:{Q}:\:{How}\:{many}\:{common}\:{three}−{digit}\:{numbers} \\ $$$$\:\:\:\:{are}\:{there}\:{in}\:{the}\:{following} \\ $$$$\:\:\:\:\:{two}\:{sequences}? \\ $$$$\:\:\:\:\:\begin{cases}{\:\:{a}_{{n}} \:=\:\mathrm{1}\:\:,\:\mathrm{5}\:,\:\mathrm{9}\:,\mathrm{13}\:,\:...}\\{\:\:{b}_{\:{m}} \:=\:\mathrm{4}\:,\:\mathrm{7}\:,\:\mathrm{10}\:,\:\mathrm{13}\:,...}\end{cases} \\ $$

Question Number 173893    Answers: 1   Comments: 1

if ∫(x) = sinx than prove that, {∫(x)^4 } + {∫(x)}^2 = 1

$$\mathrm{if}\:\int\left(\mathrm{x}\right)\:=\:\mathrm{sinx}\:\mathrm{than}\:\mathrm{prove}\:\mathrm{that}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left\{\int\left(\mathrm{x}\right)^{\mathrm{4}} \right\}\:+\:\left\{\int\left(\mathrm{x}\right)\right\}^{\mathrm{2}} \:=\:\mathrm{1} \\ $$

Question Number 173894    Answers: 2   Comments: 2

If secA − tanA = Q than prove that, cosecA = ((1 + Q^2 )/(1 − Q^2 ))

$$\mathrm{If}\:\:\mathrm{secA}\:−\:\mathrm{tanA}\:=\:\mathrm{Q}\:\mathrm{than}\:\mathrm{prove}\:\mathrm{that},\: \\ $$$$\:\:\:\:\:\:\:\mathrm{cosecA}\:=\:\frac{\mathrm{1}\:+\:\mathrm{Q}^{\mathrm{2}} }{\mathrm{1}\:−\:\mathrm{Q}^{\mathrm{2}} }\: \\ $$

Question Number 173889    Answers: 0   Comments: 1

the anser is the folowing

$${the}\:{anser}\:{is}\:{the}\:{folowing} \\ $$

Question Number 173887    Answers: 0   Comments: 0

Question Number 173888    Answers: 1   Comments: 0

Prove that, ((13x)/(x^2 −(√(13x))+1)) = (√(13)) if, x = (√(13)) + 2(√3)

$$\mathrm{Prove}\:\mathrm{that},\:\frac{\mathrm{13x}}{\mathrm{x}^{\mathrm{2}} −\sqrt{\mathrm{13x}}+\mathrm{1}}\:=\:\sqrt{\mathrm{13}}\:\mathrm{if},\:\mathrm{x}\:=\:\sqrt{\mathrm{13}}\:+\:\mathrm{2}\sqrt{\mathrm{3}}\: \\ $$

Question Number 173882    Answers: 0   Comments: 0

Question Number 173874    Answers: 1   Comments: 0

Question Number 173871    Answers: 0   Comments: 1

2^x ∙3^y =4 2^y ∙3^x =6 prove that xy=z(z+1) 6^z =2

$$\mathrm{2}^{{x}} \centerdot\mathrm{3}^{{y}} =\mathrm{4} \\ $$$$\mathrm{2}^{{y}} \centerdot\mathrm{3}^{{x}} =\mathrm{6}\:\:\:\:\:\:\:{prove}\:{that}\:\:{xy}={z}\left({z}+\mathrm{1}\right) \\ $$$$\mathrm{6}^{{z}} =\mathrm{2} \\ $$

Question Number 173870    Answers: 0   Comments: 0

(√(1+ (n+1)(n+2)(n+3)(n+4))) ∈ N

$$\sqrt{\mathrm{1}+\:\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)\left({n}+\mathrm{4}\right)}\:\in\:\mathbb{N} \\ $$

Question Number 173869    Answers: 2   Comments: 2

If, ( (((√(10)) +(√6)+2)/( (√5) −(√(3 )) + (√2))) )^( 2) = a + (√b) then. a , b = ?

