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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

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

Question Number 56697 Answers: 2 Comments: 0

Find the shortest distance between the lines L = (1, 4, 2) + N(1, 3, 2) and r = (−1, 1, −1) + λ(1, 2, −1)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the}\:\mathrm{lines} \\ $$$$\:\:\:\mathrm{L}\:\:=\:\:\left(\mathrm{1},\:\mathrm{4},\:\mathrm{2}\right)\:+\:\mathrm{N}\left(\mathrm{1},\:\mathrm{3},\:\mathrm{2}\right)\:\:\:\mathrm{and} \\ $$$$\:\:\:\mathrm{r}\:\:=\:\:\left(−\mathrm{1},\:\mathrm{1},\:−\mathrm{1}\right)\:+\:\lambda\left(\mathrm{1},\:\mathrm{2},\:−\mathrm{1}\right) \\ $$

Question Number 56696 Answers: 0 Comments: 2

Find the perpendicular distance from (1, 7, 1) to 3x − 2y + 2z = 6

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{perpendicular}\:\mathrm{distance}\:\mathrm{from}\:\:\left(\mathrm{1},\:\mathrm{7},\:\mathrm{1}\right)\:\:\mathrm{to}\:\:\mathrm{3x}\:−\:\mathrm{2y}\:+\:\mathrm{2z}\:\:=\:\:\mathrm{6} \\ $$

Question Number 56719 Answers: 1 Comments: 6

The possible value of x that satisfy : x^2 + ⌊x^2 − 2x⌋ + ⌈x^2 + 2x + 1⌉ = 2017 , x ∈ R^+ and ⌊x^2 ⌋ − ⌈2x⌉ = ...

$${The}\:\:{possible}\:\:{value}\:\:{of}\:\:{x}\:\:{that}\:\:{satisfy}\:\:: \\ $$$$\:\:\:\:\:\:\:\:{x}^{\mathrm{2}} \:+\:\lfloor{x}^{\mathrm{2}} \:−\:\mathrm{2}{x}\rfloor\:+\:\lceil{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{1}\rceil\:\:=\:\:\mathrm{2017}\:\:,\:\:{x}\:\in\:\:\mathbb{R}^{+} \\ $$$${and}\:\:\:\lfloor{x}^{\mathrm{2}} \rfloor\:−\:\lceil\mathrm{2}{x}\rceil\:\:=\:\:... \\ $$

Question Number 56694 Answers: 1 Comments: 0

A tv set was marked for sale at 760.00 cedis in order to make a profit of 20%. The tv was actually sold at a discount of 5%. Calculate to 2 s.f the actual percentage profit

$$\mathrm{A}\:\mathrm{tv}\:\mathrm{set}\:\mathrm{was}\:\mathrm{marked}\:\mathrm{for}\:\mathrm{sale}\:\mathrm{at}\:\mathrm{760}.\mathrm{00} \\ $$$$\mathrm{cedis}\:\mathrm{in}\:\mathrm{order}\:\mathrm{to}\:\mathrm{make}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\mathrm{20\%}. \\ $$$$\mathrm{The}\:\mathrm{tv}\:\mathrm{was}\:\mathrm{actually}\:\mathrm{sold}\:\mathrm{at}\:\mathrm{a}\:\mathrm{discount}\: \\ $$$$\mathrm{of}\:\mathrm{5\%}.\:\mathrm{Calculate}\:\mathrm{to}\:\mathrm{2}\:\mathrm{s}.\mathrm{f}\:\mathrm{the}\:\mathrm{actual}\: \\ $$$$\mathrm{percentage}\:\mathrm{profit} \\ $$

