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Question Number 63602 Answers: 2 Comments: 0

32x^3 −48x^2 −22x−3=0

$$\mathrm{32}{x}^{\mathrm{3}} −\mathrm{48}{x}^{\mathrm{2}} −\mathrm{22}{x}−\mathrm{3}=\mathrm{0} \\ $$

Question Number 63597 Answers: 0 Comments: 0

Question Number 63596 Answers: 0 Comments: 1

The surnames of 40 students in a class were arranged in alphabetical order. 16 of the surnames begin with O while 9 of the surnames begin with A. 14 of the letters of the alphabet do not appear as the first letter of any surname. (i) What is the probability that the surname of a child picked at random from the class begins with either A or O (ii) If more than one surname begins with a letter besides A and O. How many surnames begins with that letter ?

$$\mathrm{The}\:\mathrm{surnames}\:\mathrm{of}\:\mathrm{40}\:\mathrm{students}\:\mathrm{in}\:\mathrm{a}\:\mathrm{class}\:\mathrm{were}\:\mathrm{arranged}\:\mathrm{in} \\ $$$$\mathrm{alphabetical}\:\mathrm{order}.\:\mathrm{16}\:\mathrm{of}\:\mathrm{the}\:\mathrm{surnames}\:\mathrm{begin}\:\mathrm{with}\:\mathrm{O}\:\mathrm{while} \\ $$$$\mathrm{9}\:\mathrm{of}\:\mathrm{the}\:\mathrm{surnames}\:\mathrm{begin}\:\mathrm{with}\:\mathrm{A}.\:\:\mathrm{14}\:\mathrm{of}\:\mathrm{the}\:\mathrm{letters}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{alphabet}\:\mathrm{do}\:\mathrm{not}\:\mathrm{appear}\:\mathrm{as}\:\mathrm{the}\:\mathrm{first}\:\mathrm{letter}\:\mathrm{of}\:\mathrm{any}\:\mathrm{surname}. \\ $$$$ \\ $$$$\left(\mathrm{i}\right)\:\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{surname}\:\mathrm{of}\:\mathrm{a}\:\mathrm{child}\:\mathrm{picked} \\ $$$$\mathrm{at}\:\mathrm{random}\:\mathrm{from}\:\mathrm{the}\:\mathrm{class}\:\mathrm{begins}\:\mathrm{with}\:\mathrm{either}\:\mathrm{A}\:\mathrm{or}\:\mathrm{O} \\ $$$$\left(\mathrm{ii}\right)\:\:\mathrm{If}\:\mathrm{more}\:\mathrm{than}\:\mathrm{one}\:\mathrm{surname}\:\mathrm{begins}\:\mathrm{with}\:\mathrm{a}\:\mathrm{letter}\:\mathrm{besides}\:\mathrm{A} \\ $$$$\mathrm{and}\:\mathrm{O}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{surnames}\:\mathrm{begins}\:\mathrm{with}\:\mathrm{that}\:\mathrm{letter}\:? \\ $$$$ \\ $$

Question Number 63588 Answers: 1 Comments: 1

Question Number 63574 Answers: 0 Comments: 12

prove that Σ_(k = 1) ^∞ (1/(k(2k + 1))) = 2 − 2ln(2)

$$\mathrm{prove}\:\mathrm{that}\:\:\:\underset{\mathrm{k}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{k}\left(\mathrm{2k}\:+\:\mathrm{1}\right)}\:\:=\:\:\mathrm{2}\:−\:\mathrm{2ln}\left(\mathrm{2}\right) \\ $$

Question Number 63573 Answers: 0 Comments: 0

Question Number 63570 Answers: 1 Comments: 3

Question Number 63566 Answers: 0 Comments: 2

prove that ∫sin^n (x) dx , p∈n , p≥2 =− (1/n)cos(x) sin^(n−1) (x) + (p−1)∫sin^(n−2) (x) dx

$${prove}\:{that} \\ $$$$ \\ $$$$\int{sin}^{{n}} \left({x}\right)\:{dx}\:,\:{p}\in{n}\:,\:{p}\geqslant\mathrm{2}\:=−\:\frac{\mathrm{1}}{{n}}{cos}\left({x}\right)\:{sin}^{{n}−\mathrm{1}} \left({x}\right)\:+\:\left({p}−\mathrm{1}\right)\int{sin}^{{n}−\mathrm{2}} \left({x}\right)\:{dx} \\ $$

