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AllQuestion and Answers: Page 1799

Question Number 27942    Answers: 1   Comments: 1

Question Number 27939    Answers: 0   Comments: 1

Question Number 27936    Answers: 2   Comments: 1

Question Number 27930    Answers: 0   Comments: 2

Question Number 27920    Answers: 1   Comments: 0

∫(((x−1)dx)/((x+1)(√(x^3 +x^2 +x))))

$$\int\frac{\left({x}−\mathrm{1}\right){dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}}} \\ $$

Question Number 27922    Answers: 0   Comments: 4

a,b & c are distinct primes and x,y,z∈{0,1,2,...}. What is the number of divisors, common to the numbers a^x b^y c^z , a^x b^z c^y ,a^y b^x c^z ,a^y b^z c^x ,a^z b^x c^y & a^z b^y c^z .

$$\mathrm{a},\mathrm{b}\:\&\:\mathrm{c}\:\mathrm{are}\:\boldsymbol{\mathrm{distinct}}\:\boldsymbol{\mathrm{primes}}\:\mathrm{and} \\ $$$$\mathrm{x},\mathrm{y},\mathrm{z}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},...\right\}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\boldsymbol{\mathrm{divisors}}, \\ $$$$\boldsymbol{\mathrm{common}}\:\mathrm{to}\:\mathrm{the}\:\boldsymbol{\mathrm{numbers}}\:\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{z}}} , \\ $$$$\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{z}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{y}}} ,\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{z}}} ,\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{z}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{x}}} ,\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{z}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{y}}} \:\&\:\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{z}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{c}}^{\boldsymbol{\mathrm{z}}} \:. \\ $$

Question Number 27909    Answers: 0   Comments: 1

Question Number 27908    Answers: 1   Comments: 1

IF a_(1,) a_2 ,.....,a_(n−1) ,a_n are in AP then prove that 1/a_1 .a_n + 1/a_2 .a_(n−1) + 1/a_3 .a_(n−2) +...+1/a_n .a_1 = 2/a_1 +a_(n ) [1/a_(1 ) +1/a_2 +....+1/a_n ]

$${IF}\:\:{a}_{\mathrm{1},} {a}_{\mathrm{2}} ,.....,{a}_{{n}−\mathrm{1}} ,{a}_{{n}} \:{are}\:{in}\:{AP}\:{then}\:{prove}\:{that} \\ $$$$\mathrm{1}/{a}_{\mathrm{1}} .{a}_{{n}} +\:\mathrm{1}/{a}_{\mathrm{2}} .{a}_{{n}−\mathrm{1}} +\:\mathrm{1}/{a}_{\mathrm{3}} .{a}_{{n}−\mathrm{2}} +...+\mathrm{1}/{a}_{{n}} .{a}_{\mathrm{1}} = \\ $$$$\mathrm{2}/{a}_{\mathrm{1}} +{a}_{{n}\:} \left[\mathrm{1}/{a}_{\mathrm{1}\:} +\mathrm{1}/{a}_{\mathrm{2}} +....+\mathrm{1}/{a}_{{n}} \right] \\ $$

Question Number 27899    Answers: 2   Comments: 0

Question Number 27888    Answers: 1   Comments: 4

a and b are distinct primes and x,y∈{0,1,2,...}. What is the number of divisors common to the numbers (a^x b^y ) and (a^y b^x )?

$$\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\boldsymbol{\mathrm{distinct}}\:\boldsymbol{\mathrm{primes}}\:\mathrm{and} \\ $$$$\mathrm{x},\mathrm{y}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},...\right\}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{divisors}} \\ $$$$\boldsymbol{\mathrm{common}}\:\mathrm{to}\:\mathrm{the}\:\boldsymbol{\mathrm{numbers}}\:\left(\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{x}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{y}}} \right) \\ $$$$\boldsymbol{\mathrm{and}}\:\left(\boldsymbol{\mathrm{a}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{b}}^{\boldsymbol{\mathrm{x}}} \right)? \\ $$

Question Number 27885    Answers: 0   Comments: 1

(4) Find the term indepen− dent of x in the expansion of (x^2 −2+(1/x^2 ))^6

$$\left(\mathrm{4}\right)\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{indepen}}− \\ $$$$\boldsymbol{\mathrm{dent}}\:\boldsymbol{\mathrm{of}}\:\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{2}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\right)^{\mathrm{6}} \\ $$

