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Question Number 84364    Answers: 1   Comments: 0

lim_(x→π) (((√π)−(√(π+4x)))/(cos (((π(x+1))/2)))) = ?

$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\sqrt{\pi}−\sqrt{\pi+\mathrm{4x}}}{\mathrm{cos}\:\left(\frac{\pi\left(\mathrm{x}+\mathrm{1}\right)}{\mathrm{2}}\right)}\:=\:? \\ $$

Question Number 84359    Answers: 0   Comments: 1

Question Number 84341    Answers: 0   Comments: 1

Find the centre of symmetry of the curve: y = (1/(x + 2))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{of}\:\mathrm{symmetry}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{curve}: \\ $$$$\:\:\:\:{y}\:=\:\frac{\mathrm{1}}{{x}\:+\:\mathrm{2}} \\ $$

Question Number 84335    Answers: 0   Comments: 4

Question Number 84334    Answers: 1   Comments: 1

∫((1−u)/(−1−2u+u^2 ))du

$$\int\frac{\mathrm{1}−\mathrm{u}}{−\mathrm{1}−\mathrm{2u}+\mathrm{u}^{\mathrm{2}} }\mathrm{du} \\ $$

Question Number 84333    Answers: 3   Comments: 0

1)find without l′hopital lim_(x→0) ((2(√(x+1))−((x+1))^(1/3) −((x+1))^(1/4) )/x) 2) prove that the general solution for tbe differential equation (1+y^2 )+(1+x^2 )((dy/dx))=0 is y=((k−x)/(1+kx)),k is a constant then find the special solution if y=(2/(3 )) when x=1

$$\left.\mathrm{1}\right){find}\:{without}\:{l}'{hopital} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\frac{\mathrm{2}\sqrt{{x}+\mathrm{1}}−\sqrt[{\mathrm{3}}]{{x}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{{x}+\mathrm{1}}}{{x}} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{the}\:{general}\:{solution}\:{for}\:{tbe}\:{differential}\:{equation} \\ $$$$\left(\mathrm{1}+{y}^{\mathrm{2}} \right)+\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\frac{{dy}}{{dx}}\right)=\mathrm{0}\:{is}\:{y}=\frac{{k}−{x}}{\mathrm{1}+{kx}},{k}\:{is}\:{a}\:{constant} \\ $$$${then}\:{find}\:{the}\:{special}\:{solution}\:{if}\:{y}=\frac{\mathrm{2}}{\mathrm{3}\:}\:{when}\:{x}=\mathrm{1} \\ $$

Question Number 84330    Answers: 1   Comments: 3

lim_(x→1) (((2x^3 +x+1)−64)/(x^3 −1))

$$\underset{{x}\rightarrow\mathrm{1}} {{lim}}\frac{\left(\mathrm{2}{x}^{\mathrm{3}} +{x}+\mathrm{1}\right)−\mathrm{64}}{{x}^{\mathrm{3}} −\mathrm{1}} \\ $$

Question Number 84328    Answers: 0   Comments: 0

(3/7)×(1/2)ln[((u−1)/(u+1))]−(1/2)ln[u^2 −1]

$$\frac{\mathrm{3}}{\mathrm{7}}×\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\left[\frac{\mathrm{u}−\mathrm{1}}{\mathrm{u}+\mathrm{1}}\right]−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}\left[\mathrm{u}^{\mathrm{2}} −\mathrm{1}\right] \\ $$

Question Number 84327    Answers: 1   Comments: 0

∫((3−7u)/(7u^2 −7))du

$$\int\frac{\mathrm{3}−\mathrm{7u}}{\mathrm{7u}^{\mathrm{2}} −\mathrm{7}}\mathrm{du} \\ $$

Question Number 84326    Answers: 1   Comments: 0

∫((3−7u)/(7u^2 −7))du

$$\int\frac{\mathrm{3}−\mathrm{7u}}{\mathrm{7u}^{\mathrm{2}} −\mathrm{7}}\mathrm{du} \\ $$

