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Question Number 156186    Answers: 1   Comments: 2

Question Number 156184    Answers: 2   Comments: 1

Question Number 155810    Answers: 1   Comments: 0

Verify the identity in Excercise below 1). cos θsec θ=1 2). (1+cos β)(1−cos β)=sin^2 β 3). cos^2 x(sec^2 x−1)=sin^2 x 4). ((sin t)/(cosec t))+((cos t)/(sec t))=1 5). ((cosec^2 θ)/(1+tan^2 θ)) = cot^2 θ

$$\mathrm{Verify}\:\mathrm{the}\:\mathrm{identity}\:\mathrm{in}\:\mathrm{Excercise}\:\mathrm{below} \\ $$$$\left.\mathrm{1}\right).\:\mathrm{cos}\:\theta\mathrm{sec}\:\theta=\mathrm{1} \\ $$$$\left.\mathrm{2}\right).\:\left(\mathrm{1}+\mathrm{cos}\:\beta\right)\left(\mathrm{1}−\mathrm{cos}\:\beta\right)=\mathrm{sin}\:^{\mathrm{2}} \beta \\ $$$$\left.\mathrm{3}\right).\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}\left(\mathrm{sec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{1}\right)=\mathrm{sin}\:^{\mathrm{2}} \mathrm{x} \\ $$$$\left.\mathrm{4}\right).\:\frac{\mathrm{sin}\:\mathrm{t}}{\mathrm{cosec}\:\mathrm{t}}+\frac{\mathrm{cos}\:\mathrm{t}}{\mathrm{sec}\:\mathrm{t}}=\mathrm{1} \\ $$$$\left.\mathrm{5}\right).\:\frac{\mathrm{cosec}\:^{\mathrm{2}} \theta}{\mathrm{1}+\mathrm{tan}\:^{\mathrm{2}} \theta}\:=\:\mathrm{cot}\:^{\mathrm{2}} \theta \\ $$

Question Number 155809    Answers: 1   Comments: 0

Given that f○g = (x^2 /(2x^2 − x + 4)) and g(x) = (x/(x − 2)), find f(x) ?

$$\:\:\mathrm{Given}\:\mathrm{that}\:{f}\circ{g}\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}{x}^{\mathrm{2}} \:−\:{x}\:+\:\mathrm{4}}\:\:\mathrm{and} \\ $$$$\:\:{g}\left({x}\right)\:=\:\frac{{x}}{{x}\:−\:\mathrm{2}},\:\mathrm{find}\:{f}\left({x}\right)\:? \\ $$$$ \\ $$

Question Number 155806    Answers: 1   Comments: 0

Question Number 155803    Answers: 2   Comments: 0

proof that Σ_(n=1) ^(10) n×n! = 11!−1

$$\mathrm{proof}\:\mathrm{that}\: \\ $$$$\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\mathrm{n}×\mathrm{n}!\:=\:\mathrm{11}!−\mathrm{1} \\ $$

Question Number 155802    Answers: 1   Comments: 0

find the value of 2×1!+4×2!+6×3!+...+200×100!

$$\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\: \\ $$$$\:\mathrm{2}×\mathrm{1}!+\mathrm{4}×\mathrm{2}!+\mathrm{6}×\mathrm{3}!+...+\mathrm{200}×\mathrm{100}! \\ $$

Question Number 155801    Answers: 3   Comments: 0

{ ((m^2 =n+2)),((n^2 =m+2)) :} ⇒m≠n 4mn−m^3 −n^3 =?

$$\:\begin{cases}{\mathrm{m}^{\mathrm{2}} =\mathrm{n}+\mathrm{2}}\\{\mathrm{n}^{\mathrm{2}} =\mathrm{m}+\mathrm{2}}\end{cases}\:\Rightarrow\mathrm{m}\neq\mathrm{n} \\ $$$$\:\mathrm{4mn}−\mathrm{m}^{\mathrm{3}} −\mathrm{n}^{\mathrm{3}} =?\: \\ $$

Question Number 155797    Answers: 0   Comments: 6

a,b,c,d,e (kids) are in ascending order of heights. If they are to stand in a circle in a way so that the sum of ∣difference in heights of adjacent pairs∣ is a minimum, find this minimum.

$${a},{b},{c},{d},{e}\:\:\left({kids}\right)\:{are}\:{in}\:{ascending} \\ $$$${order}\:{of}\:{heights}.\:{If}\:{they}\:{are} \\ $$$${to}\:{stand}\:{in}\:{a}\:{circle}\:{in}\:{a}\:{way}\:{so} \\ $$$${that}\:{the}\:{sum}\:{of}\:\mid{difference}\:{in} \\ $$$${heights}\:{of}\:{adjacent}\:{pairs}\mid \\ $$$$\:{is}\:{a}\:{minimum},\:{find}\:{this} \\ $$$$\:{minimum}. \\ $$