$$ \\ $$$$\:{If},\:\:\:\left(\:\frac{\sqrt{\mathrm{10}}\:+\sqrt{\mathrm{6}}+\mathrm{2}}{\:\sqrt{\mathrm{5}}\:−\sqrt{\mathrm{3}\:}\:+\:\sqrt{\mathrm{2}}}\:\right)^{\:\mathrm{2}} =\:{a}\:+\:\sqrt{{b}} \\ $$$$\:\:\:\:\:\:\:\:{then}.\:\:\:\:\:\:\:\:{a}\:\:,\:\:\:{b}\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$

Question Number 173864    Answers: 0   Comments: 0

(√x) ≥ ((x^4 −2x^3 +2x−1)/(x^3 −2x^2 +2x))

$$\:\:\:\:\sqrt{{x}}\:\geqslant\:\frac{{x}^{\mathrm{4}} −\mathrm{2}{x}^{\mathrm{3}} +\mathrm{2}{x}−\mathrm{1}}{{x}^{\mathrm{3}} −\mathrm{2}{x}^{\mathrm{2}} +\mathrm{2}{x}} \\ $$

Question Number 174024    Answers: 2   Comments: 0

Question Number 173859    Answers: 0   Comments: 0

Question Number 173842    Answers: 1   Comments: 0

The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win: (I) only the second race: (ii) all the three races (iii) only two of the races?

$$ \\ $$The probability that an athlete will not win any of the three races is 1/4. If the athlete runs in all the races, what is the probability that the athlete will win: (I) only the second race: (ii) all the three races (iii) only two of the races?

Question Number 173839    Answers: 3   Comments: 1

x^4 +x^3 +x^2 +x+1=0

$${x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0} \\ $$

Question Number 173836    Answers: 2   Comments: 1

Question Number 173834    Answers: 2   Comments: 2

Question Number 173831    Answers: 1   Comments: 0

Solve x+y=4 x^x −y^y =13(x−y)

$$\mathrm{Solve} \\ $$$$\mathrm{x}+\mathrm{y}=\mathrm{4} \\ $$$$\mathrm{x}^{\mathrm{x}} −\mathrm{y}^{\mathrm{y}} =\mathrm{13}\left(\mathrm{x}−\mathrm{y}\right) \\ $$

Question Number 173830    Answers: 0   Comments: 0

How many positive integer multiples of 2431 can be expressed in the form 7^a −7^b where a and b are integer such that 0≤b≤a≤2022

$$ \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{multiples}\:\mathrm{of}\:\mathrm{2431} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{expressed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{7}^{\mathrm{a}} −\mathrm{7}^{\mathrm{b}} \\ $$$$\mathrm{where}\:\boldsymbol{{a}}\:\mathrm{and}\:\boldsymbol{{b}}\:\mathrm{are}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{that}\:\mathrm{0}\leqslant\boldsymbol{{b}}\leqslant\boldsymbol{{a}}\leqslant\mathrm{2022}\: \\ $$

Question Number 173829    Answers: 1   Comments: 1

If , x∈ [0 , 1] , ∣ (√(1−x^( 2) )) −ax−b ∣≤ (((√2) −1)/2) find the values of ( a , b ) a , b∈ R.

$$ \\ $$$$\:\:\:\:{If}\:\:,\:{x}\in\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\: \\ $$$$\:\:\:\:\:\:,\:\:\:\:\mid\:\sqrt{\mathrm{1}−{x}^{\:\mathrm{2}} }\:−{ax}−{b}\:\mid\leqslant\:\frac{\sqrt{\mathrm{2}}\:−\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:{find}\:{the}\:{values}\:{of}\:\:\left(\:{a}\:,\:{b}\:\right) \\ $$$$\:\:\:{a}\:,\:{b}\in\:\mathbb{R}. \\ $$$$ \\ $$

Question Number 173827    Answers: 0   Comments: 0

Question Number 173825    Answers: 0   Comments: 2

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