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AllQuestion and Answers: Page 1818

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 )?

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

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)

Question Number 27882 Answers: 1 Comments: 0

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

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

Question Number 27818 Answers: 0 Comments: 7

Question Number 27816 Answers: 1 Comments: 1

If N is perfect nth power, prove that n ∣ (d(N)−1) [Where d(N) denotes number of divisors of N] Also show by an example that its vice versa is not necessarily correct.

$$\mathrm{If}\:\mathrm{N}\:\mathrm{is}\:\boldsymbol{\mathrm{perfect}}\:\boldsymbol{\mathrm{nth}}\:\boldsymbol{\mathrm{power}},\:\mathrm{prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\mathrm{n}\:\mid\:\left(\mathrm{d}\left(\mathrm{N}\right)−\mathrm{1}\right)\: \\ $$$$\left[{Where}\:\mathrm{d}\left(\mathrm{N}\right)\:{denotes}\:\boldsymbol{{number}}\right. \\ $$$$\left.\boldsymbol{{of}}\:\boldsymbol{{divisors}}\:\boldsymbol{{of}}\:\mathrm{N}\right] \\ $$$$\mathrm{Also}\:\mathrm{show}\:\mathrm{by}\:\mathrm{an}\:\mathrm{example}\:\mathrm{that}\:\mathrm{its} \\ $$$$\mathrm{vice}\:\mathrm{versa}\:\mathrm{is}\:\mathrm{not}\:\mathrm{necessarily}\:\mathrm{correct}. \\ $$

Question Number 27815 Answers: 1 Comments: 0

∫((cos x−cos 2x)/(1−cos x))dx

$$\int\frac{\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{2x}}{\mathrm{1}−\mathrm{cos}\:\mathrm{x}}\mathrm{dx} \\ $$

Question Number 27809 Answers: 2 Comments: 0

(1) Find the term independent of x in the expansion of (x−(2/x))^(10)

$$\left(\mathrm{1}\right)\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{term}}\:\boldsymbol{\mathrm{independent}} \\ $$$$\boldsymbol{\mathrm{of}}\:\:\:\boldsymbol{\mathrm{x}}\:\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{expansion}}\:\boldsymbol{\mathrm{of}}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}−\frac{\mathrm{2}}{\boldsymbol{\mathrm{x}}}\right)^{\mathrm{10}} \\ $$

Question Number 27805 Answers: 0 Comments: 1

find ∫_1 ^∝ ((arctan(αx))/x^2 ) .

$${find}\:\:\int_{\mathrm{1}} ^{\propto} \:\:\frac{{arctan}\left(\alpha{x}\right)}{{x}^{\mathrm{2}} }\:. \\ $$

Question Number 27804 Answers: 0 Comments: 1

calculate ∫_0 ^∝ ((e^(−ax) − e^(−bx) )/x^2 )dx with a>0 b>o

$${calculate}\:\:\int_{\mathrm{0}} ^{\propto} \:\:\frac{{e}^{−{ax}} \:−\:{e}^{−{bx}} }{{x}^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$$${b}>{o} \\ $$

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