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AllQuestion and Answers: Page 1511

Question Number 56818 Answers: 0 Comments: 2

Question Number 56803 Answers: 2 Comments: 2

Evaluate: ∫_( 1) ^( 2) (A∙B × C) dt and ∫_( 1) ^( 2) A × (B × C) dt where, A = ti − 3j + 2tk, B = i − 2j + 2k, C = 3i + tj − k

$$\mathrm{Evaluate}:\:\:\:\int_{\:\mathrm{1}} ^{\:\mathrm{2}} \:\left(\mathrm{A}\centerdot\mathrm{B}\:×\:\mathrm{C}\right)\:\mathrm{dt}\:\:\:\:\mathrm{and}\:\:\:\int_{\:\mathrm{1}} ^{\:\mathrm{2}} \:\mathrm{A}\:×\:\left(\mathrm{B}\:×\:\mathrm{C}\right)\:\:\:\mathrm{dt} \\ $$$$\mathrm{where},\:\:\:\:\:\:\:\:\:\mathrm{A}\:\:=\:\:\mathrm{ti}\:−\:\mathrm{3j}\:+\:\mathrm{2tk},\:\:\:\:\:\:\:\mathrm{B}\:\:=\:\:\mathrm{i}\:−\:\mathrm{2j}\:+\:\mathrm{2k}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{C}\:\:=\:\:\mathrm{3i}\:+\:\mathrm{tj}\:−\:\mathrm{k} \\ $$

Question Number 56795 Answers: 1 Comments: 1

dy/dx=x

$${dy}/{dx}={x} \\ $$

Question Number 56801 Answers: 1 Comments: 0

The product of three consecutive terms of 4. The sum of the GP is −(7/3). Find the GP

$${The}\:{product}\:{of}\:{three}\:{consecutive}\:{terms} \\ $$$${of}\:\mathrm{4}.\:{The}\:{sum}\:{of}\:{the}\:{GP}\:{is}\:−\frac{\mathrm{7}}{\mathrm{3}}.\:{Find} \\ $$$${the}\:{GP} \\ $$

Question Number 56800 Answers: 0 Comments: 3

x,y,z are positive integers. find all solutions of x^2 +y^2 +1=xyz.

$${x},{y},{z}\:{are}\:{positive}\:{integers}. \\ $$$${find}\:{all}\:{solutions}\:{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}={xyz}. \\ $$

Question Number 56813 Answers: 0 Comments: 0

study the sequence U_(n+1) =(√((1+u_n )/2)) with U_0 =a>0.

$${study}\:{the}\:{sequence}\:\:{U}_{{n}+\mathrm{1}} =\sqrt{\frac{\mathrm{1}+{u}_{{n}} }{\mathrm{2}}} \\ $$$${with}\:{U}_{\mathrm{0}} ={a}>\mathrm{0}. \\ $$

Question Number 56785 Answers: 0 Comments: 1

Question Number 56784 Answers: 0 Comments: 0

Question Number 56777 Answers: 1 Comments: 1

Question Number 56775 Answers: 0 Comments: 1

if x,y∍ℜ,show that ∣x+y∣=∣x∣+∣y∣ iff xy≥0

$${if}\:{x},{y}\backepsilon\Re,{show}\:{that}\:\mid{x}+{y}\mid=\mid{x}\mid+\mid{y}\mid\:{iff}\:{xy}\geqslant\mathrm{0} \\ $$

Question Number 56772 Answers: 1 Comments: 0

Given that : (1/( Φ )) = (Φ/( 1 + Φ )) Prove that : without using the exact value of Φ... (( 1 )/( Φ )) = Φ − 1 Thank you

$$\mathrm{Given}\:\mathrm{that}\:: \\ $$$$\:\:\:\:\frac{\mathrm{1}}{\:\Phi\:}\:\:=\:\:\frac{\Phi}{\:\mathrm{1}\:+\:\Phi\:} \\ $$$$ \\ $$$$\mathrm{Prove}\:\mathrm{that}\::\:{without}\:{using}\:{the}\:{exact} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{value}\:{of}\:\Phi... \\ $$$$\:\:\:\:\:\frac{\:\mathrm{1}\:}{\:\Phi\:}\:\:=\:\:\Phi\:−\:\mathrm{1} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Thank}\:\mathrm{you} \\ $$

