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Question Number 55364    Answers: 0   Comments: 0

prove that ∫_(−∞ ) ^∞ f(x)dx=1 such that f(x)=(1/((√n) β((n/2),(1/2))))(1+(x^2 /n))^(−(1/2)(1+n)) and β((n/2),(1/2))=∫_0 ^∞ (x^((n/2)−1) /((1+x)^(3/2) ))dx

$$\mathrm{prove}\:\mathrm{that}\:\int_{−\infty\:} ^{\infty} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{1} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\sqrt{\mathrm{n}}\:\beta\left(\frac{\mathrm{n}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)}\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{n}}\right)^{−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\mathrm{n}\right)} \\ $$$$\mathrm{and}\:\beta\left(\frac{\mathrm{n}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}}\right)=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{x}^{\frac{\mathrm{n}}{\mathrm{2}}−\mathrm{1}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\frac{\mathrm{3}}{\mathrm{2}}} }\mathrm{dx} \\ $$

Question Number 55362    Answers: 2   Comments: 1

Question Number 55360    Answers: 2   Comments: 1

∫_1 ^( 2) (√(sin (3x−x^2 −2)))dx + (1/2)∫_3 ^1 (√(sin(((4t−t^2 −3)/4))))dt =?

$$\:\int_{\mathrm{1}} ^{\:\mathrm{2}} \sqrt{\mathrm{sin}\:\left(\mathrm{3}{x}−{x}^{\mathrm{2}} −\mathrm{2}\right)}{dx}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{3}} ^{\mathrm{1}} \sqrt{{sin}\left(\frac{\mathrm{4}{t}−{t}^{\mathrm{2}} −\mathrm{3}}{\mathrm{4}}\right)}{dt}\:\:=? \\ $$

Question Number 55359    Answers: 1   Comments: 0

Find a formula for the general term of the squence 1, 2, 2, 3, 3, 3, 4, 4, 4,4, ...

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{formula}\:\mathrm{for}\:\mathrm{the}\:\mathrm{general}\: \\ $$$$\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{squence} \\ $$$$\mathrm{1},\:\mathrm{2},\:\mathrm{2},\:\mathrm{3},\:\mathrm{3},\:\mathrm{3},\:\mathrm{4},\:\mathrm{4},\:\mathrm{4},\mathrm{4},\:... \\ $$

Question Number 55358    Answers: 0   Comments: 0

Determine all functions f : N → N satisfying xf(y)+yf(x)=(x+y)f(x^2 +y^2 ) for all positive integers x and y

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{functions}\:{f}\::\:\mathbb{N}\:\rightarrow\:\mathbb{N}\: \\ $$$$\mathrm{satisfying} \\ $$$${xf}\left({y}\right)+{yf}\left({x}\right)=\left({x}+{y}\right){f}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:{x}\:\mathrm{and}\:{y} \\ $$

Question Number 55352    Answers: 0   Comments: 0

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Question Number 55351    Answers: 0   Comments: 0

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Question Number 55344    Answers: 0   Comments: 6

A whatsapp group contains 7 women and 3 men.If they are leaving the group one at a time from the group what is the probability of a woman leaving then a man leaving and so on alternately until only a woman is remaining?

$${A}\:{whatsapp}\:{group}\:{contains}\:\mathrm{7}\:{women} \\ $$$${and}\:\mathrm{3}\:{men}.{If}\:{they}\:{are}\:{leaving} \\ $$$${the}\:{group}\:{one}\:{at}\:{a}\:{time}\:{from}\:{the} \\ $$$${group}\:{what}\:{is}\:{the}\:{probability}\:{of}\:{a} \\ $$$${woman}\:{leaving}\:{then}\:{a}\:{man}\:{leaving} \\ $$$${and}\:{so}\:{on}\:{alternately}\:{until}\:{only}\:{a} \\ $$$${woman}\:{is}\:{remaining}? \\ $$$$ \\ $$

Question Number 55333    Answers: 1   Comments: 0

The sum of all but one of the interior angles of a convex polygon equals 2525° . find the measure of the exterior angle adjacent to the remaining interior angle. can you please help if possible with diagram

$${The}\:{sum}\:{of}\:{all}\:{but}\:{one}\:{of}\:{the}\:{interior} \\ $$$${angles}\:{of}\:{a}\:{convex}\:{polygon}\:{equals}\: \\ $$$$\mathrm{2525}°\:.\:{find}\:{the}\:{measure}\:{of}\:{the} \\ $$$${exterior}\:{angle}\:{adjacent}\:{to}\:{the} \\ $$$${remaining}\:{interior}\:{angle}.\: \\ $$$${can}\:{you}\:{please}\:{help}\:{if}\:{possible}\:{with} \\ $$$${diagram}\: \\ $$

Question Number 55331    Answers: 0   Comments: 0

Question Number 55335    Answers: 1   Comments: 0

Question Number 55334    Answers: 1   Comments: 0

lim_(n→∞) (8^n /(2^(n+1) +3^(n+2) ))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\:\:\frac{\mathrm{8}^{{n}} }{\mathrm{2}^{{n}+\mathrm{1}} +\mathrm{3}^{{n}+\mathrm{2}} } \\ $$

Question Number 55326    Answers: 1   Comments: 0

find the minimum value (a/(√(a^2 +8bc)))+(b/(√(b^2 +8ac)))+(c/(√(c^2 +8ab)))

$${find}\:\:{the}\:{minimum}\:{value} \\ $$$$\frac{{a}}{\sqrt{{a}^{\mathrm{2}} +\mathrm{8}{bc}}}+\frac{{b}}{\sqrt{{b}^{\mathrm{2}} +\mathrm{8}{ac}}}+\frac{{c}}{\sqrt{{c}^{\mathrm{2}} +\mathrm{8}{ab}}} \\ $$

Question Number 55323    Answers: 0   Comments: 2

Question Number 55316    Answers: 1   Comments: 1

Question Number 55312    Answers: 1   Comments: 0

Consider the system { ((x^2 + y^2 = z)),((2x + 2y + z = k)) :} The value of xy + zk for which the system has a unique solution is ...

