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Question Number 84215    Answers: 1   Comments: 0

hi show that the following sequence is limited: U_n =((3n+2)/(2n+1)) precise the upper and lower.

$${hi} \\ $$$${show}\:{that}\:{the}\:{following}\:{sequence} \\ $$$${is}\:{limited}: \\ $$$${U}_{{n}} =\frac{\mathrm{3}{n}+\mathrm{2}}{\mathrm{2}{n}+\mathrm{1}} \\ $$$${precise}\:{the}\:{upper}\:{and}\:{lower}. \\ $$

Question Number 84206    Answers: 0   Comments: 0

a, b, c, d ∈ N (a, b, c, d) is quadruple of a, b, c, d such that b = a^2 + 1 c = b^2 + 1 d = c^2 + 1 τ(a) + τ(b) + τ(c) + τ(d) is odd number(s) . τ(k) is the number of positive divisor of natural number k . a, b, c, d < 10^6 How many quadruple of a, b, c, d ?

$${a},\:{b},\:{c},\:{d}\:\:\in\:\:\mathbb{N} \\ $$$$\left({a},\:{b},\:{c},\:{d}\right)\:\:{is}\:\:{quadruple}\:\:{of}\:\:{a},\:{b},\:{c},\:{d}\:\:{such}\:\:{that} \\ $$$$\:\:\:\:\:\:\:\:{b}\:\:=\:\:{a}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:{c}\:\:=\:\:{b}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:{d}\:\:=\:\:{c}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$$$\tau\left({a}\right)\:+\:\tau\left({b}\right)\:+\:\tau\left({c}\right)\:+\:\tau\left({d}\right)\:\:{is}\:\:\:{odd}\:\:{number}\left({s}\right)\:. \\ $$$$\tau\left({k}\right)\:\:{is}\:\:{the}\:\:{number}\:\:{of}\:\:{positive}\:\:{divisor}\:\:{of}\:\:\:{natural}\:\:{number}\:\:{k}\:. \\ $$$${a},\:{b},\:{c},\:{d}\:<\:\:\mathrm{10}^{\mathrm{6}} \\ $$$${How}\:\:{many}\:\:{quadruple}\:\:{of}\:\:{a},\:{b},\:{c},\:{d}\:\:? \\ $$

Question Number 84205    Answers: 3   Comments: 0

prove a. sin2x=tanx(1+cos2x) b. sin2x=((2tanx)/(1+tan^2 x))

$${prove}\: \\ $$$${a}.\:{sin}\mathrm{2}{x}={tanx}\left(\mathrm{1}+{cos}\mathrm{2}{x}\right) \\ $$$${b}.\:{sin}\mathrm{2}{x}=\frac{\mathrm{2}{tanx}}{\mathrm{1}+{tan}^{\mathrm{2}} {x}} \\ $$

Question Number 84202    Answers: 1   Comments: 0

prove cos^4 x−sin^4 x=cos22x

$${prove}\: \\ $$$${cos}^{\mathrm{4}} {x}−{sin}^{\mathrm{4}} {x}={cos}\mathrm{22}{x} \\ $$

Question Number 84188    Answers: 0   Comments: 4

if 2x+3y = 2020? find maximum value 3x+2y for x and natural number

$$\mathrm{if}\:\mathrm{2x}+\mathrm{3y}\:=\:\mathrm{2020}? \\ $$$$\mathrm{find}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{3x}+\mathrm{2y}\:\mathrm{for}\:\mathrm{x}\:\mathrm{and}\:\mathrm{natural} \\ $$$$\mathrm{number} \\ $$

Question Number 84182    Answers: 2   Comments: 0

Find the number of solutions for positive integers (x,y,z) satisfying x+2y+3z=n.

$${Find}\:{the}\:{number}\:{of}\:{solutions}\:{for} \\ $$$${positive}\:{integers}\:\left({x},{y},{z}\right)\:{satisfying} \\ $$$$\boldsymbol{{x}}+\mathrm{2}\boldsymbol{{y}}+\mathrm{3}\boldsymbol{{z}}=\boldsymbol{{n}}. \\ $$

Question Number 84178    Answers: 1   Comments: 0

Find (dy/dx) given that y = (sin^n x cosnx)

$$\mathrm{Find}\:\frac{{dy}}{{dx}}\:\mathrm{given}\:\mathrm{that}\:{y}\:=\:\left({sin}^{{n}} \:{x}\:{cosnx}\right) \\ $$

Question Number 84165    Answers: 1   Comments: 1

∫_0 ^1 ((ln(x+2))/(x^2 −2x+4)) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({x}+\mathrm{2}\right)}{{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{4}}\:{dx} \\ $$

Question Number 84163    Answers: 2   Comments: 0

∫ sin (50x) sin^(49) (x) dx ?

$$\int\:\mathrm{sin}\:\left(\mathrm{50}{x}\right)\:\mathrm{sin}\:^{\mathrm{49}} \left({x}\right)\:{dx}\:? \\ $$

Question Number 84157    Answers: 3   Comments: 0

if a circle having an equation x^2 +y^2 −6x−8y=0 is intersected at A and B by x+y=1.find the equation of the circle on AB as diameter

$${if}\:{a}\:{circle}\:{having}\:{an}\: \\ $$$${equation}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{6}{x}−\mathrm{8}{y}=\mathrm{0} \\ $$$${is}\:{intersected}\:{at}\:{A}\:{and}\:{B} \\ $$$${by}\:{x}+{y}=\mathrm{1}.{find}\:{the}\: \\ $$$${equation}\:{of}\:{the}\:{circle} \\ $$$${on}\:{AB}\:{as}\:{diameter} \\ $$

