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Question Number 86193    Answers: 1   Comments: 8

lim_(x→0) ((4x^2 +((6x^2 )/(√(9x^4 +9sin^2 x))))/(3x−((4x^3 −x)/(2x+1)))) = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{4}{x}^{\mathrm{2}} +\frac{\mathrm{6}{x}^{\mathrm{2}} }{\sqrt{\mathrm{9}{x}^{\mathrm{4}} +\mathrm{9sin}\:^{\mathrm{2}} {x}}}}{\mathrm{3}{x}−\frac{\mathrm{4}{x}^{\mathrm{3}} −{x}}{\mathrm{2}{x}+\mathrm{1}}}\:=\:? \\ $$

Question Number 86189    Answers: 1   Comments: 0

Given that f(x)=((2x+7)/8) and g(x)= ((3x−6)/6), find (a) g(6), (b)f^(−1) (x) (c) the value of x if f(x)=g(x)

$${Given}\:{that}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{7}}{\mathrm{8}}\:{and}\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right)=\:\frac{\mathrm{3}\boldsymbol{\mathrm{x}}−\mathrm{6}}{\mathrm{6}},\:{find} \\ $$$$\left(\mathrm{a}\right)\:\boldsymbol{\mathrm{g}}\left(\mathrm{6}\right),\:\left(\mathrm{b}\right)\boldsymbol{\mathrm{f}}^{−\mathrm{1}} \left(\boldsymbol{\mathrm{x}}\right)\:\:\:\left(\mathrm{c}\right)\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\boldsymbol{\mathrm{x}}\:{if}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right) \\ $$

Question Number 86168    Answers: 1   Comments: 0

given sin x+sin y = 2sin (x+y) x+y ≠ 0 find the value of tan (x/2) tan (y/2) =

$$\mathrm{given}\: \\ $$$$\mathrm{sin}\:\mathrm{x}+\mathrm{sin}\:\mathrm{y}\:=\:\mathrm{2sin}\:\left(\mathrm{x}+\mathrm{y}\right)\: \\ $$$$\mathrm{x}+\mathrm{y}\:\neq\:\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{tan}\:\frac{\mathrm{x}}{\mathrm{2}}\:\mathrm{tan}\:\frac{\mathrm{y}}{\mathrm{2}}\:=\: \\ $$

Question Number 86167    Answers: 1   Comments: 3

∫_0 ^(π/2) ((arc tan ((√(tan x))))/(tan x)) dx

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{arc}\:\mathrm{tan}\:\left(\sqrt{\mathrm{tan}\:\mathrm{x}}\right)}{\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}\: \\ $$

Question Number 86160    Answers: 1   Comments: 3

If (x−1) f(x) + f((1/x)) = (1/(x−1)) find f(x)

$$\mathrm{If}\:\left(\mathrm{x}−\mathrm{1}\right)\:\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{x}−\mathrm{1}} \\ $$$$\mathrm{find}\:\mathrm{f}\left(\mathrm{x}\right)\: \\ $$

Question Number 86142    Answers: 2   Comments: 0

y′ .sin t cos t = y + sin^3 t y((π/4)) = 0

$$\mathrm{y}'\:.\mathrm{sin}\:\mathrm{t}\:\mathrm{cos}\:\mathrm{t}\:=\:\mathrm{y}\:+\:\mathrm{sin}\:^{\mathrm{3}} \mathrm{t}\: \\ $$$$\mathrm{y}\left(\frac{\pi}{\mathrm{4}}\right)\:=\:\mathrm{0}\: \\ $$

Question Number 86141    Answers: 0   Comments: 5

A number n leaves a remainder of 22 when divided by 24 and remainder 30 when divided by 33. Find the least possible value of n

$$\mathrm{A}\:\mathrm{number}\:\mathrm{n}\:\mathrm{leaves}\:\mathrm{a}\:\mathrm{remainder}\:\mathrm{of}\:\:\mathrm{22}\:\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{24}\:\mathrm{and} \\ $$$$\mathrm{remainder}\:\:\mathrm{30}\:\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\:\mathrm{33}.\:\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least}\:\mathrm{possible} \\ $$$$\mathrm{value}\:\mathrm{of}\:\:\mathrm{n} \\ $$

