Question and Answers Forum

AllQuestion and Answers: Page 262

Question Number 194785 Answers: 0 Comments: 0

∫∫ x^2 +y^2 dxdy (D=x^4 +y^4 ≤1)

$$\int\int\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:{dxdy}\:\left({D}={x}^{\mathrm{4}} +{y}^{\mathrm{4}} \leqslant\mathrm{1}\right) \\ $$

Question Number 194781 Answers: 0 Comments: 0

f_(n ) the general sentence is seqiencee fibonacci. prove that : f_(2n−1) =f_n ^2 +f_(n−1) ^2

$${f}_{{n}\:} \:\:{the}\:{general}\:{sentence}\:{is}\:{seqiencee} \\ $$$${fibonacci}.\: \\ $$$${prove}\:{that}\::\:\:{f}_{\mathrm{2}{n}−\mathrm{1}} ={f}_{{n}} ^{\mathrm{2}} +{f}_{{n}−\mathrm{1}} ^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 194779 Answers: 1 Comments: 4

If a divided by b gives q remaining r Then (a/b) = q,rrr... in base b+1

$${If}\:\:{a}\:\:{divided}\:{by}\:{b}\:{gives}\:{q}\:\:{remaining}\:{r} \\ $$$${Then}\:\:\frac{{a}}{{b}}\:=\:{q},{rrr}...\:\:{in}\:{base}\:{b}+\mathrm{1} \\ $$

Question Number 194767 Answers: 2 Comments: 0

tan θ = 2 ((8sin θ+5cos θ)/(sin^3 θ+cos^3 θ+cos θ)) =?

$$\:\:\:\:\:\mathrm{tan}\:\theta\:=\:\mathrm{2}\: \\ $$$$\:\:\:\frac{\mathrm{8sin}\:\theta+\mathrm{5cos}\:\theta}{\mathrm{sin}\:^{\mathrm{3}} \theta+\mathrm{cos}\:^{\mathrm{3}} \theta+\mathrm{cos}\:\theta}\:=? \\ $$

Question Number 194766 Answers: 2 Comments: 0

1+2cot 2x cot x = 3 x=?

$$\:\:\:\mathrm{1}+\mathrm{2cot}\:\mathrm{2}{x}\:\mathrm{cot}\:{x}\:=\:\mathrm{3}\: \\ $$$$\:\:\:{x}=? \\ $$

Question Number 194759 Answers: 1 Comments: 1

∫_0 ^( 1) ∫_0 ^( 1) (((1+x^2 )/(1+x^2 +y^2 ))) dxdy

$$\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\left(\frac{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\right)\:\mathrm{dxdy}\: \\ $$

Question Number 194756 Answers: 3 Comments: 0

((x−a)/( (√x) +(√a))) = (((√x)−(√a))/3) +2(√a)

$$\:\:\:\:\: \\ $$$$ \:\frac{\mathrm{x}−\mathrm{a}}{\:\sqrt{\mathrm{x}}\:+\sqrt{\mathrm{a}}}\:=\:\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{a}}}{\mathrm{3}}\:+\mathrm{2}\sqrt{\mathrm{a}}\: \\ $$

Question Number 194736 Answers: 0 Comments: 2

( log _(sin x cos x) (cos x))( log _(sin x cos x) (sin x))=(1/4)

$$\:\:\:\: \\ $$$$ \left(\:\mathrm{log}\:_{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}} \:\left(\mathrm{cos}\:\mathrm{x}\right)\right)\left(\:\mathrm{log}\:_{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}} \left(\mathrm{sin}\:\mathrm{x}\right)\right)=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\:\: \\ $$

Question Number 194735 Answers: 1 Comments: 0

Question Number 194732 Answers: 2 Comments: 1

Question Number 194715 Answers: 0 Comments: 0

Question Number 194713 Answers: 0 Comments: 2

Question Number 194709 Answers: 1 Comments: 0

Show that in fibonacci sequence f_(3n) =f_n ^3 +f_(n+1) ^3 −f_(n−1) ^3

$${Show}\:{that}\:\:{in}\:{fibonacci}\:{sequence} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{f}_{\mathrm{3}{n}} ={f}_{{n}} ^{\mathrm{3}} +{f}_{{n}+\mathrm{1}} ^{\mathrm{3}} −{f}_{{n}−\mathrm{1}} ^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 194710 Answers: 0 Comments: 21

let p be a prime number & let a_1 ,a_2 ,a_3 ,...,a_(p ) be integers show that , there exists an integer k such that the numbers a_1 +k, a_2 +k,a_3 +k,....,a_p +k produce at least (1/2)p distinct remainders when divided by p.

$${let}\:{p}\:{be}\:{a}\:{prime}\:{number} \\ $$$$\&\:{let}\:{a}_{\mathrm{1}} \:,{a}_{\mathrm{2}} ,{a}_{\mathrm{3}} ,...,{a}_{{p}\:} {be}\:{integers} \\ $$$${show}\:{that}\:,\:{there}\:{exists}\:{an}\:{integer}\:{k}\:{such}\:{that}\:{the}\:{numbers} \\ $$$${a}_{\mathrm{1}} +{k},\:{a}_{\mathrm{2}} +{k},{a}_{\mathrm{3}} +{k},....,{a}_{{p}} +{k} \\ $$$${produce}\:{at}\:{least}\:\frac{\mathrm{1}}{\mathrm{2}}{p}\:{distinct}\:{remainders} \\ $$$${when}\:{divided}\:{by}\:{p}. \\ $$

