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Question Number 38024    Answers: 1   Comments: 0

Question Number 38015    Answers: 0   Comments: 6

Question Number 38012    Answers: 3   Comments: 1

The roots of the equation 2x^2 − x + 3 = 0 are α and β if the roots of 3x^2 + px + q=0 are α + (1/α) and β + (1/(β )) find the value of p and q.

$${The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} \:−\:{x}\:+\:\mathrm{3}\:=\:\mathrm{0}\:{are}\:\alpha\:{and}\:\beta \\ $$$${if}\:{the}\:{roots}\:{of}\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{px}\:+\:{q}=\mathrm{0}\: \\ $$$${are}\:\alpha\:+\:\frac{\mathrm{1}}{\alpha}\:{and}\:\beta\:+\:\frac{\mathrm{1}}{\beta\:}\:{find}\:{the}\:{value} \\ $$$${of}\:{p}\:{and}\:{q}. \\ $$$$\: \\ $$

Question Number 38011    Answers: 2   Comments: 0

Show that ((sin2A)/(1+ cos2A)) = TanA

$${Show}\:{that}\: \\ $$$$\:\:\frac{{sin}\mathrm{2}{A}}{\mathrm{1}+\:{cos}\mathrm{2}{A}}\:=\:{TanA} \\ $$

Question Number 38006    Answers: 1   Comments: 5

Question Number 37972    Answers: 1   Comments: 0

The distance S metres is given as a funtion f(t) where is time taken... if S = t^3 + t^2 + 4 find the velocity and acceleration

$$\:{The}\:{distance}\:{S}\:{metres}\:{is}\: \\ $$$${given}\:{as}\:{a}\:{funtion}\: \\ $$$${f}\left({t}\right)\:{where}\:{is}\:{time}\:{taken}... \\ $$$${if}\:{S}\:=\:{t}^{\mathrm{3}} \:+\:{t}^{\mathrm{2}} \:+\:\mathrm{4} \\ $$$${find}\:{the}\:{velocity}\:{and}\:{acceleration} \\ $$

Question Number 37991    Answers: 1   Comments: 4

1. Find the sum s_n =1+2x+3x^2 +4x^3 +...+nx^(n−1) Hence,or otherwise, find the sum Σ_(k=1) ^n k.2^k 2. Simplify the following i. Σ_(r=0) ^n (_(2r−1) ^(2n) ) ii.Σ_(r=0) ^n (−1)^r r(_r ^n ) iii.Σ_(r=0) ^n (−1)^r (1/(r+1))(_r ^n ) iv.Σ_(r=0) ^n (_(2r) ^(2n) ) v.Σ_(r=0) ^n (−1)^r (_(n−r) ^(n+1) ) 3.Find the sum Σ_(r=0) ^(n−k) (_k ^(n−r) ), where k=0,1,2,3,...,n

$$\mathrm{1}.\:{Find}\:{the}\:{sum} \\ $$$$\:\:\:\:{s}_{{n}} =\mathrm{1}+\mathrm{2}{x}+\mathrm{3}{x}^{\mathrm{2}} +\mathrm{4}{x}^{\mathrm{3}} +...+{nx}^{{n}−\mathrm{1}} \\ $$$${Hence},{or}\:{otherwise},\:{find}\:{the}\:{sum} \\ $$$$\:\:\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}.\mathrm{2}^{{k}} \\ $$$$\mathrm{2}.\:{Simplify}\:{the}\:{following} \\ $$$${i}.\:\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(_{\mathrm{2}{r}−\mathrm{1}} ^{\mathrm{2}{n}} \right) \\ $$$${ii}.\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{r}} {r}\left(_{{r}} ^{{n}} \right) \\ $$$${iii}.\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{r}} \frac{\mathrm{1}}{{r}+\mathrm{1}}\left(_{{r}} ^{{n}} \right) \\ $$$${iv}.\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(_{\mathrm{2}{r}} ^{\mathrm{2}{n}} \right) \\ $$$${v}.\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{r}} \left(_{{n}−{r}} ^{{n}+\mathrm{1}} \right) \\ $$$$\mathrm{3}.{Find}\:{the}\:{sum} \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{r}=\mathrm{0}} {\overset{{n}−{k}} {\sum}}\left(_{{k}} ^{{n}−{r}} \right),\:\:\:{where}\:{k}=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},...,{n} \\ $$

Question Number 37989    Answers: 2   Comments: 7

Question Number 37964    Answers: 1   Comments: 0

Question Number 37961    Answers: 1   Comments: 0

calculate ∫_0 ^∞ (dx/(x^(2 ) +(√(1+x^2 )))) .