Question Number 86140    Answers: 0   Comments: 1

Question Number 86138    Answers: 1   Comments: 1

∫ _(−4) ^8 ((∣x∣)/x) dx = ?

$$\int\underset{−\mathrm{4}} {\overset{\mathrm{8}} {\:}}\:\frac{\mid\mathrm{x}\mid}{\mathrm{x}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 86132    Answers: 1   Comments: 0

∫x^3 sin(2x^2 +6)^5 dx

$$\int{x}^{\mathrm{3}} \:{sin}\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{6}\right)^{\mathrm{5}} \:{dx} \\ $$

Question Number 86129    Answers: 0   Comments: 1

Given that f(x)=8x and g(x)=((3x−2)/4), find (a) f^(−1) (x) (b) an expression for fg(x) (c) value of x which fg(x)=20

$${Given}\:{that}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{8}\boldsymbol{\mathrm{x}}\:{and}\:\boldsymbol{\mathrm{g}}\left(\boldsymbol{\mathrm{x}}\right)=\frac{\mathrm{3}\boldsymbol{\mathrm{x}}−\mathrm{2}}{\mathrm{4}},\:{find} \\ $$$$\left(\mathrm{a}\right)\:\boldsymbol{\mathrm{f}}^{−\mathrm{1}} \left(\boldsymbol{\mathrm{x}}\right)\:\:\:\left(\mathrm{b}\right)\:{an}\:{expression}\:{for}\:\boldsymbol{{fg}}\left(\boldsymbol{\mathrm{x}}\right) \\ $$$$\left(\mathrm{c}\right)\:{value}\:{of}\:\boldsymbol{\mathrm{x}}\:{which}\:\boldsymbol{\mathrm{fg}}\left(\boldsymbol{\mathrm{x}}\right)=\mathrm{20}\: \\ $$

Question Number 86128    Answers: 1   Comments: 0

⌊2x−(1/2)⌋=⌊∣x∣−(1/2)⌋=2x−2

$$\lfloor\mathrm{2}{x}−\frac{\mathrm{1}}{\mathrm{2}}\rfloor=\lfloor\mid{x}\mid−\frac{\mathrm{1}}{\mathrm{2}}\rfloor=\mathrm{2}{x}−\mathrm{2} \\ $$

Question Number 86116    Answers: 0   Comments: 1

lim_(x→∞) ((tanx)/x)

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{tanx}}{{x}} \\ $$

Question Number 86113    Answers: 0   Comments: 5

(1)Determine the following if it is convergent or divergent Σ_(n=1) ^∞ ((sin(n))/n) (2)Σ_(n=1) ^∞ ((sin(n^p ))/n^p ), pεR,find the range of p when it is convergent

$$\left(\mathrm{1}\right){Determine}\:{the}\:{following} \\ $$$${if}\:{it}\:{is}\:{convergent}\:{or}\:{divergent} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({n}\right)}{{n}} \\ $$$$\left(\mathrm{2}\right)\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{sin}\left({n}^{{p}} \right)}{{n}^{{p}} },\:{p}\epsilon\mathbb{R},{find}\:{the}\:{range}\: \\ $$$${of}\:{p}\:{when}\:{it}\:{is}\:{convergent} \\ $$

Question Number 86111    Answers: 2   Comments: 1

sin (((3π)/2)cos x) = −(1/2)

$$\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\mathrm{cos}\:{x}\right)\:=\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 86094    Answers: 1   Comments: 1

2∫(√(x^3 +4)) dx

$$\mathrm{2}\int\sqrt{{x}^{\mathrm{3}} +\mathrm{4}}\:{dx} \\ $$

Question Number 86092    Answers: 1   Comments: 1

∫(1/(√(5−4x−2x^2 )))dx

$$\int\frac{\mathrm{1}}{\sqrt{\mathrm{5}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 86091    Answers: 1   Comments: 1

