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AlgebraQuestion and Answers: Page 267

Question Number 79947 Answers: 0 Comments: 4

Find ((ln 2)/(2!))+((ln 3)/(3!))+((ln 4)/(4!))+...+((ln n)/(n!))+...=?

$${Find} \\ $$$$\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}!}+\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{3}!}+\frac{\mathrm{ln}\:\mathrm{4}}{\mathrm{4}!}+...+\frac{\mathrm{ln}\:{n}}{{n}!}+...=? \\ $$

Question Number 79946 Answers: 2 Comments: 0

Find ∫_0 ^( n) [(x)^(1/3) ]dx=? in terms of n. (n∈N)

$${Find}\: \\ $$$$\int_{\mathrm{0}} ^{\:{n}} \left[\sqrt[{\mathrm{3}}]{{x}}\right]{dx}=?\: \\ $$$${in}\:{terms}\:{of}\:{n}.\:\left({n}\in\mathbb{N}\right) \\ $$

Question Number 79864 Answers: 0 Comments: 0

Question Number 79861 Answers: 0 Comments: 1

Question Number 79826 Answers: 0 Comments: 7

Question Number 79807 Answers: 2 Comments: 2

Question Number 79792 Answers: 1 Comments: 1

JUST FOR FUN (1/2), (2/3), 1, (8/5), (8/3), ? what do you think is the next number ? why?

$${JUST}\:{FOR}\:{FUN} \\ $$$$ \\ $$$$\frac{\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{2}}{\mathrm{3}},\:\mathrm{1},\:\frac{\mathrm{8}}{\mathrm{5}},\:\frac{\mathrm{8}}{\mathrm{3}},\:? \\ $$$${what}\:{do}\:{you}\:{think}\:{is}\:{the}\:{next}\:{number}\:? \\ $$$${why}? \\ $$

Question Number 79757 Answers: 0 Comments: 1

And equation of a circle is x^2 +y^2 −2x+4y=0. (T) is his his tangent line at M(x_0 ;y_0 ) passing by D(2;1). a) Show that y verify y_0 ^2 +y_0 =0 b) deduct others tangent′s equations to Circle passing by D.

$$\mathrm{And}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{is} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{4}{y}=\mathrm{0}. \\ $$$$\left({T}\right)\:\mathrm{is}\:\mathrm{his}\:\mathrm{his}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{at}\:\mathrm{M}\left({x}_{\mathrm{0}} ;{y}_{\mathrm{0}} \right) \\ $$$${passing}\:{by}\:{D}\left(\mathrm{2};\mathrm{1}\right). \\ $$$$ \\ $$$$\left.\mathrm{a}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{y}\:\mathrm{verify}\:\mathrm{y}_{\mathrm{0}} ^{\mathrm{2}} +\mathrm{y}_{\mathrm{0}} =\mathrm{0} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{deduct}\:\mathrm{others}\:\mathrm{tangent}'\mathrm{s}\: \\ $$$$\mathrm{equations}\:\mathrm{to}\:\mathrm{Circle}\:\mathrm{passing}\:\mathrm{by}\:\mathrm{D}. \\ $$

Question Number 79635 Answers: 0 Comments: 9

Sum: (1/1) + (1/(1 + 2)) + (1/(1 + 2 + 3)) + ... + (1/(1 + 2 + 3 + ... + 8016))

$$\mathrm{Sum}:\:\:\frac{\mathrm{1}}{\mathrm{1}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}}\:+\:...\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:...\:+\:\mathrm{8016}} \\ $$

Question Number 79571 Answers: 1 Comments: 5

Solve for x: ((x + ((x + ((x + ...))^(1/3) ))^(1/3) ))^(1/3) = ((x ((x ((x ....))^(1/3) ))^(1/3) ))^(1/3)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}: \\ $$$$\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:+\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:+\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:+\:\:...}}}\:\:\:\:\:\:\:=\:\:\:\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:\:....}}} \\ $$

Question Number 79613 Answers: 1 Comments: 0

3xy(2x−y)−3bx+3c=0 3xy(x−2y)−3by−3c=0 find non-zero, real values of x,y if b,c∈R.

$$\mathrm{3}{xy}\left(\mathrm{2}{x}−{y}\right)−\mathrm{3}{bx}+\mathrm{3}{c}=\mathrm{0} \\ $$$$\mathrm{3}{xy}\left({x}−\mathrm{2}{y}\right)−\mathrm{3}{by}−\mathrm{3}{c}=\mathrm{0} \\ $$$${find}\:{non}-{zero},\:{real}\:{values} \\ $$$${of}\:{x},{y}\:\:{if}\:{b},{c}\in\mathbb{R}. \\ $$