Question Number 63565 Answers: 0 Comments: 0

Question Number 63564 Answers: 1 Comments: 0

Question Number 63561 Answers: 1 Comments: 0

Question Number 63560 Answers: 0 Comments: 2

developp at laurent series 1) f(z) =(1/(z−2)) 2)g(z) =(3/(z^2 −3z +2)) 3)h(z) =(1/(z^2 +4))

$${developp}\:{at}\:{laurent}\:{series} \\ $$$$\left.\mathrm{1}\right)\:{f}\left({z}\right)\:=\frac{\mathrm{1}}{{z}−\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){g}\left({z}\right)\:=\frac{\mathrm{3}}{{z}^{\mathrm{2}} −\mathrm{3}{z}\:+\mathrm{2}} \\ $$$$\left.\mathrm{3}\right){h}\left({z}\right)\:=\frac{\mathrm{1}}{{z}^{\mathrm{2}} +\mathrm{4}} \\ $$

Question Number 63552 Answers: 1 Comments: 1

Calculate ∫_0 ^(1/2) x(√(x^2 +1)) dx+∫_(1/2) ^1 x^2 (√(x^3 +1)) dx+∫_1 ^2 x^3 (√(x^4 +1)) dx+∫_2 ^3 x^4 (√(x^5 +1 ))dx+...+∫_(78) ^(79) x^(80) (√(x^(81) +1)) dx+∫_(79) ^(80) x^(81) (√(x^(82) +1)) dx usingΣ_(n=2) ^(80) ∫_(n−1) ^n x^(n+1) (√(x^(n+2) +1))dx

$${Calculate}\:\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\mathrm{2}}} {\int}}{x}\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}+\underset{\frac{\mathrm{1}}{\mathrm{2}}} {\overset{\mathrm{1}} {\int}}{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}+\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}{x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{4}} +\mathrm{1}}\:{dx}+\underset{\mathrm{2}} {\overset{\mathrm{3}} {\int}}{x}^{\mathrm{4}} \sqrt{{x}^{\mathrm{5}} +\mathrm{1}\:}{dx}+...+\underset{\mathrm{78}} {\overset{\mathrm{79}} {\int}}{x}^{\mathrm{80}} \sqrt{{x}^{\mathrm{81}} +\mathrm{1}}\:{dx}+\underset{\mathrm{79}} {\overset{\mathrm{80}} {\int}}{x}^{\mathrm{81}} \sqrt{{x}^{\mathrm{82}} +\mathrm{1}}\:{dx} \\ $$$${using}\underset{{n}=\mathrm{2}} {\overset{\mathrm{80}} {\sum}}\underset{{n}−\mathrm{1}} {\overset{{n}} {\int}}{x}^{{n}+\mathrm{1}} \sqrt{{x}^{{n}+\mathrm{2}} +\mathrm{1}}{dx} \\ $$

Question Number 63539 Answers: 1 Comments: 2

The minimum value of 2x^2 −3x+2 is ___.

$$\mathrm{The}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}\:\mathrm{is}\:\_\_\_. \\ $$

Question Number 63536 Answers: 0 Comments: 1

a) if y= x^m (1−x)^n , where n∈ Z^+ , the set of positive integers, show that when (dy/dx)=0, x=(m/(m+n)) b)if y = 2(x−5)(√(x+4)) ,show that (dy/dx) = ((3(x+1))/((√(x+4)) )) c) solve the equation sinx−sin5x+cos3x = 0 for 0°≤x≤180°

