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AllQuestion and Answers: Page 1327

Question Number 80144 Answers: 0 Comments: 1

solve for x: (((√x)+1)/(√(x+1)))+ax^2 =x(a^2 +1) [a∈R]

$$\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{x}}: \\ $$$$\frac{\sqrt{\boldsymbol{\mathrm{x}}}+\mathrm{1}}{\sqrt{\boldsymbol{\mathrm{x}}+\mathrm{1}}}+\boldsymbol{\mathrm{ax}}^{\mathrm{2}} =\boldsymbol{\mathrm{x}}\left(\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\mathrm{1}\right)\:\:\:\:\:\:\left[\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}\right] \\ $$

Question Number 80116 Answers: 1 Comments: 1

Find S_m =Σ_(n=0) ^∞ (1/(Π_(k=1) ^m (n+k)))=? (m≥2)

$${Find} \\ $$$${S}_{{m}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\underset{{k}=\mathrm{1}} {\overset{{m}} {\prod}}\left({n}+{k}\right)}=? \\ $$$$\left({m}\geqslant\mathrm{2}\right) \\ $$

Question Number 80113 Answers: 0 Comments: 1

how do you simply sin (tan^(−1) (3x)+cos^(−1) (x)) ?

$${how}\:{do}\:{you}\:{simply} \\ $$$$\mathrm{sin}\:\left(\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{3}{x}\right)+\mathrm{cos}^{−\mathrm{1}} \left({x}\right)\right)\:? \\ $$

Question Number 80102 Answers: 0 Comments: 0

Question Number 80093 Answers: 3 Comments: 0

Solve for x and y x^(√y) = 64 y^(√x) = 81

$$\mathrm{Solve}\:\mathrm{for}\:\:\mathrm{x}\:\mathrm{and}\:\mathrm{y} \\ $$$$\:\:\:\:\:\mathrm{x}^{\sqrt{\mathrm{y}}} \:\:\:=\:\:\mathrm{64} \\ $$$$\:\:\:\:\:\mathrm{y}^{\sqrt{\mathrm{x}}} \:\:\:=\:\mathrm{81} \\ $$

Question Number 80088 Answers: 0 Comments: 0

When the father was son′s age, the son was ten years old; when the son will be father′s age, the father will be seventy. What are their ages ?

$$\:{When}\:\:{the}\:\:{father}\:\:{was}\:{son}'{s}\:\:{age},\:\:{the}\:\:{son} \\ $$$$\:\:{was}\:\:{ten}\:\:{years}\:\:{old};\:\:{when}\:\:{the}\:\:{son}\:\:{will}\:\:{be}\:\:{father}'{s}\:\:{age}, \\ $$$$\:\:{the}\:\:{father}\:\:{will}\:\:{be}\:\:{seventy}. \\ $$$$\:\:{What}\:\:{are}\:\:{their}\:\:{ages}\:\:? \\ $$

Question Number 80084 Answers: 0 Comments: 3

−1=(−1)^1 =(−1)^(2/2) =((−1)^2 )^(1/2) =(1)^(1/2) = =(√1)=1 what do you think about this?

$$\:\:−\mathrm{1}=\left(−\mathrm{1}\right)^{\mathrm{1}} =\left(−\mathrm{1}\right)^{\frac{\mathrm{2}}{\mathrm{2}}} =\left(\left(−\mathrm{1}\right)^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} =\left(\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} = \\ $$$$=\sqrt{\mathrm{1}}=\mathrm{1}\:\: \\ $$$$\mathrm{what}\:\mathrm{do}\:\mathrm{you}\:\mathrm{think}\:\mathrm{about}\:\mathrm{this}? \\ $$

Question Number 80068 Answers: 2 Comments: 3

Question Number 80065 Answers: 0 Comments: 0

Question Number 80064 Answers: 1 Comments: 6

lim_(x→−∞) [(√(1−xe^x ))]

$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\left[\sqrt{\mathrm{1}−{xe}^{{x}} \:}\right] \\ $$

Question Number 80057 Answers: 1 Comments: 2

Question Number 80053 Answers: 0 Comments: 4

Find integer x, y such that 2^x −y^2 =615

$${Find}\:{integer}\:{x},\:{y}\:{such}\:{that} \\ $$$$\mathrm{2}^{{x}} −{y}^{\mathrm{2}} =\mathrm{615} \\ $$

Question Number 80052 Answers: 0 Comments: 0

∫ e^(sin 2x) .cos x dx =

$$\int\:\mathrm{e}^{\mathrm{sin}\:\mathrm{2x}} .\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:= \\ $$$$ \\ $$