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{system} \\ $$$$\begin{cases}{{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:{z}}\\{\mathrm{2}{x}\:+\:\mathrm{2}{y}\:+\:{z}\:=\:{k}}\end{cases} \\ $$$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:{xy}\:+\:{zk}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{system} \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{unique}\:\mathrm{solution}\:\mathrm{is}\:... \\ $$

Question Number 55310    Answers: 1   Comments: 1

∫ 3x(√( 3x^3 + 7)) dx = . . . .

$$ \\ $$$$\int\:\mathrm{3}{x}\sqrt{\:\mathrm{3}{x}^{\mathrm{3}} +\:\mathrm{7}}\:\:{dx}\:=\:\:.\:.\:.\:. \\ $$

Question Number 55306    Answers: 1   Comments: 0

A mass of 6kg lies on an inclined plane which is smooth at angle θ to the horizontal where sin θ= (1/3).if it is connected to another mass 8kg by the same inelastic string passing over a smooth fixed pulley at the top of the plane.the partices are released from rest ,Find a) the acceleration of each mass b) the tension in the string c) the speed of the masses after 4seconds d) the distanced covered with this time in (c) above by the masses. 2017 CGCEB paper 3 Mechanics a/v

$${A}\:{mass}\:{of}\:\mathrm{6}{kg}\:{lies}\:{on}\:{an}\:{inclined}\:{plane} \\ $$$${which}\:{is}\:{smooth}\:{at}\:{angle}\:\theta\:{to}\:{the}\:{horizontal} \\ $$$${where}\:\:\mathrm{sin}\:\theta=\:\frac{\mathrm{1}}{\mathrm{3}}.{if}\:{it}\:{is}\:{connected}\:{to} \\ $$$${another}\:{mass}\:\mathrm{8}{kg}\:{by}\:{the}\:{same}\:{inelastic}\:{string} \\ $$$${passing}\:{over}\:{a}\:{smooth}\:{fixed}\:{pulley} \\ $$$${at}\:{the}\:{top}\:{of}\:{the}\:{plane}.{the}\:{partices}\:{are} \\ $$$${released}\:{from}\:{rest}\:,{Find}\: \\ $$$$\left.{a}\right)\:{the}\:{acceleration}\:{of}\:{each}\:{mass} \\ $$$$\left.{b}\right)\:{the}\:{tension}\:{in}\:{the}\:{string} \\ $$$$\left.{c}\right)\:{the}\:{speed}\:{of}\:{the}\:{masses}\:{after}\:\mathrm{4}{seconds} \\ $$$$\left.{d}\right)\:{the}\:{distanced}\:{covered}\:{with}\:{this}\:{time} \\ $$$${in}\:\left({c}\right)\:{above}\:{by}\:{the}\:{masses}. \\ $$$$ \\ $$$$\mathrm{2017}\:{CGCEB}\:{paper}\:\mathrm{3}\:{Mechanics}\:{a}/{v} \\ $$

Question Number 55301    Answers: 1   Comments: 2

find the fourth term in the expansion of (((√x)/y^2 )−(y/(√x)))^6

$${find}\:{the}\:{fourth}\:{term}\:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\frac{\sqrt{{x}}}{{y}^{\mathrm{2}} }−\frac{{y}}{\sqrt{{x}}}\right)^{\mathrm{6}} \\ $$

Question Number 55300    Answers: 1   Comments: 0

Question Number 55284    Answers: 0   Comments: 1

5^(x−2) −5^x +5^(x+1) =505

$$\mathrm{5}^{{x}−\mathrm{2}} −\mathrm{5}^{{x}} +\mathrm{5}^{{x}+\mathrm{1}} =\mathrm{505} \\ $$

Question Number 55282    Answers: 1   Comments: 1

calculatef(a)= ∫ (1+(a/x^2 ))arctan((a/x))dx 2) calculate ∫_1 ^(+∞) (1+(2/x^2 ))arctan((2/x))dx .

$${calculatef}\left({a}\right)=\:\:\int\:\:\:\left(\mathrm{1}+\frac{{a}}{{x}^{\mathrm{2}} }\right){arctan}\left(\frac{{a}}{{x}}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{1}} ^{+\infty} \left(\mathrm{1}+\frac{\mathrm{2}}{{x}^{\mathrm{2}} }\right){arctan}\left(\frac{\mathrm{2}}{{x}}\right){dx}\:. \\ $$

Question Number 55281    Answers: 0   Comments: 0

find the value of Π_(n=1) ^∞ (1+(1/n^2 ))

$${find}\:{the}\:{value}\:{of}\:\prod_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right) \\ $$

Question Number 55280    Answers: 1   Comments: 1

fint f(t)=∫_0 ^1 ((ln(1+tx^2 ))/x^2 )dx .

$${fint}\:{f}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 55279    Answers: 0   Comments: 0

find L( e^(−x) ln(1+x^2 )) with L mean laplace transform

$${find}\:{L}\left(\:{e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\right)\:\:\:{with}\:{L}\:{mean}\:{laplace} \\ $$$${transform} \\ $$

Question Number 55278    Answers: 0   Comments: 0

calculate ∫_0 ^(+∞) (dx/((x+1)(x+2)....(x+n)))

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)....\left({x}+{n}\right)} \\ $$

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