Question Number 84323    Answers: 1   Comments: 0

A particle moving in a straight line OX has a displacement x from O at time t where x satisfies the equation (d^2 x/(dt^2 )) + 2(dx/dt) + 3x = 0 the damping factor for the motion is [A] e^(−1) [B] e^(−2t) [C] e^(−3t) [D] e^(−5t)

$$\mathrm{A}\:\mathrm{particle}\:\mathrm{moving}\:\mathrm{in}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:{OX}\:\mathrm{has}\:\mathrm{a} \\ $$$$\mathrm{displacement}\:{x}\:\mathrm{from}\:{O}\:\mathrm{at}\:\mathrm{time}\:{t}\:\mathrm{where}\:{x}\:\mathrm{satisfies} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\frac{{d}^{\mathrm{2}} {x}}{{dt}^{\mathrm{2}} \:}\:+\:\mathrm{2}\frac{{dx}}{{dt}}\:+\:\mathrm{3}{x}\:=\:\mathrm{0} \\ $$$$\mathrm{the}\:\mathrm{damping}\:\mathrm{factor}\:\mathrm{for}\:\mathrm{the}\:\mathrm{motion}\:\mathrm{is} \\ $$$$\left[\mathrm{A}\right]\:{e}^{−\mathrm{1}} \\ $$$$\left[\mathrm{B}\right]\:{e}^{−\mathrm{2}{t}} \\ $$$$\left[\mathrm{C}\right]\:{e}^{−\mathrm{3}{t}} \\ $$$$\left[\mathrm{D}\right]\:{e}^{−\mathrm{5}{t}} \\ $$

Question Number 84316    Answers: 1   Comments: 1

Which one of the following sets of vectors is a basis for R^2 [A] { ((1),((−2)) ) , (((−3)),(6) )} [B] { ((1),(1) ) , ((2),(2) )} [C] { ((2),(1) ) , ((0),(1) )} [D] { ((1),(2) ) , ((4),(8) ) }

$$\mathrm{Which}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{sets}\:\mathrm{of} \\ $$$$\mathrm{vectors}\:\mathrm{is}\:\mathrm{a}\:\mathrm{basis}\:\mathrm{for}\:\mathbb{R}^{\mathrm{2}} \\ $$$$\left[\mathrm{A}\right]\:\left\{\begin{pmatrix}{\mathrm{1}}\\{−\mathrm{2}}\end{pmatrix}\:,\:\begin{pmatrix}{−\mathrm{3}}\\{\mathrm{6}}\end{pmatrix}\right\} \\ $$$$\left[\mathrm{B}\right]\:\left\{\begin{pmatrix}{\mathrm{1}}\\{\mathrm{1}}\end{pmatrix}\:,\begin{pmatrix}{\mathrm{2}}\\{\mathrm{2}}\end{pmatrix}\right\} \\ $$$$\left[\mathrm{C}\right]\:\left\{\begin{pmatrix}{\mathrm{2}}\\{\mathrm{1}}\end{pmatrix}\:,\begin{pmatrix}{\mathrm{0}}\\{\mathrm{1}}\end{pmatrix}\right\} \\ $$$$\left[\mathrm{D}\right]\:\left\{\begin{pmatrix}{\mathrm{1}}\\{\mathrm{2}}\end{pmatrix}\:,\:\begin{pmatrix}{\mathrm{4}}\\{\mathrm{8}}\end{pmatrix}\:\right\} \\ $$

Question Number 84315    Answers: 1   Comments: 0

∫xy dx

$$\int{xy}\:{dx} \\ $$

Question Number 84313    Answers: 1   Comments: 3

∫ (a^(2/5) −x^(2/5) )^(5/2) dx

$$\int\:\:\left({a}^{\mathrm{2}/\mathrm{5}} −{x}^{\mathrm{2}/\mathrm{5}} \right)^{\frac{\mathrm{5}}{\mathrm{2}}} \:{dx} \\ $$

