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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)))

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

Question Number 55276 Answers: 0 Comments: 1

calculate Σ_(n=1) ^∞ (((−1)^n )/(16n^2 −1))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{16}{n}^{\mathrm{2}} −\mathrm{1}} \\ $$

Question Number 55274 Answers: 0 Comments: 4

let ϕ(a) =∫_1 ^(√3) arctan((a/x))dx 1) calculate ϕ(a) interms of a 2) calculate ϕ^′ (a) at form of integral. 3) determine ϕ^((n)) (a) at form of integral. 4) find the value of ∫_1 ^(√3) arctan((2/x))dx .

$${let}\:\varphi\left({a}\right)\:=\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:{arctan}\left(\frac{{a}}{{x}}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\varphi\left({a}\right)\:{interms}\:{of}\:{a} \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:\varphi^{'} \left({a}\right)\:{at}\:{form}\:{of}\:{integral}. \\ $$$$\left.\mathrm{3}\right)\:{determine}\:\varphi^{\left({n}\right)} \left({a}\right)\:\:{at}\:{form}\:{of}\:{integral}. \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:{arctan}\left(\frac{\mathrm{2}}{{x}}\right){dx}\:. \\ $$

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