Question Number 155775    Answers: 0   Comments: 0

Question Number 155774    Answers: 0   Comments: 0

if x;y>0 then prove that: ((x^2 (1+xy)^2 +x^2 y^4 (1+x)^2 +(1+y)^2 )/(1 + xy + x^2 y + xy^2 )) ≥ 3xy

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{xy}\right)^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{4}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} +\left(\mathrm{1}+\mathrm{y}\right)^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{xy}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}^{\mathrm{2}} }\:\geqslant\:\mathrm{3xy} \\ $$

Question Number 155772    Answers: 1   Comments: 0

Question Number 155770    Answers: 1   Comments: 1

Question Number 155759    Answers: 0   Comments: 1

∫_0 ^( 1) ln((sin(x)+cos(x))^2 +1) dx

$$\: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{ln}\left(\left(\mathrm{sin}\left({x}\right)+\mathrm{cos}\left({x}\right)\right)^{\mathrm{2}} +\mathrm{1}\right)\:{dx} \\ $$$$\: \\ $$

Question Number 155757    Answers: 0   Comments: 3

2x^2 (dy/dx) − 2x^2 = (x−1) y^2 ; y(1) = 2 □ M

$$\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}\:−\:\mathrm{2}\boldsymbol{{x}}^{\mathrm{2}} \:=\:\left(\boldsymbol{{x}}−\mathrm{1}\right)\:\boldsymbol{{y}}^{\mathrm{2}} \:\:;\:\boldsymbol{{y}}\left(\mathrm{1}\right)\:=\:\mathrm{2}\: \\ $$$$ \\ $$$$\square\:\boldsymbol{{M}}\: \\ $$

Question Number 155745    Answers: 2   Comments: 0

Σ_(k=1) ^n ((k/((4k^2 −1)(2k+3))))=?

$$\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left(\frac{\mathrm{k}}{\left(\mathrm{4k}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{2k}+\mathrm{3}\right)}\right)=? \\ $$

Question Number 155736    Answers: 0   Comments: 0

Question Number 155735    Answers: 1   Comments: 0

Question Number 155729    Answers: 1   Comments: 0

A={(a,b)∈IR^2 / a^2 +b^2 ≤1} prove that A can′t be written as the cartesian product of two parts of IR.

$$\mathrm{A}=\left\{\left({a},{b}\right)\in\mathrm{IR}^{\mathrm{2}} \:/\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} \leqslant\mathrm{1}\right\} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{A}\:\mathrm{can}'\mathrm{t}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as}\:\mathrm{the}\:\mathrm{cartesian} \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{two}\:\mathrm{parts}\:\mathrm{of}\:\mathrm{IR}. \\ $$

Question Number 155724    Answers: 1   Comments: 0

Find: (1/(sin^2 6^° )) + (1/(sin^2 42°)) + (1/(sin^2 66°)) + (1/(sin^2 78°)) = ?

$$\mathrm{Find}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{6}^{°} }\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{42}°}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{66}°}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \mathrm{78}°}\:=\:? \\ $$

Question Number 155720    Answers: 3   Comments: 1

Question Number 155776    Answers: 0   Comments: 0

if x∈(0;(π/2)) then prove that: ((2 + (1+cotx)(tan^3 x+cot^3 x))/((1+tanx)(1+cotx))) ≥ (3/2)

$$\mathrm{if}\:\:\:\mathrm{x}\in\left(\mathrm{0};\frac{\pi}{\mathrm{2}}\right)\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{2}\:+\:\left(\mathrm{1}+\mathrm{cot}\boldsymbol{\mathrm{x}}\right)\left(\mathrm{tan}^{\mathrm{3}} \boldsymbol{\mathrm{x}}+\mathrm{cot}^{\mathrm{3}} \boldsymbol{\mathrm{x}}\right)}{\left(\mathrm{1}+\mathrm{tan}\boldsymbol{\mathrm{x}}\right)\left(\mathrm{1}+\mathrm{cot}\boldsymbol{\mathrm{x}}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Question Number 155710    Answers: 3   Comments: 1

lim_(x−oo) (1/(n(√n))) Σ_(k=1) ^n E((√(k)))

$$\mathrm{li}\underset{{x}−{oo}} {\mathrm{m}}\:\:\:\frac{\mathrm{1}}{{n}\sqrt{{n}}}\:\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{E}\left(\sqrt{\left.{k}\right)}\right. \\ $$$$ \\ $$

Question Number 155701    Answers: 0   Comments: 3

Question Number 155692    Answers: 1   Comments: 5

Question Number 155686    Answers: 2   Comments: 0

∫_0 ^(π/2) (dx/(1+tan x)) =?

$$\:\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{dx}}{\mathrm{1}+\mathrm{tan}\:{x}}\:=? \\ $$

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