Question Number 84310    Answers: 0   Comments: 0

Question Number 84309    Answers: 0   Comments: 0

lim_(x→∞) ((1−(ln ∣x∣)^(n+1) )/(1−ln∣x∣))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}−\left(\mathrm{ln}\:\mid\mathrm{x}\mid\right)^{\mathrm{n}+\mathrm{1}} }{\mathrm{1}−\mathrm{ln}\mid\mathrm{x}\mid} \\ $$

Question Number 84304    Answers: 1   Comments: 1

ΔABC AB=a BC=b AC=c and ∠C=90° r=?

$$\Delta\mathrm{ABC}\:\mathrm{AB}=\mathrm{a}\:\mathrm{BC}=\mathrm{b}\:\mathrm{AC}=\mathrm{c}\: \\ $$$$\mathrm{and}\:\angle\mathrm{C}=\mathrm{90}° \\ $$$$\mathrm{r}=? \\ $$

Question Number 84296    Answers: 0   Comments: 1

Question Number 84293    Answers: 1   Comments: 0

solve in R x^([x]) +x^(2−[x]) =x^2 +1

$${solve}\:{in}\:{R} \\ $$$${x}^{\left[{x}\right]} +{x}^{\mathrm{2}−\left[{x}\right]} ={x}^{\mathrm{2}} +\mathrm{1} \\ $$

Question Number 84288    Answers: 1   Comments: 8

Question Number 84283    Answers: 1   Comments: 0

lim_(x→a) (((∣x∣−1)^2 −(∣a∣−1)^2 )/(x−a)) = k , a>0 lim_(x→a) (((∣x∣−1)^3 −(∣a∣−1)^3 )/(x^2 −a^2 )) =

$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\frac{\left(\mid\mathrm{x}\mid−\mathrm{1}\right)^{\mathrm{2}} −\left(\mid\mathrm{a}\mid−\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{x}−\mathrm{a}}\:=\:\mathrm{k}\:,\:\mathrm{a}>\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{a}} {\mathrm{lim}}\:\frac{\left(\mid\mathrm{x}\mid−\mathrm{1}\right)^{\mathrm{3}} −\left(\mid\mathrm{a}\mid−\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{x}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} }\:=\: \\ $$$$ \\ $$

Question Number 84275    Answers: 1   Comments: 2

(a.c)b−(a.b)c = a×(c×b) ?

$$\left(\boldsymbol{\mathrm{a}}.\boldsymbol{\mathrm{c}}\right)\boldsymbol{\mathrm{b}}−\left(\boldsymbol{\mathrm{a}}.\boldsymbol{\mathrm{b}}\right)\boldsymbol{\mathrm{c}}\:=\:\boldsymbol{\mathrm{a}}×\left(\boldsymbol{\mathrm{c}}×\boldsymbol{\mathrm{b}}\right)\:? \\ $$

Question Number 84246    Answers: 1   Comments: 0

∫cos^n mx dx =

$$\int\mathrm{cos}^{{n}} \:{mx}\:\:{dx}\:= \\ $$

Question Number 84245    Answers: 0   Comments: 0

$$ \\ $$

Question Number 84242    Answers: 1   Comments: 1

∫_0 ^(ln2) (1/(cosh(x + ln4)))dx

$$\underset{\mathrm{0}} {\overset{\mathrm{ln2}} {\int}}\frac{\mathrm{1}}{\mathrm{cosh}\left({x}\:+\:\mathrm{ln4}\right)}{dx} \\ $$

Question Number 84236    Answers: 1   Comments: 0

If ax^2 +2hxy +by^2 = 1, show that (d^2 y/dx^2 ) = ((h^2 −ab)/((hx+by)^3 )).

$$\:\:\mathrm{If}\:\boldsymbol{\mathrm{ax}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{hxy}}\:+\boldsymbol{\mathrm{by}}^{\mathrm{2}} =\:\mathrm{1},\:\mathrm{show}\:\mathrm{that} \\ $$$$\:\:\:\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2}} }\:=\:\frac{\boldsymbol{\mathrm{h}}^{\mathrm{2}} −\boldsymbol{\mathrm{ab}}}{\left(\boldsymbol{\mathrm{hx}}+\boldsymbol{\mathrm{by}}\right)^{\mathrm{3}} }. \\ $$

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