∫ ((sin x−cos x)/(√(sin 2x))) dx

$$\int\:\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}}{\sqrt{\mathrm{sin}\:\mathrm{2x}}}\:\mathrm{dx} \\ $$

Question Number 86088    Answers: 0   Comments: 5

Find the sum of the series below: 1+2+3−4−5−6+7+8+9−10−11−12+13+14+15...−3020

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series}\:\mathrm{below}: \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{3}−\mathrm{4}−\mathrm{5}−\mathrm{6}+\mathrm{7}+\mathrm{8}+\mathrm{9}−\mathrm{10}−\mathrm{11}−\mathrm{12}+\mathrm{13}+\mathrm{14}+\mathrm{15}...−\mathrm{3020} \\ $$

Question Number 86085    Answers: 1   Comments: 4

If X^2 +Y^2 =10 XY=5 Find (X^2 −Y^2 )

$$\mathrm{If}\:\mathrm{X}^{\mathrm{2}} +\mathrm{Y}^{\mathrm{2}} =\mathrm{10} \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{XY}=\mathrm{5} \\ $$$$\mathrm{Find}\:\left(\mathrm{X}^{\mathrm{2}} −\mathrm{Y}^{\mathrm{2}} \right) \\ $$

Question Number 86080    Answers: 1   Comments: 1

∫(dx/(√(5−4x−2x^2 )))

$$\int\frac{{dx}}{\sqrt{\mathrm{5}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} }} \\ $$

Question Number 86042    Answers: 3   Comments: 0

solve: ⌊ (√x) ⌋=⌊(x/2)⌋

$${solve}:\:\:\lfloor\:\sqrt{{x}}\:\rfloor=\lfloor\frac{{x}}{\mathrm{2}}\rfloor \\ $$

Question Number 86041    Answers: 2   Comments: 0

1)if sin(θ−x)=k sin(θ+α) find tan(θ) and k then find θ in[0,2π] when k=(1/2) and α=π 2)if x=sin(t) and y=cos(2t) show that (d^2 y/dx^2 )+4=0

$$\left.\mathrm{1}\right){if}\: \\ $$$${sin}\left(\theta−{x}\right)={k}\:{sin}\left(\theta+\alpha\right) \\ $$$${find}\:{tan}\left(\theta\right)\:{and}\:{k} \\ $$$$ \\ $$$${then}\:{find}\:\theta\:{in}\left[\mathrm{0},\mathrm{2}\pi\right]\:\:{when}\:{k}=\frac{\mathrm{1}}{\mathrm{2}}\:{and}\:\alpha=\pi \\ $$$$ \\ $$$$\left.\mathrm{2}\right){if}\:{x}={sin}\left({t}\right)\:\:{and}\:\:{y}={cos}\left(\mathrm{2}{t}\right) \\ $$$${show}\:{that} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }+\mathrm{4}=\mathrm{0} \\ $$

Question Number 86040    Answers: 3   Comments: 0

Question Number 86039    Answers: 0   Comments: 1

∫((2x^5 −x^3 −1)/(x^3 −4x))dx

$$\int\frac{\mathrm{2}{x}^{\mathrm{5}} −{x}^{\mathrm{3}} −\mathrm{1}}{{x}^{\mathrm{3}} −\mathrm{4}{x}}{dx} \\ $$

Question Number 86120    Answers: 1   Comments: 5

(dy/dx) + ((sin 2y)/x) = x^3 cos^2 y

$$\frac{\mathrm{dy}}{\mathrm{dx}}\:+\:\frac{\mathrm{sin}\:\mathrm{2y}}{\mathrm{x}}\:=\:\mathrm{x}^{\mathrm{3}} \:\mathrm{cos}\:^{\mathrm{2}} \:\mathrm{y} \\ $$

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