$$\left.{a}\right)\:\:{if}\:{y}=\:{x}^{{m}} \left(\mathrm{1}−{x}\right)^{{n}} ,\:{where}\:{n}\in\:\mathbb{Z}^{+} ,\:{the}\:{set}\:{of}\:{positive}\:{integers}, \\ $$$${show}\:{that}\:{when}\:\frac{{dy}}{{dx}}=\mathrm{0},\:{x}=\frac{{m}}{{m}+{n}} \\ $$$$\left.{b}\right){if}\:{y}\:=\:\mathrm{2}\left({x}−\mathrm{5}\right)\sqrt{{x}+\mathrm{4}}\:,{show}\:{that}\:\frac{{dy}}{{dx}}\:=\:\frac{\mathrm{3}\left({x}+\mathrm{1}\right)}{\sqrt{{x}+\mathrm{4}}\:} \\ $$$$\left.{c}\right)\:{solve}\:{the}\:{equation}\:\:{sinx}−{sin}\mathrm{5}{x}+{cos}\mathrm{3}{x}\:=\:\mathrm{0}\:{for}\:\:\mathrm{0}°\leqslant{x}\leqslant\mathrm{180}° \\ $$

Question Number 63534 Answers: 1 Comments: 0

find the set of values of x for which y is real if y=(((x−2)(x−1))/(x+2)) , x≠−2, x∈R

$${find}\:{the}\:{set}\:{of}\:{values}\:{of}\:{x}\:{for}\:{which}\:{y}\:{is}\:{real}\:{if}\: \\ $$$$\:{y}=\frac{\left({x}−\mathrm{2}\right)\left({x}−\mathrm{1}\right)}{{x}+\mathrm{2}}\:,\:{x}\neq−\mathrm{2},\:{x}\in\mathbb{R} \\ $$

Question Number 63532 Answers: 1 Comments: 0

prove that there exist unique intergers p and s sucb that a = bp + s with −((∣b∣)/2)< s ≤((∣b∣)/2) hence find p and s given that a=49 and b=26

$${prove}\:{that}\:{there}\:{exist}\:{unique}\:{intergers}\:{p}\:{and}\:{s}\:{sucb}\:{that} \\ $$$${a}\:=\:{bp}\:+\:{s}\:{with}\:−\frac{\mid{b}\mid}{\mathrm{2}}<\:{s}\:\leqslant\frac{\mid{b}\mid}{\mathrm{2}} \\ $$$${hence}\:{find}\:{p}\:{and}\:{s}\:{given}\:{that}\:{a}=\mathrm{49}\:{and}\:{b}=\mathrm{26} \\ $$

Question Number 63522 Answers: 0 Comments: 2

Question Number 63519 Answers: 0 Comments: 4

consider the general definite intergral I_n =∫_0 ^(π/2) sin^n xdx a) prove that for n≥2, nI_n =(n−1)I_(n−2) . b) Find the values of i)∫_0 ^(π/2) sin^5 dx ii) ∫_0 ^(π/2) sin^6 dx

$${consider}\:{the}\:{general}\:{definite}\:{intergral}\:\: \\ $$$$\:{I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{{n}} {xdx} \\ $$$$\left.{a}\right)\:{prove}\:{that}\:{for}\:{n}\geqslant\mathrm{2},\:{nI}_{{n}} =\left({n}−\mathrm{1}\right){I}_{{n}−\mathrm{2}} . \\ $$$$\left.{b}\left.\right)\left.\:{Find}\:{the}\:{values}\:{of}\:\:\boldsymbol{{i}}\right)\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{5}} {dx}\:\:\:\boldsymbol{{ii}}\right)\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{6}} {dx} \\ $$

Question Number 63517 Answers: 1 Comments: 0

Given that ∣z−6∣=2∣z+6−9i∣, a) Use algebra to show that the locus of z is a circle, stating its center and its radius. b) sketch the locus z on an argand diagram.

$$\mathrm{Given}\:\mathrm{that}\:\:\mid{z}−\mathrm{6}\mid=\mathrm{2}\mid{z}+\mathrm{6}−\mathrm{9}{i}\mid, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{Use}\:\mathrm{algebra}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:{z}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle}, \\ $$$$\mathrm{stating}\:\mathrm{its}\:\mathrm{center}\:\mathrm{and}\:\mathrm{its}\:\mathrm{radius}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{sketch}\:\mathrm{the}\:\mathrm{locus}\:{z}\:\mathrm{on}\:\mathrm{an}\:\mathrm{argand}\:\mathrm{diagram}. \\ $$