Question Number 84156    Answers: 0   Comments: 2

Question Number 84152    Answers: 0   Comments: 1

Question Number 84136    Answers: 1   Comments: 0

show that: tan3x=((3+tan^2 x)/(1−3tan^2 x))×tanx

$${show}\:{that}: \\ $$$${tan}\mathrm{3}{x}=\frac{\mathrm{3}+{tan}^{\mathrm{2}} {x}}{\mathrm{1}−\mathrm{3}{tan}^{\mathrm{2}} {x}}×{tanx} \\ $$

Question Number 84135    Answers: 2   Comments: 0

find the area between the function y=2sin2x −1 and the x−axis on [−π,(π/2)]

$${find}\:{the}\:{area}\:{between}\:{the}\:{function}\: \\ $$$${y}=\mathrm{2}{sin}\mathrm{2}{x}\:−\mathrm{1}\:{and}\:\:{the}\:{x}−{axis}\:\:{on}\:\left[−\pi,\frac{\pi}{\mathrm{2}}\right] \\ $$

Question Number 84134    Answers: 1   Comments: 0

find the general solution of the equation 2sin 3θ = sin 2θ

$$\mathrm{find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\mathrm{2sin}\:\mathrm{3}\theta\:=\:\mathrm{sin}\:\mathrm{2}\theta \\ $$

Question Number 84130    Answers: 0   Comments: 2

lim_(x→+∞) x((√(x^2 +2x))−2(√(x^2 +x))+x)

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}x}\left(\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}}−\mathrm{2}\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}}+\mathrm{x}\right) \\ $$

Question Number 84126    Answers: 1   Comments: 0

∫((5−x)/(1+(√((x−4)))))dx

$$\int\frac{\mathrm{5}−{x}}{\mathrm{1}+\sqrt{\left({x}−\mathrm{4}\right)}}\boldsymbol{{dx}} \\ $$

Question Number 84125    Answers: 0   Comments: 0

is it possible to find the no. of positive integral solutions of x+y+2z+3c=7

$${is}\:{it}\:{possible}\:{to}\:{find}\: \\ $$$${the}\:{no}.\:{of}\:{positive}\: \\ $$$${integral}\:{solutions} \\ $$$${of}\:{x}+{y}+\mathrm{2}{z}+\mathrm{3}{c}=\mathrm{7} \\ $$

Question Number 84123    Answers: 0   Comments: 1

∫((5−x)/(1+(√((x−4)))))

$$\int\frac{\mathrm{5}−\boldsymbol{{x}}}{\mathrm{1}+\sqrt{\left(\boldsymbol{{x}}−\mathrm{4}\right)}} \\ $$

Question Number 84121    Answers: 0   Comments: 4

given that g(x) = { ((x + 2 , if 0 ≤ x < 2)),((x^2 , if 2 ≤ x < 4)) :} is periodic of period 4. sketch the curve for g(x) in the interval 0≤ x < 8 evaluate g(−6).

$$\mathrm{given}\:\:\mathrm{that} \\ $$$$\:\mathrm{g}\left({x}\right)\:=\:\begin{cases}{{x}\:+\:\mathrm{2}\:,\:\mathrm{if}\:\:\mathrm{0}\:\leqslant\:{x}\:<\:\mathrm{2}}\\{{x}^{\mathrm{2}} \:,\:\mathrm{if}\:\:\mathrm{2}\:\leqslant\:{x}\:<\:\mathrm{4}}\end{cases} \\ $$$$\mathrm{is}\:\mathrm{periodic}\:\mathrm{of}\:\mathrm{period}\:\mathrm{4}.\: \\ $$$$\mathrm{sketch}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{for}\:\mathrm{g}\left({x}\right)\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\:\:\mathrm{0}\leqslant\:{x}\:<\:\mathrm{8} \\ $$$$\mathrm{evaluate}\:\:\mathrm{g}\left(−\mathrm{6}\right). \\ $$

Question Number 84109    Answers: 1   Comments: 2

Question Number 84106    Answers: 0   Comments: 1

∫ (x^4 /(√(1−x^4 ))) dx = ?

$$\int\:\frac{\mathrm{x}^{\mathrm{4}} }{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}\:\mathrm{dx}\:=\:? \\ $$

Question Number 84101    Answers: 2   Comments: 0

Find the reduction formula ∫x^n e^(ax) dx

$$\:\boldsymbol{\mathrm{Find}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{reduction}}\:\boldsymbol{\mathrm{formula}} \\ $$$$\:\:\int\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{ax}}} \:\boldsymbol{\mathrm{dx}} \\ $$

Question Number 84100    Answers: 2   Comments: 0

Find the differential equations: x(dy/dx)= y− (√(x^2 +y^2 ))

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equations}: \\ $$$$\:\:\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}=\:\mathrm{y}−\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} } \\ $$

Question Number 84087    Answers: 0   Comments: 1

∫_0 ^3 (√(1+sinh^2 t)) dt

$$\int_{\mathrm{0}} ^{\mathrm{3}} \sqrt{\mathrm{1}+{sinh}^{\mathrm{2}} {t}}\:{dt} \\ $$

Question Number 84085    Answers: 1   Comments: 0

solve (d/dr)(r(dθ/dr))=0

$${solve} \\ $$$$\frac{{d}}{{dr}}\left({r}\frac{{d}\theta}{{dr}}\right)=\mathrm{0} \\ $$

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