Question Number 79560 Answers: 0 Comments: 1

(√(1+x)) ≤ ((5−x))^(1/(4 ))

$$\sqrt{\mathrm{1}+\mathrm{x}}\:\leqslant\:\sqrt[{\mathrm{4}\:}]{\mathrm{5}−\mathrm{x}} \\ $$

Question Number 79538 Answers: 1 Comments: 13

Question Number 79536 Answers: 0 Comments: 4

Question Number 79513 Answers: 0 Comments: 1

Q.solve if t^2 =n^2 cos^2 (x)+m^2 sin^2 (x) then show that: t+(d^2 t/dx^2 )=(((nm)^2 )/t^3 )

$${Q}.{solve} \\ $$$${if}\:{t}^{\mathrm{2}} ={n}^{\mathrm{2}} {cos}^{\mathrm{2}} \left({x}\right)+{m}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({x}\right) \\ $$$$ \\ $$$${then}\:{show}\:{that}: \\ $$$${t}+\frac{{d}^{\mathrm{2}} {t}}{{dx}^{\mathrm{2}} }=\frac{\left({nm}\right)^{\mathrm{2}} }{{t}^{\mathrm{3}} } \\ $$$$ \\ $$

Question Number 79512 Answers: 0 Comments: 0

Q.solve if t^2 =n^2 cos^2 (x)+m^2 sin^2 (x) then show that: t+(d^2 t/dx^2 )=(((nm)^2 )/t^3 )

Question Number 79852 Answers: 2 Comments: 1

Question Number 79424 Answers: 0 Comments: 0

Derive the width of the diffraction pattern for the case of (i)single slits (ii)double slits

$${Derive}\:{the}\:{width}\:{of}\:{the} \\ $$$${diffraction}\:{pattern}\:{for} \\ $$$${the}\:{case}\:{of} \\ $$$$\left({i}\right){single}\:{slits} \\ $$$$\left({ii}\right){double}\:{slits} \\ $$

Question Number 79417 Answers: 2 Comments: 3

For x,y∈R find the minimum and maximum of 2x^2 −3x+4y if x^2 +2y^2 −xy−5x−7y−30=0.

$${For}\:{x},{y}\in\mathbb{R}\:{find}\:{the}\:{minimum}\:{and} \\ $$$${maximum}\:{of}\:\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{4}{y} \\ $$$${if}\:{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} −{xy}−\mathrm{5}{x}−\mathrm{7}{y}−\mathrm{30}=\mathrm{0}. \\ $$

Question Number 79340 Answers: 1 Comments: 3

Question Number 79306 Answers: 1 Comments: 7

(√(3−x))−(√(x+1))>(1/2)

$$ \\ $$$$\sqrt{\mathrm{3}−\mathrm{x}}−\sqrt{\mathrm{x}+\mathrm{1}}>\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 79279 Answers: 1 Comments: 4

solve ∣x∣^3 −7x^2 +7∣x∣+15<0

$$\mathrm{solve}\: \\ $$$$\mid\mathrm{x}\mid^{\mathrm{3}} −\mathrm{7x}^{\mathrm{2}} +\mathrm{7}\mid\mathrm{x}\mid+\mathrm{15}<\mathrm{0} \\ $$

Question Number 79256 Answers: 1 Comments: 0

3s^2 −2ps−3cp−1=0 and 3s−2p−sp^2 −3cp^2 =0 find s and p both real in terms of c ∈R.

$$\mathrm{3}{s}^{\mathrm{2}} −\mathrm{2}{ps}−\mathrm{3}{cp}−\mathrm{1}=\mathrm{0}\:\:\:{and} \\ $$$$\mathrm{3}{s}−\mathrm{2}{p}−{sp}^{\mathrm{2}} −\mathrm{3}{cp}^{\mathrm{2}} =\mathrm{0} \\ $$$${find}\:{s}\:{and}\:{p}\:{both}\:{real}\:{in}\:{terms} \\ $$$${of}\:{c}\:\in\mathbb{R}. \\ $$

Question Number 79249 Answers: 1 Comments: 3

Question Number 79233 Answers: 1 Comments: 3

(1/(x(x+1)))+(1/((x+1)(x+2)))+ (1/((x+2)(x+3)))≤(3/4)

$$\frac{\mathrm{1}}{\mathrm{x}\left(\mathrm{x}+\mathrm{1}\right)}+\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}+\mathrm{2}\right)}+ \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{3}\right)}\leqslant\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 79190 Answers: 4 Comments: 13

if x^2 +y^2 =50, find the minimum and maximum of (x+y)^2 −8(x+y)+20

$${if}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{50}, \\ $$$${find}\:{the}\:{minimum}\:{and}\:{maximum}\:{of} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{8}\left({x}+{y}\right)+\mathrm{20} \\ $$

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