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Question Number 58405 Answers: 3 Comments: 2

1) lim_(x→0) ((e^(ax) −bx−1)/x^2 )=2 . find a,b ? 2) 6 balls are placed randomly into 6 cells. Then the probability that exactly one cell remains empty is ?

$$\left.\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{ax}} −{bx}−\mathrm{1}}{{x}^{\mathrm{2}} }=\mathrm{2}\:. \\ $$$${find}\:{a},{b}\:? \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{6}\:{balls}\:{are}\:{placed}\:{randomly}\:{into} \\ $$$$\mathrm{6}\:{cells}.\:{Then}\:{the}\:{probability}\:{that}\:{exactly} \\ $$$${one}\:{cell}\:{remains}\:{empty}\:{is}\:? \\ $$

Question Number 58409 Answers: 0 Comments: 3

Prove without mathematical induction that the expression (1 + (√2))^(2n) + (1 − (√2))^(2n) is even for every natural number n.

$$\mathrm{Prove}\:\mathrm{without}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{expression}\:\:\:\left(\mathrm{1}\:+\:\sqrt{\mathrm{2}}\right)^{\mathrm{2n}} \:+\:\left(\mathrm{1}\:−\:\sqrt{\mathrm{2}}\right)^{\mathrm{2n}} \:\:\mathrm{is}\:\mathrm{even}\:\mathrm{for}\:\mathrm{every} \\ $$$$\mathrm{natural}\:\mathrm{number}\:\:\mathrm{n}. \\ $$

Question Number 58402 Answers: 2 Comments: 2

The imaginary part of ((1/2)+(1/2)i)^(10) is ?

$${The}\:{imaginary}\:{part}\:{of}\:\left(\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{i}\right)^{\mathrm{10}} {is}\:? \\ $$

Question Number 58400 Answers: 1 Comments: 1

If parabola y=−x^2 −2x+k touches the parabola y=−(1/2)x^2 −4x+3 , then the value of k is ? a) 1 b)2 c)3 d)4

$${If}\:{parabola}\:{y}=−{x}^{\mathrm{2}} −\mathrm{2}{x}+{k}\:{touches} \\ $$$${the}\:{parabola}\:{y}=−\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}\:,\:{then} \\ $$$${the}\:{value}\:{of}\:{k}\:{is}\:? \\ $$$$\left.{a}\left.\right)\left.\:\left.\mathrm{1}\:\:\:\:{b}\right)\mathrm{2}\:\:\:\:\:{c}\right)\mathrm{3}\:\:\:\:\:{d}\right)\mathrm{4} \\ $$

Question Number 58410 Answers: 1 Comments: 0

Show that the sum of the cube of three consecutive number gives a multiple of 9.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{of}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{number}\:\mathrm{gives}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\:\mathrm{9}. \\ $$

Question Number 58526 Answers: 0 Comments: 1

Question Number 58390 Answers: 2 Comments: 2

write without roots in denominator if possible (1) (1/(√a)) (2) (1/((√a)+(√b))) (3) (1/((√a)+(√b)+(√c))) (4) (1/((√a)+(√b)+(√c)+(√d))) (5) (1/((√a)+(√b)+(√c)+(√d)+(√e)))

$$\mathrm{write}\:\mathrm{without}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{denominator}\:\mathrm{if}\:\mathrm{possible} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}+\sqrt{{d}}} \\ $$$$\left(\mathrm{5}\right)\:\frac{\mathrm{1}}{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}+\sqrt{{d}}+\sqrt{{e}}} \\ $$

Question Number 58387 Answers: 2 Comments: 0

Question Number 58529 Answers: 3 Comments: 4

Solve for x and y (1/x) + (1/y) = 4 ....... (i) (x^2 /y) + (y^2 /x) = 9 ....... (ii)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:\mathrm{4}\:\:\:\:\:\:\:.......\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{x}}\:\:=\:\:\mathrm{9}\:\:\:\:\:\:.......\:\left(\mathrm{ii}\right) \\ $$

Question Number 58380 Answers: 0 Comments: 3

Question Number 58378 Answers: 0 Comments: 1

Question Number 58377 Answers: 1 Comments: 0

Question Number 58374 Answers: 1 Comments: 0

Question Number 58373 Answers: 1 Comments: 1

Find the coefficient of x^6 in (2x + 1)^6 (x^2 + x + (1/4))^4

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\:\mathrm{x}^{\mathrm{6}} \:\:\mathrm{in}\:\:\:\left(\mathrm{2x}\:+\:\mathrm{1}\right)^{\mathrm{6}} \:\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{4}} \\ $$

