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Question Number 52470 Answers: 1 Comments: 1

Question Number 52459 Answers: 0 Comments: 2

let f(α)=∫_0 ^1 ((ln(1+iαx))/(1+x^2 ))dx 1)determine a explicit form of f(α) 2) calculate ∫_0 ^1 ((ln(1+ix))/(1+x^2 ))dx and ∫_0 ^1 ((ln(1+2ix))/(1+x^2 ))dx.

$${let}\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{i}\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{1}\right){determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left(\alpha\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{ix}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{ix}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx}. \\ $$

Question Number 52457 Answers: 1 Comments: 0

Two radioactive elements A and B has half lives of 100 and 50years. Samples of A and B initial contains equal numbers of atoms.What is the ratio of the remaining atoms of A to that of B after 200years?

$${Two}\:{radioactive}\:{elements}\:{A}\:{and}\:{B} \\ $$$${has}\:{half}\:{lives}\:{of}\:\mathrm{100}\:{and}\:\mathrm{50}{years}. \\ $$$${Samples}\:{of}\:{A}\:{and}\:{B}\:{initial}\:{contains} \\ $$$${equal}\:{numbers}\:{of}\:{atoms}.{What}\:{is} \\ $$$${the}\:{ratio}\:{of}\:{the}\:{remaining}\:{atoms} \\ $$$${of}\:{A}\:{to}\:{that}\:{of}\:{B}\:{after}\:\mathrm{200}{years}? \\ $$

Question Number 52450 Answers: 0 Comments: 0

let P(x)=(1+ix −x^2 )^n −1 find roots of P(x) and factorize P(x)inside C(x).

$${let}\:{P}\left({x}\right)=\left(\mathrm{1}+{ix}\:−{x}^{\mathrm{2}} \right)^{{n}} −\mathrm{1} \\ $$$${find}\:{roots}\:{of}\:\:{P}\left({x}\right)\:{and}\:{factorize}\:{P}\left({x}\right){inside}\:{C}\left({x}\right). \\ $$

Question Number 52468 Answers: 2 Comments: 4

Question Number 52466 Answers: 1 Comments: 1

Question Number 52433 Answers: 0 Comments: 4

Question Number 52428 Answers: 0 Comments: 7

Question Number 52426 Answers: 2 Comments: 1

Question Number 52421 Answers: 0 Comments: 2

Question Number 52418 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((arctan(x))/(1+x^4 )) dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{arctan}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{4}} }\:{dx} \\ $$

Question Number 52412 Answers: 0 Comments: 3

Question Number 52409 Answers: 1 Comments: 1

Question Number 52406 Answers: 2 Comments: 0

Find the cube root of 26 + 5(√3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{cube}\:\mathrm{root}\:\mathrm{of}\:\:\:\:\:\mathrm{26}\:+\:\mathrm{5}\sqrt{\mathrm{3}} \\ $$

Question Number 52397 Answers: 0 Comments: 1

Question Number 52392 Answers: 0 Comments: 0

Question Number 52391 Answers: 1 Comments: 1

2299/(7x+20)=11 sir could you help me?

$$\mathrm{2299}/\left(\mathrm{7x}+\mathrm{20}\right)=\mathrm{11}\:\:\:\: \\ $$$$\mathrm{sir}\:\mathrm{could}\:\mathrm{you}\:\mathrm{help}\:\mathrm{me}? \\ $$

Question Number 52381 Answers: 2 Comments: 0

If all the words with or without meaning are written using the letters of the word QUEEN and are arranged as in English dictionary.What is the position of the word QUEEN? a)44th b)45th c)46th d)47th

$${If}\:{all}\:{the}\:{words}\:{with}\:{or}\:{without} \\ $$$${meaning}\:{are}\:{written}\:{using}\:{the} \\ $$$${letters}\:{of}\:{the}\:{word}\:{QUEEN}\:{and} \\ $$$${are}\:{arranged}\:{as}\:{in}\:{English} \\ $$$${dictionary}.{What}\:{is}\:{the}\:{position} \\ $$$${of}\:{the}\:{word}\:{QUEEN}? \\ $$$$\left.{a}\left.\right)\left.\mathrm{4}\left.\mathrm{4}{th}\:{b}\right)\mathrm{45}{th}\:{c}\right)\mathrm{46}{th}\:{d}\right)\mathrm{47}{th} \\ $$