Question Number 56749 Answers: 1 Comments: 0

The curve y=ax^2 +bx+c crosses the y−axis at the point (0,3) and has stationary point at (1,2). Find the values of a,b and c.

$$\mathrm{The}\:\mathrm{curve}\:\mathrm{y}=\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}\:\mathrm{crosses}\:\mathrm{the} \\ $$$$\mathrm{y}−\mathrm{axis}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point}\:\left(\mathrm{0},\mathrm{3}\right)\:\mathrm{and}\:\mathrm{has} \\ $$$$\mathrm{stationary}\:\mathrm{point}\:\mathrm{at}\:\left(\mathrm{1},\mathrm{2}\right).\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}. \\ $$

Question Number 56747 Answers: 1 Comments: 0

∫ x^(2 ) e^x^2 dx

$$\int\:\mathrm{x}^{\mathrm{2}\:} \mathrm{e}^{\mathrm{x}^{\mathrm{2}} } \:\:\mathrm{dx} \\ $$

Question Number 56744 Answers: 1 Comments: 0

If R be a relation on a set of real number defined by R={(x,y): x^2 +y^2 =0}, find i− R in roster form ii−Domain of R iii−Range of R

$$\mathrm{If}\:\mathrm{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{relation}\:\mathrm{on}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{defined}\:\mathrm{by}\:\mathrm{R}=\left\{\left(\mathrm{x},\mathrm{y}\right):\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{0}\right\}, \\ $$$$\mathrm{find}\: \\ $$$$\:\:\mathrm{i}−\:\mathrm{R}\:\mathrm{in}\:\mathrm{roster}\:\mathrm{form} \\ $$$$\:\:\mathrm{ii}−\mathrm{Domain}\:\mathrm{of}\:\mathrm{R} \\ $$$$\:\:\mathrm{iii}−\mathrm{Range}\:\mathrm{of}\:\mathrm{R}\: \\ $$

Question Number 56743 Answers: 1 Comments: 0

Question Number 56738 Answers: 1 Comments: 0

The tangent to the curve y=x^3 +bx, where b is constant, at x=1 passes through the points A(−1,6) and (2,−15). Find the value of b.

$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{y}=\mathrm{x}^{\mathrm{3}} +\mathrm{bx},\:\mathrm{where} \\ $$$$\mathrm{b}\:\mathrm{is}\:\mathrm{constant},\:\mathrm{at}\:\mathrm{x}=\mathrm{1}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the} \\ $$$$\mathrm{points}\:\mathrm{A}\left(−\mathrm{1},\mathrm{6}\right)\:\mathrm{and}\:\left(\mathrm{2},−\mathrm{15}\right).\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{b}. \\ $$

Question Number 56732 Answers: 1 Comments: 1

Question Number 56763 Answers: 1 Comments: 0

(a) Determine the area of the largest rectangle that can be inscribed in the circle x^2 + y^2 = a^2 . (b) Name the rectangle so formed

$$\left(\mathrm{a}\right)\:\mathrm{Determine}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{rectangle}\:\mathrm{that}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{inscribed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{circle}\:\:\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} \:\:=\:\:\mathrm{a}^{\mathrm{2}} \:. \\ $$$$ \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Name}\:\mathrm{the}\:\mathrm{rectangle}\:\mathrm{so}\:\mathrm{formed} \\ $$

Question Number 56711 Answers: 1 Comments: 0

Find the shotest distance between the line ((x − 8)/3) = ((y − 2)/4) = ((z + 1)/1) , ((x − 3)/3) = ((y + 4)/5) = ((z −2)/2)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shotest}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{line} \\ $$$$\:\:\:\:\:\:\:\frac{\mathrm{x}\:−\:\mathrm{8}}{\mathrm{3}}\:=\:\frac{\mathrm{y}\:−\:\mathrm{2}}{\mathrm{4}}\:=\:\frac{\mathrm{z}\:+\:\mathrm{1}}{\mathrm{1}}\:,\:\:\:\:\:\:\:\frac{\mathrm{x}\:−\:\mathrm{3}}{\mathrm{3}}\:=\:\frac{\mathrm{y}\:+\:\mathrm{4}}{\mathrm{5}}\:=\:\frac{\mathrm{z}\:−\mathrm{2}}{\mathrm{2}} \\ $$

