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A and B are walking along a circular track.They start from same point at 8:00 am. A can walk 2 rounds per hour and B can walk 3 rounds per hour. How many times they cross each other before 9:30 am if they walk (i) Opposite to each other. (ii) In same direction. ?

$$\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{walking}\:\mathrm{along}\:\mathrm{a} \\ $$$$\mathrm{circular}\:\mathrm{track}.\mathrm{They}\:\mathrm{start}\:\mathrm{from} \\ $$$$\mathrm{same}\:\mathrm{point}\:\mathrm{at}\:\mathrm{8}:\mathrm{00}\:\mathrm{am}. \\ $$$$\mathrm{A}\:\mathrm{can}\:\mathrm{walk}\:\mathrm{2}\:\mathrm{rounds}\:\mathrm{per}\:\mathrm{hour} \\ $$$$\mathrm{and}\:\mathrm{B}\:\mathrm{can}\:\mathrm{walk}\:\mathrm{3}\:\mathrm{rounds}\:\mathrm{per}\:\mathrm{hour}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{times}\:\mathrm{they}\:\mathrm{cross}\:\mathrm{each} \\ $$$$\mathrm{other}\:\mathrm{before}\:\mathrm{9}:\mathrm{30}\:\mathrm{am}\:\mathrm{if}\:\mathrm{they}\:\mathrm{walk} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Opposite}\:\mathrm{to}\:\mathrm{each}\:\mathrm{other}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{In}\:\mathrm{same}\:\mathrm{direction}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:? \\ $$

Question Number 27397 Answers: 2 Comments: 0

The roots of x^2 −(a+1)x+b^2 =0 are equal. Then which of the following can be the values of a and b .

$$\mathrm{The}\:\mathrm{roots}\:\mathrm{of}\:{x}^{\mathrm{2}} −\left({a}+\mathrm{1}\right){x}+{b}^{\mathrm{2}} =\mathrm{0}\:\mathrm{are} \\ $$$$\mathrm{equal}.\:\mathrm{Then}\:\mathrm{which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:\:{a}\:\:\mathrm{and}\:\:{b}\:\:. \\ $$

Question Number 27393 Answers: 1 Comments: 0

solve tan^(−1) (2x/1−x^2 )+cot^(−1) (1−x^2 /2x)=π/3

$${solve} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{2}{x}/\mathrm{1}−{x}^{\mathrm{2}} \right)+\mathrm{cot}^{−\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} /\mathrm{2}{x}\right)=\pi/\mathrm{3} \\ $$

Question Number 27392 Answers: 0 Comments: 0

Show that the integral: ∫ e^(−x^2 ) dx Can′t be calculated trivially.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{integral}: \\ $$$$ \\ $$$$\int\:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \:\mathrm{dx} \\ $$$$ \\ $$$$\mathrm{Can}'\mathrm{t}\:\mathrm{be}\:\mathrm{calculated}\:\mathrm{trivially}. \\ $$

Question Number 27388 Answers: 0 Comments: 1

Σ_(n=1) ^∞ (((−1)^(n+1) )/n^2 )

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}+\mathrm{1}} }{\mathrm{n}^{\mathrm{2}} }\:\: \\ $$

Question Number 27384 Answers: 1 Comments: 1

let give p(x)= (((1+ix)/(1−ix)))^n − ((1+itanα )/(1−itanα)) factorize p(x) inside C[x].

$${let}\:{give}\:\:{p}\left({x}\right)=\:\left(\frac{\mathrm{1}+{ix}}{\mathrm{1}−{ix}}\right)^{{n}} −\:\frac{\mathrm{1}+{itan}\alpha\:}{\mathrm{1}−{itan}\alpha}\:\:{factorize}\:{p}\left({x}\right)\:{inside} \\ $$$${C}\left[{x}\right]. \\ $$

Question Number 27382 Answers: 1 Comments: 0

resolve inside C (((z−i)/(z+i)))^n +(((z+i)/(z−i)))^n = 2cosθ and0 <θ<π .n integer.

$${resolve}\:{inside}\:{C}\:\:\left(\frac{{z}−{i}}{{z}+{i}}\right)^{{n}} +\left(\frac{{z}+{i}}{{z}−{i}}\right)^{{n}} =\:\mathrm{2}{cos}\theta\:{and}\mathrm{0}\:<\theta<\pi\:.{n}\:{integer}. \\ $$

Question Number 27380 Answers: 0 Comments: 1

find the value of S_n = Σ_(k=0) ^(k=n) (((−1)^k C_n ^k )/(2k+1)) .

$${find}\:{the}\:{value}\:{of}\:{S}_{{n}} =\:\:\sum_{{k}=\mathrm{0}} ^{{k}={n}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} }{\mathrm{2}{k}+\mathrm{1}}\:\:. \\ $$