Question Number 194700 Answers: 0 Comments: 2

Question Number 194697 Answers: 2 Comments: 0

((tan x)/(tan x−tan 3x)) = (1/3) then ((cot x)/(cot x+cot 3x)) =?

$$\:\:\: \frac{\mathrm{tan}\:\mathrm{x}}{\mathrm{tan}\:\mathrm{x}−\mathrm{tan}\:\mathrm{3x}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\mathrm{then} \\ $$$$\:\:\:\frac{\mathrm{cot}\:\mathrm{x}}{\mathrm{cot}\:\mathrm{x}+\mathrm{cot}\:\mathrm{3x}}\:=? \\ $$

Question Number 194695 Answers: 1 Comments: 0

(x/(a+b−c)) =(y/(b+c−a))=(z/(c+a−b)) Then (a−b)x+(b−c)y+(c−a)z =?

$$\:\:\:\: \:\frac{\mathrm{x}}{\mathrm{a}+\mathrm{b}−\mathrm{c}}\:=\frac{\mathrm{y}}{\mathrm{b}+\mathrm{c}−\mathrm{a}}=\frac{\mathrm{z}}{\mathrm{c}+\mathrm{a}−\mathrm{b}} \\ $$$$\:\mathrm{Then}\:\left(\mathrm{a}−\mathrm{b}\right)\mathrm{x}+\left(\mathrm{b}−\mathrm{c}\right)\mathrm{y}+\left(\mathrm{c}−\mathrm{a}\right)\mathrm{z}\:=? \\ $$

Question Number 194693 Answers: 1 Comments: 0

if f_n =f_(n−1) +f_(n−2) ; f_1 =f_2 =1 then prove that 5∣f_(5n)

$${if}\:\:\:{f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \:\:;\:\:{f}_{\mathrm{1}} ={f}_{\mathrm{2}} =\mathrm{1} \\ $$$${then}\:\:\:{prove}\:{that}\:\:\:\mathrm{5}\mid{f}_{\mathrm{5}{n}} \:\: \\ $$

Question Number 194685 Answers: 1 Comments: 0

Question Number 194662 Answers: 0 Comments: 2

∫_0 ^(Π/2) (√(4sin^2 t+cos^2 t)) dt

$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}}\:\:{dt} \\ $$

Question Number 194654 Answers: 1 Comments: 0

Question Number 194652 Answers: 0 Comments: 0

Question Number 194649 Answers: 0 Comments: 3

calcul ∫_0 ^(Π/2) (√(4sin^2 t+cos^2 t ))dt

$${calcul}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\Pi}{\mathrm{2}}} \sqrt{\mathrm{4}{sin}^{\mathrm{2}} {t}+{cos}^{\mathrm{2}} {t}\:}{dt} \\ $$

Question Number 194648 Answers: 3 Comments: 3

Question Number 194642 Answers: 2 Comments: 0

If A= (((a b c)),((b c a)),((c a b)) ) and a,b,c >0 such that abc=1 and A^T .A=I find a^3 +b^3 +c^3 −3abc .

$$\:\mathrm{If}\:\mathrm{A}=\begin{pmatrix}{\mathrm{a}\:\:\:\:\mathrm{b}\:\:\:\:\:\:\mathrm{c}}\\{\mathrm{b}\:\:\:\:\mathrm{c}\:\:\:\:\:\:\mathrm{a}}\\{\mathrm{c}\:\:\:\:\:\mathrm{a}\:\:\:\:\:\:\mathrm{b}}\end{pmatrix}\:\mathrm{and}\:\mathrm{a},\mathrm{b},\mathrm{c}\:>\mathrm{0} \\ $$$$\:\:\mathrm{such}\:\mathrm{that}\:\mathrm{abc}=\mathrm{1}\:\mathrm{and}\:\mathrm{A}^{\mathrm{T}} .\mathrm{A}=\mathrm{I} \\ $$$$\:\mathrm{find}\:\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}} +\mathrm{c}^{\mathrm{3}} −\mathrm{3abc}\:. \\ $$

Question Number 194640 Answers: 1 Comments: 0

lim_(x→∞) ((√x) (1−cos ((1/(2(√x))))).(1/(sin ((3/( (√x)))))))

$$\:\:\: \: \\ $$$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt{\mathrm{x}}\:\left(\mathrm{1}−\mathrm{cos}\:\left(\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{x}}}\right)\right).\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\mathrm{3}}{\:\sqrt{\mathrm{x}}}\right)}\right)\: \\ $$

Pg 257 Pg 258 Pg 259 Pg 260 Pg 261 Pg 262 Pg 263 Pg 264 Pg 265 Pg 266

Contact: info@tinkutara.com