Question Number 63510 Answers: 0 Comments: 1

let f(x)=∫_0 ^∞ (t^(a−1) /(x+t)) dt with x>0 and 0<a<1 1)calculate f(x) 2)calculate g(x)=∫_0 ^∞ (t^(a−1) /((x+t)^2 ))dt 3)find the value of∫_0 ^∞ (t^(a−1) /((1+t)^2 ))dt

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}−\mathrm{1}} }{{x}+{t}}\:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$${and}\:\mathrm{0}<{a}<\mathrm{1} \\ $$$$\left.\mathrm{1}\right){calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}−\mathrm{1}} }{\left({x}+{t}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{a}−\mathrm{1}} }{\left(\mathrm{1}+{t}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 63509 Answers: 1 Comments: 1

calculate ∫_(−1) ^1 ((√(1+x^2 )) −(√(1−x^2 )))dx

$${calculate}\:\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right){dx} \\ $$

Question Number 63508 Answers: 0 Comments: 4

let f(x) =∫_(−∞) ^(+∞) (dt/((t^2 +ixt −1))) with ∣x∣>2 (i^2 =−1) 1) extract Re(f(x)) and Im(f(x)) 2) calculate f(x) 3) find olso g(x) =∫_(−∞) ^(+∞) (t/((t^2 +ixt −1)^2 ))dt 4) find values of integrals ∫_(−∞) ^(+∞) (dt/((t^2 +3it −1))) and ∫_(−∞) ^(+∞) ((tdt)/((t^2 +3it −1)^2 )) 5) give f^((n)) (x) at form of integrals.

$${let}\:\:{f}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} \:+{ixt}\:−\mathrm{1}\right)}\:\:{with}\:\mid{x}\mid>\mathrm{2}\:\:\:\left({i}^{\mathrm{2}} =−\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{extract}\:{Re}\left({f}\left({x}\right)\right)\:{and}\:{Im}\left({f}\left({x}\right)\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:\:{find}\:{olso}\:{g}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{t}}{\left({t}^{\mathrm{2}} \:+{ixt}\:−\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{values}\:{of}\:{integrals}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{2}} +\mathrm{3}{it}\:−\mathrm{1}\right)}\:\:{and}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{tdt}}{\left({t}^{\mathrm{2}} \:+\mathrm{3}{it}\:−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right)\:{give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integrals}. \\ $$

Question Number 63507 Answers: 1 Comments: 0

let U_n =∫_(1/n) ^1 ((√(x^2 +x+1)) −(√(x^2 −x+1)))dx (n>0) 1)calculate lim_(n→+∞) U_n 2) find nature of Σ U_n

$${let}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{\mathrm{1}} \left(\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}\:−\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\right){dx}\:\:\:\left({n}>\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:\:{find}\:{nature}\:{of}\:\:\Sigma\:{U}_{{n}} \\ $$

Question Number 63490 Answers: 2 Comments: 1

A father with 8 children takes 3 at a time to the garden as often as he without taking the same 3 children together more than once. The number of times he will go to the garden is

$$\mathrm{A}\:\mathrm{father}\:\mathrm{with}\:\mathrm{8}\:\mathrm{children}\:\mathrm{takes}\:\mathrm{3}\:\mathrm{at}\:\mathrm{a}\: \\ $$$$\mathrm{time}\:\mathrm{to}\:\mathrm{the}\:\mathrm{garden}\:\mathrm{as}\:\mathrm{often}\:\mathrm{as}\:\mathrm{he} \\ $$$$\mathrm{without}\:\mathrm{taking}\:\mathrm{the}\:\mathrm{same}\:\mathrm{3}\:\mathrm{children} \\ $$$$\mathrm{together}\:\mathrm{more}\:\mathrm{than}\:\mathrm{once}.\:\mathrm{The}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{times}\:\mathrm{he}\:\mathrm{will}\:\mathrm{go}\:\mathrm{to}\:\mathrm{the}\:\mathrm{garden}\:\mathrm{is} \\ $$

Question Number 63489 Answers: 0 Comments: 0

A father with 8 children takes 3 at a time to the garden as often as he without taking the same 3 children together more than once. The number of times he will go to the garden is

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