Question Number 27379 Answers: 1 Comments: 1

let give f(x)= ∫_x ^(2x) (dt/(ln(1+t^2 ))) calculate f^′ (x).

$${let}\:{give}\:{f}\left({x}\right)=\:\:\int_{{x}} ^{\mathrm{2}{x}} \:\:\frac{{dt}}{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\:\:{calculate}\:{f}^{'} \left({x}\right). \\ $$

Question Number 27376 Answers: 0 Comments: 2

Question Number 27365 Answers: 0 Comments: 0

Question Number 27345 Answers: 0 Comments: 1

prove that ∫_0 ^∞ (t^(x−1) /(e^t −1))dt =ξ(x)Γ(x) with ξ(x)= Σ_(n=1) ^∝ (1/n^x ) and Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt ( x>1)

$${prove}\:{that}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{x}−\mathrm{1}} }{{e}^{{t}} −\mathrm{1}}{dt}\:\:=\xi\left({x}\right)\Gamma\left({x}\right) \\ $$$${with}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{and}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left(\:{x}>\mathrm{1}\right) \\ $$

Question Number 27343 Answers: 1 Comments: 7

let give A=(_(2 2) ^(1 2) ) find A^n and e^A and e^(tA) . we remind that e^A = Σ_ (A^n /(n!))

$${let}\:{give}\:{A}=\left(_{\mathrm{2}\:\:\:\:\:\:\:\mathrm{2}} ^{\mathrm{1}\:\:\:\:\:\:\mathrm{2}} \right)\:\:\:{find}\:\:{A}^{{n}} \:\:\:{and}\:\:{e}^{{A}} \\ $$$${and}\:\:{e}^{{tA}} \:\:\:\:\:\:.\:{we}\:{remind}\:{that}\:\:{e}^{{A}} =\:\sum_{} \:\frac{{A}^{{n}} }{{n}!} \\ $$

Question Number 27342 Answers: 0 Comments: 1

find f(x)= ∫_0 ^∞ (e^(−x(1+t^2 )) /(1+t^2 )) dt interms ofx with x≥0 and calculate ∫_0 ^∞ e^(−t^2 ) dt .

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} }{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:\:{interms}\:{ofx} \\ $$$${with}\:{x}\geqslant\mathrm{0}\:\:\:{and}\:{calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{t}^{\mathrm{2}} } {dt}\:. \\ $$

Question Number 27341 Answers: 0 Comments: 0

prove that ∫_0 ^∞ e^(−(t^2 +(1/t^2 ))) dt is convergeny and find its value .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({t}^{\mathrm{2}} \:+\frac{\mathrm{1}}{{t}^{\mathrm{2}} }\right)} {dt}\:{is}\:{convergeny} \\ $$$${and}\:{find}\:{its}\:{value}\:. \\ $$

Question Number 27335 Answers: 0 Comments: 1

if 2 chords of ellipse have the same distance from the centre of ellipse and the eccentric angle of the end points of the chords are respectivly α β γ δ then prove that tan (α/2)×tan (β/2)×tan (γ/2)×tan (δ/2)=1

$${if}\:\mathrm{2}\:{chords}\:{of}\:{ellipse}\:{have}\:{the}\:{same} \\ $$$${distance}\:{from}\:{the}\:{centre}\:{of}\:{ellipse} \\ $$$${and}\:{the}\:{eccentric}\:{angle}\:{of}\:{the}\:{end}\:{points}\:{of}\:{the}\:{chords} \\ $$$${are}\:{respectivly}\:\alpha\:\beta\:\gamma\:\delta\:{then}\:{prove}\:{that} \\ $$$$\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}×\mathrm{tan}\:\frac{\beta}{\mathrm{2}}×\mathrm{tan}\:\frac{\gamma}{\mathrm{2}}×\mathrm{tan}\:\frac{\delta}{\mathrm{2}}=\mathrm{1} \\ $$

Question Number 27334 Answers: 1 Comments: 0

(q_1 /q_2 )=((x/(0.8−x)))^2 ; x=?

$$\frac{\mathrm{q}_{\mathrm{1}} }{\mathrm{q}_{\mathrm{2}} }=\left(\frac{\mathrm{x}}{\mathrm{0}.\mathrm{8}−\mathrm{x}}\right)^{\mathrm{2}} \:\:\:\:;\:\boldsymbol{\mathrm{x}}=? \\ $$

Question Number 27332 Answers: 1 Comments: 1

Question Number 27328 Answers: 0 Comments: 1

Question Number 27327 Answers: 0 Comments: 4

lim_(n→∞) (x^(2n) /(1+∣x∣+x^(4n) ))

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{2n}} }{\mathrm{1}+\mid\mathrm{x}\mid+\mathrm{x}^{\mathrm{4n}} } \\ $$

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