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}\:} \:+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 37957    Answers: 0   Comments: 0

Question Number 37953    Answers: 0   Comments: 1

Question Number 37948    Answers: 0   Comments: 1

If x ∈R show that (2+i)e^((1+3i)) +(2−i)e^((1−3i)) is also real.

$${If}\:{x}\:\in\mathbb{R} \\ $$$${show}\:{that}\:\left(\mathrm{2}+{i}\right){e}^{\left(\mathrm{1}+\mathrm{3}{i}\right)} +\left(\mathrm{2}−{i}\right){e}^{\left(\mathrm{1}−\mathrm{3}{i}\right)} \:{is}\:{also}\:{real}. \\ $$

Question Number 37940    Answers: 1   Comments: 0

Which of the following expressions are positive for all real values of x? a) x^2 − 2x + 5 b) x^2 −2x−1 c) x^2 +4x+2 d) 2x^2 −6x + 5

$${Which}\:{of}\:{the}\:{following}\: \\ $$$${expressions}\:{are}\:{positive}\:{for} \\ $$$${all}\:{real}\:{values}\:{of}\:\:{x}? \\ $$$$\left.{a}\left.\right)\:{x}^{\mathrm{2}} −\:\mathrm{2}{x}\:+\:\mathrm{5}\:\:\:{b}\right)\:{x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{1}\: \\ $$$$\left.{c}\left.\right)\:{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{2}\:\:\:\:\:\:{d}\right)\:\mathrm{2}{x}^{\mathrm{2}} −\mathrm{6}{x}\:+\:\mathrm{5} \\ $$

Question Number 37938    Answers: 5   Comments: 5

Question Number 37945    Answers: 0   Comments: 10

Two plane mirrors are inclined at an angle of 30°.A ray of light which makes an angle of incidence of 50° with one of the mirrors,undergoes two successive reflections at the mirrors.Calculate the angle of deviation. please help....its urgent

$${Two}\:{plane}\:{mirrors}\:{are}\:{inclined}\:{at} \\ $$$${an}\:{angle}\:{of}\:\mathrm{30}°.{A}\:{ray}\:{of}\:{light}\:{which} \\ $$$${makes}\:{an}\:{angle}\:{of}\:{incidence}\:{of}\:\mathrm{50}° \\ $$$${with}\:{one}\:{of}\:{the}\:{mirrors},{undergoes} \\ $$$${two}\:{successive}\:{reflections}\:{at}\:{the} \\ $$$${mirrors}.{Calculate}\:{the}\:{angle}\:{of} \\ $$$${deviation}. \\ $$$$ \\ $$$$ \\ $$$${please}\:{help}....{its}\:{urgent} \\ $$

Question Number 37930    Answers: 1   Comments: 1

n∈N U_(n+1) =((1/2))^(n+1) +U_n U_n =?

$${n}\in\mathbb{N} \\ $$$${U}_{{n}+\mathrm{1}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{{n}+\mathrm{1}} +{U}_{{n}} \\ $$$${U}_{{n}} =? \\ $$

Question Number 37922    Answers: 1   Comments: 2

f : N → R g : N → R f(n)=∫_0 ^(2π) x^n sin x dx g(n)=∫_0 ^(2π) x^n cos x dx ((f(n+1)−f(n))/(g(n+1)−g(n)))=?

