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Question Number 113668 Answers: 1 Comments: 0

Question Number 113667 Answers: 1 Comments: 0

In △ABC, BC=5cm AC=4cm cos(A−B)=((31)/(32)) Find the area of △ABC.

$$\mathrm{In}\:\bigtriangleup\mathrm{ABC},\:\mathrm{BC}=\mathrm{5cm}\:\mathrm{AC}=\mathrm{4cm} \\ $$$$\mathrm{cos}\left(\mathrm{A}−\mathrm{B}\right)=\frac{\mathrm{31}}{\mathrm{32}}\:\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\bigtriangleup\mathrm{ABC}. \\ $$

Question Number 113657 Answers: 2 Comments: 4

Two guns situated at the top of a hill of height 10m, fire one shot each with the same speed 5(√3)ms^(−1) at some interval of time. One gun fires horizontally and the other fires upwards at an angle of 60° with the horizontal. The shots collide in air at a point P. Find (i) the time interval between the firings and (ii) the coordinates of the hill right below the muzzle and trajectories in x-y plane.

$$\mathrm{Two}\:\mathrm{guns}\:\mathrm{situated}\:\mathrm{at}\:\mathrm{the}\:\mathrm{top}\:\mathrm{of}\:\mathrm{a}\:\mathrm{hill}\:\mathrm{of}\:\mathrm{height}\:\mathrm{10m},\:\mathrm{fire}\:\mathrm{one}\:\mathrm{shot}\:\mathrm{each}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{speed}\:\mathrm{5}\sqrt{\mathrm{3}}\mathrm{ms}^{−\mathrm{1}} \:\mathrm{at}\:\mathrm{some}\:\mathrm{interval}\:\mathrm{of}\:\mathrm{time}.\:\mathrm{One}\:\mathrm{gun}\:\mathrm{fires}\:\mathrm{horizontally} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{other}\:\mathrm{fires}\:\mathrm{upwards}\:\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{60}°\:\mathrm{with}\:\mathrm{the}\:\mathrm{horizontal}.\:\mathrm{The}\:\mathrm{shots} \\ $$$$\mathrm{collide}\:\mathrm{in}\:\mathrm{air}\:\mathrm{at}\:\mathrm{a}\:\mathrm{point}\:\mathrm{P}.\:\mathrm{Find}\:\left(\mathrm{i}\right)\:\mathrm{the}\:\mathrm{time}\:\mathrm{interval}\:\mathrm{between}\:\mathrm{the}\:\mathrm{firings}\:\mathrm{and} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{the}\:\mathrm{coordinates}\:\mathrm{of}\:\mathrm{the}\:\mathrm{hill}\:\mathrm{right}\:\mathrm{below}\:\mathrm{the}\:\mathrm{muzzle}\:\mathrm{and}\:\mathrm{trajectories}\:\mathrm{in}\:\mathrm{x}-\mathrm{y} \\ $$$$\mathrm{plane}. \\ $$

Question Number 113656 Answers: 1 Comments: 0

∫ (((1+tan (((3x)/2)))^2 )/(1+sin 3x)) dx ?

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

Question Number 113651 Answers: 3 Comments: 0

lim_(x→0) ((1/2)−(1/(1+e^(−x) ))).(1/(3x)) = ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{1}+{e}^{−{x}} }\right).\frac{\mathrm{1}}{\mathrm{3}{x}}\:=\:? \\ $$

Question Number 113857 Answers: 2 Comments: 0

find the values of k for which the line y=kx−3 does not meet the curve y=2x^2 −3x+k.

$${find}\:{the}\:{values}\:{of}\:{k}\:{for}\:{which}\:{the} \\ $$$${line}\:{y}={kx}−\mathrm{3}\:{does}\:{not}\:{meet}\:{the}\:{curve} \\ $$$${y}=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+{k}. \\ $$

Question Number 113641 Answers: 0 Comments: 1

Prove that there exists M>0 such that for any positive integers n, we have (√(1+(√(2+(√(...+(√(n+1))))))))≤M