Question Number 80108 Answers: 1 Comments: 3

a,b,c ∈R ((b+c+d)/a)=((a+c+d)/b)=((a+b+c)/d)=((a+b+d)/c)=r what is r?

$${a},{b},{c}\:\in\mathbb{R} \\ $$$$\frac{{b}+{c}+{d}}{{a}}=\frac{{a}+{c}+{d}}{{b}}=\frac{{a}+{b}+{c}}{{d}}=\frac{{a}+{b}+{d}}{{c}}={r} \\ $$$${what}\:{is}\:{r}? \\ $$

Question Number 80039 Answers: 1 Comments: 6

prove that (1+x)(1+(1/x))≥4

$${prove}\:{that} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)\geqslant\mathrm{4} \\ $$

Question Number 80037 Answers: 0 Comments: 0

A matrix A= [(a_(ij) ) ] is an upper triangular matrix if

$$\mathrm{A}\:\mathrm{matrix}\:{A}=\begin{bmatrix}{{a}_{{ij}} }\end{bmatrix}\:\mathrm{is}\:\mathrm{an}\:\mathrm{upper}\:\mathrm{triangular} \\ $$$$\mathrm{matrix}\:\mathrm{if} \\ $$

Question Number 80036 Answers: 1 Comments: 3

Σ_(n=1) ^∞ (1/((n+1)(n+2)(n+3)))=

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\left({n}+\mathrm{3}\right)}=\: \\ $$

Question Number 80027 Answers: 0 Comments: 4

find minimum value of (√(x^2 +4))+(√(x^2 −24x+153)) for x≥0 in R

$${find}\:{minimum} \\ $$$${value}\:{of}\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}}+\sqrt{{x}^{\mathrm{2}} −\mathrm{24}{x}+\mathrm{153}} \\ $$$${for}\:{x}\geqslant\mathrm{0}\:{in}\:\mathbb{R} \\ $$

Question Number 80015 Answers: 2 Comments: 2

Question Number 79992 Answers: 0 Comments: 2

lim_(x→0) [(1/x)] = ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{1}}{\mathrm{x}}\right]\:=\:? \\ $$

Question Number 80000 Answers: 1 Comments: 2

prove that lim_(x→0) ((arcsin(x/(√(1−x^2 ))))/(ln(1−x))) = −1

$$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{arcsin}}\frac{\boldsymbol{{x}}}{\sqrt{\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} }}}{\boldsymbol{{ln}}\left(\mathrm{1}−\boldsymbol{{x}}\right)}\:=\:−\mathrm{1} \\ $$

Question Number 79998 Answers: 0 Comments: 3

given 3x + 4y+1 = 3(√x) + 2(√y) find the value of (√(x.y))

$$\mathrm{given}\:\mathrm{3x}\:+\:\mathrm{4y}+\mathrm{1}\:=\:\mathrm{3}\sqrt{\mathrm{x}}\:+\:\mathrm{2}\sqrt{\mathrm{y}}\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\sqrt{\mathrm{x}.\mathrm{y}}\: \\ $$

Question Number 79978 Answers: 3 Comments: 0

Given for x,y,z>0: 2^x =3^y =5^z Arrange 2x, 3y, 5z in increasing order.

$${Given}\:{for}\:{x},{y},{z}>\mathrm{0}: \\ $$$$\mathrm{2}^{{x}} =\mathrm{3}^{{y}} =\mathrm{5}^{{z}} \\ $$$${Arrange}\:\mathrm{2}{x},\:\mathrm{3}{y},\:\mathrm{5}{z}\:{in}\:{increasing}\:{order}. \\ $$

Question Number 79974 Answers: 1 Comments: 0

Question Number 80004 Answers: 1 Comments: 2

x and y any integer satisfy equation (x−2004)(x−2006)=2^y the greatest possible value of x+y

$${x}\:{and}\:{y}\:{any}\:{integer}\:{satisfy} \\ $$$${equation}\:\left({x}−\mathrm{2004}\right)\left({x}−\mathrm{2006}\right)=\mathrm{2}^{{y}} \\ $$$${the}\:{greatest}\:{possible}\:{value} \\ $$$${of}\:{x}+{y} \\ $$

Question Number 79969 Answers: 1 Comments: 0

find the general solution for 2sin 3x = sin 2x

$${find}\:{the}\:{general}\:{solution}\:{for}\: \\ $$$$\:\:\mathrm{2sin}\:\mathrm{3}{x}\:=\:\mathrm{sin}\:\mathrm{2}{x} \\ $$

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