Question Number 56708 Answers: 0 Comments: 1

for all n ∈ N, f_n (x)=Σ_(k=1) ^n (−1)^n x^n for any −1<x<1. If f : (−1,1)→R with f =lim_(n→∞) f_(n ) of (−1,1) ∫_0 ^(1/2) f(x) dx=...

$$\mathrm{for}\:\mathrm{all}\:{n}\:\in\:\mathbb{N},\:{f}_{{n}} \left({x}\right)=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{n}} {x}^{{n}} \\ $$$$\mathrm{for}\:\mathrm{any}\:−\mathrm{1}<{x}<\mathrm{1}.\:\mathrm{If}\:{f}\::\:\left(−\mathrm{1},\mathrm{1}\right)\rightarrow\mathbb{R} \\ $$$$\mathrm{with}\:{f}\:=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{f}_{{n}\:} \mathrm{of}\:\left(−\mathrm{1},\mathrm{1}\right) \\ $$$$\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} {f}\left({x}\right)\:{dx}=... \\ $$$$ \\ $$

Question Number 56707 Answers: 1 Comments: 0

f(x)=4x^2 +1 for all x ∈ R x_n =Σ_(k=1) ^n (1/(k^2 +3k+6)) for all n ∈N lim_(n→∞) f(x_n )=...

$${f}\left({x}\right)=\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}\:\:\mathrm{for}\:\mathrm{all}\:{x}\:\in\:\mathbb{R} \\ $$$${x}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}^{\mathrm{2}} +\mathrm{3}{k}+\mathrm{6}}\:\mathrm{for}\:\mathrm{all}\:{n}\:\in\mathbb{N} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({x}_{{n}} \right)=... \\ $$

Question Number 56706 Answers: 0 Comments: 1

f : [0, 1) → R lim_(x→0) f(x)=0 lim_(x→0) ((f(x)−f((x/2)))/x)=0 lim_(x→0) ((f(x))/x) =...

$${f}\::\:\left[\mathrm{0},\:\mathrm{1}\right)\:\rightarrow\:\mathbb{R} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)−{f}\left(\frac{{x}}{\mathrm{2}}\right)}{{x}}=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{x}}\:=... \\ $$

Question Number 56705 Answers: 1 Comments: 0

the smallest value of S={^3 (√n)−^3 (√m) ∣ n, m ∈N} is...

$$\mathrm{the}\:\:\mathrm{smallest}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{S}=\left\{\:^{\mathrm{3}} \sqrt{{n}}−^{\mathrm{3}} \sqrt{{m}}\:\mid\:{n},\:{m}\:\in\mathbb{N}\right\}\:\:\mathrm{is}... \\ $$

Question Number 56700 Answers: 1 Comments: 2

find ∫ (√(x−2(√x)+3))dx

$$\:{find}\:\int\:\sqrt{{x}−\mathrm{2}\sqrt{{x}}+\mathrm{3}}{dx} \\ $$

Question Number 56699 Answers: 0 Comments: 2

let f_n (a)=∫_(−∞) ^∞ ((sin(x^n ))/(x^2 +a^2 )) dx with a positif real not 0 and n from N 1) find a explicit form of f(a) 2) calculate g_n (a) =∫_(−∞) ^(+∞) ((sin(x^n ))/((x^2 +a^2 )^2 ))dx 3) calculate ∫_(−∞) ^(+∞) ((sin(x^3 ))/(x^2 +4)) dx and ∫_(−∞) ^(+∞) ((sin(x^2 ))/(x^2 +9))dx 4) calculate ∫_(−∞) ^(+∞) ((sin(x^3 ))/((x^2 +4)^2 ))dx .

$${let}\:{f}_{{n}} \left({a}\right)=\int_{−\infty} ^{\infty} \:\:\frac{{sin}\left({x}^{{n}} \right)}{{x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} }\:{dx}\:\:\:{with}\:{a}\:{positif}\:{real}\:{not}\:\mathrm{0}\:\:{and}\:{n}\:{from}\:{N} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{calculate}\:{g}_{{n}} \left({a}\right)\:=\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({x}^{{n}} \right)}{\left({x}^{\mathrm{2}} \:+{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{3}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}\:{dx}\:{and}\:\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{9}}{dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{sin}\left({x}^{\mathrm{3}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 56698 Answers: 0 Comments: 2

find the value of ∫_0 ^∞ ((sin(x^2 ))/(x^4 +4))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} \:+\mathrm{4}}{dx} \\ $$

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