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{there}\:\mathrm{exists}\:{M}>\mathrm{0}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{integers}\:{n},\:\mathrm{we}\:\mathrm{have} \\ $$$$\sqrt{\mathrm{1}+\sqrt{\mathrm{2}+\sqrt{...+\sqrt{{n}+\mathrm{1}}}}}\leqslant{M} \\ $$

Question Number 113637 Answers: 0 Comments: 0

Montrer que pour 0<z<1 on a Γ(z)Γ(1−z)=(π/(sin(πz)))

$${Montrer}\:{que}\:{pour}\:\mathrm{0}<{z}<\mathrm{1}\:{on}\:{a} \\ $$$$\Gamma\left({z}\right)\Gamma\left(\mathrm{1}−{z}\right)=\frac{\pi}{{sin}\left(\pi{z}\right)} \\ $$

Question Number 113634 Answers: 2 Comments: 0

Bonjour besoin d′aide Calculer ∫ln(cosx)dx

$${Bonjour}\:{besoin}\:{d}'{aide} \\ $$$${Calculer}\:\int{ln}\left({cosx}\right){dx} \\ $$

Question Number 113632 Answers: 0 Comments: 0

calculate U_n =∫_(1/n) ^n (1+(1/t^2 ))arctan(1−(1/t))dt find lim_(n→+∞) U_n

$$\mathrm{calculate}\:\mathrm{U}_{\mathrm{n}} =\int_{\frac{\mathrm{1}}{\mathrm{n}}} ^{\mathrm{n}} \left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{t}^{\mathrm{2}} }\right)\mathrm{arctan}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{t}}\right)\mathrm{dt} \\ $$$$\mathrm{find}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{U}_{\mathrm{n}} \\ $$

Question Number 113630 Answers: 3 Comments: 0

explicit g(a) =∫_0 ^(π/4) ln(1+acos^2 θ)dθ

$$\mathrm{explicit}\:\mathrm{g}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\mathrm{ln}\left(\mathrm{1}+\mathrm{acos}^{\mathrm{2}} \theta\right)\mathrm{d}\theta \\ $$

Question Number 113629 Answers: 0 Comments: 0

find f(a) =∫_0 ^(π/8) ln(1+a sinθ)dθ with o<a<1

$$\mathrm{find}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}} \:\mathrm{ln}\left(\mathrm{1}+\mathrm{a}\:\mathrm{sin}\theta\right)\mathrm{d}\theta\:\:\:\mathrm{with}\:\mathrm{o}<\mathrm{a}<\mathrm{1} \\ $$

Question Number 113742 Answers: 2 Comments: 0

lim_(x→0) (((1+x)^(1/x) −e−((ex)/2))/x^2 ) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+{x}\right)^{\frac{\mathrm{1}}{{x}}} −{e}−\frac{{ex}}{\mathrm{2}}}{{x}^{\mathrm{2}} }\:=?\: \\ $$

Question Number 113614 Answers: 0 Comments: 2

Question Number 113605 Answers: 1 Comments: 1

A bag contains 6 red, 5 white and 4 black balls If twl balls are drawn , what is the probability that none of them are red ?

$$\mathrm{A}\:\mathrm{bag}\:\mathrm{contains}\:\mathrm{6}\:\mathrm{red},\:\mathrm{5}\:\mathrm{white}\:\mathrm{and}\:\mathrm{4}\:\mathrm{black}\:\mathrm{balls} \\ $$$$\mathrm{If}\:\mathrm{twl}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{drawn}\:,\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability} \\ $$$$\mathrm{that}\:\mathrm{none}\:\mathrm{of}\:\mathrm{them}\:\mathrm{are}\:\mathrm{red}\:? \\ $$$$ \\ $$