Question Number 58364 Answers: 2 Comments: 0

Let a_n = 10 × ((1/(√2)))^n a_n is a geometrical sequence S_n = a_0 + a_1 + ... + a_(n−1) S_n = 10 × ((1 − ((1/(√2)))^n )/(1 − ((1/(√2))))) Proove that : S_n = ((10(√2))/((√2) − 1)) × (1−((1/(√2)))^n )

$$\mathrm{Let}\:{a}_{{n}} =\:\mathrm{10}\:×\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} \\ $$$$\:\:\:\:{a}_{{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{geometrical}\:\mathrm{sequence} \\ $$$$\:\:{S}_{{n}} \:=\:{a}_{\mathrm{0}} \:+\:{a}_{\mathrm{1}} \:+\:...\:+\:{a}_{{n}−\mathrm{1}} \\ $$$$\:\:\:\:\:{S}_{{n}} =\:\mathrm{10}\:×\:\frac{\mathrm{1}\:−\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} }{\mathrm{1}\:−\:\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)} \\ $$$$\boldsymbol{\mathrm{Proove}}\:\boldsymbol{\mathrm{that}}\:: \\ $$$$ \\ $$$${S}_{{n}} \:=\:\frac{\mathrm{10}\sqrt{\mathrm{2}}}{\sqrt{\mathrm{2}}\:−\:\mathrm{1}}\:×\:\left(\mathrm{1}−\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\right)^{{n}} \right) \\ $$$$ \\ $$

Question Number 58362 Answers: 1 Comments: 0

∫((sin x)/(sin 3x))dx

$$\int\frac{\mathrm{sin}\:{x}}{\mathrm{sin}\:\mathrm{3}{x}}{dx} \\ $$

Question Number 58361 Answers: 0 Comments: 0

the molar heat capacity of constant pressure of a gas varies with the temperature according to the equation n c_(p=a+bθ−c/θ^(2 _ ) ) where a.b and c are constant how much heat transfered during an isobaric process in which n makes of gas undergo a temperature rise from θ_i to θ_f ? (b)the molar heat capacity of a metal at low temperature varies with the temperature according to the equation c=bθ+(a/h)θ^3 where a and b are constant .how much heat p_(γ ) make is transfered during a process in which the tempereture change from 0.01(h)to 0.02(h)

$${the}\:{molar}\:{heat}\:{capacity}\:{of}\:{constant} \\ $$$${pressure}\:{of}\:{a}\:{gas}\:{varies}\:{with}\:{the} \\ $$$${temperature}\:{according}\:{to}\:{the}\:{equation} \\ $$$${n}\:{c}_{{p}={a}+{b}\theta−{c}/\theta^{\mathrm{2}\:\:\:\:\:\:\underset{} {\:}} } \\ $$$${where}\:{a}.{b}\:{and}\:{c}\:{are}\:{constant}\:{how}\:{much}\:{heat}\:{transfered}\:{during}\:{an}\:{isobaric}\:{process}\:{in}\:{which}\:{n}\:{makes}\:{of}\:{gas}\:{undergo}\:{a}\:{temperature}\:{rise}\:{from}\:\theta_{{i}} \:\:{to}\:\theta_{{f}} \:? \\ $$$$\left({b}\right){the}\:{molar}\:{heat}\:{capacity}\:{of}\:{a}\:{metal}\:{at}\:{low}\:{temperature}\:{varies}\:{with}\:{the}\:{temperature}\:{according}\:{to}\:{the}\:{equation}\:{c}={b}\theta+\frac{{a}}{{h}}\theta^{\mathrm{3}} \:\:{where}\:{a}\:{and}\:{b}\:{are}\:{constant}\:.{how}\:{much}\:{heat}\:{p}_{\gamma\:\:} \:\:{make}\:{is}\:{transfered}\:{during}\:{a}\:{process}\:{in}\:{which}\:{the}\:{tempereture}\:{change}\:{from}\:\mathrm{0}.\mathrm{01}\left({h}\right){to}\:\mathrm{0}.\mathrm{02}\left({h}\right) \\ $$