Question Number 27884    Answers: 2   Comments: 0

(3) Find the term independ− ent of x in the expansion of ( x+(1/x))^2 (x−(1/x))^(12)

$$\left(\mathrm{3}\right)\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{independ}}− \\ $$$$\boldsymbol{\mathrm{ent}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\left(\:\boldsymbol{\mathrm{x}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{2}} \left(\boldsymbol{\mathrm{x}}−\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{12}} \\ $$

Question Number 27882    Answers: 1   Comments: 0

(2) Find the term indepen− dent of x in the expansion of (2x^2 +(1/x))^6

$$\left(\mathrm{2}\right)\:\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{indepen}}− \\ $$$$\boldsymbol{\mathrm{dent}}\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{6}} \\ $$

Question Number 27878    Answers: 0   Comments: 1

Question Number 27870    Answers: 1   Comments: 1

Question Number 27850    Answers: 1   Comments: 0

Question Number 27847    Answers: 1   Comments: 1

Find the range of f(x)=((x+5)/(x^2 −4))

$${Find}\:{the}\:{range}\:{of}\:{f}\left({x}\right)=\frac{{x}+\mathrm{5}}{{x}^{\mathrm{2}} −\mathrm{4}} \\ $$

Question Number 27844    Answers: 0   Comments: 1

For αεR, cosαcosx+siny≥sinx, then find the sum of the possible values of sinα+siny.

$${For}\:\alpha\epsilon{R},\:{cos}\alpha{cosx}+{siny}\geqslant{sinx},\:{then} \\ $$$${find}\:{the}\:{sum}\:{of}\:{the}\:{possible}\:{values}\:{of} \\ $$$${sin}\alpha+{siny}. \\ $$

Question Number 27843    Answers: 0   Comments: 1

sinx+sin3x+sin(√x)=0, then find the general solution?

$${sinx}+{sin}\mathrm{3}{x}+{sin}\sqrt{{x}}=\mathrm{0},\:{then}\:{find}\:{the}\: \\ $$$${general}\:{solution}? \\ $$

Question Number 27840    Answers: 1   Comments: 0

If xcosθ=ycos(θ+((2π)/3))=zcos(θ+((4π)/3)),then the value of xy+yz+zx.

$${If}\:{xcos}\theta={ycos}\left(\theta+\frac{\mathrm{2}\pi}{\mathrm{3}}\right)={zcos}\left(\theta+\frac{\mathrm{4}\pi}{\mathrm{3}}\right),{then} \\ $$$${the}\:{value}\:{of}\:{xy}+{yz}+{zx}. \\ $$

Question Number 27853    Answers: 0   Comments: 1

Question Number 27851    Answers: 0   Comments: 2

Question Number 27830    Answers: 0   Comments: 1

lim_(x→∝) (x−lnx)

$$\underset{{x}\rightarrow\propto} {\mathrm{lim}}\:\left({x}−\mathrm{ln}{x}\right) \\ $$

Question Number 27828    Answers: 1   Comments: 2

find the value of ∫_0 ^(π/2) (√(tanx))dx .

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{{tanx}}{dx}\:. \\ $$

Question Number 27821    Answers: 1   Comments: 0

If α and β satisfy sinαcosβ= −(1/2) then the greatest value of 2cosαsinβ

$${If}\:\alpha\:{and}\:\beta\:{satisfy}\:{sin}\alpha{cos}\beta=\:−\frac{\mathrm{1}}{\mathrm{2}}\:{then} \\ $$$${the}\:{greatest}\:{value}\:{of}\:\mathrm{2}{cos}\alpha{sin}\beta \\ $$

Question Number 27820    Answers: 1   Comments: 0

If A,Bε(0,(π/2)) such that 3sin^2 A+2sin^2 B=1 and 3sin2A−2sin2B=0, find the value of A+B.

$${If}\:{A},{B}\epsilon\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right)\:{such}\:{that}\:\mathrm{3}{sin}^{\mathrm{2}} {A}+\mathrm{2}{sin}^{\mathrm{2}} {B}=\mathrm{1} \\ $$$${and}\:\mathrm{3}{sin}\mathrm{2}{A}−\mathrm{2}{sin}\mathrm{2}{B}=\mathrm{0},\:{find}\:{the}\:{value}\:{of}\:{A}+{B}. \\ $$

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