Question Number 113604 Answers: 0 Comments: 0

if A={1,2} and let R={(a,b)∈A×A:a=3b} then R= ? help me sir

$${if}\:{A}=\left\{\mathrm{1},\mathrm{2}\right\}\:{and}\:{let}\:{R}=\left\{\left({a},{b}\right)\in{A}×{A}:{a}=\mathrm{3}{b}\right\}\: \\ $$$${then}\:{R}=\:? \\ $$$$ \\ $$$${help}\:{me}\:{sir} \\ $$

Question Number 113601 Answers: 1 Comments: 0

Question Number 113600 Answers: 1 Comments: 0

Prouver que β(a,b)=((Γ(a)Γ(b))/(Γ(a+b)))=∫_0 ^1 x^(a−1) (1−x)^(b−1) dx

$${Prouver}\:{que} \\ $$$$\beta\left({a},{b}\right)=\frac{\Gamma\left({a}\right)\Gamma\left({b}\right)}{\Gamma\left({a}+{b}\right)}=\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{a}−\mathrm{1}} \left(\mathrm{1}−{x}\right)^{{b}−\mathrm{1}} {dx} \\ $$

Question Number 113593 Answers: 2 Comments: 1

(6/(5×14))+(6/(14×23))+(6/(23×32))+...+(6/((9n−4)(9n+5)))=?

$$\frac{\mathrm{6}}{\mathrm{5}×\mathrm{14}}+\frac{\mathrm{6}}{\mathrm{14}×\mathrm{23}}+\frac{\mathrm{6}}{\mathrm{23}×\mathrm{32}}+...+\frac{\mathrm{6}}{\left(\mathrm{9}{n}−\mathrm{4}\right)\left(\mathrm{9}{n}+\mathrm{5}\right)}=? \\ $$

Question Number 113598 Answers: 1 Comments: 0

Question Number 113589 Answers: 2 Comments: 0

prove the following integral ∫_0 ^(π/2) (x^2 /(sinx))dx=2πG−(7/2)ζ(3) ∫_0 ^1 ln[((x+(√(1−x^2 )))/(x−(√(1−x^2 ))))]^2 (x/(1−x^2 ))dx=((.π^2 )/2) ∫_0 ^∞ ((e^x lnx)/(1+e^(2x) ))dx=(π/2)ln[((Γ((3/4)))/(Γ((1/4))))(√(2π))]

$${prove}\:{the}\:{following}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}} }{\mathrm{sin}{x}}{dx}=\mathrm{2}\pi{G}−\frac{\mathrm{7}}{\mathrm{2}}\zeta\left(\mathrm{3}\right) \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left[\frac{{x}+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{{x}−\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\right]^{\mathrm{2}} \frac{{x}}{\mathrm{1}−{x}^{\mathrm{2}} }{dx}=\frac{.\pi^{\mathrm{2}} }{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{e}^{{x}} \mathrm{ln}{x}}{\mathrm{1}+{e}^{\mathrm{2}{x}} }{dx}=\frac{\pi}{\mathrm{2}}\mathrm{ln}\left[\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\sqrt{\mathrm{2}\pi}\right] \\ $$

Question Number 113587 Answers: 2 Comments: 1

Question Number 113628 Answers: 1 Comments: 0

find ∫ (dx/((x+1)(√(x^2 −1))+(x−1)(√(x^2 +1))))

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

Question Number 113627 Answers: 0 Comments: 0

calculate ∫_0 ^∞ (dx/(x^4 +ix^2 +2))

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{4}} +\mathrm{ix}^{\mathrm{2}} \:+\mathrm{2}} \\ $$

Question Number 113578 Answers: 1 Comments: 2

Question Number 113576 Answers: 1 Comments: 0

lim_(r→∞) (√(4r^2 +2r)) −((8r^3 +4r^2 ))^(1/(3 )) =?

$$\:\:\:\underset{{r}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{4}{r}^{\mathrm{2}} +\mathrm{2}{r}}\:−\sqrt[{\mathrm{3}\:}]{\mathrm{8}{r}^{\mathrm{3}} +\mathrm{4}{r}^{\mathrm{2}} }\:=? \\ $$

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