Question Number 56688 Answers: 1 Comments: 0

How many odd numbers, greater than 60000, can be made from the digits 5,6,7,8,9,0, if no number contains any digit more than ones?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{odd}\:\mathrm{numbers},\:\mathrm{greater}\:\mathrm{than} \\ $$$$\mathrm{60000},\:\mathrm{can}\:\mathrm{be}\:\mathrm{made}\:\mathrm{from}\:\mathrm{the}\:\mathrm{digits} \\ $$$$\mathrm{5},\mathrm{6},\mathrm{7},\mathrm{8},\mathrm{9},\mathrm{0},\:\mathrm{if}\:\mathrm{no}\:\mathrm{number}\:\mathrm{contains}\:\mathrm{any} \\ $$$$\mathrm{digit}\:\mathrm{more}\:\mathrm{than}\:\mathrm{ones}? \\ $$

Question Number 56685 Answers: 1 Comments: 1

show that α^4 +β^4 = (α^2 +β^2 )^2 −2α^2 β^2

$$\mathrm{show}\:\mathrm{that}\:\alpha^{\mathrm{4}} +\beta^{\mathrm{4}} \:=\:\left(\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} \right)^{\mathrm{2}} \:−\mathrm{2}\alpha^{\mathrm{2}} \beta^{\mathrm{2}} \\ $$

Question Number 56681 Answers: 2 Comments: 1

Prove that sin ∣x∣ ≤ ∣x∣ ≤ tan ∣x∣ for ∣x∣ < (π/2)

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{sin}\:\mid{x}\mid\:\leqslant\:\mid{x}\mid\:\leqslant\:\mathrm{tan}\:\mid{x}\mid\:\:\:\:\mathrm{for}\:\:\:\:\mid{x}\mid\:<\:\frac{\pi}{\mathrm{2}} \\ $$

Question Number 56678 Answers: 1 Comments: 0

A box contains 9 whites, 7 red and 4 blue balls. Three balls are picked at random, one after the other without replacement. a) find the probability that: i. they are all of the same color ii. there is one of each colour iii. two of them are red b) if it is known that the first one picked is blue, find the probability that the rest are white.

$$\mathrm{A}\:\mathrm{box}\:\mathrm{contains}\:\mathrm{9}\:\mathrm{whites},\:\mathrm{7}\:\mathrm{red}\:\mathrm{and}\:\mathrm{4}\:\mathrm{blue} \\ $$$$\mathrm{balls}.\:\mathrm{Three}\:\mathrm{balls}\:\mathrm{are}\:\mathrm{picked}\:\mathrm{at}\:\mathrm{random}, \\ $$$$\mathrm{one}\:\mathrm{after}\:\mathrm{the}\:\mathrm{other}\:\mathrm{without}\:\mathrm{replacement}. \\ $$$$\left.\mathrm{a}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}: \\ $$$$\mathrm{i}.\:\mathrm{they}\:\mathrm{are}\:\mathrm{all}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{color} \\ $$$$\mathrm{ii}.\:\mathrm{there}\:\mathrm{is}\:\mathrm{one}\:\mathrm{of}\:\mathrm{each}\:\mathrm{colour} \\ $$$$\mathrm{iii}.\:\mathrm{two}\:\mathrm{of}\:\mathrm{them}\:\mathrm{are}\:\mathrm{red} \\ $$$$\left.\mathrm{b}\right)\:\mathrm{if}\:\mathrm{it}\:\mathrm{is}\:\mathrm{known}\:\mathrm{that}\:\mathrm{the}\:\mathrm{first}\:\mathrm{one}\:\mathrm{picked} \\ $$$$\mathrm{is}\:\mathrm{blue},\:\mathrm{find}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{rest} \\ $$$$\mathrm{are}\:\mathrm{white}. \\ $$

Question Number 56668 Answers: 0 Comments: 0

I congratulate Nowruz celebration to all member of this forum

$${I}\:{congratulate}\:{Nowruz}\:{celebration} \\ $$$${to}\:{all}\:{member}\:{of}\:{this}\:{forum} \\ $$

Question Number 56664 Answers: 0 Comments: 2

Question Number 56663 Answers: 0 Comments: 0

i think everybody are in deep thought...

$${i}\:{think}\:{everybody}\:{are}\:{in}\:{deep}\:{thought}... \\ $$

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