Question Number 58358 Answers: 0 Comments: 0

knowing that: cos(x)=Σ_(n=1) ^∞ (((−1)^n x^(2n) )/((2n)!)) sin(x)=Σ_(n=1) ^∞ (((−1)^n x^(2n+1) )/((2n +1)!)) prof that: cos(x+y)= cos(x)cos(y)−sin(x)sin(y)

$${knowing}\:{that}:\: \\ $$$${cos}\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}\right)!}\:\:\:\: \\ $$$${sin}\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}\:+\mathrm{1}\right)!} \\ $$$${prof}\:{that}:\:{cos}\left({x}+{y}\right)=\:{cos}\left({x}\right){cos}\left({y}\right)−{sin}\left({x}\right){sin}\left({y}\right) \\ $$

Question Number 58350 Answers: 0 Comments: 2

Question Number 58346 Answers: 2 Comments: 0

∫((cos(x))/(cos(2016°+cos(x)))×dx

$$\int\frac{\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{cos}}\left(\mathrm{2016}°+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}\right)\right.}×\boldsymbol{\mathrm{dx}} \\ $$

Question Number 58354 Answers: 1 Comments: 0

let U_n =((1^2 +2^2 +3^2 +....+n^2 )/(1^4 +2^4 +3^4 +....+n^4 )) 1)find lim_(n→+∞) U_n 2) calculate Σ_(n=1) ^∞ U_n

$${let}\:{U}_{{n}} =\frac{\mathrm{1}^{\mathrm{2}} \:+\mathrm{2}^{\mathrm{2}} \:+\mathrm{3}^{\mathrm{2}} \:+....+{n}^{\mathrm{2}} }{\mathrm{1}^{\mathrm{4}} \:+\mathrm{2}^{\mathrm{4}} \:+\mathrm{3}^{\mathrm{4}} \:+....+{n}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right){find}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{U}_{{n}} \\ $$

Question Number 58319 Answers: 1 Comments: 0

∫x^n (lnx)^n dx

$$\int{x}^{{n}} \left(\mathrm{ln}{x}\right)^{{n}} {dx} \\ $$

Question Number 58309 Answers: 2 Comments: 3

Prove that lim_(x→0) ((1−cos(x)cos(x/2)cos(x/3)...)/x^2 )=(π^2 /(12))

$${Prove}\:{that}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{1}−{cos}\left({x}\right){cos}\left({x}/\mathrm{2}\right){cos}\left({x}/\mathrm{3}\right)...}{{x}^{\mathrm{2}} }=\frac{\pi^{\mathrm{2}} }{\mathrm{12}} \\ $$

Question Number 58299 Answers: 1 Comments: 1

find ∫ (dx/((x^2 +x+1)^(3/2) ))

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

Question Number 58286 Answers: 0 Comments: 1

A small body with temperature θ and absorbtivity τ is placed in a large evaluated capacity whose interior walls are at a temperature θ_w . when θ_w −θ is small, show that the rate of heat transfer by radiation is Q^• = 4θ_w ^3 Aτδ(θ_w −θ).

$$\mathrm{A}\:\mathrm{small}\:\mathrm{body}\:\mathrm{with}\:\mathrm{temperature}\:\theta\:\mathrm{and} \\ $$$$\mathrm{absorbtivity}\:\tau\:\mathrm{is}\:\mathrm{placed}\:\mathrm{in}\:\mathrm{a}\:\mathrm{large}\: \\ $$$$\mathrm{evaluated}\:\mathrm{capacity}\:\mathrm{whose}\:\mathrm{interior}\: \\ $$$$\mathrm{walls}\:\mathrm{are}\:\mathrm{at}\:\mathrm{a}\:\mathrm{temperature}\:\theta_{\mathrm{w}} . \\ $$$$\mathrm{when}\:\:\:\:\theta_{\mathrm{w}} −\theta\:\:\:\:\mathrm{is}\:\:\mathrm{small},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{rate}\:\mathrm{of}\:\mathrm{heat}\:\mathrm{transfer}\:\mathrm{by}\:\mathrm{radiation}\:\mathrm{is} \\ $$$$\:\:\:\:\:\overset{\bullet} {\mathrm{Q}}=\:\mathrm{4}\theta_{\mathrm{w}} ^{\mathrm{3}} \mathrm{A}\tau\delta\left(\theta_{\mathrm{w}} −\theta\right). \\ $$

Question Number 58282 Answers: 1 Comments: 0

A circle tangents to :x and y axes and x^(1/2) +y^(1/2) =a^(1/2) .find its radious.

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