Question Number 52379 Answers: 1 Comments: 0

The sum of the digits in the unit place of all four digit numbers formed using the numbers 3 4 5 andL 6 without repetition is a)432 b)108 c)36 d)18

$${The}\:{sum}\:{of}\:{the}\:{digits}\:{in}\:{the}\:{unit} \\ $$$${place}\:{of}\:{all}\:{four}\:{digit}\:{numbers} \\ $$$${formed}\:{using}\:{the}\:{numbers}\:\mathrm{3}\:\mathrm{4}\:\mathrm{5}\:{and}\mathscr{L} \\ $$$$\mathrm{6}\:{without}\:{repetition}\:{is} \\ $$$$\left.{a}\left.\right)\left.\mathrm{4}\left.\mathrm{32}\:{b}\right)\mathrm{108}\:{c}\right)\mathrm{36}\:{d}\right)\mathrm{18} \\ $$

Question Number 52377 Answers: 0 Comments: 3

An iceberg is floating in sea water. The density of ice is 0.92g/cm^3 and that of sea water is 1.02g/cm^3 . What % of ice will be below the water?

$${An}\:{iceberg}\:{is}\:{floating}\:{in}\:{sea}\:{water}. \\ $$$${The}\:{density}\:{of}\:{ice}\:{is}\:\mathrm{0}.\mathrm{92}{g}/{cm}^{\mathrm{3}} \\ $$$${and}\:{that}\:{of}\:{sea}\:{water}\:{is}\:\mathrm{1}.\mathrm{02}{g}/{cm}^{\mathrm{3}} . \\ $$$${What}\:\%\:{of}\:{ice}\:{will}\:\:{be}\:{below}\:{the} \\ $$$${water}? \\ $$

Question Number 52363 Answers: 2 Comments: 0

integrate ((sin^2 xcos^2 x )/(sin^3 x+cos^3 x))dx

$${integrate}\:\frac{\mathrm{sin}^{\mathrm{2}} {x}\mathrm{cos}^{\mathrm{2}} {x}\:}{{sin}^{\mathrm{3}} {x}+{cos}^{\mathrm{3}} {x}}{dx} \\ $$

Question Number 52360 Answers: 1 Comments: 1

Question Number 52356 Answers: 1 Comments: 8

Question Number 52349 Answers: 2 Comments: 3

Determine if the series converges or diverges. (i) Σ_(n = 2) ^∞ (1/(n^2 − 1)) (ii) Σ_(n = 1) ^∞ (1/3^(n − 1) )

$$\mathrm{Determine}\:\mathrm{if}\:\mathrm{the}\:\mathrm{series}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges}. \\ $$$$\:\:\:\left(\mathrm{i}\right)\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{2}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} \:−\:\mathrm{1}} \\ $$$$ \\ $$$$\:\:\:\left(\mathrm{ii}\right)\:\:\:\:\underset{\mathrm{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{n}\:−\:\mathrm{1}} } \\ $$$$\:\:\: \\ $$

Question Number 52341 Answers: 0 Comments: 0

A sound wave of velocity 350m/s is directed towards the surface of the water. If the ratio of the wavelength of sound in air to that in water 150 ratio 475.Calculate the velocity of the wave in water.

$${A}\:{sound}\:{wave}\:{of}\:{velocity}\:\mathrm{350}{m}/{s} \\ $$$${is}\:{directed}\:{towards}\:{the}\:{surface}\:{of} \\ $$$${the}\:{water}.\:{If}\:{the}\:{ratio}\:{of}\:{the} \\ $$$${wavelength}\:{of}\:{sound}\:{in}\:{air}\:{to}\:{that} \\ $$$${in}\:{water}\:\mathrm{150}\:{ratio}\:\mathrm{475}.{Calculate} \\ $$$${the}\:{velocity}\:{of}\:{the}\:{wave}\:{in}\:{water}. \\ $$

Question Number 52340 Answers: 1 Comments: 0

A 90cm uniform lever has a load of 30N suspended at 15cm from one of its ends.If the fulcrum is at the centre of gravity what is the force that must be applied to its other end to keep it in horizontal equilibrium?

$${A}\:\mathrm{90}{cm}\:{uniform}\:{lever}\:{has}\:{a}\:{load} \\ $$$${of}\:\mathrm{30}{N}\:{suspended}\:{at}\:\mathrm{15}{cm}\:{from} \\ $$$${one}\:{of}\:{its}\:{ends}.{If}\:{the}\:{fulcrum}\:{is}\:{at} \\ $$$${the}\:{centre}\:{of}\:{gravity}\:{what}\:{is}\:{the} \\ $$$${force}\:{that}\:{must}\:{be}\:{applied}\:{to}\:{its} \\ $$$${other}\:{end}\:{to}\:{keep}\:{it}\:{in}\:{horizontal} \\ $$$${equilibrium}? \\ $$

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