$${f}\::\:\mathbb{N}\:\rightarrow\:\mathbb{R} \\ $$$${g}\::\:\mathbb{N}\:\rightarrow\:\mathbb{R} \\ $$$${f}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {x}^{{n}} \mathrm{sin}\:{x}\:{dx} \\ $$$${g}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {x}^{{n}} \mathrm{cos}\:{x}\:{dx} \\ $$$$\frac{{f}\left({n}+\mathrm{1}\right)−{f}\left({n}\right)}{{g}\left({n}+\mathrm{1}\right)−{g}\left({n}\right)}=? \\ $$

Question Number 37915    Answers: 1   Comments: 0

If y=4x^2 −1 , then find ((85)/(169))+Σ_(i=1) ^(84) (1/(y(i)))

$$\mathrm{If}\:{y}=\mathrm{4}{x}^{\mathrm{2}} −\mathrm{1}\:,\:\mathrm{then}\:\mathrm{find} \\ $$$$\frac{\mathrm{85}}{\mathrm{169}}+\underset{{i}=\mathrm{1}} {\overset{\mathrm{84}} {\Sigma}}\:\frac{\mathrm{1}}{{y}\left({i}\right)}\: \\ $$

Question Number 37914    Answers: 1   Comments: 0

In △ABC, if sin A=sin^2 B then prove 4 cos 2A−4 cos 2B=1−cos 4B

$$\mathrm{In}\:\bigtriangleup{ABC},\:\mathrm{if}\:\mathrm{sin}\:\mathrm{A}=\mathrm{sin}^{\mathrm{2}} {B}\: \\ $$$$\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{4}\:\mathrm{cos}\:\mathrm{2}{A}−\mathrm{4}\:\mathrm{cos}\:\mathrm{2}{B}=\mathrm{1}−\mathrm{cos}\:\mathrm{4}{B} \\ $$

Question Number 37913    Answers: 1   Comments: 0

Solve the diferential equatuion (dy/dx)=((2x+y+1)/(x−2y+3))

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{diferential}\:\mathrm{equatuion} \\ $$$$\frac{{dy}}{{dx}}=\frac{\mathrm{2}{x}+{y}+\mathrm{1}}{{x}−\mathrm{2}{y}+\mathrm{3}}\: \\ $$

Question Number 37912    Answers: 1   Comments: 1

Evaluate : the Integral ∫_(-(π/2)) ^(π/2) ∫_0 ^(3 cos θ) r^2 sin^2 θ. dr dθ

$$\mathrm{Evaluate}\::\:\mathrm{the}\:\mathrm{Integral} \\ $$$$\int_{-\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\mathrm{3}\:\mathrm{cos}\:\theta} {r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta.\:{dr}\:{d}\theta\: \\ $$

Question Number 37911    Answers: 0   Comments: 1

the function f(x) is defined by f(x) = { ((−x + 1 , for x≤3)),((kx − 8 , for x ≥ 3)) :} find the value of k .

$$\mathrm{the}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by} \\ $$$${f}\left({x}\right)\:=\begin{cases}{−{x}\:+\:\mathrm{1}\:,\:{for}\:{x}\leqslant\mathrm{3}}\\{{kx}\:−\:\mathrm{8}\:,\:{for}\:{x}\:\geqslant\:\mathrm{3}}\end{cases} \\ $$$${find}\:{the}\:{value}\:{of}\:{k}\:. \\ $$

Question Number 37906    Answers: 1   Comments: 0

Question Number 37902    Answers: 2   Comments: 1

ind the value of f(a) =∫_0 ^(+∞) (dx/(x^2 +(√(a^2 +x^2 )))) dx witha>0 2)calculate f^′ (a) .

$${ind}\:{the}\:{value}\:{of}\:{f}\left({a}\right)\:\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \:+\sqrt{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} }}\:{dx} \\ $$$${witha}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({a}\right)\:. \\ $$

Question Number 37901    Answers: 0   Comments: 1

let f(x)= (1+e^(−x) )^n 1) calculate f^((p)) (x) and f^((p)) (o) 2)calculate f^((n)) (0) 3)developp f at integr serie .

$${let}\:{f}\left({x}\right)=\:\left(\mathrm{1}+{e}^{−{x}} \right)^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({p}\right)} \left({x}\right)\:\:{and}\:{f}^{\left({p}\right)} \left({o}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right){developp}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$

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